**Case Study #1: Brackets.**

**Case Study #2: Quadratics.**

**Case Study #3: Equation.**

**Case Study #4: Doubling a fraction.**

**Case Study #5: Comparing functions.**

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**Case Study #1: Brackets.**

**Case Study #2: Quadratics.**

**Case Study #3: Equation.**

**Case Study #4: Doubling a fraction.**

**Case Study #5: Comparing functions.**

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You nailed it!!

amen

I just finished six months as a substitute math teacher. This sums up perfectly my experience teaching algebra to 14 y.o kids. It’s really hard to remember how it felt 30 years ago to discover for the first time these things I’ve now been using all my life.

‘What students sees’ looks familiar to me. Can’t agree more. You nailed it totally!!!

I had such a good laugh from this. I homeschool my nephews and the younger one sees it just like your drawings. I love your stuff.

Case study 1 made me realize you draw your Xs without crossing lines, which makes them look very much like two parens: )( . I expected to see a student try to solve for the parentheses: look, there’s a big one in the middle, two smaller ones on either side, and half of one at the extreme ends. π

Many of these are about the ability to control the level of granularity at which the reader perceives the written mathematics: ie the mathematician can defocus a little and see the large scale structure, deferring the fine detail to a later stage.

I recently came across a similar idea in computer coding. The presenter recommended zooming out your code so you could see the shapes and colours (assuming well-indented code, with a syntax colouring), to get the big picture (lots of little routines, some lists, a great big messy block, etc) before diving into detail.

It’s a subtle skill, but vital.

Part of problem solving – deciding what isn’t important and when it is.

This is what I was thinking when I read this post, too. I believe this process of defocusing is called “chunking” in psychology. This post also reminds me of the way I have learned to understand proofs this semester in my beginner real analysis course: (1) get a picture of the situation –> (2) work out the details / completely understand each step of the proof –> (3) step back and look at the big picture and the overall strategy. Steps (1) and (2) are necessary for (3). Similarly, only a person who has gone through the stage of being a “confused student” is able to reach the stage of seeing everything clearly as the “mathematician” does in these illustrations.

What, no logarithms!? π

I disagree slightly about what mathematicians see for case #4. I see it as (something)/8 * 2 = (something)/4. There is no way I’d distribute the 2 first!

Agreed!

You’re totally right! For this visualization, I should have picked 3 instead of 2 as the multiplier.

The proper visual for this one would be “here’s a bunch of eighths. To double the total, just make each of them into quarters.” Doubling the size of each piece rather than the number.

I agree with gasstationmanwithoutpumps on number 4. It would have been clearer with a second diagram, as you stated, displaying the doubled eights as quarter pieces. Good job on this one though.

Saving this to show to my students next year!

So true

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I love these because they illustrate (eh!?!) that kids don’t struggle because math is hard or that they are just “confused.” These bring forth the need for students to UNDERSTAND the inherent structures and meaning, not just the skill. Exactly why we need to spend as much time on students understanding what an equation IS as the skill of solving.

true

From your perpetual student desperately trying to fix her long-broken futon: this is perfect! And #2, the quadratic, yeah, I’m kind of pissed that no one ever told me that before (or maybe they did, but then they rushed on to shove h’s and k’s down my throat, so I’ve been spending all my time worrying about which one gets the negative and not enough time looking at what the expression is saying).

It’s so important for teachers to understand how things look to us on the the low-ceiling crew. Thank you!

Those skeletons though…hahaha..

Reblogged this on nderisarah and commented:

Do this problems lool familiar? What unit of maths did you dread?

I really enjoyed this.

For that (3x+1)/8*2 thing, my visual is:

We have some amount of slices of pizzas that was cut into 8. If we share them among 3 people, thereβs a slice left over. If we had twice that many, we could feed 6 people the same amount with 2 slices left over.

Great! Very funny π

Well, now I feel stupid. Thanks for the reminder, I guess.

YES!!! Learning how to ‘de-focus’ is essential and it was never explicitly taught (at least not in my experience) – I only ‘discovered’ that’s what was going on when one of my maths teachers muttered it to himself while working through a problem with me! It’s like one of ‘your’ unspoken secret tricks that somehow never gets passed-on but one is expected to find it out on one’s own. I think so many people balk at mathematical equations (and maths in general) because of the lack of this little insight.

So many great examples of Anna Sfard’s process vs. object here! Thanks!

I love you work

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May I print these and hang them in my classroom? What excellent prompts for discussion!

Sure!

Thank you!

;D

Yeah, basic summary:

Students: <:O…

Mathematicians: π

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How do you do it? Your posts are funny and awesome.

Thank you for them. π

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