In Depth Algebra Conversation – Mathematics vs. Mathematical Reasons
Sometimes, amazing, in-depth conversations come out of the Community Feedback section of the lessons on CC.BetterLesson.
For instance, an incredible back and forth was inspired by Master Teacher James Dunseith’s lesson “Solving Equations by Constructing Arguments”. We’ve included the transcript below if you would like to read the entire conversation.
In summary, a challenge that math teachers often face is making sure that their students fully understand and appreciate the math behind their solutions. The discussion below focuses on how to find a balance between students solving a problem and students actually learning the properties that helped them arrive at their solution. James and Aaron, a fellow Algebra teacher and BetterLesson user, compare notes around the complexities of this issue. What is the difference between Additive Inverse Property and Addition Property of Equality? How do you get from one to another? James explains his strategy to Aaron in detail below.
Aaron Bieniek: James – thanks again for sharing what you do. I’ve found myself in the same place you are; sort of perplexed about the best way to justify each of the steps in solving an equation. I’m always trying to balance using the properties with not focusing on them too much as if to make the actual names of the properties what is important. I also think about how not to make it too much of a burden for the students to actually have to list each properly and probably write more steps than necessary. I worry about tripping that alarm in their minds that says “why in the world are we doing this?”. On the other hand though, focusing on the properties gets the students to focus on mathematics and mathematical reasons for what they do rather than tricks they may have picked up along the way. I have to believe that in the long run this work is important. I know it’s been a while since you did this with your kids but I’m wondering if you found that writing more steps was necessary to make sure that the properties were used accurately. Like, when you add 12 to both sides in your notes jpeg you list that as Additive Inverse Property. But isn’t Addition Property of Equality a better fit? Then actually showing -12 + 12 = 0 would be the place for Additive Inverse? Then that leaves you with n/3 + 0 which is where the Identity property comes in to get you n/3 = 7. Any thoughts? | 26 days ago | Reply
James Dunseith: Aaron, I think you’ve identified an important continuum when you differentiate between “mathematics and mathematical reasons” versus the “tricks” that we pick up along the way. As students progress through their math education, some of the depth they get derives learning more of the mathematics behind a trick, and less of memorizing the trick itself. And when you learn the math behind the trick, it opens up all sorts of new possibilities. With that in mind, I’ve experienced a lot of the same issues you’ve raised: how can I give kids all the background I want them to have without losing them? The conclusion I came to in this implementation of the project is somewhere in the middle of that continuum. Your thinking is right about the Addition Property of Equality. As you note, I’ve skipped straight to saying that we start with the Additive Inverse Property. Here’s how I explain it to my students: the Additive Inverse is what helps us decide to add 12 to both sides, while the Equality property is what says we’re allowed to do that. In between the two, the Additive Identity says we can “cancel out” the -12 and 12. After that, we do call out the Equality property, but only to say that, if the original equation was true, then so must be n/3 = 7. It would be even more thorough to include the omitted -12+12 = 0 step, but that omission was the compromise I made in hopes of keeping kids at it. You’ve made me want to try it both ways next fall…if you’re teaching Algebra 1 in 2014-15, I’d be into comparing notes around this idea. | 16 days ago | Reply
Aaron Bieniek: James – I am going to be teaching Algebra next year and you’ve both inspired me and given me a kick in the pants to find ways to be more effective. I seem to have found you at the right time given that I’ve been curious about how a project based class would look. By reading your narrative I feel like I get it now and am starting to see how to craft my daily work with the kids into a project based format. I plan on using the projects you outline in your first 3 units. (Actually, I’m probably going to follow your daily plans for the first few units. I tend expect too much of the kids too fast and your narrative has helped me see how I can slow that down and “build a better student” over time.) I also plan on being more explicit about building a growth mindset in the kids, and again your narrative is helping me see how to integrate that thinking into the routine of what we do. Thanks again for sharing what you do. I hope you continue to share and know that at least one guy is getting a lot out of your contributions. | 16 days ago |Reply
What an amazing, thorough and honest back and forth! It’s great to see teachers engaging our lessons especially at this level of detail. It shows a passion for understanding and professional development towards stronger instruction in the classroom.
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