## Works in Progress

Informal thoughts and experiences from other math educators and me

Informal thoughts and experiences from other math educators and me

I read an article in the March 2023 issue of NCTM’s Journal, Mathematics Teacher: Learning and Teaching about how to extend aspects of “Number Talks” to create “Math Talk Classrooms” (pp. 164-173). This article,
“Building Equitable Math Talk Classrooms” by Karen C. Fuson and Steve Leinwand offered an example of a way to change a “Number Talk” problem to a “Math Talk” problem. They shared that there is a Grade 7 Number Talk prompt that asks students to “solve for x in the proportion 12/20=x/35”. The authors recommended turning it into a short word problem and allowing students to use whiteboards to share their strategies. They created the following prompt: “It costs $12 to buy 20 bananas. Sarah needs 35 bananas. How much will Sarah have to pay for the bananas she needs?” (p. 171). As I am currently teaching 7th graders, I decided to try it in my own classroom to see if the routine generated positive (and equitable) math discussions.I presented a group of 7th graders with the prompt from the article mentioned above and asked them to work with their elbow partner or independently on a solution. Nine students attempted to find the unit rate and then multiplied their rate by 35. Three students used various proportional reasoning strategies (e.g., $12 for 20 would mean $6 for 10) but were unable to get a solution. One student created a t-table labeling the first column money and the second column bananas and used proportions to fill in the missing values up to 35 bananas. I had one of the pairs that found the unit rate present their idea first. They explained that “If you knew 20 bananas cost $12, they you can divide 12 by 20 [writing down the improper fraction 20/12 instead] and simplify, you know one banana costs 1 2/3 of a dollar. Once you know what one banana is, you multiply this number by 35 to find out 35 bananas cost 58 1/3 dollars.” I asked if anyone had done it differently and another group showed how they divided 12 by 20 using the long division algorithm to get that each banana cost “about $1.60”. [This group also divided 20 by 12.] They showed when they multiplied 1.6 by 35, they got $56 for 35 bananas. No one challenged either of these explanations. I then asked the student who made the table to share her work with the class and explain her answer. She wrote her table on the board and said, “I worked backwards by halving until I had $3 for 5 bananas, and then I noticed [the first two numbers ]were just skip counting [that is $3 for 5 and $6 for 10] so I skip counted by 5s in the banana [column] until I reached 35, and then I skip counted by 3s in the money [column] to find that at 35 bananas, the cost was $21.” Unfortunately, she ended with, “I must have done something wrong, because everyone else’s number is so much bigger than mine. But I can’t figure it out!” I was about to ask her to say more, when the first student presenter blurted out, “Wait, I think I did something wrong because your table makes sense and $56 seems like way too much money to pay for bananas!” I asked the students to take some think time to look at their solutions again. After a few minutes, one student said, “I know $56 is too much because if I double [the initial values] the numbers, then 40 bananas would be $24 which is more than the bananas we need, so it has to be less that $24. [The student who did the table yelled out, “Wait, I am right!”]. So I must have divided wrong or something to get 1.6.” The bell was just about to ring so I did not get a chance to follow up with students directly, but I am interested in hearing more about their misconceptions around finding a solution to 12 divided by 20! This task did produce more discussion than usual. I will look for another Number Talk to turn into a short word problem to see if it also encourages more mathematical discourse! Connection to Equity: This activity covered many equity practices, but I would like to focus on how it affirmed “mathematics learners’ identities“” because it promoted the idea that “mistakes and incorrect answers are sources of learning (from the “Five Equity-Based Practices”). Only one student found the correct answer on the first try, but by reasoning through that student’s work, other students found strategies (such as estimating the $24 for 40 bananas to know $56 for 35 was too high) that will help them in the future. Furthermore, this activity included others as experts (the “I” in the ICUCARE Equity Framework). The student who presented her table was a special education student who often struggles in discussions, but by highlighting her work and strategy, it gave her confidence that she can make valuable contributions in math class too!
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In chapter 1 of their book “Teaching for Thinking” Kelemanik and Lucenta discuss three shifts to teachers’ instructional practices that support greater mathematical thinking and reasoning. In the section entitled “Focus on Thinking” (pp.6-10) they introduce the idea of using sentence frames like, “ I noticed_________ so I looked for ________” to promote student thinking. [See issue 1 of my newsletter for more details.]I have used the “notice or wonder” strategy for years and incorporating these sentence frames seemed like a natural next step. That is, the sentence frames present a way for students to take their notices and wonders and use them not only to better understand the problem, but to also develop concrete solution strategies over time. For these reasons, I used the sentence frames in the launch and closing during a 7th grade lesson on integers this week. During the launch, I presented students with a table recording a climber’s beginning and final elevation in two different columns. I ask students to think about how we could represent the difference between the final and beginning elevation and then share their ideas with the sentence frame “ I noticed_________ so I looked for ________”. At least during this initial attempt, my students did not share any strategies. I heard responses such as, “I noticed the first column only had two different numbers, so I looked to see how many different numbers were in the second column.” While true, this does not get at a useful strategy for finding the difference between two elevations. However, I had better luck with using “We noticed_________ so we ________” and “They noticed_________ so they ________” in the closing. One group said, “We noticed that finding the change was easier when we had a picture, so we put the numbers [from each column] on a number line that went up and down like the thermometers.” I also felt the students were rephrasing each other’s work with more detail when they used the “They noticed_________ so they ________” sentence frame compared to when I just asked them to restate a peer’s thinking.Next time, I will only use the frames in the closing because I think it is easier for students to understand how to share a strategy once they have an answer. However, I will record their responses over several days and then have a class discussion about how we could use some of these ideas in the launch. I think this will lead to more meaningful strategies being shared at the beginning of the activity. |
## AuthorI am the Math Instructional Leadership Specialist (ILS) at the Conservatory of the Arts in Springfield MA. All opinions and suggestions are my own and do not necessarily represent Springfield Public School policy. ## Blog Archives## Categories## Newsletter Archives |