In terms of reading the notation out loud, if there is verbal ambiguity, remove it?

For instance, for your example: "X plus B quantity squared" or "x + b squared"

You can also do things like change the pitch of speech as symbols are nested, play specific tones to represent symbols, pan things across the stereo field to represent groupings, and otherwise make the symbolic equation into a multimodal experience.

But of course, you can't do any of this if the semantic representation of the equation is lost and it is rendered as strictly graphics or whatever.

I'm a bit confused to your original point about visualization because how ever in the world could I program if I couldn't abstractly manipulate symbols? I suppose not visually, but there's something non-word-oriented happening in my head.

>I'm a bit confused to your original point about visualization

I guess I find that programming is somewhat more word-oriented than pure math. For instance, how do you think about complex exponentiation, or a rotation matrix? Do you bring to mind the sensation of spinning around? For myself, I bring to mind the image of the entire complex plane rotating and stretching along a spiral, but I'm led to believe that those who are blind from birth aren't really capable of doing that.

Harder examples might include fourier series, convolution, gradient descent, etc.

I think you could almost consider the visualization of these things like a crutch. I wonder if not being able to visualize them might remove preconceived notions about how they behave, and give you different insights.

I had a math graduate student teaching my linear algebra class. He taught dot and cross products entirely algebraically, never drawing vectors as arrows, but as arrays of numbers. When I suggested after class that teaching the visual representation might help some students, he pushed back. Visual understanding, he explained, was a crutch best avoided, because visual intuition could break down in higher dimensions. I thought that was a surprising perspective from a math graduate student, of all people.

This is quite true, especially true when talking about direction (gradient) in high dimensional space. I don't think this can be avoided, since after all we are creatures living in 3D space where left right up down are quite well-defined, just need to make a mental note every time you have to deal with more than 3 dimensions.

Seems like an odd choice when talking about the cross product, since the cross product is only a thing in 3D. You can define analogous things in other dimensions but it becomes clearer and clearer that it’s not meaningfully a ‘product’.

So it doesn’t matter if your visual intuition for a cross product breaks down in higher dimensions - a cross product is only a thing in three.

This is very off topic, but the wedge product absolutely is "meaningfully a product", generalizes fine to arbitrary dimension, and has a perfectly reasonable visual/spatial/geometric interpretation.

(Indeed, we should entirely scrap the cross product in undergraduate level technical instruction and replace it with the wedge product; one happy effect will be replacing students' misleading spatial intuitions with better ones.)

Or bivectors, or k-vectors, or blades…

Good point; you're right. I might be misremembering the cross product. I do remember that he didn't even teach the geometric interpretation of vector addition.

I can see where he's coming from. Geometric intuition is an useful tool but in this semester's linear 2 class I developed more because I stopped using it. It's too strong of a tool and blots out "dryer" intuition and methods, and also as you progress you find more and more places where it's not useful. What's the geometrical intuition for whether two circles intersect in Q^2? who knows?

The way I've heard those distinguished in spoken math is x + b^2 is said "x plus b squared" and (x + b)^2 is said "x plus b all squared. There's a similar approach for divide "x plus b over 8" vs "x plus b all over 8". That was often enough but if it wasn't you'd be reduced to pronouncing brackets.

Using postfix operations in the Way of Forth would be unambiguous and of course otherwise superior as well as is well known [citation needed]. “x b plus squared vs x b squared plus”. Well, at least as long as it’s agreed on whether “x b” means two variables or one with a two-letter name. But the latter don’t really exist in math. You just expand to new alphabets when you run out of letters.