## Rounding Numbers

Over the past few years, my approach to teaching rounding has followed a similar path. The choice of methodology was often influenced by the adopted textbook within a school district. I would generally guide students to place numbers on a number line and determine their proximity to the nearest value. The class would focused on rounding to the nearest tens, gradually progressing to hundreds and eventually expanding to larger place values.

We would introduce a rule that involved underlining the digit to the right of the one being rounded. The rule states that if the underlined digit is “5 or more, round up; if it’s 4 or less, keep it the same.” I believe this has even been turned into a song at some point. This linear progression served as the foundation for teaching rounding at the early elementary level.

This year brought about a change in our district as elementary teachers adopted a new resource. The emphasis shifted to exploring the value of numbers even before delving into the concept of rounding. Students were tasked with identifying the value of specific digits and ordering them, whether from greatest to least or vice versa. We incorporated the use of base-ten blocks and relied on expanded form extensively to help students develop a deeper understanding of number formation and the impact of place value on value itself. Students became more familiar with the value associated with each digit within a four or more digit number.

Students began identifying the closest multiple of a given number. For instance, they would determine the nearest multiple of 10,000 for a number like 432,000. This transition was seamless because students had already explored multiples and factors earlier in the year. Understanding how multiples relate to rounding was a novel approach and it resonated well with the students. Now, when asked to round to a specific digit, students consider the nearest multiple, which seems to make more sense to them. I am excited to continue using a similar rounding teaching strategy moving forward.

## Assessment Retakes

Student assesment retakes have been a controversial topic among educators and parents alike. Some argue that giving students the opportunity to retake a test is necessary to ensure that they have mastered the material. This seems to be more prevalent around the circles that embrace standards-based practices. Others believe that it creates an atmosphere where students are not held accountable for their initial performance. I have seen first-hand how the idea of a retake plays a role in how students approach a test knowing they have a second attempt if the first goes awry.

There are several questions that must be addressed when considering implemeting as assessment retake policy. Where should students retake the test? Some schools may have designated retake days or a flex time, while others may allow students to retake the test during a designated study hall or after school. If it is after/before school, tranportation considerations need to be taken in to account. This can be an issue with invidividual teachers if no time exists for the retake. I know of some schools that build this “flex” time in to their master schedules while other schools leave it up to the teacher to decide if it happens.

Another important factor to consider is how much practice students should have before taking the retake. It is important to ensure that students have a thorough understanding of the material before retaking the test. This may involve additional practice materials or targeted review sessions with a teacher or someone else.

Additionally, it is important to determine how long the period should be between the initial assessment and the retake. Some schools may require a certain amount of time to pass before allowing students to retake the test, while others may allow students to retake the test immediately after receiving their initial grade. My thinking is that a certain amount of time is needed for error analysis and practice to occur before another attempt. Missing classtime for the retake can cause issues down the road.

Leaders should consider whether the retake policy should be implemented school-wide or on a classroom-by-classroom basis. Some schools may choose to have a consistent retake policy across all subjects and grades. Other districts or schools may leave it up to individual teachers to decide whether to allow retakes.

I believe the goal of any retake policy should be to promote student learning and achievement. I wrote about this same topic a few years back and am still refining my thiking on how to make retakes more effective.

## Area and Math Mosaics

I believe that it is important for students to perceive math beyond mere digits on a page. Math is often seen as a subject that requires excessive computation, but it encompasses spatial reasoning and artistic aspects as well. Recently, I conducted a class where I blended the skill of multiplying unit fractions with creating a math mosaic and it turned out to be an enriching experience for the students.

To introduce the task, I collected different colored poster paper, glue sticks, scissors, and pencils. I divided the students into random groups and asked them to select an image that was traced onto the larger poster. The teams then decided on the colors and the quantity of each color required to complete the entire mosaic. Students estimated the amount needed and then used 1-2″ strips to find the exact amount. The gluing part of the project was the most time-consuming and required precision.

I was impressed by the critical thinking displayed by the students as they worked on the project. They were meticulous in their approach and made necessary edits to ensure that their work looked aesthetically pleasing. Most groups had to revise their total area numbers as well as number models. Additionally, the groups had to determine which medium to use to create their project, such as stone, tile, or glass. They calculated the entire amount required and added it to their total.

Once the project was completed, the class had a gallery walk where they examined all the creations. This project proved to be an excellent opportunity for students to practice their fraction multiplication skills while infusing an artistic element.

If you’re interested in exploring this specific project further, you can find detailed instructions by clicking here.

## Transitioning Back to School

The first four school days of 2023 are officially in the books. This school year students and teachers had about 2 1/2 weeks off from school. Teachers came back for an institute day on Monday, and students returned on Tuesday. I find that every year, the transition from winter break back to a regular school routine can be rough. Students and teachers alike make a hard stop and transition back to commuting, eating at certain times, sustaining attention for a certain amount of time, and remembering expectations, etc. Having an institute day on Monday before the students arrived back was helpful in preparing to gradually move students back into school mode. The planning of the first few days reminded me of the first few days of school. They are actually similar in acclimating students to a routine, building a classroom community, and putting together expectations. I made an extended effort to build these in place as students entered the building on Tuesday. This post is primarily used to remind myself of what to do next school year and to share what seemed to work/didn’t work.

On Tuesday studetns came back and I gave time for them to discuss their break with their peers. Most of the students did not have a chance to talk with each other over break so this was a time to reconnect. After that students worked on filling out a 2023 reflection sheet that was created by @druinok.

Students had no problem coming up with 2 good things that happened in 2022 and 2 things that they were looking forward to in 2023. They had a bit of trouble with something to stop and the three goals. The class brainstormed a few ideas about what to stop and a common theme was procrastinating and having a positive attitude. Students then took the sheet and made a few edits after thinking it over. I mentioned that we will be revisiting this later in the school year.

After completing the sheet students added their responses to a Desmos deck that had similar questions. Students logged in using their Google credentials so I could provide feedback.

Students filled out the deck and confirmed their selection on slide seven. Later that evening I went into each submission and wrote a few comments.

One was related to what they did over break and the other was about their goal(s). Students reviewed the feedback the next day. I need to remind myself to do this next year as most students enjoyed this time and I was able to reconnect with them individually.

The second activity involved teams and involved blending math, puzzles and teamwork. Fortunately over break I found a terrific 2023 puzzle by @mathequalslove. I printed out the puzzle at home and tried it out. The “easy” puzzle was a perfect fit for my class as the pieces went horizontal and vertical. Students were randomly placed in groups and assigned the task of putting together the puzzle. I mentioned that the pieces could go horizontal or vertical. I didn’t realize that (or didn’t read it carefully enough) when I put it together at home and had to reach out to Sarah to find a solution. Some students had a challenging time putting together the puzzle. I had a few groups that thought it was impossible, but then they prevailed. Students cut out the final product, put a few designs on it and I put it on the wall. My hope is that when students see the wall it will bring back positive memories of persevering and working through a challenge.

The third task to help with the transition involved order of operations and collaboration. I have to give props to @seewins for putting together the 2023 year game challenge. I alwsy look forward to this amazing resource as Craig as been creating them for years.

Students worked in stations to find as many solutions as possible. The class worked on this for around 20 minutes and there were cheers when the class found a solution – talk about teamwork! I left the task open this week and some students even got their sibilings involved. One kid with the help of an older sibling was able to get 100.

On Friday students finished off the week by reflecting on the last four days. They reviewed their goal sheets and filled out a simple deck on how they were feeling.

The results indicated that many students were in the easy or not there yet. Only a few indicated that it was really tough. I believe we are making progress, but not fully in a routine yet. I feel like using activities like these mentioned inthe post has helped make the transition a bit easier and I will most likely use someting similar after long breaks moving forward.

## Math Assessments and Self-Reflections

As 2022 ends I’m starting to think about next year. I’m now in the middle of a school break and reflecting on the progress that was made this past year. I’ve had some time to think about the last few weeks of school and what will come in January.

Before break my fifth grade students finished a unit on decimal multiplication and division. During the first three assessments I kept on finding that students made simple mistakes or didn’t completely answer questions before turning in the test. I feel like part of that is due to the increased staminia needed as we traversed from remote to hybird and eventually to in-person learning. The simple mistakes or incomplete work pieces were overlooked and impacted their marks – especially related to written mathematical responses.

To address this I decided to created a test checklist. The checklist included a line and a task. For example, __ I made an estimate before using an algorithm. The sheet was about 4″ x 4″ and printed out on colorful paper. All students filled out the sheet and checked-off each line before stapling it to the front of the test. I’d say most followed-through on checking and it reminded students to check their work in the process.

After the assessment I had students self-reflect on their performance. Students completed a Desmos task and here is the deck.

Students then checked-off correct vs incorrect answers. Students saw a list of concepts that might need bolstering and strength areas based on thier initial responses.

Students spent a good deal of time on this particular slide. They had to made a judgement call regarding where they were compared to the standard. Some students felt like they should’ve been placed in a different category because of a simple mistake. The next slide added an opportunity for students to provide context for their analysis or ask questions.

For the most part students found the process useful. I’m looking forward to using a similar self-reflection process for the next unit assessment.

## 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.

## 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.