7 = 1, huh?
Check out my post over at the other blog for what I did about this. Comment here or there, whichever you prefer.
Month: October 2013
I don’t have a picture for this, but every single one of my 4th Graders thought that “average” meant “the most common thing.”
(a) Where do they get this idea from?
(b) Is it a big deal misconception?
(c) How do you create a need for something besides “most common”?
(I think I have my own answers for (a) and (c), but I’m more curious to know what you guys all think.)
I think that this might be my favorite conversation since I started hanging out (as a math assistant) with 3rd Graders. File this one under “place value.”
The question that this kid was grappling with was to write “8,000 + 500 + 20 + 6” in standard form. I’ve forgotten most of the details of the conversation, but all of the interesting stuff is right there on the page.
First, note how she first writes 25,03 to that first question, at the top of the image. And though it’s harder to see, she’s done that for 60,47 as well. When I came over to her, I thought that she was just confused about the convention, and I offered a correction.
Then she started working on the “8,000 + 500 + 20 + 6” problem. First she wrote 26, and then she tried to figure out how much more it was. She ended up with what you see in the image, at which point I realized that there was something about the way she was thinking that I hadn’t anticipated. She was chunking the number into 26 and everything else.
She really struggled to figure out what to do with 8,000 and 500. The boxes around the numbers are part of my attempt to push her to tell me what each numeral represented (“5 whats? Shouldn’t this just be 8 plus 5 plus 2 plus 6, so 21?”).
But there was no way that was going to work. The fact that she was chunking “26” together means that she doesn’t really get place value for 10s either. Those of you with more experience will hopefully help me out in the comments, but I’d imagine she sees 26 as a single number, not as composed of any parts.
This was confirmed when I asked her what the 2 in the 26 meant. She thought I was nuts. She said that 26 is the number right after 25. I repeated the question, and she thought it was ridiculous.
To me, this really speaks to the value of activities that defamiliarize place value for students. See Anna’s Ba-na-na or Christopher’s Orpda for activities that do this.
The kid also answered the “How do you know?” question:
“Because 5 is half of 10, and 50 is the same number as 5, just with a 0, and 10 is the same number as 100 just with another 0.”
What does this mistake say about the way kids see numbers and multiplicative relationships?
https://twitter.com/MTChirps/status/390185033631137793/
https://twitter.com/MTChirps/status/390201190085971968
https://twitter.com/daveinstpaul/status/390198645884088321
@mpershan @MTChirps @MrPoliquin But rules of implication may be at heart of the error rather than knowledge of angles. And maybe guessing.
— Christopher Danielson (@Trianglemancsd) October 15, 2013
@MTChirps Does she know that the converse often doesn't follow from the conditional? Maybe she thinks the converse is always trouble.
— Michael Pershan (@mpershan) October 15, 2013
Your thoughts?
Oh man, this is going to be tough for kids. Good mistake.
What makes this so hard? Or am I over-estimating its difficulty?
Thanks Matt!
We here at Math Mistakes are always happy to share thoughtful writing about mistakes and student work.
First up is Nicora Placa, whose Bridging the Gap is one of my favorite new blogs. She asks, “Is a Careless Error Really Careless?” Here’s the headline news from a 1982 experiment:
However, as he continued his analysis of the interviews, he realized there was another explanation. Students who incorrectly answered the questions were doing something that made sense to them. Their intuition was to place the multiplier next to the letter associated with the larger group. Although incorrect, it was meaningful to students.
I’m a bit ambivalent about this. I worry that coherent explanations of mistakes are often post-facto explanation of intuitions in disguise. But I have very little evidence for my claim that doesn’t come in the form of extensive surveys about exponents.
One of my favorite things about her blog is how well (and simply) annotated it is. She’s added so many things to my reading list. Go check her out.
Next up is Evelyn Lamb with a nice piece about a university-level mistake. There is a precise mathematical definition for the terms “closed” and “open” as they describe sets of numbers, but blah blah anything I could do to explain this would be worse than the way Evelyn does, so just go and check out her piece:
I had underestimated the power of the English language to suggest mathematically incorrect statements to my students. In mathematics, “open” and “closed” are not antonyms. Sets can be open, closed, both, or neither. (A set that is both open and closed is sometimes called “clopen.”) [ … ] But in English, the two words are basically opposites (although for doors and lids, we have the option of “ajar” in addition to open and closed). My students used their intuition about the way the words “open” and “closed” relate to each other in English and applied that intuition to the mathematical use of the terms.
Maybe we can change that vocabulary? Is there a better way to describe those sets, at least during students earliest exposure to these concepts?
Finally we’ve got Michael Fenton who posts about an interesting interaction he had with a student who completely nailed a set of rational exponents questions and was baffled by some square roots:
He writes lots of interesting things that are worth thinking about, including a sketch of how he thinks learning rational exponents might go better. Go check him out.
Categories
(-9) – (-4.8) = …
More and more these days, when I look at student work I’m just using it as a jumping off point for anything that I find interesting. When we started this project last June, I was only looking to explain how the student ended up writing what she did, but these days that requirement seems sort of restrictive. Different pieces of student work are interesting for different reasons, and what interests us is going to vary anyway.
To me, this mistake raises the possibility that it was a reading error. Reading errors tend to get poo-pooed by teachers — along with procedural errors, “stupid” mistakes, and guesses — as the results of non-mathematical issues. Either the kid was rushing, or the kid wasn’t thinking, or the kid was sloppy, etc.
Maybe that’s right. But it also seems to me that as you get better at math you get better at noticing the structure of these sorts of questions. You know what details are crucial, you eyes start to dart in different ways, you chunk the expression differently.
In other words, you learn how to read mathematically. And while some people would prefer to distinguish between mathematical knowledge and mathematical conventions and language, such distinctions don’t really do much for me. Being able to parse mathematical language seems bound up with mathematical knowledge.
In summary: A lot of the things that we call “reading errors” or “sloppiness” are really issues in mathematical thinking.
In this case I’ll offer a testable hypothesis: People who don’t really get how negative numbers work don’t see a distinction between subtraction symbols and negative signs, and will tend to elide them in reading a problem. People who do get negative numbers immediately read the numbers, along with their sign, and then read the operation between them.
(Three cheers to Andrew for the submission!)