My fourth grade students finished up a project involving area last week. Students were asked to find the area of different playing areas for certain sports. They first calculated the areas of the playing field by multiplying fractions and then found the product.

The next step involved creating a visual model on anchor chart paper. Students worked in groups to put together their athletic park involving the field areas. They were given the area of the park and then had to place the fields where they wanted according to the team’s decision. Students also added additional facilities for their athletic field and then presented their projects to the class.

While presenting, students in the audience were required to either 1) ask a question or 2) provide a constructive comment. Most of the questions that were asked related to why certain fields were placed in specific areas on the field. One question stood out more than the others … does the distance make sense?

Students were looking at the length of the fields and observing whether it was reasonable or not compared to the total length. The class then had a conversation about the terms reasonableness and proportions. The discussion involved how a double-number line and a grid could’ve helped visualize how the distances match.

I’m hoping to revisit this idea during the next few weeks as the school year finishes up.

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This year I’ve been using number sense routines* with my 3rd-5th grade classes. The routines have specifically been put into place to help students strengthen their place value and estimation skills. The routines last around 5-10 minutes and generally occur during the first part of class The routines is the first thing on the board as students enter. Students use a template, complete the routine independently and we discuss the results and process as a class.

Two of the more productive routines this year have been Estimation180 (3rd grade) and Who am I (4th). Both ask students to use hints or models and then use those visualizations to solve problems. Students document their thinking on an individual page and then we discuss it as a class through a debrief session. While working with students this year I noticed that not all students participated to the extend that I’d like. The conversations were decent and students were engaged, but the reflection piece wasn’t as thorough. So this year I’ve decided to add an individual reflection component for these specific tasks. The reasoning actually came from a book that I read back in April that emphasized how sentence stems can be used to help students reflect on their mathematical thinking.

I put these sentence stems into practice and added them to a reflection sheet. I added extra space after the “because” to help encourage students to write more about their own thinking process.

Students complete each one of these around 2-3 times a month. Students complete the reflection sheet, discuss the writing with partners and eventually put them in their folders. The sheets are revisited throughout the year to see the growth over time.

* The images from this post are from a math routines presentation on 5/3. Feel free to check out the entire presentation here.

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There’s around one month of school left and it feels like the home stretch. The next month is full of changes. The weather changes from chilly temps to sunny days (at least in the Chicago burbs), class lists and sections are starting to take form, driving to/from school with the windows down is the norm, and planning for that final month is in full swing. The majority of my math classes just finished a unit assessment and there’s one unit remaining. So often I find that students perceive the end of a math unit to “close out” the learning on a particular skill set. I observe that this idea often gets pushed out as grade deadlines approach.

As my classes start a new unit I’m pausing to reflect on how my practice has changed. Last year I read How to Make it Stick and I intentionally planned to use more retrieval practices. This year I’ve incorporated more review opportunities through online formative quizzes and by trying to make implicit connections to past learning. I’ve often asked students how today’s objective connects to this week’s learning.

While digging through my resource materials early this year I found optional mid-year and cumulative assessments. Generally, I find that there’s not enough time to complete all of the assignments/tasks in the resource so these particular tests aren’t used frequently. This year I decided to use them to help with spaced retrieval practice. Instead of using a mid-year and cumulative assessment directly following a unit I decided to space out these assignments and take off the grading emphasis. These types of assignments take multiple days to complete and I often have students work with partners to reflect on their progress. So far I’ve seen positive progress as students this year are referring back to past skills more quickly and bridging the connections on a frequent basis. I’m looking forward to using a similar strategy next year.

]]>One of my classes is in the middle of a unit on geometry and measurement. They’ve identified shapes before, such as rectangles, squares, triangles and hexagons. Earlier in the year they found the area and volume of shapes involving rectangles, squares and triangles. The current unit investigates how polygons (specifically triangles and quadrilaterals) are similar and the study of shapes progress as students create hierarchies.

- CCSS.MATH.CONTENT.5.G.B.3

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

- CCSS.MATH.CONTENT.5.G.B.4

Classify two-dimensional figures in a hierarchy based on properties.

In order to dig deeper into the above standards the students starts the classification process. This was fairly new for most of the students. I explained what classification meant and gave a few examples related to the characteristics of triangles and quadrilaterals. Students were given a sheet of quadrilaterals to cut out and classify. The next question I was asked was related to how each shape should be categorized. The class reviewed different vocabulary words associated with polygons and then I left the students create their own categories.

After discussing equal side lengths and parallel sides two of my students created the classifications related to those terms.

Almost every student had a different way to organize their shapes. Students went to different tables and observed how their peers classified the shapes and then the class discussed similarities. Next week students will classify the shapes with a hierarchy chart. I’m looking forward to seeing what they create.

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My fourth graders are in the midst of math project. They’ve been studying measurement and are completing a project involving creating nets, assembling them and finding the volume.

I’ve used this task in years past and students spend a decent amount of time planning and putting together their rectangular prism cities. It’s generally one of the “favorite” activities of the year as indicated by student surveys that I give in June. The engagement is great and students are proud of what they create at the end. Now to the challenging – It takes an LARGE amount of time to complete these types of projects. Students have the potential to lose focus and stray from the concept/objective. I’m certainly not a pro with math projects, but I’ve found certain things work, while others don’t. The bullet list below could apply to other long-term (>3 class sessions) projects beyond math. I’m tackling the points below before I plan out a fifth grade project that’s scheduled to take place in April.

**Clearly define directions, expectations and criteria**- I spend a good 15-20 minutes explaining the project and directions with the students. During this time I’ll answer students’ questions and elaborate on the criteria for success. I tend to also reinforce the expectations of how teams should work together (because all teams works great, right??) and what goals they’ll accomplish by the end of the project

**Objectives … Objectives … Objectives**- I remind the students of the objectives and skills that the project will be addressing. The projects are fun and engaging for the students, but I want to ensure that they understand the reasoning behind the project. Teachers understand why the project is happening, but it’s also good to have a list available when an admin stops by your room and students look like they’re creating something massive with paper, iPads, scissors, glue and other materials. Also, the SMP‘s can play a huge role here. I personally find it challenging to pinpoint exactly where the SMP’s become directly evident in lessons (it’s usually a vague “hey look we’re using attend to precision here” type of statements. Math projects are full of the SMP’s and this aspect can be part of the objectives and emphasized in a self-reflection activity – see last bullet point.

**Eliminate specific models/examples**- This might irk some people, but I’m not a fan of showing examples of what their project should look like. Providing really vague or general examples are okay in my book. I tend to get questions asking if a certain aspect of the project could look like _____. I tell students that if it follows the criteria it’s good to go. Ideally, I’d like students to work together and create something original, not copy what I show as the example. This allows students an opportunity to focus on the criteria and not “what the teacher wants” type of mentality.

**Create a timeline**- I find creating a timeline is one of most important pieces when introducing the project. Adding in checkpoints along the way where teachers “check-in” on what’s happening gives students (and me) an added accountability piece to make sure we’re sticking to what’s expected.

**Be flexible**- Sometimes timelines need to be changed. Assemblies, snow days, fire drills, (insert an event that impacts your instruction) happen. Be upfront with the students that the time will need to be extended. Most of my students give a sigh of relief when I tell them that they’ll have an extra period to work on the project – so do I as I want to make sure that they make a quality product.

**Include self-reflection**- Students need time to process the math that they’re using while completing a project. I like to give students time to write down how they’re using the time that they’re given and what was accomplished during that session. I find providing this time gives me an insight to how each group is progressing and also adds an emphasis on what skills are being addressed. For math projects, I find that adding a reference to the SMPs can be an added bonus as most of them become apparent as students create their projects.

**Sharing is caring**- After everyone has finished I like to share the projects to people outside of our classroom community. I might share a link out on Twitter and have the students submit their projects to SeeSaw. Sharing with other classrooms in the district has an added bonus. Plus, students are creating their projects for an authentic audience and they have the potential to receive feedback. That adds another quality component in my mind.

I find that having these components in place before assigning a long-term project to be helpful. It makes the project worth the time as students are more efficient during that time and the quality of what’s created tends to be better.

]]>This post has been marinating for a while and I’ve been waiting to write it up. State testing is just around the corner and I feel like this is a good time to press send.

One huge emphasis that I see in schools is related to the idea of student growth. This is communicated in schools, during teacher pd meetings, when talking about Hattie’s next best effect-size list and can even be part of teacher evaluation criteria. I see this when school districts use MAP, state testing, or a similar type of tool that measures growth over time.

Schools, teachers, and parents want students to grow. In schools the focus of growth primarily related to academic content. How is that measured? Well that depends on the teacher/district/organization. Some teachers go the route of using a pre vs. post-test. Others give multiple formative checkpoints and then use them along with student reflection components to show growth on the summative. Case in point – there are multiple methods to show student learning and growth.

Here’s my not so small gripe. In an effort to show growth some educators may feel rushed to “get through” as much content as possible. I hear this a lot more in math classes than other content areas at the elementary level. Math has a subjective linear vibe that I think some teachers hold onto. This idea is often reinforced through the structure of some of the adaptive standardized tests that communicate math growth to teachers and school administrators. This can be a bit troublesome if these types of tests are used for evaluation purposes as it brings along additional pressure.

I find students make meaningful math connections when they’re given time to process and apply information. I believe rushing through concepts or stretching to just expose students to higher-level concepts that aren’t part of the lesson isn’t as beneficial as it seems. Moving off the pacing guide or lesson to stretch to other concepts might not be the best idea. When I first started teaching I remember having a teacher talk about exposure all the time. The teacher would say, “If they’re just exposed… then they’ll complete those problems correct and that’ll push them to the next concept.” This teacher was truly amazing (and made some great coffee in the mornings), but I questioned this then and still do now. If we’re exposing students to math concepts so they’ll score well on an adaptive test then that’s another issue altogether.

If a lesson is looked through a linear math lens a teacher might feel as though they should introduce fraction multiplication if students are doing really well with multiplying whole numbers. Is that the right move? Should a teacher stray from the lesson plan to possibly reach a few kids that seem like they’re ready? I’m not saying yes or no because it depends on the station and students, but I’m more in the no camp. Stretching math concepts in a lesson/task for exposure sake doesn’t last.

Last summer I was able to reading Making It Stick and came away with some applicable ideas related to changing my study guides and how retrieval practice benefits those wanting to learn. I find that there’s sometimes pressure to stretch to another concept for exposure sake. Instead of stretching concepts, there should be opportunities for students to enrich their understanding through connections. This looks different and is more challenging in my opinion than just pressing the accelerate button temporarily. I believe that taking time for students to process, reflect and engage in meaningful math tasks will last more than a glimpse optimistic exposure that may soon be forgotten.

]]>Last Tuesday I was able to participate in the ICTM chat. The chat was on the topic of math writing in the classroom. My teaching journey began as general fourth grade teacher and writing was often placed squarely in the writing/language arts block. This year I’m trying to find ways encourage students to strengthen their writing in the math classroom.

When asked to explain their writing I find students at the upper elementary level sometimes struggle to find the right words. Some students have the perspective that writing should be minimal to non-existent in the math classroom. Mathematical writing at the 3-5th level is often expected on state and district assessments. Beyond that it’s a bit inconsistent – depending on the teacher. Generally the math writing prompts ask students to explain why or the steps involved in solving a problem. When students approach these problems I find that they’re sometimes unsure of where to start or are very brief with their submission – often too brief.

Over the past few years I’ve been working for ways to chip away at this issue. I think part of the concern is due to exposure and practice. I believe students have opportunities on a daily basis to explain their math reasoning. Quality math tasks can give students opportunities to engage in explaining their math reasoning. I see it more so in classrooms through verbal interactions – not so much in the written realm. Students may present their ideas to the class and/or engage in a dialogue with others about why their answer makes sense to them. I also see students participate in these types of sessions with partners or a small group. I find that students can take those meaningful dialogue experiences, but lose some of the substance as teachers attempt to connect the transfer to written form. Bottom line, I feel like students need more meaningful math experiences with writing and revisions in a math class setting.

How do students get there? The article discussed in the chat covered four different types of math writing:

- Exploratory Writing
- Informative/Explanatory
- Argumentative Writing
- Mathematically Creative Writing

I find students aren’t expected to write unless they have a prompt to answer – which generally falls into informative or argumentative. For some students this is the only practice they get in strengthening their mathematical writing skills.

Part of the questions discussed in the ICTM chat revolved around the different types of math writing that educators currently see in the classroom. Organizing this math writing (along with a criteria) into the four categories was fairly new to me and I dug deep into trying to find ways to apply this in my own classroom. Moving forward I plan on using my student math reflection journals with more frequency. Right now students use them to reflect on their math unit assessments, set goals and progress monitor. They’re revisited every 1 – 2 months. I find that’s a valuable use of time, but I’d like to expand and have students more frequently reflect on weekly topics or the skills that are highlighted in class. I’m also planning on looking at more of the creative math writing component. I find that interesting and there’s a self-motivation piece that could be helpful. Planning on using this will take some time, but again, I feel like students will benefit from analyzing the skills discussed in class and applying them to their life.

I think the more practice that students have with writing in math class the better prepared they’ll be to explain their mathematical thinking in written form. I’m looking forward to seeing how this idea turns out.

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Last week I had the opportunity to attend the IAGC conference. I don’t attend every year, but this year I went with a colleague because we both were presenting. For those that might not be aware, this conference is really geared towards gifted education in the state of Illinois. Administrators and teachers primarily attend this conference. It’s a multi-day conference that’s hosted in a Marriot in Naperville. I believe last year there was around 500 people (I’m estimating) attend – so it’s more of a smallish conference compared to some of the other education powerhouses in the the state. The freezing temperatures impacted attendance this year as driving conditions probably kept some away.

Most of the attendees that I saw last week are teachers that work with students with exceptional needs. Many of the teachers created sub plans and then school ended up being canceled because of the cold weather. Go figure. Whenever I’m asked to be out of the classroom for a meeting, conference, workshop, etc … I always am wondering if it’s truly worth the sub plans – kinda joking here.

What I enjoyed more particularly about this conference was that some of the presenters were teachers. Certainly not all, but there were a few sessions that I attended that involved teachers presenting to an audience. Some of the themes were related to acceleration, leadership, curriculum, social-emotional and equity with identification. I attended on Thursday and was able to attend a couple sessions – mostly related to curriculum and differentiation.

There’s a practicality component that is involved when a teacher is having a conversation with another educator. Most of the sessions were small enough that you could engage with the presenter by just raising your voice. Love that. I find it similar to an edcamp experience as individuals have a similar understanding of what’s expected in schools. There’s a large difference (at least from what I see) between teachers currently in the classroom and those that aren’t.

Now onto the sessions … I attended a session related to using adapted software in the classroom. Hearing positive ideas about how to use these in the classroom was different since I’m usually not too fond of this type of learning. The presenter gave a brief overview of a couple paid/free adaptive programs and spent a good amount of time answering questions about how to use them in the classroom. Another session emphasized the use of Jacob’s Ladder scaffolding techniques. The questions from the participants spurred additional questions – love when that happens. Later in the day Cheryl and I presented on using math routines in the classroom. Participants that attended offered their own ideas on routines that they found helpful.

It was truly an engaging experience to hear how others use and differentiate math routines. It was also refreshing to connect with other educators outside of our local area about how to create more meaningful math experiences for students. I think teachers need to explore outside of their district boundaries from time to time. Discussing some of the routines afterwards was also encouraging.

All in all, it was decent conference and worth the sub plans. I left with a better understanding of how certain tools and strategies can help meet the needs of exceptional students. I’m looking forward to the conference next year.

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My second grade students just started a unit on decimals. Based on the pre-assessment, most students have no problem with identifying the value and place value position of digits in the ones – hundred-thousands place. It’s a different story for numbers to the left of the decimal point.

Earlier in the week students explored the tenths and hundredths place. Students connected money concepts to place value and fractions. They compared 1/2 with 0.50 and $1.50 with 1.50. They completed similar activities where they needed create benchmarks on number lines and place numbers. Some were still having trouble and I believe this is partially due to exposure. Also, I was finding that their were issues with spatial awareness. Students were looking placing able to approximate benchmarks of half, but placing 0.1 close to the half. Student practiced using number lines and using benchmarks. The most tricky piece was looking at the differences between the hundredths and thousandths. This challenge reminded me of how students develop an understanding of the magnitude of numbers.

Today I grouped students into teams and they used dice to create different decimals. The decimals ranged between 0 and 3. Students were given a horizontal and vertical number line on a 11 by 17 paper. This gave students room to work. The two number lines were different sizes. An indicator line was placed at the beginning and end of each line.

After students created their decimals they started to place benchmarks. Some students had to get out the erasers as realization set in that the maximum would be three instead of two. Students also reevaluated their benchmark placement. Groups noticed that the two number lines were different sizes and had to adjust their benchmarks accordingly. I found it interesting that some students used the vertical number line top down, while other went bottom to top.

We’ll be reviewing the number lines on Monday. I’m looking forward to the discussion and we might even break out the rulers to evaluate the reasonableness between benchmarks.

]]>Last night I was fortunate to attend ICTM’s chat on feedback. It was a productive chat and Anne had some great questions cued up for us. I came away with a few new tools that I need to research. Chats like these are motivating as the frigid cold of the midwest is ever-present this time of the year and new ideas can spark my planning process.

Teachers know that student feedback is important – it’s everywhere in schools. It’s on every teacher evaluation tool that I’ve experienced. ASCD describes it as “Basically, feedback is information about how we are doing in our efforts to reach a goal.” Teachers give feedback all the time – most without even labeling it specifically as feedback.

The chat was still on my mind this morning as my colleagues and I were having a conversation about math units. After reviewing multiple student papers, I started thinking about feedback in more detail. Specifically, I started thinking about how feedback takes on different forms and the tools that are used to give that feedback can vary from class to class. In all cases that I’ve come across, educators want students to actually USE the feedback.

Technology can be used for this although the reliability of the feedback might not match the need. I’ve also seen cases where the automated feedback is disregarded by students in an effort to score more points. It depends on what’s needed. In some cases, a quick verbal prompt might be the feedback that’s needed. For others, a conversation with a partner can help students identify misconceptions or spur thought.

Let’s take this problem:

This particular students was able to identify the rule and complete everything but the bottom problem. Being able to anticipate misconceptions can lead to better student feedback. There are a few questions that I might have before approaching the student and giving feedback.

- How can this student divide 14 by 7, but still have trouble with the bottom problem?
- Does this student think of “divide by two” as half of the in?
- Was this a simple mistake?

Or here’s another one:

- I notice that there isn’t any work or model here
- Did the student notice that the denominators weren’t equal?
- What strategy was used here?
- Was this a simple mistake?

Last one:

- Did the student miscount the boxes?
- Is the students missing pieces? (yes, this has happened before)
- How did the student get 6 as the numerator?

In all of these cases a simple mistake is probable. I’m working with K-6th grade math this year and sometimes rushing leads to simple mistakes. I try (as much as I can) to limit that option when deciding to give feedback. In all three of the cases I could ask the student to recheck their work. Some students will, while others won’t. I could also write on their paper a statement or question about wondering what strategy they used. I could also have the students meet with a peer and discuss the problem in more detail.

There are so many ways to communicate feedback and it’s not a simple issue. Some students are more responsive to written feedback, while other students want to have a conversation with another peer to discuss their strategies. As students get older the type of feedback also changes. Many of my upper elementary students prefer a brief comment on a paper or a quick underline, question mark, or specific arrow to help them move towards a goal. Having a 1:1 feedback conversation with a student is my number one option because then I can see how receptive they are and answer any follow-up questions. If you don’t have time to do that with every kid (who does?) then you use other options.

There are a ton (I mean a TON) of apps out there that “help” students along their math learning journey. I tend to be a bit caution when deciding to use them in the classroom. Is the feedback appropriate for their needs? Is the feedback helping them in their efforts to reach a goal? In some cases it may, but I think it’s worthwhile consider the ways in which feedback is given.

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