Categories

## The Mistake

Students don’t know how to read equations, and when they see two numbers they habitually add them together. Any blank is there as the result of an operation. The equals sign just means “make sure that you do this operation.”

My goal is to help students connect equations to the notion of equivalence — something that students in my experience already come into my classes with a decent understanding of, whether from experience or school.

I show this image, and talk about how we know that the pairs of apple buckets have the same number of apples (you can move one apple from one bucket to the other).

Then, we move to puzzles. I tried to leave different buckets “missing,” because I know that these are really four different types of problems. In particular, students are the most confused when the third bucket is missing (since they just tend to sum the first two numbers and put that in the third bucket).

Then, I want to nudge students towards connecting the arrow symbol to the equals sign and buckets to boxes of missing numbers:

This might also be a good time for this activity:

For an extra challenge, I ask students to only use the digits 0-9 each once.

## Commentary

Every year I see this mistake in 3rd Grade. I’ve tangled with it over and over again, and I’ve also tangled with the research on the equals sign:

Does Understanding the Equal Sign Matter?

Equations and Equivalence in 3rd Grade

Ultimately, I don’t think the problem is in interpreting the equals sign exactly. It’s more about reading an equation, which is hard.

That said, I don’t want to ram directly into their rigid understanding of equals signs and equations, so I use arrows and buckets to help describe equivalence. Then, I just casually slide into using the equals signs and abstract equations in a similar way. That’s my current approach to making a change.

I used to have a big conversation in class about what the equals sign meant, but ultimately I became dissatisfied with that and moved to this approach.

Categories

## 8 + 32 = 40 <--- wait, what's the mistake??

Greetings from Singapore!

This is from my son’s Primary 1 workbook. He had written 38 instead of 32 in the top row and 2 instead of 8 in the second row.

The submitter thinks that this teacher’s feedback was probably not so effective, and I’m inclined (from a distance, obviously) to agree:

The teacher’s marking indicates that my son should fix his mistake in the parentheses provided. Yet seeing the “mistake” made me pause. I suppose that if the focus is on simply getting the numbers transferred correctly, then it’s a mistake. Yet if the focus is on finding the sum, and stating the relationship between the two numbers, then perhaps it shouldn’t be considered a mistake. This is interesting to me not so much because it reveals what’s going on in the kid’s head as it does how often teachers have narrow ideas of what’s correct. As for what’s going on in the kid’s head, he could’ve simply been rushing. He correctly transferred the numbers in the other problems on the worksheet. This “mistake” could’ve been used to spark a discussion about why the correct answer was obtained and when it’s appropriate to shift quantities around and when it isn’t.

Categories

## How Many More Miles Total?

The submitter of this mistake notes,

This mistake brings up the concept of teaching with keywords to me.  I asked the student to tell me how he got his answer; he pointed to the word “total” and said that he needed to add.  I’d like to know what ideas educators have to intervene when a student is already clearly looking for keywords and is not making sense of the problem.
What do we mean by “make sense of a problem”?
Are we imagining an all-math skill, tools that can be used to make sense of any math problem no matter the topic or age of a student? Something like “read the problem carefully!” or “draw a picture!”?
Or are we imagining a local skill, some way to make sense of this problem and problems like it? Something like…well, I’m not exactly sure what would help someone make sense of this problem. Maybe, “if you see names, you might be comparing!” or “if you see a lot of numbers in a problem, rewrite them in a list so you can focus on what the problem is asking”?
I suppose that I’m inclined to think of keyword mistakes as a lot like applying the distributive property where it doesn’t belong or other “over-extending” mistakes. Generalizations are smart things to do, and a keyword generalization is a smart thing to think also, and it’s usually correct. Knowing that this mistake exists, I might create a set of problems all that contain the word “total” with some being summing up problems and others being compare problems and so on. My reasoning is sort of simple: this kid thinks that “total” means add ’em all up, so let’s provide him with counterexamples and then urge him to make a new generalization.
Thoughts?
Categories

## “46+30+2=58”

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’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?
Categories

## Some Numbers and Operations in Base 10

Explanations? What lessons are there about the way kids think in this work?

Categories

## 210 + 50 = 710

How do you teach kids to add numbers? Does it help kids avoid this mistake?

Thanks to Tom Fleming for the submission.