3. U = {1, 2, 3, 4}
T = {1, 4, 9, 16}
V = {1, 4}; yes, V is a subset of U; yes, V is a subset of T; no, U is not a subset of T, since 2 is not an element of T

4. K = {1, 2, 3, 4, 9, 16}
K is not a subset of U; U is a subset of K; T is a subset of K; U is a subset of U (by definition of subset).
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I feel sad about New Math’s failure. There was some hubris to the undertaking, but otherwise it was a good faith attempt on the part of mathematicians to improve math textbooks at the time.

Today’s Common Core seems like a straight power grab/graft move on the part of Bill Gates and Friends to run American education their way and not coincidentally make money.

My high school text came from SMSG (School Mathematics Study Group). It was part of the post-Sputnik educational reforms. I make fun, but it was a noble undertaking, albeit deeply flawed:
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Almost half the nation’s high school teachers of mathematics attended at least one such [SMSG Training] during the 12 year life of SMSG; but an equivalent seeding was impossible for elementary school teachers, who outnumbered the high school math teachers ten to one. While there were some Institutes for elementary school teachers, these were mainly for experimentation. The SMSG books themselves achieved unexpectedly wide circulation, and were indeed, as Begle had urged, enthusiastically if often ignorantly imitated, even (or especially) at the more elementary levels. And the research literature produced in the colleges of education, and the journals of classroom practice written and read by teachers, were for the entire decade of the sixties dominated by obeisance to the SMSG program.

–“Whatever Happened to the New Math?”
https://people.math.rochester.edu/faculty/rarm/smsg.html
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It wasn’t just Stanford. I recalled one SMSG working group there. Authorities from Harvard, Yale and others were involved.

Great rundown on where the New Math came from and how it ended.

More Spanky

https://www.youtube.com/watch?v=qMGZhV0hn80

Who remembers the 60s New Math? The big brains at Stanford put together a hot new textbook, so students would understand algebra based on set theory from the ground-up.

Here are the third and fourth problems on the very first problem set of the New Math textbook I had in the 9th grade of a forward-thinking Catholic school:
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3. Find U, the set of all whole numbers from 1 to 4, inclusive. Then find T, the set of squares of all members of U. Now find V, the set of all numbers belonging to both U and T. (Did you include 2 in V? But 2 is not a member of T, so that it cannot belong to both U and. T.) Does every member of V belong to U? Is V a subset of U? Is V a subset of T? Is U a subset of T?

4. Returning to problem 3, let K be the set of all numbers each of which belongs either to U or to T or to both. (Did you include 2 in K? You are right, because 2 belongs to U and hence belongs to either U or to T. The numbers 1 and h belong to both U and T but we include them only once in K.) Is K a subset of U? Is U a subset of K? Is T a subset of K? Is U a subset of U?
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Are we having fun yet?

I read this now and I’m perplexed. I mean, I could work it out with some diagrams, taking it slowly. But this was literally my first math assignment in my first week of high school.

My mom located a product she could use with her computer to transfer videotape onto DVD, and did that for a lot of them. I now have the whole collection, and am (slowly) getting them into digital/shareable form.

I also have the photo collection…much harder to understand the context of a particular photo when I don’t remember it, since it’s not part of a continuous image stream.

“Sandy and Bill leave from their homes 15 miles apart at 10 AM. Sandy walks 5 mph, Bill walks 3 MPH. Where do they meet?”

…could be made a lot more interesting if the book substituted:

“You are a train dispatcher for a railroad. You have two trains leaving from cities 90 miles apart, a 60mph passenger train and a 45mph freight. Single-track line. You have various sidings along the way where you can have one train wait for the other, so you need to know where they will meet (and collide) if you don’t turn one of them off into a siding. Where is that point?”

You could even show them the way real dispatchers sometimes did this, using strings and graph papers, vividly illustrating the mathematical concept of Slope.

I guess I was lucky as I had first read about Newton and Leibnitz and the problems they were trying to solve that led to the “discovery” of Calculus. Areas under curves and “fluxions” as they called them. So when it came time for the math I understood the necessity and purpose.

When my kids struggled with a mathematical concept I always started by walking them through a reason for the process and, when applicable, a little history about how humans discovered it. “Let’s say you’re an Egyptian stone quarry foreman and you need to know how much stone you need to build a 50′ tall pyramid…” People tend to remember tangible, visual examples.

“It was not until my junior year at the Naval Academy, when we started to do differential equations, that the light came on. Eureka! Drop a wrench from orbit, and over time it would accelerate at a determinable pace, up until the moment when it entered the atmosphere, where friction would impede the rate of acceleration at an increasingly greater rate (based on air density, interpolated over a changing altitude) and that wrench struck someone’s head at a certain velocity, that any of this applied in the real word. By then it was too late, I was too far gone, and an opportunity was lost.”

There is a group at Marshall University that has constructed a mechanical differential analyzer and believes that the direct mechanical analogy to differential equations can be a very useful educational tool. See my post here:

https://chicagoboyz.net/archives/57194.html