Like Victor I have had a growing uneasiness with math being such an important subject for a vast majority of people not being interested in it, and like him I felt the problem was with math and not with the students. Leaving a general tendency of sloth in the Western world out of the equation, I too believe math should be replaced in the curriculum with other subjects (programming, experimental data analysis, formal philosophy, music) which lead to similar objectives (logic, scientific language, culture, beauty) AND with a more immediate appeal, not only for coolness sake, but for applicability to the real world.

Most contras worry about obscuring math for those who now specialize in it, like us. I believe this is false, much like having driving lessons on the curriculum, instead of mechanical engineering, has not led to shortage of mechanical engineers making cars. Math specialists will still find their way to it, when intrigued by the mechanics under the hood of applications like Victor’s. It’s those other 99% which never grow into a math career but still find it on their curriculum, who would benefit of killing math in favour of a more intuitive scientific language, in particular a more visual one.

I think we are watching a genius at work and should not treat his visions lightly.

But I wonder if we’re not discussing a point here that is tangential to the real issue. I have no doubt that Victor is quite capable of “doing the math” that we would ask our own students to do, whatever that means. However, I suspect that some of his ideas would be really positive ways to get _younger_ students into mathematics and general quantitative thinking quicker and more fluidly. Many of the comments I’m seeing here are focusing on high school-age students, implicitly or otherwise, and perhaps that’s not where the power of his ideas lies.

Which would do more damage to 5-9 year olds, in terms of their ultimate attitude toward and appreciation of the power of quantitative thought: giving them worksheet after worksheet after timed test, or letting them work on real problems, and teaching them how to use something like the “Scrubber” that Victor proposes to figure out real answers? I don’t really care that there aren’t lots of people who can write simulations and program calculators – what I do care about is the poor attitude of most towards math. I’d be willing to bet that an improvement in attitudes would be joined by an improvement in understanding.

You have to give me specifics so I can understand, and then once we’re on the same page, then we can talk about the abstract case! No abstractions till I can picture it in my head, and I’ve never seen a county with n towns in it. Talk about, I dunno, eight towns instead. And for heaven’s sake, if we’re going to be talking about a flow of 3 trains or 4 trains per hour, do not label the towns 3 and 4 and so on! That’s confusing! Give the towns some fricking names. Amsterdam, Berlin, Copenhagen.

Don’t use glyphs when you can use words. But I guess then he wouldn’t be talking to mathematicians, who doubtless don’t have to unpack a sigma the way I do, and would instead be talking to me, which he doesn’t want to do.

I have some sympathy for Bret here in that he’s worried about “symbolic overload” which has glazed the eyes of many a student. I’ve lately trying to use design to alleviate the problem, whereas he wants to do away with symbols altogether. I’m worried his solution doesn’t work in general (for reasons other people have already brought up). I’m especially concerned about second-level variation — like having students experiment try to match an exponential decay equation ax^b with a particular set of data, you have one kind of variable with the x but also varying the parameters a and b, so you’ve got one level of scrubbing with the x but an entirely different type of scrubbing for the a and b.

I guess Bret’s solution to that would be more like his dynamic exploration demonstration, but how would students set the problem up in the first place WITHOUT referring to at least x?

I would argue that replacing “simulations” with “calculators” in his quote isn’t the same thing. Calculators are a general-purpose tool that replace manual algorithms with digital ones. Bret’s examples have all been incredibly problem-specific, not general-purpose at all, and as such don’t replace general-purpose descriptors.

Also, I dunno, isn’t almost everything he’s talking about already being done? Is he even aware of the Wolfram Demonstrations site?

I love graphing and visual representations, I will always use them as part of my teaching wherever I can, and more actual tools to create those visualizations would be fantastic. But I don’t see how his ideas actually solve the problems he’s talking about.

First, in looking at Victor’s examples, the inherent question to me is, how did he set up the “equations” in the first place? Understanding that the total height in pixels needed to be 768, and that the height of the bars could vary to fill the given space indicates an understanding of a “variable”. While my introductory algebra students may not immediately use an x to indicate the variable, they do understand that it is changing. They also understand “guess and check” methods for solving such a problem. But isn’t the beauty of algebra that we don’t have to guess and check, but that using variables will yield an exact answer in very few steps?

Additionally, if students have the ability to set up an equation that uses a “scrubbing” type of solution strategy, aren’t they already quite developed in their understanding of Algebra, or in algebraic thinking, if not solving problems algebraically?
Or does the “scrubbing calculator” come across as more “mathmagic” to students in introductory Algebra?

Without understanding properties such as order of operations, students are once again facing the challenge of trusting a calculator without knowing for sure if the solution is reasonable. Sure, anyone can say, look at the equation, the numbers add up to 768! But that means we have to believe the equation was set up correctly in the first place and that the program we used to “scrub” followed the steps correctly. Too often we trust answers on calculators when they don’t see the nuances in what we intended, assumed parentheses, the difference between a negative and subtraction, etc. Using a scrubbing tool all students will arrive at the same answer, and you lose the rich mathematical discussion on: 1. How different people could arrive at different answers. 2. How to solve problems in a variety of ways and 3. What strategies are mathematically valid and why they are so.

Looking at his “double scrubble” example of the car trip problem, understanding that 2910-1000 and 426 +1000 needed to be equal seems like a fairly sophisticated problem solving solution for this situation. Again, students with this understanding could probably solve the 3-step equation he started with, and would have set it up themselves anyway. I suspect, though I can’t be sure, that most of my students would have added the two amounts together, divided by two and then found the difference. Oh, yes, that’s what the calculator does for you, without the same understanding of what is happening. Seems to me that his method of “removing the symbols” is still more sophisticated than the basic computation that most of us would use. If we are going to advocate for simplicity, let’s recognize that many times using Algebra is NOT necessary for routine problem solving situations.

While I am all about making math accessible to all, I do feel that both of the problems Victor presents are ones that all of us should be able to solve by implementing a strategy slightly more advanced than guess and check (often times without a variable). While we don’t want students to get so bogged down in details that they lose both interest and confidence in their ability to do math, can’t we advocate for a balance between mathematical understanding and real-world practicality? Why does it have to be one or the other?

As always, Dan, you raise such thought-provoking questions, thank you for the opportunity to reflect on what is best for our students.