My classes have been using a WODB board this year. The board has been a permanent fixture in my room and it has been up since August. I came across the idea last year after reading Christopher’s idea and Joel’s example. I’m finding that it has been a great routine for my 3-5th grade math students. My goal was to change my WODB bulletin board every week, but it’s really being changed around 2-3 weeks or so. My boards started out as mainly shapes, but has moved to numbers and equations recently. Changing it less gives kids time to see other options and add more notes.

My third grade students are in the middle of a unit on fractions. They used number lines to multiply fractions by whole numbers earlier in the week. The students are becoming better at multiplying fractions using visual models, although some are more wanting to multiply the numerators and denominators.

Today, the students completed an individual Which One Doesn’t Belong task. I’ve heard of other classes doing something similar, so I thought it might work well with my kids. Students were given a criteria for success page and then asked a bunch of questions.

Students were asked to create four different fraction multiplication models. Students then created two different solutions for the WODB prompt. After a brief amount of modeling, students started to create their own WODB boards. Many students had questions about what could count for solutions? I put it back on the students to figure out if their solutions were appropriate or not. For the most part, students did a fine job finding two different solutions.

Students then wrote down their solutions and folded the paper to hide them.

The students took pictures and put them in their SeeSaw accounts. Next week, the kids will look at each others’ responses and see if their solutions match. I’m looking forward to what they observe.

]]>My third graders have been studying fractions for the past few weeks. Last week, students represented fractions as multiples of a unit fraction. In one of the lessons they broke apart 4/5 into four 1/5 pieces. They used number lines to show all the different unit fraction pieces.

Afterwards, students represented fraction as multiples of a unit fraction. They showed how addition equations and multiplication equations are related. Students reviewed how repeated addition is similar to multplication. Students are becoming better at understanding how to multiply a whole number by a fraction and show the progression on a number line. I had some students want to jump to multiplying the numerators and denominators , but these students had trouble when explaining why they used that process.

The next task really seemed to stretch their thinking. It also showed me how well students grasped the idea of combining fractions with similar denominators. The students were asked to create a three-fruit salad. They were given fruit and the typical weight for each item. Students needed to combine three of the fruits to total exactly five pounds.

Students needed to create two different recipes for this task. Students needed to also show how they combined the fruits withe a visual model and display a number model. For the most part, students were able to combine the fruit accordingly. An interesting tidbit was that many of them overemphasized one fruit over another. For example, I had a few students that took 16 cups of grapes to get a total of four pounds. Then they just found two items that totaled one pound. I asked the students what those types of fruit salads would taste like. They didn’t have a response, so I’m assuming we’ll have to look at the context a bit more next time. This could also be used for a ratio/proportion lesson somewhere down the road.

Later on in the week, I gave a similar task related to a four-fruit salad with a weight of exactly six pounds. This time the weights weren’t unit fractions.

Students first noticed the strawberries 7/8 weight. A few students were pretty sure that the 7/8 might not be compatible with the rest of the fractions. Others disagreed with this idea and said that the 1/3 didn’t fit. Students worked out this task and needed some help along the way. Students moved towards creating just number models, as some decided to not go the visual model route. Overall, I’m impressed with how they tackled this problem. We’ll be discussing it tomorrow afternoon.

]]>My fourth grade class has been exploring geometry and polygons. They’ve been comparing polygons and looking closely at how to classify them. Students are familiar with these shapes and can classify them by their looks. Students were confident early in the week as they were able to label polygons based on side length, angles, and sides. Things started to change when the word hierarchy was introduced. The word is new to most of my students so the class first reviewed the term.

Students were then given a paper full of shapes. They cut out the shapes and used their desk to classify them. Students classified the shapes according to their attributes. Some students sorted the polygons by angle size, while others used symmetry or side length.

Students were finished in about ten minutes. The class took a gallery walk and reviewed all the other ideas. Students discussed which polygons met or didn’t meet the category title. Students then went back to their seats and reviewed the term hierarchy again.

The class took the cards and taped/glued them to make their own hierarchy. Students are starting to see the characteristics of polygons in a different light. This is good news as later on in the year students will use polygon characteristics to find area measurements. They’ll also be transforming the polygons on a grid in about a month.

I’m looking forward to the polygon discussion tomorrow.

Side note: This is my 300th post. I had no idea that I’d be writing so much over the years, but it’s been an amazing journey.

]]>My third grade class finished up a cumulative assessment last week. This particular assignment was completed independently and covered skills from January – March. The assignment spanned the last two units of study and reviewed topic of factors, multiples, composite/prime numbers, area, fractions, decimals, measurement conversions, using standard algorithms, and angles. There was a hefty amount of content found in fairly large assignment. It took around two classes to complete the task.

It’s my personal belief that an assessment should be worthwhile to the student and the teacher. Why take the time to give the assessment in the first place?? Well …. don’t answer that – especially when state standardized testing is right around the corner. : ) There are some assessments that teachers are required to give and others that are more optional.

My assessment for learning belief stems from past experiences that weren’t so thrilling. I remember being given a graded test and then immediately moving on to the next topic of study. There wasn’t a review of the test or even feedback. A large letter grade (usually in a big red marker) was on the front and that was that. This left me salty. All teachers were students at some point and this memory has stuck with me.

I like to have students review their results and take a deeper look into what they understand. In reality the assessment should be formative and the experience is one stop along their math journey. It should be a worthwhile event. It’s either a wasted opportunity or a time slot where students can analyze their results, use feedback, and make it more of a meaningful experience.

So back on track … These third graders took the cumulative assessment last week. I graded them around mid-week and started to notice a few trends. Certain problems were generally correct, while others were very troublesome for students. Take a look at my chicken-scratch below.

As you can tell, problems 2, 4, 8, 11 and 22 didn’t fare well. It seemed that problems 3, 17, 18, and 21 didn’t have too many issues. My first thought was that I might not have reviewed those concepts as much as I should have. There are so many variables at play here that I can’t cut the poor performance on a particular question down to one reason. That doesn’t mean I can’t play detective though. My second thought revolved around the idea that directions might have been skimmed over or students weren’t quite sure what was being asked. So, I took a closer look at the questions that were more problematic. I looked in my highlighter stash and took out a yellow and pink. I highlighted the problems that were more problematic pink. Yellow was given to the problems that were more correct.

The next day I was able to review the assessment results with the class. I gave back the test to the students and reviewed my teacher copy with the pink and yellow with the class. I used the document camera and made a pitstop each pink and yellow highlight and asked students what types of misconceptions could possibly exist when answering that particular question. I was then able to offer feedback to the class. For example, one of the directions asked students to record to multiplicative comparison statements. Many students created number models, but didn’t use statements.

Students also mixed up factors and multiples

Many students forgot to include 81 in the factor pair and thought they didn’t have to include it since it was in the directions. Hmmmm…. not sure about that one.

Some of the problems required reteaching. I thought that was great opportunity to readdress a specific skill, but I could tell that it was more than just a silly mistake. I think the default for students is to say that 1.) they were rushing or 2.) it was a silly mistake. Sometimes it’s neither. I had a mini lesson on measurement conversions.

I also reviewed how to use the standard algorithm to add and subtract larger numbers. Some students had trouble lining up the numbers or forgot to regroup as needed.

I offered up some graph paper to students that needed to keep their work organized.

After the review, which took about 10-15 minutes, I gave students a second opportunity to retake the problems that were incorrect the first time around. I ended up grading the second attempts and was excited as students made a decent amount of progress. The majority of pink highlighted problems from earlier were correct on the second attempt. #Eduwin! The feedback and error analysis time seemed to help clarify the directions and ended up being a valuable use of time. I’m considering using sometime similar for the next cumulative assessment, which will most likely occur around May.

Now, I don’t use this method for all of assessments. My third grade class has eight unit assessments a year. After each assessment I tend to have students analyze their test performance in relation to the math standard that’s expected. Students reflect and observe which particular math skills need bolstering and set goals based on those results. There’s a progress monitoring piece involved as students refer back to these goals during there next unit.

Side note: I had trouble finding a title for this post. I was debating between misconception analysis and assessment analysis. Both seemed decent, but didn’t really reflect the post. So I tried something different – I wrote the post and then created the title. I feel like error analysis fits a bit more as the errors that were made weren’t necessarily misconceptions. Also, this post has me thinking of problematic test questions. That could be an entirely different post.

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I find that each year brings new ideas and this year is no different. I always tend to ask question about making relevant connections to the content that I teach. This year my students are studying Hamlet. They’re not delving too deep into the original text. In fact, we’re reading this book and have been exploring Hamlet for the past month. It’s been an exciting journey. Along with reviewing the play, the class used a character map, learned about Will, and viewed clips from a contemporary portrayal of Hamlet with David Tennant.

The class is now in the final stretch of our Hamlet unit. So, for the last unit I decided to try something different. Ideally, I’d like to have students remember Hamlet when they encounter it again in a few years. I decided to use a menu board approach. Each student picked one project below.

The class then reviewed a criteria for success rubric. Honestly, the rubric seems quite intense at first. But in all fairness, I needed to have a rubric that actually encompassed all of the menu items.

I made sure to have the students review the part on the left side. In that past, I’ve found that sometimes students might pick a project that is less challenging. I was hoping to be proven wrong with this project.

Students “signed-off” on the project and were committed. I find value in having students actual write that they agree to the criteria. I think it adds an ownership element that isn’t always there. It also reminds me what resources to pick up before next class.

I was pleasantly surprised to see that all of the menu items were picked – some more than others.

Students were then given the remainder of that Monday to work on the projects. Near the end of class I told the students the plan for the rest of the week. They had the next four days to complete their menu item. My job was to gather materials and the technology that was needed. I had to find more technology since my school isn’t 1:1. I begged and borrowed from the other teachers in my building to get enough Chromebooks and iPads to make the projects feasible. Priority for iPad and Chromebook use was given to the stop-motion-video and board game creators. I was pleasantly surprised to find that some of my kids wanted to create a video game using Scratch. One of my favorites was a duel between Hamlet and Laertes, where Hamlet always wins.

Click to view slideshow.Near the end of the week most students were finished, although a few voluntarily came in during their recess to finish up the project. The next Monday was designed for feedback.

Over the weekend I created a Google Form for student feedback. Students scanned the code when they entered the class. Each student filled out the feedback form and reviewed another student’s project.

You can view the sheet here. Currently, the class is halfway through giving feedback because we’ve had a slight interruption because of Parcc testing. Tomorrow the class will be giving additional feedback. My plan is to print out the feedback and give the responses (without the names) to each student. The authors will then have an opportunity to analyze the feedback and give responses as needed.

The student engagement for this project was top notch and I was impressed with the quality of work produced. This reading menu has me wondering how a menu system could be applied in the math classroom. So far, I haven’t had as much success with a menu in the math classroom. I’ve used choice boards, but they haven’ been anything spectacular. Anyone have success with this? This topic is something to ponder before heading off into spring break next week.

]]>I shifted the discussion to the meaning of the fraction bar. One of the students mentioned that the fraction 1/2 is the same as 1 divided by 2. Another student said that is the same as 0.5. This conversation was productive and moved the discussion back on course. Students started to build upon each response and were able to start thinking more about their own understanding of fractions. I then introduced the idea of fraction as division. This resonated well with students and I could tell that they were really thinking about how they view fractions. I then put this problem on the board.

Students thought for a little while and then decided to split up each fraction into three pieces. They then counted up the pieces to find 9.

I then introduced students to a common numerator and denominator model. Students thought about this problem and then started making a few guesses.

One thing that seemed to shift this thinking was to look at fraction as division. In my years of teaching this seems to make quite a few connections Many students know that a half of a half is a quarter, but are a confused when it comes to dividing a half. One student mentioned that they both have common denominators and that might be useful when dividing. Another student said that a fraction is division, so you could divide the numerators and denominators.

The class agreed that this will work as long as the denominators are the same. They also concluded that if the denominators aren’t the same, we can find an equivalent fraction to create ones that are. This conversation lasted for about five minutes. It was productive and not once was there mention of a “fast” method to divide fractions. I’m hoping that students hold on to visual models and using a variety of strategies when dividing fractions in the future. Next week, we’ll be investigating how to divide mixed numbers. That’ll most likely happen after our week long PARCC adventure.

]]>My fourth graders are starting a new unit on fraction computation this week. Last week, students finished up month long unit on volume and area. After grading the tests, I started to reflect on a few different activities that seemed to help students understand volume a bit better. One particular task will be highlighted in this post. I’m not going to lie, this task was quite challenging for kids, but I feel like the students were able to make some amazing math connections in the process.

So last week, I brought the students to the front of the room and we discussed area and volume. Students provided examples of area and volume and referenced the city that they created earlier in the year. Students then randomly came up to the room and drew out a slip of a paper. The slips indicated a particular volume task. The tasks were all related to making a 3D shape that matched a certain dimension range.

Students drew the small little sheets out of a cup. It was exciting as students weren’t quite sure which sheet they were going to get. Students were then given the direction sheet, where they were asked to create the net, tape/glue it together, place it on the sheet, and then take a picture and send it to their digital portfolio.

Students were then given the centimeter grid and were off to the races. Some students had to take multiple grid sheets as they missed the required dimensions on many different attempts. Eventually, most students calculate the volume that they needed and used a formula. Students then used the formula to calculate the volume before creating the net.

Students didn’t seem to have too many problems with rectangular prisms or cubes, but cylinders and cones were a bit more challenging. Students were able to create the base fairly quickly. The curved surface was an issue for some. Many students had trouble creating a large enough curved surface to match the cones and cylinders. One student mentioned that the curved surface needed to be around 3 1/4 of the length of the circumference. I enjoyed hearing that as a couple students had a conversation on how to make their shape fit a required dimension. That’s an #eduwin in my book. Students then attached their constructed structures to the direction sheet.

Students then put the different structures on a map and created a small city. I’m hoping at some point the students will be able to create a short stop-motion-video using the volume structures. It might fit in perfectly with our rate/ratio unit that will be coming up after PARCC testing.

]]>My fourth grade students have been exploring volume and area for the past few weeks. Lately, they’ve investigated different methods to find the volume of prisms, pyramids, cones, and cylinders. Through this process, they created their cities of volume and have been studying this topic extensively. This fourth grade crew has made a lot of progress in finding the volume of objects when given the dimensions. This particular unit of study is more focused on making spatial connections and using formulas to find volume. Although the kids have been showing a better understanding, I’m observing very similar errors when I give checkpoints.

- Using inappropriate units (squared vs. cubed)

Students need constant reminders to show appropriate units. When I whiteout the unit line it’s interesting which students automatically write down the correct units and those that leave it blank. Lately I’ve been bringing out the base-ten blocks to show the difference between linear measurements, area, and volume. Students tend to not have any issues with telling the difference at that time, but when concentrating on formulas, the units are sometimes omitted. I’m currently looking at different ways for students to show their understanding of the differences between square and cubic units. I don’t want to heavily focus on this, but I’m noticing it as more of a student afterthought than something that they think of while answer a question.

- Find the lengths of a side or the circumference with volume is given

Students seem to be efficient when trying to find the volume of prisms and cylinders. When given the measurements of each side, students tend to perform the calculations correctly. It’s a bit of a different story when students are given the volume and are asked to find other dimensions. Some students rock this and do well, others not so much. The class reviewed these types of problems by using a variable for the missing side or circumference. We then created a few different steps that can be taken when tackling these types of problems. I’d say the majority of issues with this specific problem came when students were given the volume of a cylinder or cone and needed to find the volume. This is something that the class is still reviewing.

- Remember that in r^2 actually means r * r and not r * 2

I’m going to chalk this up to not having enough practice with exponents. At this level, students have used exponents, but more so to show Scientific notation. When students hear “to the second power”, some hear that the word second and just multiply the radius by two. Some students also problematically use the diameter and call it the radius. Digging deeper into this issue has also revealed that some students aren’t using the Order of Operations to solve for volume. Next week I’m planning on co-creating an anchor chart to address this. Also, Pi Day (3/14/18) is coming up soon and the class will definitely address the vocabulary and formulas associated with that soon.

These three issues have come up fairly consistently during the past week. I’m looking forward to addressing them next week, but also having the students become more aware of what fixable mistakes exist so we can be more proactive. I’m also looking into having students create a culminating volume activity. Putting that together is in my plans for tomorrow.

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I’m grading tests this weekend. My third grade group just finished up an assessment on fractions and multiplication. It’s been about a 1-2 month journey full of investigations on this particular topic. Many formative checkpoints were planned along the way and the unit assessment was scheduled for last week.

While reviewing the student work, I’ll sometimes write questions or direct students to a part of the question that would’ve made their answer more complete. There are moments of pride and moments where simple mistakes drive me a bit crazy. You see, there are only eight units with my new resource, so the assessments influence grade reporting quite significantly.

After grading the assessments, I have student analyze their results. They comb through the test and look at how each question aligns with certain skills. They also determine if a missed question was a fixable mistake. I want students to be able to recognize when this occurs and fix them when they can.

In my experience, 9/10 times students believe that the reason they missed a question was because it was a fixable mistake. That’s not always the case. There’s a certain amount of self-reflection and humility that’s involved in this process. Being able to be a bit more honest and communicating what a simple mistake is and what it isn’t might be in order before the next assessment. So, what steps do students take if a missed questions isn’t a fixable mistake? It’s one step in the right direction to admit that it isn’t fixable, but then what happens next? Do students and teachers have plan for this, or do we move on to the next unit?

So, was it a simple mistake or something more? This question comes up more often than not while I’m grading student work or reflecting back on a class conversation. Some of the answers are more positive than others. A simple calculation error can vastly impact an answer, but it may be a simple mistake and the student has a solid conceptual understanding of that skill. But, a number model that doesn’t match the problem tells me that the student might not be certain about the operation that needs to be completed. Was the simple mistake putting the wrong operation sign in the number model? I guess you could go down many different paths here.

The question type can influence how well and thorough a student responds. Some questions are quite poor in giving educators quality feedback to help inform instruction. Right off the top of my head, multiple-choice and true/false questions fit that bill. They sure are easy to grade by human or a machine. Hooray! But, they don’t give me quality feedback that I can use immediately.

This also has me wondering about the quality of the assessments that are given. Measuring how proficient a student is on a particular concept doesn’t always have to come from standardized or unit assessment results. Classroom observations and formative checkpoints are beneficial and give teachers insights to what students are thinking. I want to make students’ math thinking visible. Whether that’s using technology or not, making that thinking visible puts the teacher in a better position. From what I observe, some of the best math task questions are open-ended and tend to have a written component where students are asked to explain their thinking. The quality of the question and openness of the answer helps educators dig deeper into how and what students are thinking. I think that’s why teachers are always looking out for better math tasks that help students demonstrate their understanding more accurately.

How do you help students determine if a mistake was simple or something more?

]]>About a month ago my teaching team was asked by administration to map out our new sixth grade unit assessments. By mapping, I mean that we had to review the questions on the tests and determine if the question was was multiple choice, short answer, essay or a performance assessment. We then wrote what type of objective was present, the domain for learning, and the DOK level.

This mapping took a long time and the entire team was knee-deep in our teacher guides. It was a productive session, but we all needed some more caffeine afterwards. While reviewing the assessments, we noticed how the guides emphasized the standards and the Standards for Mathematical Practices. Both were given somewhat equal allocated text boxes in the guides. Both seemed to be highly valued by the publisher and our district math coaches. While the team was matching up questions and standards, I noticed that the SMPs weren’t getting any love. They sat there unhighlighted and under appreciated. This had me internally asking questions about how teachers actually incorporate and communicate the SMPs. So I went to visit the #Mtbos community and came across a Tweet by @cmmteach.

I completely understand that the standards are important, but what about the SMPs? These practices are part of our lessons, but I’m wondering how teachers address that importance. I asked a bunch of teachers this same question (I think they’re tired of me talking about it) and I generally get the same generic response. That response generally is, “I know what they are and they are part of the lessons” or “I sometimes mention them when moments come up to use them.” I see the SMPs briefly reviewed during math pd opportunities. I also observe posters of the practices hanging in the classrooms around schools. I even think there’s a Jedi one roaming the Internet as I’m writing this. I wonder how often they’re referred to and what students think of them. A few years ago I even had my students personalize the SMPs, but haven’t revisited them in as much detail since them. Are they really engrained as part of the daily math lessons or do they need to be outright communicated. Maybe there isn’t a right answer here.

I’m curious to how other educators communicate the SMPs. What’s your favorite strategy or technique?

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