IMG_3658

I find this fascinating. This student clearly knows how that multiplying the base and the height of a rectangle gives you its area. She even knows how to multiply fraction. But when it comes to part (d), she adds the numbers instead of multiplying them.

In earlier writing I hypothesized that, when put in unfamiliar situations, students often default to an “easier” operation. This idea now seems problematic to me. What, after all, is an “easier” operation any way? And what exactly would trigger this default to some other operation? And how do we explain why competent adults — like me — make similar mistakes on my own work?

It now seems more likely to me that we associate certain pairs of numbers with certain operations. Think about the numbers 100 and 1/2. I’d suggest that most people have an association of “50” with 100 and 1/2. After all, how often have you been asked to add 100 and 1/2 together? How often have you been asked to subtract 1/2 from 100? In contrast, how often have you been asked to find 1/2 of 100?

How often have you been asked to multiply 5 1/2 and 2 1/4 together? My guess is that you — and the student above — have been asked to add these sorts of mixed numbers more often than multiply them.

The idea here is that the pairs of numbers themselves come with associations.

There’s a hard version of this claim that I don’t mean to make. I don’t mean to say that, no matter the context, you’d expect a student to add 5 1/2 and 2 1/4 together. I think a division problem with mixed numbers is unlikely to trigger associations with addition. Maybe I’m moving towards a two-part model? The sorts of mistakes we make with numbers depends both on the associations with the operation and also associations with the numbers? And things get really bad when these two associations point in the same direction?

This theory feels very testable, but at the moment I’m having a hard time articulating a possible test of it. But we should be able to mess with people’s associations with numbers and see if that changes the sorts of mistakes that they make. Ideas?

array

This child made it clear that

  • She knew that an array was a rectangle
  • That this was technically a rectangle
  • These super-long folks were not arrays, or at least she didn’t think they were, because they didn’t look like a rectangle
  • The 2 x 17 was an array

To what do you attribute this perception? (You can check your answers in the back of the book.)

When I was a kid, a friend asked what my dad does for a living. “He’s a dank,” 18-year old Michael said. What I meant to say was that my dad worked at a bank, but I was distracted or tired and I mixed up the two words.

I thought about this while looking through a 3rd grader’s addition work. “He’s a dank” seems a lot like saying “46+30+2=58” to me.

mm1

 

I’m not sure what to call this sort of mistake. I’m tempted to call it a memory overload error, but I have no idea if that’s (a) psychologically apt or (b) meaningful to other people.

The crucial thing, though, is not to simply disregard these sorts of mistakes as silly errors, or as a sign that the student is lacking some general cognitive skill like “attention to detail” or “being careful with their work.” That would be a bad misdiagnosis.

To start building the case for why, pay attention to the “stupid” arithmetic mistakes that adults (and teachers and mathematicians) make while they’re working on a problem. Here’s one I made last summer while trying a matrix multiplication, when I did 1*2+2*3 and ended up with 10.

 

 

Do I suffer from a general sloppiness in my work? A lack of attention to detail? Nah, I was just distracted by making sure that I kept track of a bunch of others things that weren’t automatic for me. My attention was elsewhere.

What causes these sorts of errors? Any sort of distraction, but it’s important not to trivialize distraction. Distraction can come from any number of places.

  • Distraction can come from various non-mathematical things, like friends, chatting, not caring about the problem, etc.
  • Distraction can also come from mathematical factors. If I were better at the matrix multiplication part of matrix multiplication, I would be less likely to mess up some quick arithmetic that I’d otherwise get right.

What about my 3rd grader? There are two possibilities, and both are worth considering:

  • The kid might have been distracted by whatever non-mathematical thing happened to be drawing her attention away at the moment.
  • She might have found keeping track of the tens and ones difficult, and paying attention to the decomposition used up the mental resources that were needed to keep track of everything. She ends up adding 2 and 3 for the tens digit, 6 and 2 for the ones digit.

One of the themes of this blog has been a desire to dig deeper than “stupid mistake.” This is one sort of error that teachers often identify as a “silly” mistake, but labeling it as “silly” probably misses out on some truth about a kid’s mathematical thinking.

Questions:

  1. What do memory overload mistakes look like in geometry? In non-computational contexts?
  2. What other categories of “silly” errors are there? (I’d toss “mathematical habits” into the mix. Or maybe we should call that “fluency with a falsity”?)
  3. What sort of feedback would you give my 3rd grader?

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A note from the submitter:

Along the lines of one I sent you awhile back. This is one of my best students, and several other students gave answers with similar misconceptions. I pretty much ignored it last time it came up, thinking that it was an anomaly, but I think it’s a significant hole in my students’ understanding. Students were using calculators today.

What’s going on here in the student work? What’s the connection to the earlier post?

Photo (3)

 

A reflection from the submitter:

I think this 10th grader is saying .174>.34>.5.  I wonder what she would have concluded if she’d followed the directions and rounded to 3 decimal places? Many kids were tripped up by the .5, maybe she’d say they were increasing except the .5?

Do you agree with the submitter’s assessment? How do you help a student learning trigonometry nail this down?

I think it’s important to say something more subtle than “this kid doesn’t understand decimals.” One thing that this site has documented is that kids can understand something at 1:00 and then do something entirely different at 1:01. It’s best to see this not as a failure of decimal knowledge, but maybe a failure to use decimal knowledge in this situation. (Some people would say this kid’s knowledge of decimals in a certain context failed to transfer to this problem.) The difference is in how we respond. This kid probably doesn’t need the “basics” of decimals. We just need to make a connection to somewhere where she knows about decimals, I’m speculating.

Here’s the breakdown of student thinking about double-digit multiplication that I’m seeing as we begin our unit in my 4th Grade class.

Direct Modeling:

Directly Modeling

 

Direct Modeling With Composition Into Groups:

Composing

 

Breaking The Numbers Apart With Addition:

Breaking The Numbers Apart

 

Breaking The Numbers Apart 2

 

Breaking The Numbers Apart With Arrays:

Arrays Arrays - Breaking Numbers Apart

 

Use of Standard Algorithm:

Standard Algorithm Standard Algorithm - Wrong

 

No Real Strategy, But Knowledge Of Multiplication by Multiples of 10:

Not Sure