Categories
Addition Division Multiplication Numbers & Operations in Base 10 Subtraction

Some Numbers and Operations in Base 10

IMG_0043

 

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

Categories
Numbers & Operations in Base 10 Subtraction

Subtracting Diameters of Planets

planets table

photo (10)

BTW, the original answer was 7,700.

What think you all?

Categories
Fractions The Number System

1 divided by 2/3 is…

1 divided by two thirds

I’m not sure that I agree with the teacher’s diagnosis that this sort of mistake is procedural. Presumably, she means that the mistake was the result of remembering some sort of algorithm, a wrong algorithm. But what sort of division algorithm would lead you to multiply 2 and 3? in this situation?

On twitter, someone suggested (was it you?) that the issue here was that 1 divided by 1/3 is 3. And then twice that would be 6.  This would then be an instance of a more common pattern of error, the “it’s always linear” error. (Was it Dave who said this? Maybe this was Dave.)

This actually fits pretty well with the student’s explanation. It’s not a bad take.

Do you agree? The first three times that I wrote this post I said something like “The ‘linearity’ hypothesis is a pretty good one, but it doesn’t quite fit with what the student said. I’d suggest that the student had an association between the numbers 1, 2, 3 and 6, and it’s that instant judgement that her explanation is aiming to justify.” But now I’m not sure if I can really find any fault with the linearity hypothesis.

People who have more experience with fractions than I have: is the linearity explanation one that resonates with your experience of kids learning to divide fractions?

(P.S. I’ve got 4th and a 5th grade assignments, along with my high school classes this year. I’m excited to bulk up my understanding of little person stuff. Any of your submissions would help me test the theory that analyzing student work is a solid way to help build pedagogical content knowledge. SUBMIT!)

Categories
Expressions and Equations Fractions Solving Linear Equations

Fractions and Solving Equations

timon

 

Offered to you without comment. Say something interesting in the comments.

(Thanks Timon!)

Categories
Decimals Numbers & Operations in Base 10

Multiplying Decimals

CR parallelogram

How did the student get from 0.8 times 1.6 to 8.0?

What tendency is this an example of? (Or is the mistake unique to the context?)

How would you test your theories?

Thanks to Chris Robinson for the work.

Categories
Fractions Numbers & Operations -- Fractions

Summer’s over, and we’re back with some fractions

2 activities

 

Summer’s over. Let’s get back to work here.

A sprinkling of thoughts:

  • Nicora, on a  recent post, came out against using “x out of y” to introduce fractions. She writes: “One of the most common ways to introduce fractions to young students is to talk about ¾ as “3 out of 4. And it’s part of the reason why students have so much trouble with fractions later on.” I wonder whether this is the sort of mistake that you’d see more of if you use this sort of language in class.
  • I bet that this student would have trouble if asked to place these fractions on the number line.
  • I don’t really get the task. Are these supposed to add up to 1? What’s the unit here?
  • I don’t think that the student believes that 3/5 is more than 2/3. I’d bet that this student just isn’t paying care to the relative sizes of boxes. This aspect of the representation just doesn’t strike her as significant.
  • I also notice, though, that within each activity there’s an attempt to make the boxes the same size.

I think that this last observation might be a way into a line of questioning that could help. I’d point to a shaded in box (maybe the kickball one) and ask, “What does this mean?” And then I’d point to another box and ask the same question. This would force us to bring out the unit, and the comparisons between the shaded boxes would force us to have a conversation about the relative amount of time spent at each activity. This would naturally bring us into ranking, which I think would be a good follow-up activity.

There are a bunch of interesting things here — please comment on them — but one moral I’ll take out of this is that learning math often involves becoming sensitive to nuances that would otherwise seem irrelevant.

Categories
Multiplication Numbers & Operations in Base 10

Multiplication Shortcut Fail

fawn mult

 

  1. What exactly is the shortcut?
  2. Why does this shortcut seem reasonable?
  3. We’d all agree, I think, that 2 x 6 = 26 is not a result of this kid not understanding what multiplication is. I’ve made that mistake before, and I bet that you have too. So, why is it a  common mistake? What does this say about how a mind works while working on math?

[Any advice on how to tag this, CCSS-wise?]

Categories
Multiplication Numbers & Operations in Base 10

Lattice versus Partial Products

triangleman1

 

I’d love to hear anything that you’re thinking about this in the comments.

I wonder: should we be giving kids explicit guides for these sorts of algorithms at all? Algorithms ought to be patterns in thought. Does a mistake like this begin to make a case against these sort of printed aids?

Thanks to Professor Triangleman for the submission.

Categories
Decimals Elementary School Measurement & Data Money

Money and Decimals

Brown2

 

What strikes me about this piece of student work is how clean and predictable their mistake is.

Is this sort of mistake the rule or the exception? Does a mistake like this reflect the fact that many/most student errors are due to coherent mental models, or is it the rarer exception in a world dominated by stormy minds that fling ideas at math less predictably?

Thanks again to Dionn!

Categories
Fractions Number & Operations -- Fractions

Subtracting Fractions

Brown1

I’ve been reading some constructivist stuff lately, so…

What resources does this student have for refining their understanding of fractions?

(Thanks for the submission, Dionn!)