I made an observation that many students were making a mistake in one problem that they weren't making in the next problem. So I tweeted about it.

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It's astounding how many kids are making this mistake (or -3) in #3 but doing #4 just fine. @mpershan #mathmistakes pic.twitter.com/FEemgRRbcX
— Tina Cardone (@crstn85) October 20, 2015
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And because my online math community is spectacular, I got some feedback.

Inspired by this conversation, I did an experiment last Thursday. I gave students these two problems for a My Favorite No (without talking to them about the test first). The results weren't good, but they weren't particularly consistent either. This is what happened:

- 8 and + 3 (3 students)

+ 8 and + 3

- 8 and *3

*8 and *3

*8 and *(-3)

Dropped the 8 entirely, *3

- 8 and dropped 3

Distributed (both)

*(-8) and incomplete (3 students)

Incomplete, *3

both incomplete (2 students)

I repeated the experiment this Thursday and I only had one student add! A couple distributed (we spent the past week working on solving equations and some were with distribution) and a couple multiplied by negative two. I didn't have as many students participate this week (alternating day schedule means this Thursday was my 15 student support rather than my 20 student Algebra class, and a fire alarm during my contained class meant they didn't all do it) but despite completely flawed data I'm allowing myself to feel a sense of accomplishment. If I'm a good researcher I'll do this again in a few weeks to see if they retained it. Remind me?

Tina,

ReplyDeleteThis is such a great example of being curious around what students are thinking. I loved how you involved the students in conversation around the mistake to see such an interesting difference. I am sorry I missed the original post because I have a bit of a different thought around the mistake in the first question versus the second just based on what my the elementary students may have experienced....

We do a lot of work with small arrays combining to form larger arrays such as (3x2)+(3x6)=3x8 and use dot images to nudge students to thinking about groupings of dots to bring out the properties of operations. In all of this work, we use the notation similar to what I did above to reflect students' thoughts on "I did this first and then I did this second, and then I combined the two outcomes." So, I see this notation feeling a lot like your second problem, simply because of "doing something first" with the first set of parentheses, then "doing something next" with the next set and then combining or subtracting after.

Thinking about that process in the first question, I immediately think that students lose that sense of order somehow. Again, with the elementary, if they do add or subtract from "something they solved as a set" it is typically after because their thought process lends to that. For example, in figuring 9x4 a student may do (10x4)-4 or (5x4)+(4x4). In neither case do they need to deal with a number standing alone before the parentheses.

I don't know if any of that makes any sense at all or if it is really even relevant...heehee! I was just thinking of the primary exposure to the notation that my students would have when they leave elementary.

Great stuff to think about! ~Kristin