In recent discussion about Bret Victor’s Kill Math project Ben Blum-Smith brought up the books *Visual Complex Analysis* and *Visual Group Theory* which as he puts it “all arguments are geometric and illustrated by diagrams”. (I’m not familiar with the latter, but *Visual Complex Analysis* is fantastic and I highly recommend it.)

I feel like these sorts of books will eventually create a revolution in upper-level mathematics — I’m eagerly awaiting someone to write *Visual Linear Algebra* — but could we re-conceive lower level mathematics in the same way?

By Visual Algebra I’m not meaning graphs, I’m meaning the more mundane symbolic “solve for *x*” manipulation.

Solve for *x*: 2*x* + 3 = 5.

In the same vein as my puzzle equivalent to solving a quadratic, solid lines mean multiply, dotted lines mean add.

Solving for the highlighted circle is equivalent to solving for *x* in 2*x* + 3 = 5.

(I swear I have seen something closely resembling this elsewhere for equations, and I think it even has a buzzword attached — anyone know?)

I originally thought of these sorts of puzzles as a gentle introduction to the topic, but would it be possible to integrate this kind of visual-symbolic thinking in every part of an algebra course?

**ADD**: Here’s an image where the puzzle is closer in look to the equation:

This sort of thing is risky because rather than applying inverses and so forth students may make it a general method to draw circles and arrows everywhere.

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Filed under: Education, Mathematics, Puzzles, Visual Design

josh g., on June 2, 2011 at 12:04 pm said:The three-page thingy I made up last year included a number of examples that were either “solve for x”, or two or more variables.

What struck me about using these a dozen or so times is the blatantly fantastic success rate. Kids not only could figure out what the problem required, but they weren’t intimidated by it and so they just dove in thinking and came out victorious. This included a room of “Essentials” streamed kids (ie. the lower achievers); some of them didn’t bother, but a number of them ended up doing the equivalent of factoring trinomials in the space of 20 min or so.

So, yeah, I’m tempted to integrate these more generally into teaching as a visualization tool too, but I dunno how far I’d take it. At some point they don’t scale well and they turn into a mess, but if you’re using them as a scaffold to understanding the symbolic algebra then that shouldn’t be a problem.

Also, I have *no* idea how well students would fare at actually creating these graphs themselves based on some reality-based problem.

Visual Algebra? « The Number Warrior | Solve Math & Science Problems - Solveable.com, on June 2, 2011 at 2:15 pm said:[...] highlighted circle іѕ corresponding tο solving fοr x іn 2x + 3 = 5. More: Visual Algebra? « Thе Number Warrior #dd_ajax_float{ background:none repeat scroll 0 0 #eaeae3; border:1px solid #eaeae3; float:left; [...]

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xiousgeonz, on July 1, 2011 at 1:24 pm said:I’m working on Visual Algebra — but I’m going to attempt to make it visual and reasonably concrete (as opposed to using prettier symbols).

I think there’s a risk of having short-term success in teaching methods that involve havingt things look and feel right as well as make math sense, if the math sense part doesn’t happen. I have lots of students who will gleefully change mixed numbers into improper fractions — because they remember the “dance” of the pencil. They’ll still say 1/2 + 1/3 is 2/6, though…

Anonymous, on July 22, 2011 at 1:24 pm said:On one of these pages you showed 2 bricks, then 4 stacked and connected it to the mathematical teaching. This is a huge method for teaching math challenged students. When I teach visually, the become mathematical sponges. PLEASE continue with your project. Even if the more difficult aspects of, say Algebra I, the visual foundation you are presenting, I think, would be enough of a sound basis to get them to be interested in the solutions of problems.

Some prototype pages of Visual Algebra « The Number Warrior, on October 5, 2011 at 1:59 pm said:[...] I theorized a book called Visual Algebra it was non-obvious what such a beast would look like; perhaps this prototype from my work in [...]

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Greg Weiz, on October 13, 2012 at 3:28 pm said:I think it’s been done before – http://www.mathedpage.org/manipulatives/alhs/alhs.pdf

Jason Dyer, on October 17, 2012 at 7:07 am said:This is more extensive than any manipulatives-based curriculum I’ve seen before. Thanks!

» Computable algebraic expressions in tree form Gyre&Gimble, on October 24, 2012 at 8:24 am said:[...] tree expression makes the invisible structure explicit. Some math educators such as Jason Dyer and Bret Victor have experimented with the idea of students working directly with a structured form [...]