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“We focus on the rules”
Help! I just moved to middle school from 4th grade, and I’m working with a great group of people, but I have a hard time being the new guy and going with the flow.
In my district, teachers are to work with their grade-level departments, collaboratively planning remote lessons. I only get to design one of every six or so lessons for math, and coming up, I am starting off a mini unit on multiplication and division of rational numbers. When I asked if I should/could have an introductory lesson that focused on reviewing multiplication of positive integers, connecting it to multiplying positive and negative integers, this is the response I got:
“In the past we have modeled number lines and barely with tiles for multiplying and dividing integers just as a quick why, but we focus on the rules. (same sign = positive, different signs = negative). The standard doesn’t ask for them to model.
In this unit, we should be including fractions and decimals.
I would have used the bottom of [a step-by-step worksheet] to model grouping with tiles but that is about as far as I would have gone. I tried years past to go deeper (and it could have been how I explained it) but I had students who were more confused because of that.”
What would you do?
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3 Replies
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As a middle school teacher this sounds very familiar to me. I had a teacher once try to tell me that he thought the algebra tiles were too abstract for the students to understand. The problem is that many teachers only have an abstract understanding of rules and procedures and they too get confused when they try to understand a math concept more deeply and with a conceptual context. It’s not easy being new on a team but you need to try to push back on that thinking. That is the why districts push to have teachers collaborate. It has the potential to move everyone forward as teachers discuss their different approaches. Here are some resources that I hope will help you.
Here is a blog post that I wrote specifically about subtracting integers using integer tiles and some of the road blocks I ran into and what I tried. It speaks specifically to the issues with rules and rule following teachers trying to teach conceptually.
https://burnsmath.blogspot.com/2016/11/concrete-models-are-they-actually.html
Here is a Desmos Hot Air Balloon activity. The hot air balloon is a great context for connecting integer operations to the number line. I can show you how to extend the context to multiplication if you’d like.
https://teacher.desmos.com/activitybuilder/custom/5efe58da277abd3653440339
And this is actually my latest favorite context for working with integers (I’m an interventionist and it is by far the most intuitive for students that I have found). Army Men
http://mrpiccmath.weebly.com/blog/my-battle-with-the-integers
Here is a presentation I used with students. The army men are great because they combine the integer tile idea and can be used to think about the number line. You put the men in cars and BAM multiplication.
https://docs.google.com/presentation/d/1n04y6ej9xeKelMhsDAGe9QH4_uPLci9QqGSsIh_5EEw/edit?usp=sharing
Let me know if you have any questions or if there is anything I can do to help you with this situation.
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Thank you for sharing! We actually did that Desmos activity, which was great. I’m looking at recreating James Tanton’s older YouTube video for my students, where he looks at using the area model to show how multiplication of negative numbers works.
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I don’t know if I’m familiar with that one. I’ll have to look it up. I actually had a student take a video in class when we felt like we had a way to explain dividing a positive by a negative. That is the one that I have not found a context/explanation that I think is intuitive and not something I have to memorize. It’s an issue with subtraction too but the Hot Air Ballons and Army men do well for that.
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