Better Quality Math Discussions

This year my district decided to switch to a new math resource. After using Origo for more than a decade we are now using Illustrative Mathematics. Besides the change in materials, teachers have had to navigate a new platform and instructional approaches that are significantly different compared to the last adopted resource. This change in expectations has been a challenge. The shift in a different math instruction approach was discussed during curriculum night earlier in the year. One of the larger foundational shifts involves the increased amount of math discussions that are expected to occur throughout a lesson. This year students are asked to engage in quality math discussions at least a few times every lesson. There are many “what do you notice, what do you wonder”type of prompts as well as others. The conversations are usually around 3-5 minutes and then students share their discussions with the whole class.

Along with other teachers, I observed that the math conversation opportunities were far from perfect. Some groups had one particular student that spoke for the entire time. Other groups didn’t stick to the prompt or jumped into the conversation before the partner was ready to discuss. After reflecting a bit I felt that students needed a routine for math discussions. That structure, just like many of the routines at the beginning of the year, would hopefully pay dividends as the year progressed. My goal was and still is to improve the quality of the math discussions happening in the classroom. I re-read this book to get a few ideas bout the process. Then I started to build a Desmos deck to help communicate the process that the classes were going to use moving forward.

The deck started off by asking students about past math conversations.

Students picked an option and we discussed it as a class. The consensus was that the class should analyze the picture or problem first. That led to the next slide related to what happens after we analyze the prompt.

Moving on the next slide gets the students talking about the process after analysis. Students will give a non-verbal signal showing that they’re reading to discuss. The class had a fun time creating their non-verbal signals, although I had to repeat more than once to make sure they were appropriate (ah fifth graders!). As students progressed through the deck we reviewed who should go first in the group and why.

We went with the alphabetical approach since the groups will change throughout the year. The class also discussed how to non-verbally show that you agree with the statement from your partner to make sure the conversation continues without interruption. We also discussed sentence stems that can be used to help start the conversation.

The next few slides reviewed the process discussed earlier in the deck.

The class went through a review of the process and tried out a practice round with their current partner. The entire class deck took about 25-30 minutes including the practice round. Feel free to use the Desmos activity by clicking here. The class completed this activity on Tuesday and we used the process daily since then. So far I’m seeing positive results and better quality math conversations. Of course there are hiccups, such as students still using more time than anticipated and/or students finishing too early, but I’m glad to see the conversations moving in the right direction. Later in the week the classes reinforced the math conversations procedure with this quiz.

I’m curious to see what others use to emphasis quality math conversations in the classroom.

First Four Days

School is open! The first week of the 22-23 school is in the books. It has actually been four days but l will round up because it felt like more than a week. It has been a whirlwind of a first few days as students headed back to school and are starting to settle into new routines. Teachers are doing the same and navigating new instructional resources. This is the first time in three years that we have had a somewhat normal of a start and the overall excitement being seen around the school is telling.

I find that students are excited and at the same time anxious to be back in school to see their friends and begin a new journey. Class discussions about community, expectations, routines, and drills flooded the halls this week. Clubs and sports are in session and the usual community builders are back in action. This year has been a bit different as there is a higher than usual emphasis on social and emotional learning as well as drills regarding safety. Beginning of the year professional development was geared towards bringing awareness to the need for students to feel like they belong. Staff kept this in mind when thinking about community builders early this week. The list below includes a few items that were used during the first four days.

Sara’s name tents. I missed a couple days and will need to make them up next week. I continue to be amazed with how receptive students are to the correspondence and it is a great way to build rapport.

Class puzzle. Each student receives the same piece (about 6″ x 6″ that includes information about likes and dislikes. It also allows for an opportunity for students to use art tools to create a background image. The pieces fit together to make the class. I usually hand this up on the door for the year since each piece tessellates.

Getting to Know You Quiz. Students give the teacher a two question multiple choice quiz. The students get a kick out of creating a couple questions for the teacher to complete. Even more, they find joy in becoming the teacher and grading the teacher’s response.


Along with the feeling of community I ended up getting a couple items in the classroom that are geared towards make a positive difference. These items are intended to help to contribute to the environment throughout the year.

Paper roller coaster. This is a yearly hit with the kids and without social distancing guidelines there will be more collaboration involved with the builds. I wrote about this in the past and am looking forward to using it with my 3rd and 4th grade crew over the coming months.

Kolam tiles. I am always game for math puzzles and games and this seems like a winner. I usually include a math station for students that displays different math puzzles. The Kolam tiles are unique to the classroom and will be a great addition.

Light covers. I have heard parents, students and staff discuss the impact of bright florescent lights in the classroom. Being under the lights for a prolonged period of time has caused me headaches before and this will give a different vibe to the classroom setting. I have already had a few students mention how they like it. My classroom is on the second floor this year so I refer to them as skylights.

I hope all of those in the classroom are off to a great 22-23 school year!

New Faces in the Teachers’ Lounge

I hear from time to time that things in education are difficult to change or the status-quo goes. It is possible to make a shift but that takes time, leadership and often a large amount of support. One thing that has changed quickly is the amount of teacher movement this summer. The sheer amount of teacher and admin position movement this summer is on the rise and I do not think we have seen the end of it. Political and systematic issues have created a challenging atmosphere in schools. More than a few teachers that I have interacted with over the past decade have moved to different positions and/or have switched careers altogether. While I am sad to see them go I am also excited for the new adventures that await them.

Now here is the situation. I am assuming that there will be many new faces in teachers’ lounges across the nation – recent hires and transitioning teachers. More so now than in most year. How will established school communities embrace these new employees? I truly believe having a staff with diverse backgrounds benefits a school district. The new hires have strengths and talents that should be highlighted. I wonder what will be in place to encourage new staff to feel empowered to bring their ideas to the table? What supports exists to sustain and retain teachers for the 22-23 school year?

I do not have clear answers to these questions, but it is worth digging deep to find solutions to make teaching a more sustainable profession. Optimistically, I would like to see the education transitioning tides change and to be able to look back and remember the 22-23 school year as one of the better ones in recent memory.

First Name Math and Art Project

It is official. The 21-22 school year is in the books. It was a year like no other as teachers navigated remote, hybrid and in-person learning throughout the school year. One of the last projects of the year involved a first name coordinate grid task. During the last week of school my class explored coordinate grids and plotting points. Students found the midpoints of lines and solved problems involving scale models on coordinate grids. I thought this task might be a way to help reinforce coordinate points and quadrants. In addition, it was great to have students encounter math in a unique way. Kudos goes to Cathy for sharing the deck which helps guide students through a learning process of graphing points on a grid.

The instructions were key for students as they built confidence before starting the final project. Some students decided to use scratch paper and create their name there first as a draft. Other students dove right in and started the creation process. Students were given about 30 minutes to work on the project and they were asked to submit it to Canvas once finished. About half of the class finished within that time. During the next class the projects were shared and students explained how they made the different shapes and borders using tables.

I was impressed with the time many of the students put in to make this project a reality. During the last day of school I handed each student their printed out copy of the project. I am hoping they brought it home and it is something that brings back a positive math experience memory. Looking forward, I am planning on doing this math and art combination activity next school year.

Area of Complex Shapes

One of my classes has been exploring area lately. They started by counting squares and differentiating between what is considered area and perimeter. Students were able to add halves and reasonably estimate what the area of a rectangle, parallelogram and triangle would be based on a brief observation. Becoming precise was not valued early on in the process but proved to be a tough transition as students were expected to use formulas later in the unit. Late last week students were asked to find the area of the shape below.

At first students were fairly confident in being able to find the area. They quickly counted up the squares that were fully visible. Then added the halves or what they perceived as half.

Students knew that there were at least 15 full squares covered and then added the halves. Estimates were given based on the full squares visible and ranged from 20 to 45. Confidence waned during this time as some students erased the numbers and started to deconstruct the shape into smaller shapes.

Earlier in the unit students made the connection that the area of a triangle can be found by using a rectangle method. Students also explored how parallelograms can be modified and rearranged into a rectangle.

Using that understanding, a number of students tried different methods to find the area of the shape. Students worked in groups to find a common understanding of where to start and how to dismantle the shape into parallelograms, triangles and rectangles.

This group decided to split apart the shape into triangles and rectangles. They specifically used the rectangle method to find the area of the triangles and counted the middle.

Another group tried a hybrid approach with mostly triangles and two parallelograms. The problem that this group had was trying to decide what constitutes the base and height of each triangle.

The other group decided to split one side of the shape into triangles and the other side into parallelograms. When I showed this to the class I received a few shocked looks. They were amazed at how simple this looked and yet they came up with the correct answer.


Overall, this was a time consuming task, but I feel like it was worthwhile. Students were able to think about math and measurement a bit differently. There are more efficient ways, but not one right way to complete the task. I am hoping that students remember this task and build upon their understanding as we move towards additional measurement concepts next school year.

Stock Market Game Simulations

My fifth grade math class has been participating in a stock market game simulation this year. In years past, this has been a culminating math extension activity for students where they can see how math and economics are related. Students use spreadsheets, gather data related to revenue/expenses, use math terms such as interest, rate, and explore world events that impact markets. All-in-all, it is a fun session that students tend to remember as they move onwards towards middle and high school.

Each year I have 5-6 teams consisting of 3-4 students on each team. Each team is given 100k and asked to invest at least 30% of their money in equities. The game occurs January – April. There is a brief introduction to the stock market and the metrics used to determine whether something is a good buy or not. My teams are only able to purchase/sell during class time and after a consensus is made.

This year the stock market game has been a wild ride. The invasion of Ukraine has directly impacted markets and students’ portfolios. Some of the teams are near 120k while others are hovering around 80k. Teams are getting their information from a variety of sources. Hot stock tips from someone at home (this happens every year!) or carefully researching and then deciding on what to purchase. The decision tree in what to purchase runs the gamut. Once students purchase a stock the emotional highs and lows are quite significant – especially this year.

At the end of the game there is usually some type of reflection. Students analyze their holdings and trading history. They reflect on what could have been done differently to optimize their overall equity in the end. While doing this, I tend to also reflect on how the game was organized and decide on what changes might be needed for the next session.

Even before this session ends I have come to the conclusion that a change is needed. Although I believe school stock market game simulations are fun and applicable, the game itself does not encourage students to look long-term. While reading students reflections in past years, I rarely hear comments about long-term investing because it is not part of the goal. Usually the comments involve regretting not buying at the right time or selling too early. There is generally a lot of emotional buying and selling going on during these simulations. I would say it is much better to do this with a fictional 100k and at 10 years old compared to 30. I have to wonder though what is being taught indirectly during the stock market game simulation process?

I would like to see simulations last longer than a few months and involve applicable situations. This year I heard the terms Bitcoin, cryptocurrency, Meta, Apple, and Netflix multiple times. I did not hear mutual fund, index fund or fees once. When is it appropriate to invest or save? How does investing look depending on these situations?

  • Plan on making a downpayment on a house in 5 years
  • Create a college fund for a daughter that is currently 7 years old
  • Plan a retirement fund for someone that is 35 years old

There are plenty of other situations that could be used. This adds a different dynamic to the game, but also allows students to see how investing involves planning depending on the situation. Instead of going with a gut-feeling or gambling, students could look at the risk involved in the time horizon and manage their investing accordingly. This type of simulation would involve more up-front time and education. I think it would pay off though as investing is not as simple as what is currently being used during stock market game simulations. I assume that students would see that investment risk depends on the context and that would influence their decision making process.

Fraction Division Strategies

My classes have been recently exploring fraction division. Students completed word problems involving dividing fractional pieces and they were finding the idea challenging. In order to gain clarity, I worked with students in small groups to determine where the trouble spots seemed to developed. I started to notice a couple things: 1) students were relying on a fraction division algorithm without context 2) students were not sure how to determine the dividend, which made creating a number model problematic.

Relying on the traditional fraction division shortcut ended up causing problems for more than a few of my students. Students were not able to explain their reasoning for flipping the second fraction. This become even more apparent when students attempted fraction division word problems. Because you have to “flip” the second fraction students were not sure how to identify the dividend. This caused confusion. I planned out a small fraction bootcamp for students to explore fraction division through visual models. Students started out with problems like 2 ÷ 1/4 and progressed to where a fraction is in the divisor and dividend. Students were making progress and relying less on the shortcut method, although some used that to check their work.

After our mini camp, students were given prompts to show their understanding of fraction division.

1.) Juliane has 12 bags of confetti to spread on 16 tables. She wants to put the same amount of confetti on each table. How much of one bag of confetti should she put on each table?

This was the first problem and achieved the highest accuracy. Students drew out the 12 bags and spread it on 16 tables, finding the answer to be 12/16. Some showed a number model of 12 ÷ 16 = 12/16 and others drew a picture.


2.) Write a number story that can be modeled by 4 ÷ 5 = 4/5

This was more challenging. The number stories indicated whether a students could determine what was being shared and in how many pieces. It was interesting to read the responses and revealed an understanding of what is being split equally. Here are a few response:

There were 4 candy bars and 5 children. How much of the candy bars will each child get?

I have 4 boxes of apples and I wanted to put them in 5 bags and all the bags have the same amount of apples. How much of the box of apple go into the bags?

Tyler has 4 rats and 5 carrots for his rats to each get equally fed how much will each rat get?

There were 4 oranges jamal and his four friends wanted to spilt the oranges to a even amount how much of and orange does each person get?


3. Explain using words and the process you would use to complete the problem 5 ÷ 1/3.  Give the reason why you completed each step.

This problem caused a few student headaches – but in a good way. Students that relied on the shortcut were confused in how to explain the reasoning for flipping the second fractions. Out of all of the problems, this one highlighted the conceptual understanding of fraction division the most. Some students sent in pictures with written explanations while others created number models. Here are a few of the responses:

First I would do 5 ÷ 1/3 This works, because it is the same question just written in a different way. Next I would see how many 1/3 can fit in 5. To do this  I would  do 5*3. This works, because there is 3 1/3’s I one. And there is 5 ones in 5*3 = 15. So the answer is 15. (appreciate the thorough thinking behind this response!)

1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3 

First I switched 5 to 5/1 and then 5/1 to 15/3. Why I did this is to make the denominators the same same number. Then I divided across numerators and denominators to get 15/1 then I simplified 15/1 to get 15. Why I divided across numerators and denominators is to get the answer. Why I simplified to make the number a whole number.

I think the answer is 15 because you can think about how many 1/3 are in 5 and that answer is the answer to your problem. 

First I converted 5 to 5/1 then I did 5/1 divided by 1/3 to get 5/1/3 then I did 5/1/3 X 3/3 to get 15/1 which I simplified into 15


I was pleasantly surprised to see the improvement in being able to navigate fraction division. Being able to conceptually understand fraction multiplication/division can sometimes be a roadblock for students. I am hoping to break that and looking forward to discussing and highlighting a few student examples with the class next week.

Math and Multilingual Students

This year I have been trying to intentionally read more books. Some have been educational while others have been more non-fiction wonderings. During the last couple weeks I have had the opportunity to read Teaching Math to Multilingual Students with a group of Illinois educators brought together by the Metro Chicago Mathematics Initiative. We read a few chapters and meet online to discuss our thinking. We are about halfway through the book right now and this post will document some of my takeaways as I think about math through a different lens.

Positioning

“Contrary to popular belief, student silence is often the result of unfair or inequitable positioning in content classrooms” p. 27

To be honest, the idea of positioning multilingual learners as classroom leaders has not been at the forefront of my mind. Positioning is is a concept that involves identity and access. Teachers are required to make many decisions lesson by lesson and they impact positioning within their classrooms based on what is being communicated and who is being a spectator. Positioning can have students’ competencies recognized or ignored by highlighting certain work/strategies and dismissing others. Intentionally planning out phrases that can be used might be one way to think about positioning differently moving forward. In the moment this can required a large amount of patience as the pace of the class has the potential to be disrupted. Hello wait time! Teachers should refocus students’ attention if disrespectful behavior occurs. It might be helpful to revisit norms to ensure everyone is on the same page.

Encountering Unknown Contexts

“How will you identify factors that hinder participation for multilingual learners in your mathematical classroom?” p. 43

Teachers tend to engage students in learning through contexts that are understandable. Many of the problems in district-adopted resources involves a few problems related to sports. From what I see, those sports at the K-5 level in math class are primarily basketball, football, baseball and occasionally soccer. Understanding the games themselves is a prerequisite to answering the question. These may be unknown to multilingual learners. Put the shoe on the other foot. I doubt many students in my class would be able to complete a math word problem about the game cricket without understanding the game first. This also applies to the vocabulary terms used to describe the game.

Group Work

“… One student grabbed Julia’s pencil out of her hand to complete her mathematical work for her.” p. 45

Many math classrooms are instructionally moving in the direction of having students work together to discuss their mathematical thinking. Communicating understandings and having to defend them is an important tasks and group dynamics play a role here. Teachers should discuss with their class what productive partnerships look and sound like. This might also be an important time to revisit math station norms. I have noticed that groups may sometimes show that patience is lacking and a particular students will complete the work for the entire group. I am assuming most educators have seen this type of behavior. I have also seen students take pencils out of the hands of others to write the answer. This is an act of positioning and the behavior should be addressed. This year has been trying in having consistent quality discussions in small groups. The last couple years of elearning and hybrid instruction has significantly decreased the amount of opportunities students have had to work with others outside of a Zoom breakout room. Getting back into the groove of being able to facilitate a conversation and possibly encouraging students to use sentence starters can go a long way in helping.

I am hoping to learn more as the book study continues.

More than Numbers

Before winter break 2021, my 3-5th grade students started an isometric name design project. I found this idea a few years ago on the bird app and was reminded after taking a look at Adrianne’s Desmos task. Since most students that I teach are in-person this year, I thought it’d be beneficial to expand on my geometry and measurement unit by having students explore the connections between math and art. To introduce the activity I showcased isometric art and grid work. Students were especially fascinated with optical illusions. Students were given directions.

Students were handed two pages of isometric paper and an example letter page.

After reviewing the examples, the students were off to work independently. Some students created draft drawings and other immediately started on the isometric grid. There were errors – many as expected, and the students took it in stride and persevered. I heard a few comments related to how this was definitely different than “regular math” and some students even brought a few pages home to practice. I’d say most students used 2-3 pieces of isometric grid paper. The shading was key to make the letters pop. If I was doing this project again I’d probably spend additional time having the students watch this video. Students needed to look at the 3d letters and pick which side to highlight to show perspective. This took a different type of thinking. Students also were asked to find the volume of their name using cubic units.

After the projects were posted, one student mentioned that math is more than just numbers. I’m more than inclined to agree!

Box Plots and Spreadsheets

One of my classes has been exploring box plots and data landmarks lately. Earlier in the year the class created histograms and found data landmarks on line plots. Box plots was not as easy as a transition as anticipated. There were a few roadblocks as students analyzed and created their own box plots while determining Q1 and Q3. Some students picked up on the concept quickly while others took more time. To help reinforce the concept I thought about bringing in a spreadsheet activity. I have been using spreadsheets quite a bit this year and it has been another medium in which students can experience statistics.

Students were first asked to create a question that they would be asking the class. The numbers could range between 1-51. I gave students free rein on what questions to ask and held my breath.. Here were a couple of the survey questions:

  • What is your favorite number between 1-51?
  • How many hours of sleep do you get per night?
  • On a scale of 1-50, what do you rate a cheese burger?
  • How many movies have you watched this year?
  • On a scales of 1-50, how well do you like dogs?
  • How many digits of pi can you recite?

Once students created questions they went around and surveyed everyone in the class. I gave each student a roster list so they could check-off who answered This took a good chuck on time – 10-15 minutes. Once the data was collected students grabbed a Chromebook and copied a spreadsheet that I had pre-populated.



Students took the data from the survey collection sheet and transferred it to column A. The data landmarks in row three were placeholders and awaiting formulas. Students then entered the minimum, median, maximum and mean formulas. They were familiar with those formulas as we explored them earlier in the year. I discussed with the class about quartiles and we put together a formulas for Q1 and Q3. We made predictions of what the vertical box plot might look like before finalizing. Students then entered the formulas for the quartiles and analyzed the box plot to see if it matched the data.

It was interesting to hear the conversations that students had as they compared the data to the box plot. The class had a discussion about interquartile range and variability. It was time well spent. From there, students shared their spreadsheets with me and I took a closer look to see how the data matched and if the correct formulas were in the appropriate places. Students seemed to grasp the concept fairly well. Feel free to use a copy of the spreadsheet by clicking here.

During the next day the class reviewed box plots and the spreadsheets that were created earlier. Students then complete the Desmos task Two Truths and a Lie. This is one of my favorite tasks for students to discuss box plots and use math vocabulary while doing so.

The spreadsheet and Desmos task took about 2-3 days to complete. The class took a unit assessment on Friday and I will be checking out how they did over the weekend. I put these two activities in a digital folder for next year.

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