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.

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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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We chatted on twitter about this question for a bit. What predictions can you make about the predictions that were made? Click through to check your answers!

 

Here are the results from the 59 students who answered this question on an exam:

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How would you give feedback to the students who wrote “infinity”?

Imagine that you were to give feedback to the students who wrote “-3/7.” What feedback would you give?

In a previous post, lots of commenters said that they didn’t feel that you could give helpful, written feedback because there wasn’t enough evidence of student thinking on the quiz. Given that complaint, it might be interested to see how those same teachers would give written feedback on a quiz that gives significantly more evidence of how a student is thinking.

Here’s another quiz: what sort of written feedback would you give? (The checkmarks are from the student, who was provided with an answer key and checked her own work, ala this.)

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As before, imagine that you don’t have to write a grade on this paper. Some things I’m wondering about:

  • Would you give comments on every solution, or only some of them?
  • Will you ask kids to “explain why you said _______”? When is it helpful to ask for an explanation? When isn’t it?
  • Will you give your kids specific next steps, or will you mostly point out the good and the bad of their work?
  • Will you throw up your hands and say “You really need to have a conversation with the kid!” for this sort of quiz also?

 

In a lot of ways, it’s much easier for me to come up with helpful feedback to give on rich, juicy problems (see here) than it is for your typical quiz or test. I find it much harder to think about how to give feedback that helps a kid’s learning when (a) the quiz is full of non-open questions and (b) the kid’s solutions don’t show a lot of thinking. But a lot of classroom assessments end up like that, and it’s important to figure out how to deal with those tough situations effectively.

So: What would you write as feedback on this quiz?

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Some constraints/notes, that you should feel free to reject or challenge:

  1. Assume that we’re dealing with written feedback here. Not a conversation.
  2. Assume that we don’t have to write a grade on this piece of work. (If we wrote a grade on here, some research indicates that would ruin any feedback we gave.)
  3. You might decide to give feedback on every question of this quiz, you might not.

I’ll jump in with my thoughts in the comments. Here are some questions about your choices that I’m wondering about:

  • Would you choose to mark the questions as right/wrong?
  • Would you try to find something to value about this kid’s work in your comments, or will you be all hardass instead?
  • Would you ask questions or give suggestions?
  • Would you write one, several, or many comments?
  • Would you reject the constraints in some way?
  • Would you ask the kid to explain himself?

Excited to read your thoughts!

What mistakes would you expect to see in the proofs of the problem below? Take a moment and make some predictions. You might find it helpful to know that this was part of an end-of-the-year exam, and that kids were able to use whatever proof representation they wanted. In other words, they could write two-column proofs, paragraph proofs or flowchart.

proof pic1

 

Here are the mistakes, pulled from the class set. Or maybe you feel uncomfy describing these proofs as “mistakes”? Maybe it’s better to say that they contain mistakes? Or that they are proofs that aren’t where we probably want them to be?

mistake2   mistake3

 

Do you feel comfy calling these “mistakes”? What would you call them?

What are the next steps for these kids? What would you recommend that their teacher do?