Educational Journey of a TechnoBabe

ATA Summer Conference - Calling Statement

noreply@blogger.com (Sandi Berg) — Thu, 15 Aug 2019 22:58:00 +0000

I am currently attending the ATA Summer Conference - "Initiatives in Leadership" program in Banff, AB.

This is the view from the bed in my hotel room.

I know...it sure makes it tough to leave in the morning!

Our task is to develop a calling statement that I will implement this year. I feel like I'm cheating a little as my calling statement is centered around work that began last year but will continue this year.

As this post is only meant to be limited to about 700 words, I will outline the project for now. More in-depth details will be provided in a future post which I will link here when it's ready.

Background of The Project:

Edmonton Public School Division developed a Mathematics Intervention / Programming Instrument (MIPI) which is an assessment for Grades 2-10 that is implemented at the beginning of the school year. In my school division, many of the teachers have implemented or will implement the MIPI.

I have concerns about some aspects of the MIPI. First of all, there are issues with some of the questions not quite hitting the outcomes I think they were meant to hit. However, my main 2 concerns are that 1) the questions are low-complexity questions and 2) they are all multiple-choice and numerical response. I understand that the intent was to get some basic data for a starting point however I think it's important to dig deeper. Why did they get that question wrong? What are the misconceptions? So, we developed / are in the process of developing what we lovingly call (for now) the MIPI+. (Yes Edmonton Public knows...)

We created 5 questions for each grade from grades 2-10 that delve deeper into students' conceptual understanding and deliberately try to pull out misconceptions. The first question (which is identical for grades 2-10), for example, is "Make the following statement true: 8 + 4 = __ + 5. Convince me." (This question was taken from a research study which I will link when I have more time...this is due tomorrow morning after all). The students will take about 5 minutes to answer this question. The teacher, using the rubric, will take about 5 minutes to sort all the responses. As a result, the teacher will have a deeper understanding of what their students truly understand about equality and what misconceptions they have. The electronic version of the rubric, will be hyperlinked to our math framework which will answer the questions:

Why is this a misconception?
How is this misconception developed?
What is the research around this misconception?
How do I deal with this misconception once students have it OR how do I prevent students from developing this misconception? ie. What are the simple classroom activities and conversations can we have to develop this understanding?
What are grade appropriate questions I can ask my "grade 5" student to continue this work?

So what does this mean for this year?

I am going to work on the continued development of these resources (teacher guide, rubrics, student exemplars, training tools, etc.) this year.

Although teachers are not required to use this tool, in PLC's, we will support teachers in

understanding the purpose and intent of the MIPI+
working through the administration, assessment and analyzing of the MIPI+ (Even if they aren't assessing their own students, they will have an opportunity to analyze results)
determining next steps in student learning
exploring best practices in assessment

I will know I have made a difference with this project when I see

teachers administering, assessing and analyzing the results to benefit student learning.
teachers creating and sharing new questions that can be added to our math framework to continue the work.
teachers talking and sharing best practices around assessment.

The one key measure of success will be that when students move into the next grade and the teacher administers the MIPI+, the misconceptions, that were previously demonstrated, will be gone.

I will need help from people outside of my "echo chamber". We've heard a lot of "this is great!" I'm hoping to hear those thoughtful questions that challenge me to think more deeply about my reasoning behind the decisions we make. This will help me make the tool meet the needs of the teachers better.

For example, I was asked, "You said we should do this at the beginning of the school year. (ex. beginning of grade 5) Why not administer it at the end of the school year (end of grade 4) and pass the information right away on to the next (grade 5) teacher? This would give the grade 5 teacher time to plan."

In the beginning my response was that you want the data of the students you actually have in front of you. If you have new students, you'll have to re-administer it anyways. Then after I had time to reflect, I changed the recommended implementation process. Although implementing all 5 questions at the beginning of the school year is still one of the options we recommend, we now suggest that the first question (Equality) is done right away. Then, you can appropriately plan for the equality work you need to do. Then 3 or 4 weeks later, you can do the second question (which focuses on place value in grades 3+). Teachers would do place value work, if needed. Then, the fraction question (Grades 4+) would be done about a week or so before you start fractions...and so on. This way, the work that you do with equality will help build their understanding of place value...and fractions...and so on.

I still don't recommend doing it in June! Though we did do one field test of this so we will see how those same students do when they right it again in September. Maybe there is no difference. We'll see.

I will seek mentoring and coaching from...anyone who is willing! I've received one suggestion but I'm still looking for others! I am an executive member of the Math Council so of course I'll be accessing them for lots of support as well.

My peer coach(es) will be...all of you! In a future blog post, I will share more specific details about the project so that you can provide that constructive feedback.

Oh, and did I say that we will be sharing ALL of these resources freely? You just have to ask...once I've shared and you've explored the resources in more detail! I will post more information about that in my next blog post.

This project was a big part of my work last year and I can't wait to keep working on and revising it through all your amazing constructive feedback!

Sandi

Do we have time for this? Analog clocks

noreply@blogger.com (Sandi Berg) — Sat, 30 Mar 2019 17:10:00 +0000

There's some controversy out there around analog clocks. I've seen articles claiming that schools are taking them down because students can't read them. This causes an uproar. Two different sides are shared.

Let's look at some of the arguments proposed from both sides.

Students always have their phone which displays a digital clock. This is often true...but not always. Some students/adults don't have cell phones. Secondly, though you can set up your phone to display a digital clock, you can also display an analog clock. So we should tweak that statement to say: Many students have a phone that displays a digital clock.
Reading analog clocks teaches students how to skip count. It certainly helps build that skill. Even better would be if teachers also focused on multiplication rather than adding. That's one 5, two 5's, three 5's, etc. as that aligns better to the numerals on the clock.
Reading analog clocks teaches students about base 60. True, if a teacher has talked about that concept. It also teaches a student base 13...think about a 24 hour clock.
It teaches them how to read airline tickets. They have to know how to read a 24 hour clock. This is the only one that I think fails as an argument. Neither a digital nor an analog clocks teaches that unless it has 13:00, 14:00, etc on it. I'm not saying they don't need to know how a 24 hour clock works nor understand how to convert between the two formats. The fact that it's analog doesn't, by nature, teach that.

Those are just a few of the arguments for and against teaching analog clocks. I know there are more out there. I don't know if this is the reason analog clocks are being removed. I know that many have to be replaced due to old age and failure. Maybe they are taking them down because students can't read them in that school. I don't know. I haven't had an opportunity to delve deeper into the stories and it doesn't influence me in any way. My goal for this post is not to convince you either way because I'm not convinced either way.

The bottom line is that I will teach students how to read both digital and analog clocks. Why? Because it's in my curriculum. In Alberta, it is addressed specifically in Grade 4. I am required to teach it. What clock type students choose to put on their cell phones or buy as a watch is totally up to them.

So, how do I teach it? The same way I teach everything....by starting slowly, with visuals. Teaching time is NOT a separate unit that you dedicate a week to in January. Start it right away in September but go slow...very slow.

At the very beginning of the year.

First, take a clock that you can remove the inside circle from - the one that the numbers are written on.

Remove the inside circle. You can either do the next step right on that circle OR you can use the circle to trace out a brand new circle and write just the numbers 1-12 on it. I would prefer to draw a new circle with just 1-12 on it so that it removes the other info that's not needed yet. Honestly, either way is fine, though.

Draw lines from the center to each of the numbers 1-12. You'll be cutting it into "pie" shapes. Shade each of the sections in different colors. Write "ish" by each number. You'll have to place it in a way that the "ish" is in the correct portion.

I apologize that my numbers are so light. I was just working with a digital image.

Now, if possible, remove the minute and second hand. Or, if you can't or don't want to, color the hour hand so it stands out.

You may even want to draw arrows to help for each number.

Do a mini lesson on how to read this clock. The clock above shows that it's 10ish. Go through a few times. What would 4ish look like?

Yes, it's not very specific After a while, you can say something like, "it's 2ish almost 3."

Do this for two weeks, three weeks, four weeks. However long it takes. Do not rush this. You have all year to teach them about reading a clock. Once students are comfortable, you can move on to the next activity.

Activity 2:

Did you know that you can tell what time it is within about 5 minutes even without the minute hand? Want to see? Open the following google slide presentation. Do not put it in presentation mode. On the second slide, place the clocks in the correct time order. You'll notice that one is at the bottom. During the actually activity, this would be sorted normally with all the other ones. While you are doing it, feel free to consider it during or after you are done the rest. Further on in my blog post, I'll talk about how I address that one with students. Then, move to slide 3 and add the times.

Now that you've done that, I'll explain the process I use with students.

I hand out 9 images of clocks that only have the hour hand drawn and hour numbers listed on it. These are the same images as you saw on slide 2. These have been cut out as separate images so they can be individually moved around.

In pairs, students work together to sort them into time order.

Once they are done, they meet with another pair and compare their answers, sharing their reasoning, and moving clocks around some more if they choose.

I open up my google slides version and display slide 2 (not in presentation mode). It shows all of the clocks at the top (and one at the bottom). One student will come up and move a clock into the grey box. Another student will come up and place a clock that should come before or after it. They explain how they know it goes there. We repeat for all until we finally get to the clock that I placed at the bottom of the screen. Where does this one go? One student will come up and place it. I ask, "who else placed it here?" "Who placed it in a different spot?" That student comes up and places it in a different spot. "Interesting. We have two different answers. Talk to your partner. Which do you agree with and why?" The discussion then becomes: where does 12:00 go? Both spots!

Now, I had out the cut out times from page 3. Students use these to match up to their clocks. They compare to a different group. As a class, we discuss.

The final page in the handout is an activity students will complete individually. Use this as a formative assessment to inform your teaching practices.

Now, use these types of phrasing when you are talking about the time in class. Again, don't rush this. Don't worry about using precise times yet. You'll notice that none of the examples referred to 3:15.

Activity 3:

Moving on to minutes? Don't refer to minutes yet. Give students an image of a clock that has the hour numbers written on it and the tick marks for minutes. Have them count the ticks around the clock. What patterns do they notice?

Do they notice that it's in groups of five? Re-count the circle. one two three four five. That's one group of five (notice it's at the 1). 12345. That's two groups of five.. (notice that it's at the 2). How many minutes is that? 10 minutes. Interesting. Does that's always work? Keep checking. 12345. That's 3 fives. 15 minutes. Students figure out the rest. Now you're connecting it to multiplication.

Spend time working with naming regular times. 3:15 (not quarter after).

Activity 4:

Have students draw a circle and break it into quarters. What do they notice about that and how it relates to a clock? (If they drew it so the lines could be imagined as going through 12 and 6, 9 and 3, they should be able to make connections to quarter after.)

Make sure to talk about 15 minutes out of 60 minutes. 15/60. etc.

Activity 6:

Adding seconds? Have students figure out how high they can count in a minute. Time it according to just the minute hand. (Use a clock that doesn't have a second hand if possible and a really obvious shift from minute to minute). They count in their head. Share responses. Some will have counted really fast. Some counted really slow. That's ok. Talk about how we've decided to set a specific counting speed to help us be consistent. (It's humans who decided that there are 60 seconds in a minute.) Show a clock with a second hand and the minute ticks. Have them recount. Share responses. They're probably more consistent now. Then, have them focus on the second hand as it passes each of the ticks. That should bring them a lot closer to accurate timing.

Now that they've done this activity, give them their own clock with tick marks to look at. What relationships can they find between the second hand and minute hand and hour hand. They might say, hour hand is shorter in length, second hand is longer in length. The longer the hand, the faster it moves. 60 seconds in a minute. 60 minutes in an hour.

You can even give a puzzler: If I count to 60 that's 1 minute. How would you figure out how many minutes it would take me to count to 180. Interesting enough, some may argue that it will take more than 3 minutes. Why? It takes longer to say one hundred seventy nine than it takes to say "one". Don't discount those arguments because they are absolutely correct!

I can not stress enough: Go slow. Build understanding. Don't move on until they UNDERSTAND what's going on. You have all year.

Instructions and Handout:

Google doc
Word Doc
PDF

Presenting File:
Google Slides
Smart Notebook

Question for Self-Reflection:

How do you help students understand the concept of time? Not just reading a clock, but actual understanding of what time is?

Ditch C.U.B.E.S. when solving word problems!

noreply@blogger.com (Sandi Berg) — Sat, 17 Nov 2018 01:00:00 +0000

No I'm not referring to manipulatives when I say ditch the C.U.B.E.S. when solving word problems. Full Disclosure: In the past, I probably fell into the C.U.B.E.S. supporter bunch. I've learned a lot over the past few years and now focus on other ways of teaching students how to solve word problems.

First of all, some of you may be asking, "What is C.U.B.E.S.?" C.U.B.E.S. is an acronym that you may have seen posted in math classrooms throughout the years. It may take the form of a different acronym but they all get to the same basic idea: when solving a word problem use C.U.B.E.S.

Circle all key numbers.
Underline the question.
Box any key words.
Eliminate unnecessary information.
Solve and check.

Rather than telling you why I no longer use this, let's just jump right in and try an example. As you look at each stage (before skipping ahead to the next image), ask yourself "What's the answer?"

You see the following word problem that makes no sense to you...

so you faithfully follow the steps...

Step 1: Circle all key numbers

What's your gut instinct for the answer?

Step 2: Underline the question:

Do you feel pretty confident about that gut instinct?

Step 3: Underline the key words:

Still feeling pretty confident?

This is where most students stop when they don't know how to solve a word problem. What answer do you think comes to mind?

Let's look at the full question. Were you right?

What operation is used in this question?

Did you know that most students who aren't sure what to do when solving a word problem will take all of the numbers they see and just add them?

Do you have a math word wall? Have you taught students to identify key words? What have they learned "in total" means?

Let's try another example:

Let's start with that key word: "in total"

Circle all the numbers:

What do you think the answer is?

Are you trying not to add? But if it's not adding, then what is it?

Let's look at the question:

Woah...did that 5 just get removed from the calculations? Oops! How many students might get thrown off by that?

Let's look at the full question....

What operation is used in this question? Were there any other "key words" that students might have identified? Would they have helped or hindered their understanding?

Let's look at another example:

Here are the numbers...

Here's the question...

What are you going to do with the information?

Again, let's look at the full question. Does identifying key words help here?

You know me...here comes another example.

Let's pull those numbers out.

What's the question?

What's the full set of information?

So, what's the commonality in these four questions? They all focus on the key word "in total" which is typically taught as "addition" but every question actually addresses a different operation and some have extraneous numbers.

Here's a word problem for you to try: "There are 125 sheep and 5 dogs in a flock. How old is the shepherd?" I'll wait while you figure it out.

Did that seem like a silly question to you? Obviously, you can't figure it out.

Robert Kaplinsky asks 32 students to respond to this question. How many do you think agree with you and how many do you think attempted to solve it? Once you've made your prediction, watch the video.

Were you surprised? Why did so few students make sense of this problem? How do we help students dig deeper into the word problem rather than relying on following steps like C.U.B.E.S.?

Brian Bushart introduced me to Numberless Word Problems and I fell in love.

As always, let's begin with an example.

There are some mice on the field. Some more mice come.

What do you notice?
What do you know to be true?
What do you think to be true?

Take some time to think about the three questions in relation to the two statements above.

I have used this example in many of my workshops and I'm always fascinated by the discourse that occurs. If the participants/students have never experienced a numberless word problem before, they might experience confusion - there are no numbers! They sometimes start by retelling me the information that's on the board. I have had to prompt them to think about what "numbers" are in the question. What words might imply numbers? They will focus on the word "some". What does some mean? This is where it gets interesting. In my experience, most will say "more than one" but some will disagree and state that it means at least three. Why three? If the writer meant two, then they would have said "couple". We often spend quite a bit of time talking about the differences between "couple", "some", and another examples they might bring up. It's really very interesting. I'll then ask the question, "If -some- means -more than one- what do we know?". Students will have time to talk about that. They'll naturally start to think about the question "How many mice are on the field now?". Invariably, I'll hear the answer 3 but then they'll discuss how more than 1 and more than 1 has to be more than 3. They'll share their reasoning. I'll follow up this question with "if -some- means -more than two- what do we know?". After we have fully dissected this question, I will replace one word:

There are 7 mice on the field. Some more mice come.

What do you notice?
What do you know to be true?
What do you think to be true?

We'll discuss what's changed and how that changes the rest of the information we talked about. After a full discussion, I adjust the question again.

There are 7 mice on the field. 4 more mice come.

What do you notice?
What do you know to be true?
What do you think the question is going to be?
Solve that question.

At this stage, students solve the question.

Questions like this are easy to create. Just pull out a question that you would give them anyways, replace the numbers with generic words like "some", "many", etc. Check out #numberlesswp on twitter.

CAUTION! This is great at the beginning of a class. Do NOT spend an entire class talking about addition and then pull out this question. Why? If we do 40 minutes of addition questions and then we have a "word problem", students are just going to assume that it's addition. The thinking stops and they revert to running a procedure rather than delving into the meaning behind the words.

Can you run a numberless word problem in other formats? Absolutely! Brian gave me permission to turn his numberless graph into a presentation I can share with teachers. I've embedded it below. Click through it, thinking carefully about each slide, asking yourself the same 3 starting questions as in the mouse question above. I've inserted a blank slide between each image mainly so I didn't accidentally click too many times in a row when presenting. :) You can download the slideshow from my account and you definitely should check out Brian's "original blog post" on the activity.

Once students have spent time making sense of a problem without numbers, when they get to a question they are not sure how to solve, they will slow down and think more deeply about the information provided. They won't just take the numbers, throw in an operation and "solve" the question.

Question for Reflection:
How do you think your students would do in the "How old is the shepherd" problem? Try it in your class and see how they do. Are you surprised by their responses?

Activity to Try:
There are many words / phrases like "in total" that have many mathematical meanings depending on the context of the problem. Can you create 4 word problems that each use that word/phrase but utilize a different operation (like my examples above)?

The equals sign does not mean the answer comes next!

noreply@blogger.com (Sandi Berg) — Thu, 08 Nov 2018 19:14:00 +0000

As part of the team who worked on developing Elementary Mathematics Professional Learning Opportunities (EMPL), I spent a significant amount of time learning about the importance of Equality and have presented many times across Alberta on this topic using the resources and research we compiled through the project. A recent comment on Facebook made me aware that I should write a blog post about this too! If you've visited the Equality portion of the EMPL site, you will see that this post is a compilation of a part of those resources. Note: All images that contain the EMPL logo are pulled from the presentation document that we developed.

Our team defined 3 big ideas around Equality.

Mathematically, equality refers to a relationship between objects that can be quantified.
Mathematically, equality can refer to a relationship between units of measure.
Operations emerge as a way to balance an inequality.

This post will focus on the first big idea: "Mathematically, equality refers to a relationship between objects that can be quantified."

I use the term "objects" loosely. In the beginning, students will be considering and using concrete objects (bears, counters, beans, etc.). As their understanding progresses, they will move on to more abstract representations of those same objects. Even in higher grades, when you are referring purely to a symbolic representation such as an algebraic expression, it still refers to objects of some kind. You’ve abstracted the objects out of the equation. Please don't get hung up on the terminology or lack thereof that I have used here. Let's look at the bigger picture and then we can always return to the tendency, preference and need we have to define terminology more explicitly. :)

Before we start, please jot down a simple addition or subtraction equation that you would ask students to solve in a K-4 classroom. We will come back to this question later.

Take a moment to think about the following:

How would your students answer this question? "What does the equal sign mean?"

I would recommend going into your classroom and asking your students that question. Ask it in grade 2. Ask it in Grade 6. Ask it in high school. Just ask. I'll wait. Come back when you are ready...

Researchers asked students this very question. They looked at students' responses and determined that there were two kinds of understanding.

One is an "operational understanding". Students who have an operational understanding of the equals sign might respond by saying...

These students believe the equal sign means "add the numbers" or "the answer".

How might an "Operational" student respond when asked to complete the following statement to make it true?

3 + 2 = __ + 1

Most operational thinking students will put a 5 here because 3 + 2 = 5.

Some will even continue and tell you the answer, therefore, is 6 because 5 + 1 = 6.

The other type of understanding is a "relational understanding". Students who have a relational understanding of the equals sign might respond by saying...

These students see the equal sign as an equivalence relation between two quantities.

How might a "Relational" student respond when asked to complete the following statement to make it true?

3 + 2 = __ + 1

Relational thinking students will write a 4 here because 3 + 2 = 5 and 4 + 1 = 5

Researchers asked students to do the following:

Make this statement true.

8 + 4 = __ + 5

What percent of your students do you believe would get this right? What other answers do you think you might get? Now, without pre-teaching anything or having a prior discussion, give this question to your students and see how they respond. Their answers will tell you a lot about their understanding of equality and the equals sign.

Let's take a look at the results.

How would "Operational" students respond?

They may have answered 12. The numbers below indicate the percent of students who gave this response.

"Operational" students may have also responded with 17. How do you think they arrived at that answer?

Some even wrote both 12 (in the first blank) and then = 17 at the end.

What percent responded correctly with the answer 7?

"Other" refers to any answer given that was not one of the four previously discussed.

If we choose to believe the results from the research study are still accurate today, about how many students in your class truly understand the equals sign? I strongly recommend that you give this question to your students to see how they respond.

So what do we do?

Let's look at Equality as a quantity through a few True or False questions. I've included images from the EMPL presentations, but you should use concrete objects for students to look at when doing this activity in your class.

This one should be pretty easy for both you and your students. I would say something like, "The first set has a quantity of 3. The second set has a quantity of 3. They both of a quantity of 3. They are equal. Let's replace those quantities with their symbols. There are 3 objects on each side.

3 = 3

This is when it gets interesting for students. If they just see 3 = 3, many will tell you this is formatted incorrectly. "We don't write 3 = 3." We need to make sure that students see such basic equations in order to make sure we are not perpetuating the misunderstanding. This is one of the first equations they should learn.

What changed? Are these sets still equal? Let's replace the quantities with their symbols.

3 = 3

We are still comparing quantities. Therefore the color of the cubes does not change the relationship. This is why it's important to make sure students understand that we are talking about quantity not other characteristics such as color.

What changed? Are these sets still equal? We are still comparing quantities.

3 = 3

Therefore the elephant still represents one object. This is why it's important to make sure that students understand that we are talking about quantity not other physical characteristics.

What changed? Are these sets still equal? 3 = 3 for the same reasons as above.

What about these sets? Are they equal? Yes, these are still sets. They are just sets of nothing. Students need to understand that it's ok to say 0 = 0.

I think you get the idea. Let's move on to the next stage of building understanding of the equals sign.

Let's jump back to one set for now. You could do this at this stage or before completing the previous activity.

How many dots are there? How do you know? Johnny may have seen it as 3 +3 (write that) while Lisa saw it as 1 + 4 + 1 (write that beside the 3 + 3). Did they see the same amount? Yes. Write the equals sign between the two. Now you have 3 + 3 = 1 + 4 + 1. Reinforce that they are both the same quantities.

Let's move back to comparing two sets.

What's the relationship between these sets? The relationship is equal. How do we record that in math?

We'll start with the total in each. 6 = 6 (same as before). However, we want to take it to the next level. (This should sound familiar...) How did you know there were 6 in the first? Did you just know (subitize) that it was 6? Did you see 3 + 3? 4 + 2? Let's say you just knew it was 6. You subitized it because it looks like the dots on a dice.

What about the ones on the right? Did you know it was 6? Did you see 3 + 3? Did you see 1 + 4 + 1? 1 + 2 + 2 + 1? Let's say you say it as 3 + 3.

How would you record this?

You say the first set as 6 and the second set as 3 + 3 so 6 = 3 + 3.

Remember, students who have an operational understanding will say the equation written in the form 6 = 3 + 3 as written "wrong".

Students could also write this as 4 + 2 = 1 + 4 +1 or 3 + 3 = 3 + 2 +1 or...it all depends on how they saw the first set and how they saw the second set. There are many options.

Now, go back to that question I asked you to write down near the beginning of the blog post. Did you write it in the form of 5 + 6 = ? or 9 - 4 = ? with the "question" on the left and the "answer" as the blank on the right? Most of our resources show questions in this format. This is one of the reasons why students are led to believe that the equals sign means the answer comes next...because this is the way they have always seen it.

What are some simple ways that you can change this?

Compare how two students saw the same image. Write it as a comparison: 3 + 4 = 2 + 5
Instead of always saying "5 + 6 is equal to" try "5 + 6 is the same amount as", "5 + 6 is the same as", and "5 + 6 has the same value as". Use a different phrase for the equals sign often. You may be worried that students will be confused about the changes in terminology but they won't. Trust them. You can always say "has the same value as...is equal to" the first few times to help them understand the term "value" in this context.
Instead of always writing your questions as "5 + 6 = ?", turn it into "? = 5 + 6". You can phrase it as "What is 5 + 6?" as you point to the question mark?
When looking at the first set of 6 in the image above, say "There are 6 here." (Write down 6). Ask, "how did you see it?" If they say 3 on the top and 3 on the bottom, say "You saw six as" (write the equals sign) "3 and 3" (write 3 +3 so you now have 6 = 3+ 3)

There are many other activities that will take this to the next level.

Misconceptions about Equality are not easily undone.

What do you do though, when a student says 3 + 2 = 5 + 1? How do you respond? Ask, "What materials could you use to demonstrate this?" Have them build it concretely such as unifix cubes.

Do the blocks prove this is true or not true?

Does this make it easier to see?

Does turning the blocks change your thinking?

Build the left side of the equation: 3 + 2

Consider the rest of the equation. Using the same blocks, can you represent 5 + 1?

How does this help prove or disprove 2 + 3 = 5 + 1.

All of these ways are valid. Students need more than one way to explain it. Make sure the equation is always in front of them as they manipulate the materials. A student who can conserve 5 should understand this. It's important that they also understand that they can't add extra blocks. Equality is about representing the same quantity in different ways.

If 2 + 3 = 5

and

4 + 1 = 5,

then

2 + 3 = 4 + 1

So, 2 + 3 can not be equal to 5 + 1

If students are struggling in math class, give them the question 8 + 4 = __ + 5. How they answer will tell you if you need to stop what you are doing and deal with any misconceptions they have around the equals sign. Spend the time! How can we expect them to manipulate two step equations if they don't understand that it's about relationships?

Reflection for the Reader:

Are you surprised by the results of the research study? How are you dispelling misconceptions around the equals sign regardless of the grade level you are teaching?

Placing the focus in Place Value

noreply@blogger.com (Sandi Berg) — Thu, 18 Oct 2018 12:38:00 +0000

I have broken my once a week posting rule but I do have a good excuse...honest! I started a new role this year. I'm back with Chinook's Edge School Division. I worked there for several years before I was seconded to Central Alberta Regional Consortium for 7 years. I accepted a position as a Learning Services Coordinator and have many new roles. I started August 20th with my head down, hair back, going full tilt as I was adjusting to my new role and it's just starting to slow down a little.

I'm sitting here wide awake at 5:30 a.m. which is highly unusual for me. I am so not a morning person. However, I am in a hotel and have nothing else to do so I thought I would write a short post on place value. Why place value? I know that years ago, I had little understanding of how little understanding I was providing my students with the ways I "taught" place value. I know better know and have spoken about its importance within sessions. Yesterday, I saw a twitter post from one of my favorite math people James Tanton and it reminded me that I should take some time here to write about it.

Let's start with a question. How many tens are there in the number 234?

Some text to distance you from my answer. You can skip the rest of this paragraph and move on to the next one once you have answered the question above. Nothing important to read here. Honestly. Why are you still reading this? You don't follow directions very well do you? Neither do I so I'm not surprised. Ok. Enough chit chat. Let's get to the "answer".

Did you say that there are three tens? Twelve years ago, that's what my students would have said too. Why? Because when I asked that question in class, I was really asking the question "what digit is in the tens place?". At the time, I didn't really understand that the question I was actually asking has a very different answer. Now, I am much more careful about how I talk about place value before asking that question.

Let's explore one of the activities I use to build a better understanding of place value. You have base ten blocks in front of you and we have named them tens and ones. We aren't using the hundreds right now. Let's focus on building the understanding with smaller numbers first. Once you have that, you will see how it transfers to larger numbers.

Here is the process that I generally use when introducing place value now. Please keep in mind that this isn't meant to imply that this is the way you should be doing it but it is what has worked for me with the students and teachers I have worked with. Stop at the end of step 3 to see if you can answer the question I pose to students at that point.

Step 1: Make sure that students have had prior experience using base ten blocks. They must understand the relationships between the unit block, the tens and the hundreds. I am not spending time here discussing those relationships however.

Build me the number 23 using the base ten blocks. Let me clarify that. I don't want you to place the blocks on the table so they look like the digits 23. (Yes, I say this...it has happened more than once). I want you to represent the number 23 as a value. When I count your blocks, I will know that they have a total value of 23.

Step 2: Give students time to work. Walk around to formatively assess students. Which ones are struggling to build 23 in any way? Which ones are showing it just one way? Which have moved on to find a second way? Have any explored the idea that there is more than one way? Watch for a third representation (You are not going to call on that student. You just want to be aware that they already have that concept) Let's say that Johnny has represented them using all ones and Patty has represented them using two tens and three ones. I will make sure to let them I will ask them to share their representation with the class.

Johnny, how did you represent 23? Johnny will say "with 23 ones." Class, does 23 ones represent 23? Yes. Who else represented 23 using 23 ones? Write 23 ones on the board.

Patty, how did you represent 23? Patty will say "with 2 tens and 3 ones." Class, does 2 tens and 3 ones represent 23? Yes. Who else represented 23 using 2 tens and 3 ones? Write 2 tens and 3 ones on the board.

Class, who is correct? They both are because they both represent 23.

Step 3: I tell this story if I haven't told it to them before.

Class, I love math. I think about it all the time. I start walking around the room, looking dreamily up in the sky. Ok, you don't have to look dreamily...you can look up thinking hard or with a pondering look on your face. Your call.

Because I think about math all the time, I'm a little clutzy. Right now, I have 23 ones in my hand. Have a bunch of ones in your hand and wander around looking up...however you wish.

I'm wandering around, thinking about math...and I trip and fall. Don't actually fall...Just stumble a little bit. :) and a I drop my 23 ones. (Don't actually drop them. You don't want to be cleaning up that mess!)

Now, I have 2 tens and 3 ones in my hand. Grab those...doesn't have to be exact. No one is counting.

I'm wandering around, thinking about math...and I trip and fall. Which one's worse? Dropping the 23 ones because it's a lot to pick up and you probably won't find them all.

Because I'm clutzy, I will usually...BUT NOT ALWAYS...represent my numbers using the fewest blocks possible but I know that I can represent 23 either way. It will depend on what I want to do with that number. So, typically, if I ask you to represent a number, please use the fewest blocks possible but know that you can change that to suit whatever you are doing with the number as long as you represent it accurately.

Now, We have represented the number 23 using the fewest blocks possible: 2 tens and 3 ones and we have represented the number 23 using the most blocks possible: 23 ones. I wonder if we can represent the number 23 a different way. I'm going to give you 3 minutes to work on this puzzler. You might not figure it out in that amount of time and that's ok as long as you keep working on it. Who can you work with? Seat partner.

Note to reader: Remember to stop here and solve this question yourself!

Step 4: Wander around to formatively assess students as they work to find the answer. Choose a student who figured it out to share with the class and then ask who else found that representation.

Lise, how did you represent 23? Using one ten and thirteen ones. Who else represented 23 this way? Write one ten and thirteen ones on the board.

Reinforce that all three representations are accurate and acceptable and that it will be up to them to decide which one makes the most sense when working.

Question for the reader: When does representing the number 23 using 1 ten and 13 ones most often show up?

When you are working with the "traditional" algorithm for subtraction. On a side note, in Alberta, students are required to explore the standard/traditional algorithm for subtraction beginning in grade 2. Some cautions:

This should NEVER be introduced until they have a solid understanding of place value.
This should NEVER be introduced until they have had an opportunity to work concretely with the concept of subtraction. Spend lots of time building concrete understanding before representing any concept abstractly. If they truly understand the concept, you can tie that understanding to the abstract representation and it will then have meaning for them.
You should NEVER use the term borrowing. What does borrowing imply? That you will pay it back. Do you ever pay back that ten you borrowed? No. I'm ok with the terms exchange or trade. I would love to hear any other terms you use.
NEVER say that you can't take a bigger number away from a smaller number (this is usually said to help stop kids from subtracting upwards). You CAN take a bigger number away from a smaller number: 3 - 8 = -5. We just haven't learned how to do that yet. If they are subtracting upwards, that tells you they don't understand the concept. They are just trying to use the algorithm without understanding it.

After we have dissected the number 23, we move onto a number like 32. How many different ways can you represent this number? Since this will require a lot more blocks, they typically don't have enough to leave all representations on their desk so I have them build one representation and write down the "words" on their whiteboard (ex. 3 tens, 2 ones), or take a photo of it, etc.

Lots of time needs to be spent at this level with small numbers. Once they have this, move on to bigger numbers which is harder to represent using base ten blocks as you need too many. Can they find all the ways and write the "words" on their whiteboards without the blocks?

Afterwards, I might give them some questions to play with on a sheet or on the board. I let them choose an "easy one, a medium one and a hard one" to try.

When students are feeling confident and need a challenge, or if I am working with a grade that needs to move on to place value in hundreds, we look at three digit numbers, representing small quantities first (like 123) using base ten blocks (if needed as some students might not need them) and "words" (1 hundred, 2 tens, 3 ones). Here are some questions I let them choose an "easy one, a medium one and a hard one" to try.

Question for the reader: Now, go back to the original question I asked at the beginning: "How many tens are there in the number 234?" What would you say now? There are actually 23 tens.

Earlier, I had mentioned that James Tanton's tweet reminded me that I wanted to talk about this. He reminded me that we do this naturally for certain numbers such as twelve hundred, or six hundred fifty-seven thousand.

Reflection for the reader: How do you ensure that your students understand the difference between the questions "what digit is in the tens place" and "how many tens are there?"

Puzzle Me This...Is it a Word Problem or Problem Solving?

noreply@blogger.com (Sandi Berg) — Fri, 06 Jul 2018 14:00:00 +0000

Before I get into the nitty gritty of this post, let's start with a little self assessment.

When I do this in person with teachers, I have them close their eyes. I can't ask you to do that, however. You won't be able to read what I'm writing! :) Instead, just take a moment to reflect on the question before moving on.

I am going to give you a word to think about. This is not meant to be in a specific context. You don't have to think about how it relates to math. When you think about this word, focus on the feelings it invokes in you. There is no right or wrong answer or feeling. Just be honest.

The word: Problems

Reflect. Think about it. Don't read the next paragraph until you are ready.

How did you feel about the word "problems"? Was it overall a positive or negative emotion for you? In my experience, many people (but not all) think about problems negatively. These people see problems every day as negative things. Maybe they are unsolvable. Maybe I'm having money trouble or my child is misbehaving or there are issues with my boss. Some people see problems as positive, as a challenge to be solved.

Let's look at another word. Complete the same reflection. What emotions does this word invoke in you?

The word: Puzzlers

Reflect. Think about it. Don't read the next paragraph until you are ready.

How did you feel about the word "puzzlers"? Was this one a positive or negative emotion for you? In my experience, more people (but not all) felt a positive connection to this word. Puzzlers can usually be solved. They are challenging. You can ask others for help but often you feel more satisfaction when you can solve it on your own.

We are going to come back to the term "puzzlers" later. Let's move on for now.

In Alberta, there are 7 mathematical processes attached to our mathematics curriculum. The following is an exact quote from the Alberta Mathematics 2016 Program of Studies.

Mathematical Processes	There are critical components that students must encounter in a mathematics program in order to achieve the goals of mathematics education and embrace lifelong learning in mathematics.
Communication [C]	communication in order to learn and express their understanding
Connections[CN]	connect mathematical ideas to other concepts in mathematics, to everyday experiences and to other disciplines
Mental Mathematics and Estimation [ME]	demonstrate fluency with mental mathematics and estimation
Problem Solving [PS]	develop and apply new mathematical knowledge through problem solving
Reasoning [R]	develop mathematical reasoning
Technology [T]	select and use technologies as tools for learning and for solving problems
Visualization	develop visualization skills to assist in processing information, making connections and solving problems

The mathematics curriculum is meant to be taught through these 7 processes. I find that the most misunderstood process is Problem Solving. In my experience it has been used synonymously with the term "word problems" in many classrooms. However, the typical way that word problems are used in the classroom is not as problem solving.

What is Problem Solving?
Just because it is a word problem, does not necessarily mean it is problem solving.

“Learning through problem solving should be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the type How would you …? or How could you …?, the problem-solving approach is being modelled. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies.

A problem-solving activity must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learnings in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement.” (Alberta Education)

Is this problem solving?
My students have learned how to add two digit numbers with regrouping. I give them this question: Johnny has 25 cents. He finds 35 cents. How much money does he have now?

This is just a word problem. Remember, "If students have already been given ways to solve the problem, it is not a problem, but practice." Of course, students need practice. There is nothing wrong with the question as long as we understand its purpose - to practice a skill that has already been learned/taught. However, we need to find ways to help students learn through problem solving as well.

Why use Problem Solving?
“Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. Creating an environment where students openly look for, and engage in, finding a variety of strategies for solving problems empowers students to explore alternatives and develops confident, cognitive mathematical risk takers.” (Alberta Education)

We need our students to develop problem solving skills. Not just in math class but every where. The problem solving skills they learn in math class can be used in many non math contexts. It's about perseverance. It's about knowing that you can figure it out given enough time and the right access to resources when needed. I truly believe that there are many missed opportunities to teach math through problem solving.

What do I do differently?
Let's return to the idea of Puzzlers that we touched on in the beginning. I use the term puzzlers instead of the word "problems"or "problem solving" because of the emotional connections students have with the words "puzzlers" and "problems". I use Puzzlers in my class all the time. Every day.

Using “Puzzlers”
One way you can do this is to provide students with a question that is slightly above their skill set. For example, you are working on 2 digit addition. You start with a simple question such as 21 + 33. This is a question without regrouping. You notice that students have mastered this so you tell them you are giving them a “Puzzler”. A “Puzzler” is a bit more challenging. A “Puzzler” is going to make you think. And, you may not finish the “Puzzler” in the amount of time you are given and that’s ok as long as you persevere for the entire time you are given. You give students the new question 28 + 33. This question has regrouping. Students are provided time to work in a Think-Pair-Share process where they work on their own (typically no more than 2 minutes so that frustration doesn't have time to set in), share their strategy with a partner and then some students share with the class. You can talk about how they solved it. How is it similar to what they've done? What makes it a puzzler? Then give them an opportunity to practice by providing another question that involves simple regrouping. This allows the students who didn't figure it out an opportunity to try the strategies that they heard. When students are comfortable, you can give them a new puzzler such as 48 + 95 or 138 + 26. You decide what the next more complex question would be. By using this processes, I only have to "teach" the first easy step. After that, they learn to puzzle their way through it.

I love this method because it teaches students that they can figure things out on their own. They build perseverance. They learn that it's ok to struggle in the beginning and to just keep plugging away at it. They learn that sometimes they may have to get outside help to understand and that's ok. Students can also work at their own rate on these. Often I will have several puzzlers written out on the board in different colors. They will try the first one, self assess and tell me if they are ready to move up in complexity or if they would like me to give them another of the same color type to try again. I've never had a student keep working on the same color if they should be moving up to the next color. They want to try a harder puzzler. They want to get up to the super duper ooper puzzler (that is always above grade level). When a student is confidently working above grade level, I just get them to create their own puzzlers. They solve it and trade with other students who are working at that level as well. This gives me time to support all of the students who need more help.

I created this simple poster to put up in my classroom. Download it and feel free to use it as is or edit it.

Problem Solving is NOT a unit of study...
Problem Solving is not a unit of study you do in June. It has to be embedded every day all year. You don't develop problem solving skills by limiting problem solving time to solving a word problem of the week or as a unit at the end of the year or even by doing a project at the end of each unit. It has to be developed EVERY SINGLE DAY ALL YEAR.

Problem Solving does not have to be a word problem...
Is this problem solving? 25 x 14
It depends. Did you already teach your students how to multiply two 2 digit numbers? If so, then it is not problem solving. It's practice. OR, have they just learned the distributive property as it relates to 2 digit by 1 digit multipliction? Then, yes it is problem solving because they are taking that understanding and applying it to a new context. Keep in mind, though, you may have students who knew how to do this before they came to you so for those students, it is practice.

Is this problem solving? 2 x 6
It depends. Have you already been working on these facts? If so, it is not problem solving. Even if they don't know the answer, they still have learned the skills to figure it out. OR, have you learned only up to 2x5 and students have to figure out how to use the facts they do know in order to figure it out. In this case, it's problem solving.

Where do I get the ideas for puzzlers?
The same place you get all your questions...your brain, the internet, textbooks, etc. Just stop pre-teaching everything. Help them understand the lowest level (21 + 7) and then give the rest as puzzlers. Start with the word problem that you would have given them at the end of learning.

Another simple method I've used...
When I taught high school math, I used to teach the steps for everything. Do this first. Do this next. Rinse and repeat. One day, in the middle of a lesson, I looked at the next question and thought to my self - this question is similar to what we have been doing but a little bit different - so I tried something new. This is how the activity progressed:
T: Ok class. We are going to try something different. Put your pencils down. Don't touch them. Now, I'm going to give you a minute to read the next question and think about what you might try first to solve this. Don't do anything though. Don't touch your pencil. Just think about it. Don't worry about doing the question. I'm not looking for an answer...just a possible first step to try.
I gave students a minute or two to think.
T: Turn to your partner, tell them what idea you had for a first step and why you think that would be a good first step. It's ok if you're not confident that it will work or that it's right. Just share.
Partners share ideas.
T: Who would like to share either their first step or their partner's first step?
Students share different first steps.
T: You've heard several possibilities for a first step. Let's decide on one to try first.
I had them vote. If I am worried that they are leaning towards one that won't work and this is the first time they are playing in a problem solving context, I will choose for them.
T: We are all going to do this first step together. Walk me through it.
I wrote on the board as students helped walk me through it. Many students also wrote it down but I'm a big fan of focusing on the process rather than copying when we are learning.
T: Now that we have the first step, take a moment to think individually about what our next step could be.
Repeat the Think Pair Share process until the problem is complete.

I loved this process because it taught my students what they needed to do in order to solve a question that appeared overwhelming in the beginning. Focus on the first step. Try it. Now what? Try it. Now what? Keep going.

After students were comfortable with this, I would make sure that we tried a recommendation that would not lead to a successful solution. This way, they learned how to deal with that situation. Oops. That didn't work. What did we learn from that path? Ok. Let's go back and try something different.

The biggest change I noticed? When students were given a question to try on their own, I stopped seeing hands going up immediately followed by "Mrs. Berg. I don't know what to do! You didn't teach us how to do this!" They had built the confidence to just try...something....anything...and see where it led.

All students are problem solvers.
Our job is to help them become better problem solvers.

Self-reflection question for you: What strategies do you use to build a problem-solving environment in your classroom?

In the Count I Trust!

noreply@blogger.com (Sandi Berg) — Fri, 29 Jun 2018 15:00:00 +0000

Until I began working on the Elementary Mathematics Professional Learning Opportunities (EMPL) project, I had no idea what "Trust the Count" meant nor did I understand how much it underpinned everything in mathematics. This blog post will focus on this idea - what is it? - how do I know if students trust the count?- what do I do if they don't trust the count?

The EMPL working group set one of the big ideas for Additive Thinking as
"Once students trust “the count”, they can flexibly manipulate numbers in order to make solving problems easier by

using Parts and Wholes
decomposing / Recomposing
partitioning
compensating
using Constant Difference"

Additive Thinking is about being flexible with numbers. This big idea says that before they can be flexible with numbers, they must trust the count.

What is Trust the Count?

In order to explore this idea, you are going to engage in a little formative assessment. Don't worry. It's not going in your gradebook!

Look at the image below. How many dots are there? How do you know?

Now, click on the video below and watch the image change.

How many dots are there now? How do you know?

Click on the video below and watch it change again.

How many dots are there now? How do you know?

Let's take a moment to assess your responses.

For the first image, you might have counted out every dot. Maybe you saw the 5 dots on a dice in the center and one on each side. Or, did you see 1-2-1-2-1? Or, 3-1-3? Or, something else? How you figured out that there are 7 dots in the first image is irrelevant.

After you clicked and saw the first and second animations, how did you answer? Did you know it was still 7 dots or did you recount the dots? Someone who recounts all of the dots does NOT trust the count. They have not yet internalized the idea that the quantity stays the same if all you did was move them around.

This is a simple test that you can use when your students are struggling with operations. Use a quantity of items that they have not yet subitized - 9 is a good number for this. If they recount the dots, then stop everything you are doing and focus on helping build the idea of trusting the count. Operations on numbers will not make sense if they don't trust the count.

Students who don't trust the count are most likely missing mastery of the following grade 1 Alberta outcomes:

Number outcome 2: Subitize (recognize at a glance) and name familiar arrangements of 1 to 10 objects or dots.
Number outcome 3: Demonstrate an understanding of counting by

indicating that the last number said identifies "how many"
showing that any set has only one count
using counting-on
using parts or equal groups to count sets

Demonstrate an understanding of conservation of number

So what do I do to help students trust the count?

These activities should be started in kindergarten when students are learning what numbers are all about and then continue on throughout the grade levels. Begin with the lowest level that makes sense for the student you are working with.

Activity 1:

Step 1: Place some objects in front of the student. Ask "how many are there?" Remember, it doesn't matter if they subitize or count each item. This step isn't about trusting the count. Ask "how did you know?" Did they count each one? Did they see groups of 2 and skip count? etc. Their skills here let you know if they are seeing the dots as individual pieces or can visualize grouping them together.

Step 2: Move the objects around the table. Ask, "how many are there now?" If they count each one, that's ok for now. Ask, "how do you know?"

Step 3: Move the objects around the table. Ask, "did I add any?" No. Ask, "did I remove any?" No. Ask, "how many are there now?" If they count each one, that's ok for now. Ask, "how do you know?"

Step 4: Move the objects around the table. Ask, "did I add any?" No. Ask, "did I remove any?" No. Ask, "Before checking, can you guess how many there are now?" It's ok if they are wrong or unsure of the answer. Say, "Let's check." Give them time to check. Ask, "how many are there now?" If they counted each one, that's ok for now. Ask, "how do you know?"

Repeat Step 4 several times but not too many times in a row. Spend about 5 minutes on this activity. Spending too long on it won't benefit them. It is better to spend smaller chunks of time on it but do it often. Don't change the quantity during a session, especially during these early stages.

Don't be surprised when (not if) at the end of a mini session, they are confidently repeating the number but then the next day, they are reverting back to counting each one several times. Building this idea takes time. Do...not...rush...this! I can not stress this enough.

Once students have done this activity with you a few times, you can have them work in partners, spreading the items out and pushing them tightly together but never changing the quantity.

Focus on building sense of very small numbers - 2 through 9 - before moving to larger numbers. Seriously, we rush this piece because we think students are getting it when they aren't there yet or that we have to get so far so fast. You have all the time in the world. Go slow.

CAUTION! I once had a teacher tell me that she did this activity with a student. She placed some objects in front of the student, had them figure out how many there were and then told the student to close his eyes! She rearranged the items, told the student to open his eyes and say how many there were. This is NOT about trusting the count. This is about trusting the TEACHER. They must KNOW that you didn't add anything or remove anything otherwise and they can only do that if they see you move the items.

Activity 2:

The situation: The student is subitizing all of the sides of a dice. They see six dots and say 6 without counting. They see 4 dots and say 4. When you show them a six and a four and ask them "how many are there in all?", they start counting each dot 1-2-3-4-5-6 7-8-9-10.

A different situation: You give them a word problem - There were 7 hot dogs ordered on Monday and 8 hot dogs ordered on Tuesday. How many hot dogs were ordered in all? They draw out the 7 hot dogs and then the 8 hot dogs and then count all - 1-2-3-4-5-6-7 8-9-10-11-12-13-14-15. This is a counting all strategy.

What do you do? Let's consider the dice showing six dots and four dots. Point to the six and ask "how many?" 6. Cover the six with your hand and ask, "how many?" If a student is unsure, lift your hand so they can see it. Make sure they understand that it's not a trick. You are not sneakily turning the dice while they can't see it. Cover the dice and ask "how many?" again. Repeat this, covering and uncovering until they confidently know that there is six under there. Again, be patient. The key word in the phrase "trust the count" is "trust". Trust always takes time to build. When they are confidently saying six, point to the first dot on the four and wait patiently. More often then not, they will say "one". Point to your hand covering the six and ask, "how many?" six. Point to the dot and say "sssssssssssssssssssssssseeeeeeeeeeeeeeeeeeeeeeeevvvvvvvvvvvvvvvvvvvvveeeeeeeeeeeeeeeeennnnnnnnnnn". Seriously, drag it out. If they say "seven" at any point while you are doing this stop and say "seven". If they don't, that's ok. They are probably looking at you like you have grown two heads but that's because they aren't sure where you are going with this. If they don't say 'seven" when you have dragged out saying it, just say "seven" yourself. The goal is for them to figure it out themselves but you may need to help a bit but not too much. You can point to your hand covering the six again and ask, "how many?" and then point to the first dot on the four and wait for a couple of seconds and, if needed, say "seven". Now, point to the second dot, wait a moment and then stretch out the word eight. Repeat this process until all four dots have been accounted for. Ask, "how many altogether?" Repeat the count if needed. Roll the dice again, and repeat this process.

This is called "Counting on". Students start with the total of one number (6) and then continue adding on from there. 6 -7-8-9-10. This is NOT an additive thinking strategy. It is a counting strategy but it will help move them towards additive thinking.

This is an activity students can do with partners once they understand it. Give them two dice and a cup. One student rolls the dice. The other student names them. The first student covers up one dice and the other student figures out how many. The first student checks to confirm.

You can also place a number of objects on the desk with a container that is big enough to cover many of them. Cover 5 of them and say, "There are five under the container. How many are there altogether?" When the student successfully counts on, uncover the container and check. Cover a different amount but don't change the total quantity. They might count on. They might add the two numbers. This helps build the understanding that you can separate a quantity into different parts but you still have the same quantity (trust the count). It also builds toward mastery of different addition facts. If a student is working on 9, and they see it as 1+8, 2+7, 3+6, 4+5, 5+4, 6+3, 7+2, 8+1 over and over again, they will add those facts to their basic fact toolbox.

A student completing the hot dog question might still draw out 7 hot dogs and 8 hot dogs but, if they have mastered counting on, they will say 7 -8-9-10-11-12-13-14-15 OR 8 -9-10-11-12-13-14-15. If they are still counting all, point to the first 7 (or 8 - whichever they drew first) and ask if you have to count all of these hot dogs again. Draw a circle around all of them and write the number 7 below it. Cover up the image of the hot dogs but let the 7 show. Ask, "How many hot dogs are hidden here?" 7. Point to the 8th hot dog and repeat the same process completed above with the dice.

Teachers will often tell me that they tell students to put the first number in their head and then keep counting. This is great. Visualization is an extremely important skill. I do this AFTER they have had lots of time to understand it first with manipulatives. I find that this makes the transfer to visualization easier if they already have something to picture in their head.

Activity 3:
Here is one more mini assessment for students working with numbers higher than 20. Have them build 24 with base ten blocks using two tens and 4 ones. Then have them trade/exchange a ten for ten ones. Ask them what their new number is. You might be surprised how many might have to recheck. They've learned to trade; most do it automatically. However, they don't necessarily understand that the total value hasn't changed.

In a later blog post, I will be speaking a lot about place value and how to dig deeply into it. Alberta teachers, please remember that place value is currently introduced into the curriculum in grade 2. This is NOT an activity for grade 1 students.

A Support Document

The EMPL group created a document (just scroll down a bit after following the link) explaining "trust the count", some misconceptions about number sense, and everything I have outlined above plus more. It can be printed and sent home to parents if you wish.

Source

Trusting the count is something we have internalized ourselves and we may be surprised by how many students come into classrooms without this understanding. For me personally, I know that I was playing board games and card games all the time before I even started Kindergarten. I was already subitizing. I could count on. These activities had me playing with numbers all the time. Now, I watch students come into school who have grown up on devices instead of on physical games. Many of them are coming in never having played a board game or pulled out a deck of cards. I can't control what happens before they get to me. I make sure they get all of those experiences in my class. We play lots of board games, card games and dice games. I try to make sure we spend at least 5 minutes every day on a game.

Self Reflection question for you: What strategies have you used to support students who do not trust the count?

Turning "Mad Minutes" on their head!

noreply@blogger.com (Sandi Berg) — Fri, 22 Jun 2018 13:38:00 +0000

Warning: This post will push buttons and I am ok with that. Every time I share this story during a workshop, I have push back from one or two people. That's ok. Everyone is at a different stage in their learning based on their own experiences. I am simply sharing my experiences and my observations with the students I have worked with. Some of you may question whether or not I expect my students to memorize their basic facts. The Alberta Curriculum requires students to recall their facts.

When I was in elementary school, I was great at memorizing. I could learn facts very quickly. People told me that I must be smart because I had memorized so much and so easily! I was very proud of this ability.

My teachers would give us Mad Minutes. These both thrilled me and terrified me. I knew my facts so I was pretty confident. How fast could I answer all the questions this time? At the same time, I was always terrified that I would make a mistake. If I made a mistake, nobody would think I was smart any more.

In my current role, I get to speak to a lot of adults. Some have a loving relationship with math. Others hate it. When I ask them why they hate math, more often than not, I hear horror stories regarding Mad Minutes. This is not the only reason they hate math but often it was the beginning of the downfall. Mad Minutes had taught some of them that you could only be "good" at math if you were fast and if you could memorize. We know that this is not true. When I am thinking deeply about math, I am not fast. I am making connections to other ideas. I am looking for nuances. I am analyzing. I am trying to find patterns. This can not be done quickly. I spend a lot of time talking with adults about the misconceptions that math is all about being fast and a good memorizer.

Let's take a quick peek into the past...
When I taught grade 3, I knew I wanted to do something different. I didn't want to put the time pressure on students but I still wanted to use Mad Minutes. So, my first change to Mad Minutes occurred when I displayed a "count up" timer on my board. My students all started with papers that displayed 10 questions on the 2 times table. They all started when I started the timer. When they finished, they would look at the timer and record the time they had finished at. I collected all the papers and marked them. The next day, we repeated this process...and so on and so on. When a student answered all 10 questions correctly in 30 seconds, I upgraded them and gave them the 3 times table questions. Eventually students were working on all different levels of facts so I put many copies of each of the facts into separate folders so they would come in from recess and grab their applicable mad minute, wait for me to tell them to start, write it, rinse and repeat.

I was very proud of the change that I had made. It was still about memorizing the facts. It was still about getting faster. I felt that my change had taken away the stigma though of "fast and memorization = good math student".

Fast forward 10 years. I give a "mad minute" and mark my students work. Suzy does all the questions and gets them all right. What does that tell me? Suzy knows her facts. Did I already know that? Yep or Probably. Jill does all the questions and gets them all wrong. What does that tell me? Think about it for a moment. Did you say that Jill doesn't know her facts? That might be true. It's also possible that Jill's dog died last night or her parents had a fight or she's mad at you or... There are many reasons that she might have received the mark she did. What about Patty who finished half the questions but got them all right? Take a moment to think about her. Is it because she needed more time to finish? Was she counting on her fingers? Was she working merrily along and then saw a snowflake out the window? I had an epiphany. I realized that those mad minutes weren't giving me any information about my students that I probably didn't already know OR I had to go talk to them anyways to get more information.

Before I share how I turned mad minutes on their head, let me share some concerns that are always shared with me when I talk about changing mad minutes.

Supporters of mad minutes will tell me that students need to learn to deal with pressure and mad minutes teach them that. I agree that we all end up working under pressure in our lives and knowing how to do that effectively is an important skill. I'm not sure that it's something that an 8 year old needs to learn under these circumstances. I can find better ways to help them learn that.

Supporters of mad minutes will tell me that mad minutes motivate students to learn their facts. They want to beat their score. Yes, this is true for some students. Is some students good enough for me in my classroom? Nope. I want more for all my students.

Some teachers will tell me about the changes they've made either to how they frame the activity or how they implement it. I love hearing about those changes!

Do I still want my students to master their basic facts? Yes! It's part of the Alberta curriculum that students master those basic facts. It has always been in the curriculum. It was never removed, despite what some people claim. All you have to do is go to the grade 3-5 curriculum and look for those multiplication facts. The clarifications introduced in 2014 more explicitly stated the need for students to recall the facts. Interestingly enough, even though mastery and recall are specifically stated in the curriculum, some will argue that mastery and recall is not the same thing as memorization.

What is mastery? They are able to recall their facts when needed AND they understand those facts and what they mean. What is recall? Recall occurs when students commit the facts to memory and retrieve them when needed. (Source)

I think of it this way...everybody has two toolboxes. One contains the "basic facts" that you have mastered. 3x5 = 15, 4x2 = 8, etc. You are able to retrieve these facts without thinking about it or figuring them out. You just know.

Your other toolbox is your strategy toolbox. This box contains all the strategies that your understand and use. It may include the traditional algorithm, doubling, knowing the distributive property, etc.

Now think of these two students: Joey has a very large basic fact toolbox. He's in grade 3 but has memorized everything up to 12x12. However, his strategy toolbox is very small.

Steven, on the other hand, has a very small basic fact toolbox. He is still working on committing the grade 3 requirements to memory. However, his strategy toolbox has several well used, well understood strategies that he accesses whenever he needs to.

Which student will be more successful? Steven. He has the strategies mastered to help him figure out anything he doesn't know. Joey will often struggle to figure out what he doesn't have memorized because he doesn't have a large strategy toolbox.

Where do I want my students to be at? I want them to have a healthy strategy toolbox AND a healthy basic fact toolbox.

Ok ok. Enough background. This isn't a groundbreaking, earth shattering idea. It is a small change that I made.

I'm going to use a simple addition mad minute. I apologize that all of the questions are vertical. I typically like to have about 50/50 vertical/horizontal layouts. I figured for this blog post example, it would work.

Everything written below in normal text is something that I say during the activity. If it is in italics, then it is something I do or notice.
--------------------------------------------------------------------------------------This week, we have been working on facts of 10. Can someone tell me what a fact of ten is?
Students explain that two numbers that add to ten would be considered a fact of ten.

Can someone give me an example?
7+3, 2+8

I display just the top row of questions on the SmartBoard for students to see.

I have placed some questions on the board. If it is a fact of ten, we are going to draw a rectangle around it. Is the first question a fact of ten? Yes. Let's draw a rectangle around it.
I draw a big enough rectangle that it leaves me some space inside to answer the question but I do NOT answer it.

Let's look at the next question. Is it a fact of ten? No. Do I draw a rectangle around it? No. What about the next one? No.

I am going to hand out your sheet. Please keep it face down on your desk for now.
When they are all handed out...
When I say go, you will turn over your sheet and find as many questions as you can that are facts of ten. What are you going to do when you find one? Draw a rectangle around it.
I am only going to give you one minute though. You might not find them all and that's ok as long as you keep looking for the entire minute. Remember, do NOT answer anything! Just draw rectangles. Ready? Go.

Students have one minute.

Stop! Now, I am going to give you one minute to answer JUST the questions you drew rectangles around. If you don't finish, that's ok as long as you keep going until your time is up. If you finish before the time is up, you can look for any you might have missed. Go.

Students have one minute to answer the questions.

Let's go through the questions. Is 9+1 a fact of ten? Yes we already drew a rectangle around it.

How much is 9+1? 10. I write 10.

What about 9+5? No.

3+3? No.
Keep going until you have gone through all the questions.
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Another week, I would give them the same sheet, either a fresh copy or this used copy. Maybe we've been working on doubles (ex. 3+3, 4+4). This time we will circle the doubles questions for 1 minute and then answer for 2 minutes. I only gave 1 minute on the "facts of 10" activity because all of the answers were 10. All other strategies will need more time so I give them 2 minutes of working time instead of 1 minute.

This process helps students look for questions that specific strategies are most useful for. They are quickly analyzing a question before solving it. They aren't panicking about getting all the questions done, resulting in the silly mistakes that often occur.

As I am typing this, I am thinking about extensions that I could introduce. So here is a new one for you...if a student finishes before the 2 minutes is up and they have checked to make sure they have found all questions, they can create and answer their own questions based on the strategy they are working on. For example, we might be working on doubles addition to 9+9. The student might extend to 10+10, 11+11, etc on their paper.

When I am finished sharing this idea with teachers, I still have some "yeah butters". This is one of the areas that some people are unwilling to change in. All I ask them to do is try it with students. See what happens. Listen to the conversations about their thinking. Check their anxiety levels. If the teacher is willing to try it 3 or 4 times with a positive mindset, I feel confident that it will be a win for the teacher and for the students.

Reflection question for you: How do you assess students' mastery of basic facts? What do you like about it? Dislike?

How I use "Number Talks" in the classroom

noreply@blogger.com (Sandi Berg) — Fri, 15 Jun 2018 14:31:00 +0000

Number Talks have become quite popular in the math class recently. You can purchase several books by Sherry Parish (NOT an affiliate link) that teach you how to run them or read many blog posts about them.

A friend of mine tweeted about his experience running a Number Talk and through follow-up conversations, we realized that I run mine a little differently than the format shared in the books. Although I talk briefly about my process in my Parent Night blog post "What are these stupid strategies kids are learning days? Why can't they just learn the #@$#* way I learned it?", I thought I would focus a single blog post on it so that I can go into my process in more detail.

How many of you have students who are effective at communicating their mathematical thinking? They won't be amazing at communicating their thinking in the beginning (unless they have been practicing this a lot in previous years) and that's ok. This is a skill that students develop over time. They must be provided with opportunities to practice it in meaningful settings.

You can use number talks at any point in students' learning and understanding. Maybe you are introducing 3 digit addition and want to see what strategies they have for 2 digit addition. Maybe you have been exploring 3 digit addition for a while and want to see what they will do if given a 3 digit plus 4 digit number.

I typically have students complete their work using as much mental math as possible, although they know they can always access manipulatives if they need it or use a mini whiteboard to record their thinking as they work through it. However, I have noticed that people often use different strategies when given paper and pencil versus solving it mentally. When I ask adults to solve an addition question in their head, they often use alternate strategies but admit that if I had said to use paper and pencil, they would have used the traditional algorithm instead.

Everything written below in normal text is something that I say during the process. If it is in italics, then it is something I do or notice.

------------------------------------------

Step 1: Post a meaningful question for students to solve on the board so that all can see it. This might be a "How many dots are there?" or "Add 324 + 493" or pretty much anything else that you would like to focus on.

I am going to give you 2 minutes to work on figuring it out. It's only 2 minutes though so will everyone finish? (No.) Is that ok? (Yes as long as you keep working on it the entire time.) If you figure it out and still have time left, see if you can confirm your answer by solving it a different way.

While you are thinking, place a closed fist in front of your chest. That will tell me you are thinking. When you have solved it one way, put your thumb up. If you have time, solve it a different way. If you do, put another finger up.

Step 2: Students work. My "2 minutes" are up when I see that almost every student has their thumb up.

Turn to your partner and explain to them how you solved the question on the board. If you didn't finish, that's ok. Share as far as you got. I should hear things like "First I did this. Then I did this."

Students have time to share their strategies with their partner. During this time, I wander around the classroom listening for a variety of strategies that I would like to have shared during the group sharing time. I will let students know that I think they have explained an interesting strategy and will ask them to share it with the rest of the class. I try to focus on the students who may not typically share in class and the students who don't often feel success.

Step 3: Class Sharing. During this time, I will call upon students to share strategies, starting with the least efficient / easiest strategy first and then moving onto more complex strategies. I will ask students who want to share their strategies, and of course the students whom I talked with earlier always put their hands up because they know I am going to call on them anyways.

Who would like to share how they solved this question or how their partner solved this question? If you didn't have enough time to finish, that's ok. You can still share your thinking as far as you got.

I phrase the first statement that way so that students know they have permission to talk about the strategy that their partner used. I started saying it that way because, especially in the beginning, I would have students who didn't / couldn't solve it on their own until after they talked to their partner. This gave them permission to share even if it wasn't their own original idea. In fact, after students have had experience with Number Talks, I will require them to explain the strategy their partner used rather than their own strategy. This really helps them focus on listening to understand and to ask clarifying question.

I phrase the second statement this way because it's important that students realize that it's not about solving it quickly. Even if you didn't finish, you still have ideas and strategies you can share.

As students are explaining their strategy, I am asking them clarifying questions, ensuring that they are being very specific and clear about their thinking. For example, if they are solving 24 + 95 and they say 3 + 9 = 12, I will ask them - Where did the 3 came from? They will tell me that it's from 1 + 2. I will ask them - is that really a 9? They will tell me - No, that's 9 tens - or - No, that's 90. As they are explaining their strategy, I am writing it out the board. This helps students see the thinking represented symbolically.

Step 4: After a student has finished explaining their strategy:

Who else solved it the same way?

Students are going to say - I solved it that way too - so you might as well ask. This also shows this student that other people thought about the question the same way they did.

Step 5: Does it make sense to others?

Let's take a moment to think about that strategy. If that strategy made complete sense to you, give me a thumbs up. If that strategy made no sense to you, give me a thumbs down. If that strategy makes some sense to you, give me a thumb in the middle. It's ok to give me a thumbs up, thumbs down, or thumb in the middle.

Having students self assess this way gives you an opportunity to see how students feel about this particular strategy. Does it make sense to most students? Who doesn't it make sense to? When students see a strategy for the first time, you might see a lot of thumbs down. The next time they hear it, you will probably see some more thumbs in the middle. The more they see it or hear it, the more their thumbs will move.

Step 6: Repeat with the other strategies. Don't do too many strategies, though. Focus on about three. Any more than three and students tend to lose focus.

Step 7: Reinforce the idea that there are many ways to solve a single question.

We just heard 3 ways to solve this question. Which way was correct? (They all are). Take a moment to think about the one that makes the most sense to you today. Wait a few seconds. Turn to your partner and tell them which one makes the most sense for you today and tell them why. Give students a minute or two at most to share.

------------------------------------------

Number talks shouldn't take up more than 10 minutes in the class. I wouldn't do these every day but I like using them about 2-3 times per week, regardless of the grade level. They have so many benefits to them that the time invested is well spent. Every student shares their thinking with their partner so every single student is practicing communicating their thinking. They are seeing their thinking represented symbolically. They are listening for understanding to others' strategies. They are seeing strategies move from less complex to more complex. They are reinforcing the idea that there are many strategies for solving the exact same question.

Reflection Question for you: Have you ever used a process similar to Number Talks in your classroom? What benefits have you seen? Are there parts of the process you have struggled with?

I would love to hear your thoughts in the comment section!

Homework and "My Favorite No"

noreply@blogger.com (Sandi Berg) — Fri, 08 Jun 2018 19:30:00 +0000

Let's step back in time to 10 years ago. I am teaching high school math. Every day, I diligently teach the class, assign practice questions and then assign homework which I would check the next day for completion. 0 for incomplete. 5 for partially complete. 10 for complete. This mark was not based on understanding. It was based on compliance. What did I notice? My compliant students were completing it. The students who were struggling or who weren't engaged, didn't complete it. Writing a mark in my book changed nothing. Their compliance was affecting their grades since I was following the typically standard "Homework counts for 10% of the grade" designation.

I quickly realized that I was completely and utterly wasting my time. This did nothing to help me understand what my students knew and didn't know. So I changed. At the end of class, I gave students questions out of the book but told them to do as many as they felt they needed to do in order to demonstrate understanding. I wasn't checking it though. I created videos for each of those questions where I solved them and posted them online for students to view if needed. If they were stuck, they had the video they could look at. I encouraged them to google for help as well. Anything they were still stuck on, they could still come to me for help.

So, if I removed homework checks from my daily routine and deleted it from my gradebook, was there something I added? Absolutely. At the beginning of each class, I handed out a "mini quiz" with one question from the previous day on it. They had about 5 minutes to answer the question. At the end of the class, while students were working, I went through the mini quizzes, calling students up as I marked each one to return it, having a quick conversation or "mini lesson" to fix understanding. If I felt that enough students were struggling, I planned to rework the lesson for the next day. These mini quizzes were worth 5 marks each. Students could come and rewrite mini quizzes over lunch hour or during their end of class work time if they felt they understood it better. I was writing a lot of new mini quizzes.

One day, I saw a video entitled "My Favorite No". This teacher started out by giving a one question quiz (just like I did) but here's where she differs. She immediately, in front of all those students, went through the quizzes and sorted them into "Yes" (the answer is correct) and "No" (the answer is incorrect) piles, saying "Yes" or "No" as she went. When she reached the end of the quizzes, she went back through the "No" pile looking for her favorite "No". She looked for a common misunderstanding / error that students predictably had. Rewriting the students' work (to protect their identity), she told students she had found her "Favorite No". Yes, there is a mistake but there is good math happening in it as well. Then, she asked students to find all the good in the work. Students shared all the things that this student had done correctly. Then, she asked them to find the error. Once the error was identified, she asked them to help her fix it.

I love this process. Instead of waiting until the end of the class to go through mini quizzes individually with students, errors and misunderstandings are addressed immediately. Students who made mistakes are seeing it solved correctly...not at the end of class but right away. Students are celebrating mistakes in class, looking for the good work that occurred and looking for errors that need to be fixed.

For me, this was powerful and a game changer. It allows me to immediately address misconceptions. I can refocus my lesson for the day if need be. The time at the end of class became usable for other support. Students weren't spending their lunch hours in my classroom rewriting mini quizzes and I wasn't spending my own time marking them.

Did I enter marks into my grade-book? No. This wasn't about compliance. This wasn't about compiling numbers to show that students had done work. This was about finding and addressing mathematical misunderstandings. Did I spend time looking through each paper to identify students with specific issues that needed to be addressed? Absolutely, but I only really needed to deal with students with major misunderstandings as most errors were dealt with during the classroom discussion.

I love how one little change can make a huge difference in the culture of the classroom.

Reflection Question for you: What do you do to build a classroom culture that allows for students to respectfully discuss errors that have been made?

What are these stupid strategies kids are learning days? Why can't they just learn the #@$#* way I learned it?

noreply@blogger.com (Sandi Berg) — Fri, 01 Jun 2018 18:48:00 +0000

Wow! This turned into a huge post. I apologize! You may want to break this reading up into several sittings. Sorry! I have included the slides that I share so you could read the slide and then, if interested, read the details below it and if not interested, just skip ahead to the next slide.

I chose to start my blog reawakening with this question because I get this question a lot from parents...not all parents but many. It's imperative that we have parents on our side when we are teaching in a way that is often different that the way they learned. If parents understand that these changes are good, then they are more open to them.

In order to help with this, I hold parent nights so that parents can come and learn more about the pedagogy of our curriculum. They are encouraged to bring in their children with them. (Preferably no more than one child per parent) Some of these sessions take place where I work (CARC) and others have taken place within a school that has arranged for me to hold it there. Typically, I set each session to be grade specific or a very small span of grades. This helps narrow the examples and activities that are provided and to focus of our conversations.

Here is the link to the Google Slides I use/adapt. You have my permission to use/adapt this presentation to meet your needs!

Let's see how a typical parent night runs.

Everything written below in normal text is something that I say during the workshop. If it is in italics, then it is something I do or notice.

--------------------------------------------------------------------------------------

The parents arrive, some with their children and some without. They all have different expressions on their faces when they come in. Some are excited to be here. They are ready to learn and to have conversations around math. Others look like they are ready to bolt. It is obvious that they have not built a positive relationship with math. Others look like they are here to challenge everything. They don't like the way that their child is being taught and have come prepared to put an end to that. The rest of the parents are somewhere in between.

During our time together, we will be looking at math in a variety of ways. Everything we do follows the philosophy of the current Alberta Mathematics Program of Studies. However, your teacher may be teaching strategies that are different that the ones we do today.

We are going to explore strategies that may be new to you. When you see them or hear them, keep an open mind. Ask yourself, do they make sense to you? Have you ever seen them before?

We are going to have an opportunity to make connections between what you learned in school and how your child is experiencing math now.

At the end of tonight, I hope that you feel like you have a better understanding of the philosophy of the Mathematics curriculum.

We only have 1+ hr together which means that our time together is just a teaser. We could spend days and days on these ideas.

Each of these questions was posted on the screen one at a time so that they could focus their discussion.

Question 1: Parents, turn to your child and ask them this question. Listen carefully to their responses. If you don’t have a child here tonight, write this question down so that you can ask them later.

Question 2: Parents, tell your child one thing you think THEY are really good at in math.

Question 3: Parents, ask your child “What is one thing you feel you really struggle with in math?” Again, listen carefully to their responses.

After conversations: You may or may not have been surprised about some of the answers shared during this conversation. Often, we have different beliefs about our strengths and areas for growth. Have this conversation several times throughout the year. Expect the answers to change over time.

These two questions show up separately.
Question 1: Parents, do not answer this question out loud. When you were in school, what was your least favorite subject?

I talk with a lot of people around the world and often ask them this question. What do you think most would say? Pause Unless they are a high school math teacher, university math professor, engineer or in another math related jobs, many adults (but not all) have told me that math was their least favorite subject. Why?

Many people were shown algorithms and procedures and told to follow them blindly. "When dividing fractions, invert and multiply. Don't ask why." Often when asked how the algorithm works, they were told just to follow it. Some people had no issues with the algorithm. They could follow it. Many who could follow it also understood it while others followed it blindly but could get the answer. However, many people need or want to understand how it works. Without that understanding, they couldn't follow the algorithm.

People often identify their memorization skills as the reason they are good or bad at math. If you have all your basic facts memorized, you were often perceived as good at math. If you didn't have them all memorized, well...you obviously weren't good at math.

How fast were you at finding answers? Could you do it in your head? If you thought slower than others, if it took you more time to find an answer, if you needed paper to solve it, you were often perceived as being bad at math. Speed became a division at math.

If you were good at running algorithms, following procedures and memorizing basic facts, you were successful in math which made you feel like you were good at math. If you weren't good memorizing, you probably felt you were bad at math.

Question 2: What area in math do you think people often identify as being a huge issue? Algebra is often identified as the biggest issue for many people. Even though we use it all the time, people don't see it expressed in variable form very often so they assume they are not using it. Algebra in symbolic form can be intimidating. People have told me that they don't understand how there can be letters in math, or why the value of those letters change. They "learned" algebra by jumping to abstract concepts too quickly rather than exploring it concretely first.

It is through our experiences with math that makes us feel that we are either good at it or not good at it.

If, in your head, you said math was your least favorite subject or if you feel like you are bad at math, the good news is that you were NEVER bad at math. You just weren't taught the way you needed to be taught in order to understand it. And that's what we are trying to do in schools. Teach math in such a way that it makes sense to them rather than through rote memorization.

There are three things teachers want you know about their mathematics classes. The first goal is for your child to love and understand mathematics. Both goals need to be attained for future success. If a kid understand math but never loves it, they won’t think about all the opportunities they may have in their future that could involve mathematics. For those that love math but don’t understand it, there may come a time that they dislike math due to their limited understanding. And we want kids to keep as many options open as possible and the reality at this point is that some students fear of mathematics prevents them from being as successful as they can be or pursuing certain areas of study

Secondly, you are going to see a lot of concrete manipulatives in your child's class. Tools such as base ten blocks, integer tiles, unifix cubes, cuisenaire rods, and other tools help students build a concrete understanding which allows them to make connections to abstract representations. The use of manipulatives or other materials does not mean that the mathematical understanding is any less, and often it means the opposite. If a student can understand an algorithm AND explain that with the use of manipulatives, they have a stronger foundational understanding of the concept.

Math class is no longer sitting in rows, working on worksheets. It is an active learning process. They will not be talked to for a period of time and then sent home with a ton of homework to practice what they heard. They will be doing activities and group work along with some individual work. They will build mathematical ideas using manipulatives. They will talk to people about their understandings and strategies. Some of the activities may spill into their outside of school life and will hopefully engage family or friends.

I hear a lot of people complain about the "New" math. The math has not changed. They still need to learn to add, subtract, multiply and divide. They will still learn math in school. However, the WAY your child is taught math will not be the same as the way you were taught. And change isn’t a bad thing. There will still be times that some things seem familiar, but there are other times that things will look or sound different. What we hope you will do is encourage your child to work on mathematics, to ask your child to explain what they are doing and why, to involve your child in as many ways as you can think of that involve mathematical thinking (ie grocery bills, filling up with gas, counting out money) and tell your child that they are capable and can be a successful mathematical thinker.

Two questions are posted on the screen for everyone in the room. These are adjusted based on the grade level. Choose one question to solve in your head. Do not use paper or a calculating device other than your brain. Hold a closed fist in front your chest while you are thinking. When you have solved it, put up your thumb. If you have time and can think of another way to solve it, put up another finger.

I give enough time so that almost everyone has at least one thumb up If both of you have your thumbs up, please explain to the other person exactly how you figured out the answer. I should hear things like, "First I did this and then I did that." Parents, you will go second. If you are still thinking, go ahead and finish thinking before sharing.

Participants are given time to share their thinking. When it looks like everyone has had a chance to share their thinking... Let's share your thinking with the rest of the group. Who would like to explain the strategy they used or their partner used to solve one of the questions on the board?

Someone shares their thinking. I ask questions to clarify their thinking for others and make sure that steps aren't skipped. Throughout their explanation, I am recording their thinking on the board. Did anyone else do it the same way? Cool. If that makes sense to you give me a thumbs up. If that makes no sense to you, give me a thumbs down. If that makes some sense to you, give me a thumbs in the middle. Who did it a different way?

The traditional algorithm will always be shared by a parent. Sometimes they are embarrassed that they used it. Others are proud. Is it ok for students to use the traditional/standard algorithm in school?Yes! There is absolutely nothing wrong with using the traditional/standard algorithm. It is absolutely a strategy that works.

We run through several examples So, we have heard many different ways to solve those two questions. Who's strategy is correct?

Participants always respond that all of them are correct. If you only walk away with one understanding from this night, I want it to be this - there are many ways to add numbers. The "grade 3 outcome" says that students must be able to add "3 digit" numbers. It doesn't say how they must do it, just that they must be able to do it. They must master a strategy for adding numbers but what they master may or may not be different than the one we are most familiar with and that's ok.

There are many different strategies for adding, subtracting, multiplying and dividing. There are the traditional methods we learned in school but there are also many other strategies students are exposed to like the ones shared today. You may find your child using strategies that are very different from the ones you learned and that’s ok. We want children to understand that there are many strategies but…

They don’t need to master all of them for a particular concept.

There is no right or wrong strategy as long as the strategy is efficient and effective for your child. It might not be the most efficient strategy available but it may be the most efficient one for them at their level of understanding at this moment in time. With experience, students will develop more efficient strategies.

So what does it look like when a student has mastered a strategy? If a student has mastered a strategy, then for that student, the strategy is efficient, effective and explainable. If it is efficient, the student can use it to find the correct answer in a reasonable amount of time. Although there is no time limit, I think about a minute would be a fair time estimate. When a student is first introduced to a new strategy, it might take them 10 minutes to use it to answer a question and that's ok. They just haven't mastered it yet. Through more experience with a particular strategy, they will usually become more efficient at it. What is efficient for a grade 2 student to add numbers will look very different than what a grade 6 student may use. Efficiency increases over time and experience. If a student has mastered a strategy, then for that student, the strategy is effective, it must get you the right answer every single time. Sometimes students make up a strategy that flukes into the right answer but doesn't work in any other situation. So, it must be effective every single time. If a student has mastered a strategy, then for that student, the strategy is explainable. They have to be able to explain why the strategy works, why they chose that strategy to use for this particular question and whether it will work for really big numbers and really little numbers.

We have to remember that many of us were taught math focusing on learning the traditional/standard algorithm. If we felt we were successful in math, we often were able to learn those algorithms easily and feel comfortable using them. We have had many years of experience working with that algorithm. If you were one of the adults who, in your head, said that you don't feel like you were good at math in school, you may have said the algorithms and procedures didn't make sense to you. Alberta Education dictates the outcomes we must teach, such as "adding three digit numbers", but they do not tell us how to teach those strategies. That has always been up to the professional judgement of teachers. Nor does it dictate which strategies students must use. That is up to the student.

At this point, we engage with a math activity that is specific to the grade level of students who are attending. For example, if it's grade 5, I will show them how to use base ten blocks to build a conceptual understanding of 2 digit by 2 digit multiplication then work through representing that as an array model. I will be sharing these activities in later blog posts.

So what can you do at home to help your child see more success in math? Play games! Board games, dice games, card games, etc. It’s better to play for a short amount of time but more often than playing for a long time once in a while.

It's important to understand the difference between a game and a practice activity. "Games" that don't require any kind of strategy are simply tools for practicing basic facts. For example, roll two dice and multiply them together to get an answer fits in here. These aren't games. These are pounding a nail in over and over until the nail is embedded in the wood. Children need some of this but it doesn't need to be the main focus in order for them to master those facts.

A game requires some sort of strategy. Being strategic will help them build deeper mathematical thinking. They need many opportunities with this.

So let's take a look at some examples of games that we have played in the past.

The card game war. Is it a practicing activity or a thinking game?

War is definitely a practicing activity. There is absolutely no strategy involved. You simply flip over cards and figure out who is larger. It's about practicing comparing numbers. One person doesn't even have to think if they rely on the other person to decide who is bigger each time. So how do we turn War from a practicing activity to a strategy game"?

Divide a deck of cards so that each player receives one suit of cards. (ie. You get all of the Clubs) Place the rest of the cards back.
Players fan out their cards so they can see them but the other player can not.
Players each choose one card to play and place it face down in front of them.
When both players are ready, they flip their cards. Highest card wins. If there is a tie, players replace the card back in their hand.
Cards that have been claimed go into a discard pile. Not back in their hand.

Participants are given a few minutes to play "War...with a Twist".

Variations:

Player A takes all the red cards. Player B takes all the red cards. Each player chooses two cards for each round. Decide at the beginning of the game which operation will be applied.

Variations:

Player A takes one black suit AND one red suit. Player B takes the other black suit and red suit.
Black cards are positive numbers.
Red cards are negative numbers.

Bump Games are something I have added to my classroom over the past few years. First, I'll teach you how to play and then you can decide if they are a practice activity or a strategy game.

Instructions: Player 1 rolls the two dice and add together. For example, I roll a 1 and a 3. What do I get? 4. This is Alien Bump - Multiplication by 6. What do you think I will multiply the 4 by? 6. What is 4 multiplied by 6? 24. I cover the alien beside the number 24 using a tile (or a coin, or a bingo chip - whatever you have at home.) Do not cover the number 24 otherwise you won't be able to see it. Make sure to put it on the alien. My partner rolls. They get a 2 and a 5. She adds those together to get 7. Then she multiplies 7 by 6 to get 42. She places her different colored tile on the alien beside 42. We keep going back and forth one at a time. What happens if I roll a 7? My partner has already covered the 42! I bump her chip off and give it back to her and then cover the alien beside the 42. What happens if I roll another 4? I'm already on the 24! This time, since I am already on the spot, I get to put my tile on top of the number 24, claiming this spot. Now that I have a tile on both parts of the spot (the alien and the number), it can't be bumped off! Play until someone uses all their tiles or the entire board is covered.

Give participants time to play.

There are many different Bump Games that you can play.

So, is a Bump Game a practice activity or a strategy game?

Bump Games do not require ANY strategy. It’s complete luck. This is a good way to introduce the game however. Once they are familiar with the process, then switch it up! Make it more challenging.

Variations:

Add an extra die. Ex. When the game asks for TWO 6 sided dice, give them THREE 6 sided dice.
Roll all THREE dice.
Player chooses which of the TWO dice he/she will use.

Variations:

After rolling, player may add ,subtract, multiply or divide the two dice to get a number that is useful.

There are many games that you can play to help students practice basic facts but still allow for strategic choices.

Parents, it is extremely important that you never say "I am/was bad at math". Women, this is especially true for you. Research has shown that when a mother tells her daughter that "I was bad at math", the daughter's belief about their own ability drops and as a result, so does their achievement.

It is ok to say, "I don't understand this...yet" or "I didn't learn this way. Please help me understand."

Your child is going to come home using strategies that you may not have seen before. That's expected and it's ok if you don't understand them right away. Ask your child to explain it to you. Remember though, they may not have mastered it yet so be patient. Remember that they will often explore the strategies using simple examples and then move on to more complex questions. I don't know how many times I've seen a strategy presented on Facebook using an over simplified example. Adults get upset because you would never use that strategy on that particular example. Please keep in mind, we only do that when introducing a strategy. We are not saying it makes sense to use that strategy on such an over simplified example. However, it helps students explore the strategy before it they have a chance to use it with more appropriate questions. Try that strategy out yourself. Remember, it might not be your most efficient strategy and that’s ok. If you are stuck, google it or ask your child's teacher to help you understand it. I will be creating blog spots exploring many different strategies and how they can be introduced, moving from simplified questions to more complex questions.

There are many opportunities to work with numbers at home. At this point, I share activities that are appropriate for the age level of the students in the session. Baking is a popular one for all age levels. Giving children two few forks to put out at supper and asking how many they are short. Counting toys. There are too many to list here.

My "dumping grounds" website contains some of the activities that I have created for different grade levels (based on Alberta Curriculum), strategy games and practice activities that you can access to help your child. Videos are slowly being added for many of the ideas as well. Feel free to use this to help you understand and explore math!

One final reminder: My goal is to help my students build deep, conceptual understanding, through concrete experiences and making connections to abstract representations. They are still expected to master basic facts. They will still learn how to add, subtract, multiply and divide. They are just not doing it through rote memorization, flash cards or mad minutes.

--------------------------------------------------------------------------------------
When the session is over, parents always come up to me to talk about their math experiences and how they connected with the math in this session. Here are some comments that I always receive, regardless of the grade levels I focused on or whether they believed that they were good or bad at math.

"Why wasn't I taught this way? If I had been taught this way, I would have understood math!"

"When I came here tonight, I didn't understand why my son was drawing boxes to add. Now I understand." [This was after I showed them how to add using base ten blocks]

"I always thought I was cheating. I couldn't do math the way the teacher showed me so I had to figure out my own way of doing it. Now, I know I wasn't cheating."

"When you said that I was never bad at math...that I just wasn't taught the way I needed to be taught in order to understand, that opened the world to me. I'm not stupid! I can learn this."

"My child might use a strategy that's different than I do and that's ok!"

"When's the next Parent Night?I want to learn more math!"

Parent nights have evolved so that parents who have participated in this beginning "theoretical" portion come just for the new activities and parents who are new to the event come earlier to learn the "theoretical" part as well as all of the activities.

When parents have an opportunity to experience and understand what is going on in the classroom, they will support you. I believe that it's imperative that they have an opportunity to participate in a parent night the first few weeks of school. This will set the stage for the rest of the year. I encourage you to create a class website where students create videos explaining all of the strategies you explore. What does it look like to solve a question using concrete materials? What does it look like to solve it using each of the different strategies they have learned? What a great assessment tool for students as well? Do they understand their chosen strategy well enough to explain it to someone who has zero understanding?

Wow! Did you make it to the end of this post! Great job! That was a lot of reading. I won't be making all of my blog posts that long. I won't be able to keep up with my one post a week if I do that every time. :)

If you're not too exhausted from all that reading, I have a Reflection Question for the you: What opportunities have you provided to help parents understand the strategies that their children are exploring in class? I would love to hear about them in the comments below!

If you have any questions about how I ran parent nights that I didn't address, please feel free to ask! I will respond to all questions as best as I can.

Have a fantastic day and I will see you in a week!

Sandi

Welcome Back from an Extreme Hiatus!

noreply@blogger.com (Sandi Berg) — Fri, 25 May 2018 15:21:00 +0000

Oh my how time flies. I was recently asked by a publisher if I had ever thought about writing a book. I love the idea but realized if I am going to write a book, I should probably get back to writing on my blog. So here I am!

Here's a little update about what has been happening in my world over the past 6 years.

Through my current role as a Curriculum Implementation Support Consultant for Central Alberta Regional Consortium, I have done a ton of work around and reading on research on K-12 mathematics. I have been working with K-12 math teachers, exploring ways to teach math in a way that makes sense to students, rather than through rote memorization of rules, procedures and facts. I'll be writing about this alot as this is the question I am always asking myself: "How do I teach this outcome for deeper understanding so that students will make connections, think critically, problem solve, often through concrete experiences building towards abstract thinking and mastering the outcome.

I started posting all of the resources I have created for use in workshops, for modelling lessons in classrooms and at teachers' requests for specific activities, into an online "dumping grounds" using OneDrive. These resources can all be downloaded an edited as needed by anyone as they are in Word, PowerPoint, Smart Notebook, etc. If you check it out, you will notice that there is a folder for every grade which contains grade specific materials. Remember, these are based on Alberta Curriculum so if you are not from Alberta, you will have to explore other grade levels to find appropriate materials. Some folders, like fractions and algebraic equations, have their own folders as they span many grades. Many of the activities include "scripts" as part of the teacher instructions. They will be written out in detail: You say this, you write this, you ask this, they should respond this...etc. This isn't to script you! This is so you know what is going on in my head. I've done many of those activities 20-30 times in classes and I am constantly going off script. Every time I teach it, students interact with the lesson differently and make me think of the idea/activity in a different way. Maybe they needed more scaffolding. Maybe they needed to be pushed further. When I say or do something that makes the activity better, I go back to the instructions and modify those so they are constantly being updated. I will be focusing many of these blog posts on the activities that are already in those folders. There is a games folder which contains a Basic Facts games folder. Inside that folder, you will find "games" that help students practice basic facts. These aren't about helping students build understanding but rather an opportunity to build fluency with those basic facts.

Through my role at CARC, I have been part of the development of the Elementary Mathematics Professional Learning which focuses on supporting K-6 Alberta Math teachers. The content is organized around Equality, Additive Thinking, Multiplicative Thinking, Instructional Practices and Assessment. Although connections are made to Alberta curriculum, these big ideas are applicable to everyone. Many of the blog posts will focus on these ideas.

I have so many ideas that I look forward to blogging about. Big ideas will be explored over several blog posts, building upon previous ideas. I will share activities that I have shared throughout many of the workshops I have provided over the past 7 years. I encourage you to share ways that I could make these activities even better for students.

I will always connect my activities to the current Alberta curriculum but in the comments, please feel free to share how it connects to the outcomes where you are teaching.

I know that I will push buttons. After 7 years in this role, I have changed my teaching dramatically! Some things have been dropped, added and remained the same. My goal is to reach each and every student!

I know that some of the posts will push buttons and I am ok with that. I know that you may not agree with everything I have to say and I am ok with that. I know that some of you will share your ideas and beliefs about how mathematics should be taught that will challenge my own ideas and I am ok with that. I ask you to read each of these posts with an open mind. I encourage each and every one of you to share your ideas in a respectful way and I promise to read them with an open mind as well.

When I share an activity with you, please know that I have taught it in a classroom to students and in workshops with teachers many times, sometimes up to 40 times over the past 7 years. Also remember, that I know that I can always make them better. Each class, each student is different and that means the lessons must adapt to meet their needs, usually on the fly.

I look forward to taking on a more active blogging role. I am setting a goal of blogging once a week. That is my promise to myself and to you.

Sandi

Minute to Win It Math Circuits

noreply@blogger.com (Sandi Berg) — Thu, 13 Dec 2012 19:13:00 +0000

I'm supposed to be planning for a grade 8 manipulative workshop I'm holding in January so I've been researching hands-on/student centered ways to guide student learning. The problem always ends up that I find all these fabulous blogs that I immerse myself in reading and I get totally off topic. But, that's also a good thing, right? I am totally immersed in reading fabulous blogs! Well, today I came across Reflections of a High School Math Teacher and couldn't stop reading.

At this moment, my favorite entry is Minute to Win It Circuits...Move and Learn. Here's an overview of the activity:

He set up 5 stations around the class, allowing for two groups at each station at once. Which means, of course, that there was actually two copies of each station.

Station 1:

Math problem written very large on a full sheet of paper. "Write the phrase as a power and then evaluate. four to the third power."
Cut up the paper into 6 equal pieces (approximately 3x3).
Randomize and clip together.
Make sure there are two sets of this problem at the station. One for group 1 and one for group 2.

Station 2-5: Same idea. Different problems.

Process:

Groups are assigned to stations.
Set timer for 1 minute.
Partners work together to unscramble the pieces and then solve the problem.
Write answer down on paper.
Check answer based on answer provided at station.
After minute is over, they rotate, moving to the next station.

In the comments, someone (A. Nonymous) shared that he/she puts the answer to the problem they completed at the NEXT station.

I think this is very cool! If you have 5 stations your time would be

1 min to work
15 seconds to clean up
15 seconds to move to the next station

Total time used: 7.5 minutes.

I'm throwing out some ideas here for implementing in my own classroom (when I return back to the class).

What if I had students show work and write answers on index cards (or regular paper) and at the end of the minute, they put answer in container and move on. Then I've collected it.
Hmmm....to make sure they remembered name, I could number papers and assign students numbers. Give all students a "booklet" of papers with their number on it ahead of time. They answer, rip of paper and put in container.
Answer provided at next station (as per A. Nonymous). They would have to switch stations quickly to get time to read answer.
Focus each question on an outcome I am specificially assessing. But then each students would have to work individually. Still would work. More question sets at each station then. Don't think I'd want to do more than 5 stations for time.
Could have some stations as "group" effort. Find a partner at the station you're at and answer the following. Write partner's name on bottom of paper.

We'll see which ideas I actually keep when it comes down to it.

I love, Love, LOVE this idea and hope that I don't forget about it when it comes down to actually trying the activity. Maybe I'll even have the Grade 8 teachers who are attending my session in January attempt something like this. Ooooh....brainwave! <evil laugh>

If you try this, I would love to hear how it goes, what questions you used, what course you tried it in. What worked. What didn't. Anything and everything (except what you had for lunch).

Rewrites

noreply@blogger.com (Sandi Berg) — Thu, 29 Nov 2012 19:55:00 +0000

When I was still in the classroom, I had students complete daily mini quizzes (one or two small questions based on the work done in the previous class). This allowed me to see what they got and what they didn't get right away. I "assigned" homework each day but told them that they had the answers in the book so they could check it themselves. If they needed help, they could come see me before school or at lunch.

My students were allowed to rewrite these mini quizzes as many times as needed to get a full 5/5. Now that I look back, I know that there are a few things I would change. Number 1: I would allow students to review with me and then write the new mini quiz right away. One blogger said that her students can't review with her and rewrite on the same day. Very cool. Another teacher has his students email him a request for assessment detailing

What they want to reassess
Reasons why they didn't do well the first time
What they have done (not plan to do) to make sure they get it now
What date/time they want to rewrite

They must complete these steps for each and every reassessment they want to complete. If a student does not sufficiently answer the questions or they give lame responses (ie. I looked over the mini quiz), he replies to the email detailing where they need to be more specific.

You can find his specific email form letter on his blog.

I would also have students sign up for a rewrite during class but not actually show up to do it. grrrr... This would definitely help. I can't imagine that they would spend the time writing all the detailed information in the email and then not show up for the rewrite. Also, I'm not going to call it a rewrite anymore. I'll be using the term reassessment as I love the connotations that this word evokes. It even sounds more serious.

I love it! I will definitely be implementing the email Request to Reassess when I return to the classroom next year. I think I will add this to the course outline that I ~~stole~~ borrowed from him in yesterday's post. Either that or put it on my class blog. Actually, that's probably the best place for it so that students can just do a copy paste and focus on filling in the details. Oooohhhh....maybe even a google form. I will have to play with those two ideas and see which one better fits the situation. (All my students have a google education account so I don't have to worry about creating google email addresses.)

Course Outlines

noreply@blogger.com (Sandi Berg) — Wed, 28 Nov 2012 19:22:00 +0000

This morning, I spent quite a bit of time looking for activities/lessons on rational expressions. I got distracted and veered way off topic when I discovered Continuous Everywhere but Differentiable Nowhere's blog. After reading through a few posts, I found links to his course outlines. I have only read the first one but absolutely love how it begins. I definitely plan on ~~stealing~~ borrowing the introduction. Don't panic...he gives permission for anyone to do this on his blog!

Here's the portion I love but feel free to check out the course outline in it's entirety here.

Greetings!

Howdy! This year we are going to discover beautiful, useful and extraordinary islands of knowledge. I’m going to be challenging you consistently as you build bridges from island to island – from lines to quadratics to polynomials to matrices. These islands will provide resting places for our adventures, while we explore what great things these paradises have to offer. Still, any adventure isn’t an adventure unless there is uncertainty, unexpected perils. I can promise you that you will not be immune from confusion, wandering dazed and confused.

But don’t worry! Yes, at times it will be hard – all good adventures are – but rest assured that I’m always going to be right there with you. I am here to tell you now: we are in this together and we can conquer all. In fact, I’m going to make sure we make it to the end of the year with lots of sparkling mathematical treasures to your name: graphs of neat functions, complex numbers, quadratic functions, the remainder and factor theorems, compound interest, and other surprises.

teaching goes both ways
With this said, you are now at a point in your education where you are responsible for your own learning. You are old enough to know what you need to do when you are having difficulty. Wait, are you? Pop quiz.

When you are feeling lost in class, you should:
(a) wait until the next class and hope that it will all begin to make sense.
(b) not do anything… it’s only one concept and you know you’ll be tested on a bunch, so it won ‘t be a big deal to not learn it.
(c) ask someone else for help – whether it be Mr. Shah, your desk partner, or a friend.
(d) watch America’s Next Top Model and hope that the concept will be explained during a photo shoot.

You are in this class to learn some math – and even though we are in this journey together (remember: I am always on your side), that does not absolve you of responsibility. For this class to operate smoothly, for us to have a good time, to get all we need to get accomplished in mere months, you need to
… come to class prepared every day
… spend quality time working on your homework daily
… not be afraid to ask questions about concepts or homework problems you are struggling with
… be an engaged participant in every class
… be kind and respectful to the other members of the class

If you keep your end of the bargain, I guarantee you that your mind with be brimming with intellectual riches at the end of the school year. You will have learned a lot.

Just as I expect only the best from you, I want you to expect the best from me. I promise to come to teach class well-prepared, ready to embark on our daily adventures. I promise to try my best to make my presentations clear and interesting. I promise to respect you.

I can't wait to get back in the classroom and add this to my course outlines. LOVE IT!

Modelling One to One Correspondence Lesson in Kdg

noreply@blogger.com (Sandi Berg) — Fri, 19 Oct 2012 17:31:00 +0000

Yesterday, I had the pleasure of modelling a One to One Correspondence Lesson in Kindergarten. I love kindergarten. They are so eager to learn! Wouldn't it be fabulous if we could bottle that eagerness and implant it in all students?

Outcome: Number 5: Compare quantities 1 to 10, using one-to-one correspondance

Activity #1:
Materials:

Dice
Bingo Chips
Plastic Container

The Process:
I started by rolling a die to obtain a random number (although sometimes if a number showed up too many times, my die would accidentally and quietly get knocked over onto a different number). Let's pretend a 3 showed up. I would loudly and slowly clap three times. Students would hold up fingers to show me how many times they had clapped. Then, we would clap and count the same number together (so, if you're following along...that was 3 times). We repeated this activity several times until everyone knew what they were supposed to be doing.

The next stage was exactly the same except for one difference. This time, they had their eyes closed while I was clapping and while they were holding up their fingers. This allowed me to see which students were having difficulty as they weren't able to look at other students' responses for support anymore. I would sometimes repeat the claps if it looked like too many had trouble. "I'm going to repeat the claps again so that you can check your work!".

At the end of the session, the classroom teacher told me that she liked the idea of having students close their eyes while listening. I had to admit to her that I stole the idea from HER. Two days prior to this lesson, I had been in her classroom modelling a 3-D shapes lesson. I was early and caught the tail end of her rhyming lesson. She had her students close her eyes and named two rhyming words, then had students repeat back those rhyming words. I thought it was a great idea to remove the visual so I ~~stole~~ borrowed it.

Activity #2:
Materials per group of 2:

Brown paper lunch bag
Numerals 1-9 (9 fit nicely on a piece of paper, 10 did not otherwise I would have done 10). (One side had the numeral, the other side had an arrangement of circles or squares matching the numeral)
Envelope (to hold the numerals)
9 bug foam stickers

The Process:
Obviously I modeled this process before having students do the activity.

Students removed the envelope containing the numerals and placed it off to the side. They emptied the bag so that all the bug stickers were on the table. Each student counted the bugs.

Partner 2 closed their eyes or turned their back.

Partner 1 pulled a card out of the envelope. They counted out the correct number of bugs to match the numeral and placed them in the bag. They hid the numeral card.

Partner 2 opens their eyes, empties out the bag and counts the bugs.

Partner 1 checks their work.

Now, the roles reverse.

When the activity was finished, I told students they could each keep one bug foam sticker. They cleaned up and returned everything else to me.

The classroom teacher mentioned that this would be an easy activity to adapt to other themes, using different objects, stickers, etc.

Activity #3: I did not have time to get to this activity but thought I would share it anyways.
Again, prior modeling would have to be done.

Materials per group of 2:

Numerals 1-9 (9 fit nicely on a piece of paper, 10 did not otherwise I would have done 10). (One side had the numeral, the other side had an arrangement of circles or squares matching the numeral)
Envelope (to hold the numerals)

The Process:

Partners sit one behind the other, partner 1 in front; partner 2 behind them.

Partner 2 pulls out a card and then taps slowly, gently but firmly on partner 1's shoulder.

Partner 1 counts and reports the number of counts.

Switch.

This activity was meant to be a repeat of the first activity. I thought clapping would be too difficult to distinguish once all the groups were doing it. Obviously there would have to be a conversation about quality of tapping. Wouldn't want any injuries!

I hope that you find these low tech activities useful! I'm sorry that I don't have pictures to share of the students in the class but the FOIP forms would have been complicated.

Introducing Probability to 30-2

noreply@blogger.com (Sandi Berg) — Thu, 06 Sep 2012 20:45:00 +0000

I will be modelling a lesson in two weeks, introducing probability to Math 30-2 students. I brainstormed a bit and knew that I wanted to do a hands-on activity based on the Monty Hall problem. I raced to the internet, did some research and came up with a rough-draft below. I would love some input regarding a) better questions I could be asking and b) the ordering. It seems a little weird to throw the card question in the middle. Not sure if I want it there are at the end. Would love your input!

Note: I still need to check that it's ok to bring candy for students. I've never been told no in the past as I'm a guest but...I haven't been to this school yet and better dot my i's and cross my t's and all that.

My rough draft lesson plan.

30-2 Probability Intro

Monty Hall Problem

Based on:

· Monty Hall problem

· And blog write up and comments listed at Point of Inflection

· And lesson plan shown here: http://www.ms.uky.edu/algebracubed/lessons/Monty_Hall_Lesson.pdf

Materials:

· Candy - would be a toy car if not allowed to use candy

· Bowls with lids, labeled 1, 2, 3

· Blocker so they can’t see which one I’m changing

· Cards

· Envelope you can’t see through

· 3 cups per group

· 1 object per group (represent candy) – or candy

Instructions:

· Place 3 bowls on table. One has candy inside of it.
Ask “Who wants to play?”

o Don’t give out rules yet

· “One of these three bowls has candy in it. If you can guess which one, you get the candy.”

· Play

· “Who else wants to play?”

· Play a few times.

· “What’s the probability of winning?”

· Next time, say “Your guess was as good as any other. But I want to give you another chance. I will show you that this bowl <lift an empty one> is empty. Do you want to keep your original guess or switch?” Finish

· Play again but this time stop after showing an empty bowl.

· Ask “what are the odds now?” They will probably answer 50%.

· “Interesting” Finish game.

· Play again, stopping again after showing the bowl.

· Ask “when you chose at first, you had a 1/3 chance of winning. Now you say that the same bowl you picked has a 50% chance of winning. How could flipping over an empty bowl over here improve the chances that this bowl is a winner?”

· Allow for discussion.

Switch to cards:

· “Let’s try this with cards. <Student> pick a card but don’t look at it.” Put in envelope.

· “Who do you think probably has the Ace of Spades?

· “What’s the probability that it’s the Ace of Spades?” 1/52

· “I’m going to get rid of a card that is not the Ace of Spades.” Look at 3 and discard 1.

· “Do you think you’re more likely, equally likely, or less likely to have the Ace of Spades now?”

· Run through deck, discussing probability as you go.

Back to the bowls:

· When finished, return to 3 bowl problem.

· Ask students “We’re going back to the question - Should you switch or stay? In partners, I’d like you to discuss how to PROVE if the contestant should switch or stay.”

Provide groups of two with student handout, 3 cups and candy/other object. Explain that they will be creating data based on the Monty Hall problem. Make sure they understand that they need to do the test 50 times each (a total of 100 times) - 50 for switching and 50 for staying.

Student Handout

Names:_________________

_________________

Let’s Make a Deal

1. Do you think you will win more often if you switch or stay? Explain.

2. Play the Monty Hall Game with a partner. Record your results in the table below. Be sure to play 50 times WITH switching doors and 50 times WITHOUT switching doors.

Strategy	Switch Doors	Stay
WINS
LOSSES
Winning Percentage

1. Do your results display a difference in your chance of winning based on whether or not you switched doors? Explain.

2. How do your results compare with your prediction?

3. What is the probability of winning the Monty Hall Game? Explain.

4. Do you believe that you have a better chance of winning if you switch doors? Why?

Exploring the Laws of Logarithms

noreply@blogger.com (Sandi Berg) — Fri, 15 Jun 2012 21:48:00 +0000

I was looking for a more student directed way of learning the Laws of Logarithms for Math 30-1 and 30-2 students in Alberta when I found a tweet by RobAnthony01 talking about his process. I used his 140 characters and turned it into something I can give to my students to work on independently, with my guidance as necessary of course.

You can download my activity at my Teachers Pay Teachers store (for free). You have to sign up for a free account but then you can download the file. I've added other math products (for free) in there as well. Of course, you can just download it using File==> Download as... This is my first time embedding a google doc within blogger so we'll see how well it works.

It's not a very complicated activity but sure would be a heck of a lot more interesting than me saying "When adding two logs...do this. When subtracting two logs...do this."

Reflection App - Well sort of...

noreply@blogger.com (Sandi Berg) — Wed, 13 Jun 2012 20:26:00 +0000

Yesterday, I discovered a cool new tool and just had to share it. It's called Reflection and can be found on the Reflection App website.

What it works on:
Devices: iPhone 4S, iPad 2, iPad 3,
Platforms: PC (Windows XP or greater), Mac (OS X 10.6.8)
I was testing on a Windows XP using my iPhone 4S.

What does it do? It's a projection "app". It allows you to wirelessly project what is happening on your iPhone or iPad onto your computer.

Right now, if you want to show what's happening on your iPhone or iPad, you have a few choices.

You can AirPlay it to an Apple TV. Not sure how many classrooms have that. Mine didn't.
You can also purchase the Apple dongle at a cost of about $35. I've used the dongle. Honestly, I had trouble with it. I would have to take my cover off because I couldn't get it to attach securely. Even with the cover removed, the second I moved around it had a tendency to jiggle loose. Then, if it had jiggled loose too often, it would stop projecting my iPad completely. So frustrating. Especially when you're doing a presentation about how great iPads can be if used properly. Thirdly, I was tied down and limited to the length of my VGA cable. Frustrating.
I've heard people talking about other apps that will project as well but have not researched them. I was really looking for something that would allow a class of 20-30 iPads to project inexpensively.

So far, this is my favorite...Let me tell you about it.

I installed the software on my laptop. It was fast and easy, about a 9Mb download. Then I went to my iPhone 4S, (make sure your laptop/desktop and iPhone/iPad are on the same network) accessed AirPlay and started projecting. That's it. There's nothing to install on the iPhone or iPad. That means anyone on the same network as my computer can instantly project through my computer to the projector. Awesome!

Wait! You scream. Does that mean ANYONE can tap into my computer and start projecting? Well technically yes unless you take the following action. You can set a password. This way, you can either 1) turn it off at the beginning of class or 2) give it to your students for that class and then change it afterwards. Again, easy-peasy to do.

What can you use it for? Let students show videos they've found, work they've done, anything they've created, brainstorming they've completed in groups. It even projects sound! So, if they're showing a video, the sound automatically transfers with it. I did notice a slight lag when I ended the YouTube video. It showed up on the computer screen for about 5 seconds after I closed the YouTube App. It will not display the video on your device at the same time, however. Oh, and apparently multiple devices can connect to the computer at once. The website states that it will start to slow down as you increase the number of connections. I didn't bring my iPad to work today so I will have to test it and see how it goes. This spot here will be updated as soon as I have tested multiple devices.

As far as I can tell, only two things didn't work on it. 1) Facebook wouldn't play embedded videos or at least not the one I had tried. 2) Skype didn't transfer. There may be other apps that don't work but so far so good with the ones that I've tried.

One other small glitch: Once when I had told it to connect, I had this blog open in edit mode. When it displayed my iPhone it was itty bitty on the screen. I just canceled the airplay and reconnected and it was fine. So far that has only happened once.

Cost: The free version only allows you a total of 10 minutes time and that's it to explore the app but within that time, you can pretty much guarantee that you'll love it. Full Disclosure Here: Once I ran out of my free time, I contacted the company, explained who I was and that I work with teachers across Central Alberta sharing resources, activities, apps and ideas that they may find useful within their math classroom and then I very politely requested a free copy of their app. I want to be able to actually SHOW people how easy this app is to use. The current cost is $14.99 per license or $49.99 for 5. Contact them for more than 20 licenses for another discount.

Check out the Reflection App website to watch the video and find the download file. Let me know if there's anything serious that I missed when playing with the app - either good or bad.

Edited to add: I was asked where I'd heard about the Reflection "app" and I couldn't remember at first but now I suspect it was because of this post. Not sure why I would have kept reading about it since she states that only works on Macs but it definitely works on PC's now. So, thank-you, Kathy, for pointing me in this direction!

Assessment - Killing the Overkill

noreply@blogger.com (Sandi Berg) — Tue, 12 Jun 2012 17:13:00 +0000

Students Can Demonstrate Understanding with a Shorter Assessment Piece

I've been working on changing my assessments. Marian Small has been a huge influence on this ever since I attended her session on the High School Math Institute. Now, every time that I am creating a new assessment, I search "Marian Small" +topic in google to help generate some ideas rather than always recreating my own. I always try to change the question so that I'm not just stealing her ideas, especially since I share so many of the things I create with the teachers around Central Alberta during Assessment sessions. However, I certainly use her framework as it makes the questions so much more interesting and checks for a deeper understanding.

When I present a session on assessment, one of the first slides I show asks

What Does...

Solve 3⁴

on a test tell us about the student?

The inevitable response is that "They can operate a calculator". It terrifies me that when I first started teaching, I would strongly populate my exams with questions like this...ones that tested their ability to operate a calculator. Of course, I would have to have many questions like this to test whether they could do it in multiple settings. So, my students were answering a bunch of repetitive questions and I was stuck marking a bunch of repetitive questions. Of course, it was easy because a simple answer key worked but what did it tell me about my students' understanding? Pretty much nothing.

Once I began my assessment journey, I realized how easy it would be to change these questions to better get at their true understanding. Yes, it makes my answer key pretty much useless. Yes, it might take me extra time to mark the question. However, I was then able to cut down 20-50 questions to just a few deep questions so it balances out in the end.

Giving up a little bit of control during the assessment process allows students to demonstrate their knowledge more fully.

I wanted to know if students truly understood that 3⁴ is really 3x3x3x3. Why not let them pick the numbers they used? By wording the question carefully, I could prevent "easy outs". This is how I changed the first question: (The bold portions would not be bolded on the test. They're bolded so you can see how I tried to ensure I didn't get situations like 1¹
. The others are samples of how I would assess the other outcomes.

Question 1:
Choose a base greater than one: ____________

Choose an exponent greater than the base: ______________

Create a visual representation for the power you produced.

Question 2:
You simplified an expression. The result was (3/4)^1/2. What could the original expression have been if you were using the following operations?
(a^m)(aⁿ) = _____________________

a^m/aⁿ = _______________________

(a^m)ⁿ = _______________________

Question 3:
Choose one of the following operations: (ab)^m OR (a/b)ⁿ

Choose your own bases and exponents, however "m" and "n" can NOT be a whole number.

Explain TWO ways to determine the result without using any "rules".

Question 4: (From Marian Small)
Write an expression that you would likely use three power laws to simplify. Use a variety of exponents.

Simplify it.

Question 5:
Jamie states that a⁰ is a. Pat says that it is 1. Jessie says that it is 0. Casey says that none of them are correct. Who is right and how would you convince everyone else without saying "The rule is..."

Like I said, when I first started teaching, I would overkill on questions. I believed an assessment was supposed to take all class which meant I would have to put a LOT of questions on it. Then, of course, when you have a class of 16-30 students, you had to MARK all those questions. I would now use this assessment to check students understanding in Math 9:

Side Note #1: These types of questions are excellent discussion starters, review questions, exit slips, "homework checks" and formative and summative assessment questions.

Side Note #2: Students should NOT see this style of question for the first time on a final assessment! They should experience it ahead of time. I would worry that they would feel like they were being "tricked" otherwise.

I've considered including the outcomes on the assessment to give students and parents a better idea of students' success within each outcome. I'm trying to decide between two formats:

Option 1:
I could attach the table included below to their paper after the assessment. I would highlight the questions they were successful (or not successful) at.

Number
1 Q1 Q5 Q4	Demonstrate an understanding of powers with integral bases (excluding base 0) and whole number exponents by Representing repeated multiplication, using powers Using patterns to show that a power with an exponent of zero is equal to one Solving problems involving powers
2 Q2 Q2 Q2 Q3 Q3 Q4 All	Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents (a^m)(aⁿ) = a^m+n a^m÷aⁿ = a^m-n (a^m)ⁿ =a^mn (ab)^m=a^mbⁿ (a/b)ⁿ=aⁿ/bⁿ, b ≠ 0

OR, I could add a column to the assessment (something like below) but then I would have to make sure to blank out the pieces that give answers.

Exponent Laws Assessment

Question 1: Choose a base greater than one: ____________ Choose an exponent greater than the base: ______________ Create a visual representation for the power you produced.	Demonstrate an understanding of powers with integral bases (excluding base 0) and whole number exponents
Question 2: You simplified an expression. The result was (3/4)^1/2. What could the original expression have been if you were using the following operations? (a^m)(aⁿ) = _______________________ a^m/aⁿ = _______________________ (a^m)ⁿ = _______________________	Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents (a^m)(aⁿ) = ___ a^m÷aⁿ = ___ (a^m)ⁿ = ___
Question 3: Choose one of the following operations: (ab)^m OR (a/b)ⁿ Choose your own bases and exponents, however "m" and "n" can NOT be a whole number. Explain TWO ways to determine the result without using any "rules	Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents (ab)^m= ___ (a/b)ⁿ= ___
Question 4: (From Marian Small) Write an expression that you would likely use three power laws to simplify. Use a variety of exponents. Simplify it.	Solving problems involving powers
Question 5: Jamie states that a⁰ is a. Pat says that it is 1. Jessie says that it is 0. Casey says that none of them are correct. Who is right and how would you convince everyone else without saying "The rule is...	Show that a power with an exponent of zero is equal to___

So there it is. A short assessment piece I could use to check my students' understanding of the power laws. What do you do in your classroom to assess students' understanding of the power laws? I would love to hear ideas for a performance assessment piece, other assessment questions (ie. a fabulous word problem), etc. How would you share the connections between their responses and the outcomes (like I was attempting in option 1 and 2)?

Share and link in the comment section!

Chomping Down with "Screen Chomp"

noreply@blogger.com (Sandi Berg) — Fri, 08 Jun 2012 15:00:00 +0000

Free
Ipad

I absolutely adore Screen Chomp. Any time anyone asks me for a great ipad app, it's the first one I share. Why? First of all, it's so easy my Kdg niece can use it without help. The easiest way to use it is to just hit record and then draw on the screen and speak. When finished, you'll have a perfect recording of what you drew and said! You can even prepare the screen ahead of time by adding images if you want. Preview the recording immediately after making it. Then it's next to nothing to share that recording on twitter, facebook or just grab the link and share that!

What a fabulous tool for schools working with digital portfolios.

How would I use this? Digital portfolios, assessment for learning, assessment of learning, students creating tutorials for school website, projects, and so on and so on... Can you imagine having a student reading a passage of text and posting it beside how they did on the same text a month ago? Students don't even have to be in the room to share their understanding or thinking. Struggling student? Watch them thinking through a math problem pre-written on the screen. My brain goes into overload every time I think about how it can be used in the classroom.

The second reason I absolutely love this program is that it is completely and utterly free!

Third, as long as you have a wifi connection, you don't have to worry about how to download that video.

So, if you don't add any other app to your ipad, make sure to add this one!

I'll Be Back...

noreply@blogger.com (Sandi Berg) — Tue, 05 Jun 2012 15:53:00 +0000

next year in my role as CARC's Mathematics Lead Teacher/Facilitator. Woot Woot! I miss my students but look forward to working with K-12 Math teachers in Central Alberta again. I had so much fun running around to different schools providing PD and modelling lessons. However, I have a whole new respect for substitute teachers. I had no idea how hard it would be! I've never subbed a day in my life and this year I modelled lessons in classrooms. The only difference was that I had to create my own lessons rather than following a predetermined one. When I return to my classroom, I'm going to make sure to leave chocolate and let them know how much I appreciate their hardwork, especially those who have to come in to my -1 courses with little or no math background.

Summary for this year:

Favorite Session I offered: Hands on and Virtual Manipulatives
Although this often became a contentious issue when dealing with high school -1 teachers, I really enjoyed sharing some of the ideas I have gathered over the years. My favorite has always been "Using Algebra Tiles to teach Factoring Polynomials and Completing the Square". You can find the information about this in my previous post. I also learned a lot from the teachers who were willing to share their ideas for using the manipulatives. Many teachers at the high school level are concerned about the lack of basic skills (such as multiplying in their heads) so I also created a lot of games for practicing basic skills at any level.

Favorite Learning:
When I started this job, I spent a day running around to schools meeting teachers, taking notes about sessions they would like, questions they had, etc. By the end of the day, my hand was cramped and you couldn't read my handwriting. (I wasn't hauling my laptop around and my phone was too small to take extended notes). That weekend, I went out and bought the iPad 2 with bluetooth keyboard. What a lifesaver! Now, I can take notes and still read my writing. I'm also a much faster typer than writer so I could keep up much better.

Just after I bought the iPad, I was asked by a school to come in and help with an Early Numeracy Intervention Program using iPads that they wanted to set up. Perfect! So, off I went to explore and download a variety of iPad apps for improving math skills. Luckily, almost every developer I contacted was willing to give me free samples! Woot woot! You can find a list of my favorites on my delicious account.

My Goal for next year:
To post more on my blog. Not sure what happened this year. I guess it had a lot to do with learning and becoming comfortable with my role.

I'm also going to work on posting as much as possible specifically about Alberta 30-1, 30-2 and 30-3 math curriculum. So, if you have something to share, I would love to see it!

Well, I should probably get back to planning for Grade 12 math now. Will see you again soon!

Completing the Square Smart Notebook Activity

noreply@blogger.com (Sandi Berg) — Fri, 03 Feb 2012 00:00:00 +0000

I've uploaded another Smartboard lesson to Teachers Pay Teachers entitled "Completing the Square".

Students will use algebra tiles and the student handout (found in the attachments of the file), guided by the teacher using this file, to create the formula for Completing the Square.

Below I have included three sample pages from the file.

There is an intro story - I had created worksheets for today's lesson but accidentally spilled ink all over it. Since I didn't have time to remake the worksheets, you need to help me figure out the missing numbers. I set up all the expressions so that you could build a perfect square out of each expression. What number is missing? We would build the first factored form together once they have built the square. After than they should have no problem writing it for the perfect squares.

A discussion would then need to occur regarding patterns. Students are then asked to...

Since they can't build it using algebra tiles (I remind them they are not allowed to deface school property by breaking any of the pieces), they should be aware they have to do it just with paper and pencil although they might try to draw it.

Then we start to deal with expressions that have too many and too few pieces. The story is that a little munchkin snuck into my classroom and either added to or took away unit pieces out of the algebra tile bags that I had created again for class. They originally started with the exact number needed to make perfect squares. Help me figure out how many extra/too few pieces there are. Once they have built it and determined the number of extra pieces (write that in the circle), we would discuss that it is a perfect square - which would be (x + 3)^2 but has 2 extra pieces so we represent it as (x + 3)^2 + 2.

We repeat the discussion again for "missing" pieces but when we are missing pieces we have a perfect square minus whatever we're missing.

Aligns to Alberta Curriculum: Math 20-1

This lesson addresses the process for "Completing the Square" in order to address the following outcome:

*Analyze quadratic functions of the form y=ax^2+bx+c to identify characteristics of the corresponding graph, including vertex, domain and range, direction of opening, axis of symmetry, x and y intercepts and to solve problems.

This resource can be located on my Teachers Pay Teachers store AT NO CHARGE! Make sure to post a review in my store. Let me know if you have any suggestions for improvement!

Factoring Polynomials Intro

noreply@blogger.com (Sandi Berg) — Wed, 01 Feb 2012 18:51:00 +0000

As CARC's Mathematics Lead Teacher Facilitator, I create many resources for teachers within my region. Several months ago, I discovered Teachers Pay Teachers which is a website where teachers can share and download resources of all kinds, for all grades, for all subjects for free and for a fee.

I thought it was silly that I'm sharing just a small portion of my resources with only the teachers I meet through workshops. Why not post them on this website so that anyone who wants to can download them!

The first Smartboard lesson I've uploaded to this website is entitled "Factoring Polynomial Expressions".

This Smart Notebook file, along with Algebra Tiles, can be used to help students learn how to factor and multiply polynomial expressions. It leads them through the concrete (using the tiles) to the abstract (without the tiles). This file has been used with students, modified, presented to teachers, modified, and modified some more. I converted the student pages to a pdf (found in the attachments of the file) to be handed out to students for record keeping.

It's also an excellent refresher for students who will be working on "Completing the Square."

Aligns to Alberta Curriculum: Math 10C and 20-1

Below I have include two sample pages from the file.

This resource can be located on my Teachers Pay Teachers store AT NO CHARGE! Make sure to post a review in my store. Let me know if you have any suggestions for improvement!

Factoring Polynomial Expressions by Sandi Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Hands-On Math: Number Sense

noreply@blogger.com (Sandi Berg) — Fri, 14 Oct 2011 15:38:00 +0000

$1.99
iPad
Levels: I would say any student who needs to develop number sense would benefit.

I've been exploring a lot of apps lately, looking for just the right apps to share with other teachers and using my 3 nieces, Kdg, Grade 2, and Grade 8 to test them out on. I have found many apps that allow students to practice what they already know but I've come to realize that there are few apps that actually teach number sense.

I am happy to say that this app tops my list! On the left hand side, students will find a set of blocks sized 1-10 that they can easily change from 1-10 to 0.1-1 to 1/10 - 1.

The teacher can download a teacher guide pdf with instructions from the website or through the app on how to use the program or sample activities the students can try.

Students can use the program to explore addition, subtraction, fact families, multiplication, division, greater than, less than, equal to, the commutative property, comparing and ordering whole numbers.

By clicking on the calculator, students are provided with a randomly generated question. They can use the program to solve the question. If they enter the correct answer, they receive acknowledgement from the program verbally. If the answer is incorrect "Try again" appears.

There's also a number line that can be turned on and off and converted to whole number, fraction or decimal format.

What a great program to use if you don't want to pull out all of the manipulatives for one or two students, if you don't actually have the manipulatives in your classroom due to budget cuts, or for students use at home or on the road or anywhere they happen to be when they want to practice!

I found this features list on the Ventura Education Systems website and stole it to include here. Shhhh. Don't tell!

Hands on Math: Interactive Number Sense creates a virtual math playground where students explore, investigate and discover mathematical concepts. When students are interacting with the Number Sense Playground colored number bars are used to represent whole numbers, fractions and decimals.

Using [this app] students can develop important mathematical concepts:
ordering and comparing whole numbers, decimals and fractions;
adding, subtracting, multiplying and dividing whole numbers, fractions and decimals;
locating whole numbers, fractions or decimals on a number line.

Sound Effects: Can be turned off.
Speech: Can be turned off.

There's a slider bar titled Whole Number View Options. You can set it so that the blocks show up individually, as a single block with the number written in the center or as a single block without a number. This allows students to move from the concrete visual to a little more abstract.

Other than that, I didn't run into any kind of technical difficulties while running this app.

As one of my nieces would say "Two thumbs WAAAAAAAAY up" for this app!

The Review I added to the app:
I've been exploring a lot of apps lately, looking for just the right apps to share with other mathematics teachers and using my 3 nieces, Kdg, Grade 2, and Grade 8 to test them out on. I have found many apps that allow students to practice what they already know but I've come to realize that there are few apps that actually teach number sense. This app would be a great addition to the home and the classroom. It's colorful, easy to use and I didn't have any technical glitches which is always a bonus!

This app allows students to explore number sense on their own or based on questions provided by the calculator. No rods to lose! This is an app that will remain on my ipad forever regardless of the ages of my nieces or the grades I am teaching.