I still feel this way about math rules, however, reading Tracy Zager's book, Becoming the Math Teacher You Wish You'd Had, gave me a new perspective on how I handle these situations in my classroom. Mathematicians look for patterns. Mathematicians make generalizations to extend what they know to other problems. The problem isn't the math rules that my students know, it is that they overgeneralize (use them when it is not appropriate) or have lost the mathematical reasoning that led to the generalization.
Tracy suggests making a "Claims Wall". As students start to make these generalizations add them to the claims wall and continually revisit them to be sure they continue to work in other contexts and with other number sets. A colleague recently told me, we need to play to their strengths. If rules are what students know, then that is our entry point into the conversation.
With this in mind, I started my review of fraction multiplication and division by having students make claims. I gave each team a post it to write their claims and made it clear they did not need to explain them at this time. Then we looked at everyone's claims and made a combined list. Here is what we had:
The other claim that was not on a post it from students is the last one on the division poster. It is one I added. After we had the claim that you can use multiplication, I put forward the claim that you can use division. I asked the students if they thought that was true. They all thought that seemed like a reasonable claim, but with only a minute or so to think about it none of them felt they knew how to use division to divide fractions. I can't wait for that conversation!
Starting the unit this way was a big risk for me. I started class knowing we would do claims and had no idea of where it would go from there. I did not know what I was going to need to have prepared for our exploration of these claims (because I didn't know what claims we would have). I had a few things on hand based on claims I thought would come up. It was a bit of an unsettling feeling. I decided to analyze the claims in an order I was comfortable with, not the order they are on the sheet. That allowed me to plan our approach and prepare more for certain claims.
In case you don't read beyond this I want you to know this risk was definitely worth the pay off for me. Because we were starting with what students knew they all had access to the activities. Then as we explored, they were able to self assess and realize what they didn't know about the topic. The learning about fractions became authentic (without made up "real-world" problems to explore).
Here is the breakdown of what we did and what I plan to do:
- Multiply across: I had the students go to the white boards (I have some new non-permanent vertical surfaces in my room). They needed to come up with mathematical reasoning why this claim works. Many groups drew unit rectangles and used an area model. One group used a coordinate grid (still an area model). Even though they knew the visual representation, through discussion we realized they didn't know how that proved their claim. They just knew that they had used it before in math class. (This is an issue I have with visual representations. We cannot just teach them to kids or it is just more math to memorize. Students need to be coming up with authentic representations and making sense of the math.)
- Cross canceling: We used this worksheet to analyze why this strategy works. I have used it in the past and had it prepared in hopes that there would be a claim about this strategy. This strategy helps as we work with proportions throughout the year and it's a great discussion about the commutative property.
- Multiplying mixed numbers: I sent students back to the whiteboards to show how to multiply without converting to fractions greater than 1. I want to pull some different problems to focus on efficiency with fractions greater than one compared to keeping them mixed numbers. We didn't get a chance to discuss yet, but here are some pics of what they created. I am very excited for these conversations (Spoiler Alert: when we talk distributive property we will also discuss their love for the order of operations and how it doesn't need to be followed if you have mathematical reasoning.)
- Multiplying by numbers less than 1: For this we will play the role of skeptic and see if we can come up with counterexamples. Then try to prove it will work for every case without having to list every problem that exists.
I will try to add to this blog next week. Last year I took a course through the US Math Recovery Council that deepened my understanding of fraction division. We used this book:
I want to revisit the chapter on fraction division to prepare for analyzing our division claims. It is a fabulous book and I highly recommend it for all ages of fraction work.
Here are my thoughts after this class activity.
In class we give students tasks to allow them to explore a concept and build their mathematical reasoning behind it. Then we pull together everything that has been discovered and come to these generalizations. Some students get to these math rules because they work through the mathematical reasoning and it leads them there. Others understand the mathematical reasoning but do not connect it with the rule that the class has determined. (I have many students who feel they struggle with math but actually have great math reasoning.) Others never understood either piece but continue to work to memorize the rules because the class has moved on and they need to be able to solve the problems. In the end most students know the rules. They do not care about the mathematical reasoning because they can answer the problems. How do we make them care about the mathematical reasoning when they can successfully solve problems? Does it matter? (I would argue yes.) Everything we do in my class is about mathematical reasoning. Explain. Justify. Prove. Always, Sometimes, Never. Notice/Wonder. Question. The focus is never on the answer. But if the answer is what the student can do, I do not want to minimize that. For them to want to know the mathematical reasoning I need to ask them to do things that require more than just calculation. If I can get students to continue to question, then they will not be satisfied with simply knowing an answer or a rule.
If you are interested in my other Becoming the Math Teacher You Wish You'd Had blog posts you can find a complete list here.