What was the point of Set Theory?

[Tully Mars]

I remember being introduced to Set Theory in the second grade. All I remember about it was the examples of unions and intersections of sets.

With most other concepts from elementary education, I would later understand why the earlier concept was important. This is especially true with math, where I would discover a practical, real-world use for it or I would see where a concept was a building block for something more important.

But, all these years later, I still don't see a point behind the Set Theory from the second grade. Where were "they" trying to go with that?

[don't ask]

I once saw a comedian do a really funny bit about the Christian hymn "All Things Bright And Beautiful" using set theory with big set diagrams he drew on sheets of cardboard. So, if nothing else, it provided lots of laughs to many audiences.

[DSYoungEsq]

Originally Posted by Tully Mars

I remember being introduced to Set Theory in the second grade. All I remember about it was the examples of unions and intersections of sets.

With most other concepts from elementary education, I would later understand why the earlier concept was important. This is especially true with math, where I would discover a practical, real-world use for it or I would see where a concept was a building block for something more important.

But, all these years later, I still don't see a point behind the Set Theory from the second grade. Where were "they" trying to go with that?

Well, the basics of it you can read at Wikipedia: New Math. In short, in the early '60s, there was a theory that we needed to emphasize mathematics teaching in order to not be left behind technologically. New Math was based on the theory that it would be easier to learn higher-level mathematics if students were introduced to concepts like set theory at an early age; it would be part of their foundations, so to speak. The fad lasted a relatively short time, fortunately, and was followed by a strong "back to basics" movement in the '70s.

Now, while the concept was rightly abandoned, it sprang from a correct thought, in my opinion (and I'm in the process of being educated to teach math, so it's a subject near and dear to my heart, having had to suffer through "New Math" as a child). In short, the idea behind New Math was that students shouldn't simply be taught rote approaches to arithmetic, followed by rote approaches to more complicated arithmetic problems, until they reach high school and suddenly get introduced to algebra, geometry, trigonometry, etc. In short, students at an early age should be learning mathematical processes. It's equally important to know why 4 + 3 = 7, or 5(6 + 4) = 50. And they need to be able to articulate this to others, and make connections between these concepts and other concepts, both mathematical and non-mathematical. Theory is important.

But there was a feeling that developed among the general population that this newfangled emphasis on sets and functions and such was taking away from the rigorous teaching of arithmetic facts (the addition and times tables, basically). I'm not certain how that got started: in my school, we used new math books (I remember the set theory part quite well), but we damn certain knew our addition and multiplication facts (I remember how long it took me to get past thinking 7 x 8 = 56 but 8 x 7 = 54). So I'm not certain the criticism was fair. Personally, I suspect it had more to do with our country's usual conservative reaction to anything different, and changing education is always controversial anyway.

Why set theory? Well, along with logic, it's the basic underlying foundation of mathematics. After all, what do we really mean when we say, "one?" Set theory answers that question, as well as explaining what is meant by addition, etc. However, just because it's the foundation doesn't mean that you need it to understand arithmetic concepts. Plenty of children learn that "two and two are four" without bothering to understand what "two" is beyond the picture in their mind of two things.

[Tully Mars]

Thanks, DSYoungEsq.

I can see where it would be good to try to communicate the concepts of addition, but Set Theory sure whiffed on that for me. But, that was 1969. If they had been able to relate it to something practical, maybe I could have gotten more excited about it. I do remember that in a class of 20-25 kids, I was the only one who got the concept and could do the problems. I just didn't see the point in it.

Now, Logic Theory, specifically digital logic, managed to soak in a little better. But I didn't encounter that until college and I could see the practical applications for it by then.

Maybe it would have been better, at the second grade level, to teach us "1 + 1 = 2. Trust us for now, we'll explain why later."

[robardin]

It's not true that there's no "real world" application for the kind of "set theory" taught in grade school. For example, you mention taking Digital Logic in college, so you were likely an EE or a CS major. In which case you should appreciate "set theory" as being fundamental to modeling abstract classes in software design.

Another software related example would be database design. SQL queries are expressed in terms of data sets. Every intro DB class sees a few confused students who can't grasp that there's no "looping" in a SQL query (not counting the advanced topic of building a stored procedure involving query cursors).

Why first raise the subject in second grade? That seems maybe a bit early to me, but I think it's good to introduce young minds to a formal way of thinking of how the world is organized, and in particular to non-boolean (good/bad, black/white, true/false, it is or it isn't) logical thinking at an early age, while it still may "soak in" to an intuitive level.

[Tully Mars]

robardin, I had to laugh (at myself) at this answer.

What do you mean, non-boolean logical thinking. There is no such thing.

[Thudlow Boink]

Originally Posted by Tully Mars

What do you mean, non-boolean logical thinking. There is no such thing.

Fuzzy logic?

[Thudlow Boink]

Originally Posted by Tully Mars

But, all these years later, I still don't see a point behind the Set Theory from the second grade. Where were "they" trying to go with that?

Okay, here's an application of Set Theory that a second grader could appreciate:

You and a friend are going to order pizza. There's a set of topping that you like, and a set of toppings your friend likes. The toppings you actually order should be chosen from the intersection of these two sets.

Now, suppose you and a friend are going to make sandwiches, and you're going to the store to buy the ingredients. There's a set of things you like on a sandwich, and a set of things your friend likes on a sandwich. The things you buy should be chosen from the union of these two sets.


Set theory also gives you a way to organize your thinking when you're classifying things. For example, Harvey is a member of the set of hamsters, which is a subset of the set of rodents, which is a subset of the set of mammals, which is a subset of the set of animals.

[ultrafilter]

If you go on to study any kind of higher mathematics, second grade set theory features very prominently. It also shows up in a lot of other disciplines (computer science in particular).

Originally Posted by Tully Mars

robardin, I had to laugh (at myself) at this answer.

What do you mean, non-boolean logical thinking. There is no such thing.

There are quite a few logics with more than two values out there. Even computers use more than 0 and 1.

[Hari Seldon]

Originally Posted by Tully Mars

robardin, I had to laugh (at myself) at this answer.

What do you mean, non-boolean logical thinking. There is no such thing.

Of course there is. There are many kinds of logic that are not boolean. For one thing, quantum logic is totally different and booleanness hardly makes sense. Closely related is linear logic. There is also intuitionistic logic, less radical than the others. But it is non-boolean, since in intuitionistic logic, something can be neither nor false.

But I meant to respond to the OP. In my opinion it was a serious mistake to introduce the "new math". A French mathematician named Jean Dieudonne who went around the US preaching against teaching of euclidean geometry and replacing it with set theory. In my opinion, both changes were disastrous. Euclidean geometry has its foundational flaws, but then so does all mathematics taught before advanced undergraduate mathematics. More importantly, geometry has an immediate appeal that set theory lacks. One important reason that set theory was a disaster was that it was put in second grade where the teachers were at a loss. The only way it could have possibly been successful would have been if it had been taught by teams of specially trained math specialists. I recall my daughter's 3rd grade class being asked how subsets does a set of 3 elements have; express your answer as a power of 2. The obvious answer was 2^3 and there was a reason behind that answer. The answer the teacher gave was 256. She had obviously gotten the right answer and, having misunderstood not so much the question as the thinking behind the question. The class jeered at my daughter because she had, for once, gotten the "wrong answer".

That said, I am all in favor of teaching arithmetic conceptually and not by rote. I once asked a class of undergrads taking history of math whether they could explain the significance of the digits 3 and 4 in the number 34. Very few could and that is a shame. I was irritated when the kids were made to memorize words like "commutative" and "associative". Whatever the importance of the concepts, the names are unimportant, but can be tested easily. And at that level, the concepts are unimportant since it is only when you meet a non-commutative or non-associative algebra that the concepts even have any real meaning. And you might that in third year college. Matrix multiplication is non-commutative and vector cross product is non-associative.

One of the ironies of this is that the kids learn no arithmetic any more; arithmetic is what comes from a calculator. I once witnessed a grad student in math pull out her calculator to work out 8*75. It took her maybe 10 seconds to get an answer that just jumped into my head instantaneously. Is this worng? I honestly don't know. I feel it as a loss, but I cannot really justify why. After all, other things have fallen out of the curriculum. Back in the days that every guild used its own measures, conversion was studied in school. Once upon a time, British students learned arithmetic of pounds'shillings'pence and I see no loss in the fact that don't any longer.

[Tully Mars]

Originally Posted by ultrafilter

If you go on to study any kind of higher mathematics, second grade set theory features very prominently. It also shows up in a lot of other disciplines (computer science in particular).



There are quite a few logics with more than two values out there. Even computers use more than 0 and 1.

I've been treating tri-stated signals as "1" for years. I didn't realized there was another purpose for them. Thanks, I'll study this some more.

[gazpacho]

Originally Posted by ultrafilter

If you go on to study any kind of higher mathematics, second grade set theory features very prominently. It also shows up in a lot of other disciplines (computer science in particular).



There are quite a few logics with more than two values out there. Even computers use more than 0 and 1.

Hi impedance is not a logic state. It is an output driver condition.

[ultrafilter]

Originally Posted by gazpacho

Hi impedance is not a logic state. It is an output driver condition.

Is it unreasonable to treat it as a logic state for modeling system behavior? Even if not, that doesn't change the point that there are a lot of many-valued logics out there.

[Thudlow Boink]

Originally Posted by Hari Seldon

And at that level, the concepts are unimportant since it is only when you meet a non-commutative or non-associative algebra that the concepts even have any real meaning.

I don't know about that. It seems reasonable to teach second-graders (with or without using the word "commutative") that addition is commutative but subtraction isn't.

[Wendell Wagner]

I think that it's very reasonable to teach the basics of set theory just before teaching algebra, which would mean in seventh to ninth grade (i.e., at 12 to 14, for those of you who don't understand American educational terms), depending on how advanced the school's math program is. I also think that toward the end of the year of algebra, the school should teach the basics of combinatorics (like permutations and combinations). Basic probability and statistics should also be taught in high school. There's no reason to teach any set theory, combinatorics, probability, or statistics in first or second grade, but there's also no reason why they shouldn't be learned by most high school students, since they're no harder than algebra, geometry, or trigonometry. Basic statistics is more important to most people than trigonometry, I think. Everyone should be able to understand the techniques of opinion polling.

[Mathochist]

Originally Posted by ultrafilter

There are quite a few logics with more than two values out there. Even computers use more than 0 and 1.

And even when you use two truth values you can have non-boolean property lattices, as in quantum logic.

[Monty]

Set Theory is also quite useful in Semantics.

[chappachula]

Originally Posted by Monty

Set Theory is also quite useful in Semantics.

A subject every 8 year old needs to master, right?

[Stranger On A Train]

Originally Posted by DSYoungEsq

Now, while the concept was rightly abandoned, it sprang from a correct thought, in my opinion (and I'm in the process of being educated to teach math, so it's a subject near and dear to my heart, having had to suffer through "New Math" as a child). In short, the idea behind New Math was that students shouldn't simply be taught rote approaches to arithmetic, followed by rote approaches to more complicated arithmetic problems, until they reach high school and suddenly get introduced to algebra, geometry, trigonometry, etc. In short, students at an early age should be learning mathematical processes. It's equally important to know why 4 + 3 = 7, or 5(6 + 4) = 50. And they need to be able to articulate this to others, and make connections between these concepts and other concepts, both mathematical and non-mathematical. Theory is important.

Although I missed New Math by a few years, I was fortunate to be introduced to some of the more abstruse concepts of algebra and analytical geometry relatively early in life by virtue of being gifted a few old college-level texts on the subject, and found them relatively easy to understand in concept (though it was a few more years before I found access to applications of said theories). I think the real problem with New Math was two-fold: it was so focused on theory that relationship to real-world applications of set theory and combinatorics was underdeveloped, and both the teachers and texts failed to really understand the concepts and therefore presented them in a manner that was unclear and unintuitive; this combined with the fact that most parents were not familiar with the material made the whole effort futile and counterproductive. Richard Feynman has an anecdote in "Surely You're Joking, Mr. Feynman" about serving on a board to review math and science texts for the state of California in the 'Sixties, and finding a math text that had a problem involving adding up the temperatures of various colors (classes) of stars (some of which don't exist) to get the "total temperature", which is a complete nonsense concept. He also reported a science text which repeatedly invoked the phrase, "Energy makes it go!" without ever explaining conceptually what energy is or where it comes from.

Set theory really isn't that complicated, and the concepts behind it should be accessible to any eight year old of average intelligence. Then again, the basic concepts of geometry and calculus are not beyond the grasp of a reasonably sophisticated child, nor is fundmental symbolic logic. And despite the mistaken belief that there are no real world applications of this outside of computer science and esoteric sciences, there are plenty of ways in which we apply set theory, logic, statistics, et cetera implicitly on a daily basis; only, we're not conscious that this is what we're doing any more than most people realize that in catching a baseball they're solving a complex nonlinear problem in differential equations.

People get freaked out by math and the physical sciences because they see numbers and get intimidated, but a conceptual understanding of the basics is no more complicated than grammar or music, and certainly more simple and consistant than phonics and spelling. Like those topics, the appropriate way to teach math and science is to introduce a concept, develop it sufficiently to make use of it, show applications of it (and go through the rote exercise of applying it to increasingly more abstract situations), and then move onto the next concept; repeat ad nausum.

There is great merit in training students to think in terms of logic and quanta, rather than the murky, thick-headed way most people cope with the world. Certainly, an adult that is to be trusted with an automobile, household cleaning solvents, and the vote should have a basic understanding of statistics so that they know when to cry, "Bullshit!" to obtuse claims and misdirections by pundits and politicians. The real problem, in my estimation, that educators themselves are, by and large, not trained in such thinking, and especially not the pedagogy of teaching the same to students.

And so, if you come up with a clever way of doing long division from left to right, you get penalized and humiliated because the teacher can't cope with something she doesn't understand and has no willingness to absorb. But hey, I'm not bitter and resentful, and I learned a lot by spending hours facing the corner.

Stranger

[Monty]

Originally Posted by chappachula

A subject every 8 year old needs to master, right?

Well, it would help for the student to have been introduced to Set Theory prior to college.

[Monty]

Originally Posted by Stranger On A Train

And so, if you come up with a clever way of doing long division from left to right, you get penalized and humiliated because the teacher can't cope with something she doesn't understand and has no willingness to absorb. But hey, I'm not bitter and resentful, and I learned a lot by spending hours facing the corner.

We must have had the same Math teacher in elementary school. To this day, I still do long division from left to right, on one line instead of the standard top-to-bottom method. When the teacher told me I was doing it wrong, I asked her to prove my method didn't work. I spent the rest of the period in the corner wondering why my method worked if it was wrong.

[DanBlather]

This is a hot topic fopr me. I was one of the early guinea pigs for new math. It meant spending the first few weeks of each school year on sets, intersections, and unions. It never led anywhere, and them when I actually used set theory in CS it was taught all over again. It's not that friggin hard to learn the basics when you needed them.

The idea that you need to understand that zero is the null set, and so on, in order to grasp basic mathematical ideas is absurd. People were learning and using math concepts for years before set theory.

I think math should be taught as a series of tools needed to solve classes of problems. For example: enumerating and counting items (I have two oxen and three barrels of wine) requires that you learn the integers, then you have comparison (6 oxen is more than five), followed by addition and subtraction (I have 3 oxen and you have 2, together we have 5). Go on like this to show how you need multiplication to compare two pieces of land that have different shapes. Calculus would be really intersting if you showed young gearheads that you can find the speed of a car by taking the slope of the curve at any point in a graph showing distance as a function of time, and acceleration as the slope of a point on the graph of speed over time.

I know that there is some small number of people who love math theory just for it's own sake, but I think the rest of us want to know why we need to solve a quadratic equation, or how to use the Pythagoreum theory.

[gazpacho]

Originally Posted by ultrafilter

Is it unreasonable to treat it as a logic state for modeling system behavior? Even if not, that doesn't change the point that there are a lot of many-valued logics out there.

Every simulator I have use have modeled it as a driver state. If a node only has drivers driving Z it will be shown as Z of X to indicate the node is not driven because that is important to know in simulation but the physical gates will intemperate this as either high or low. Perhaps not consistently either high or low but the gates will not use three levels of logic. The only places I have seen that use multi level things if flash memory. There is not really any multi level logic. There are digital to analog converters storing multiple levels in the cells and analog to digital converters after or as part of the sense amps. Nobody does any thing like logic gates with anything but two levels. I would be interested in seeing other than two level logic gates.

[gazpacho]

Originally Posted by DanBlather

The idea that you need to understand that zero is the null set, and so on, in order to grasp basic mathematical ideas is absurd. People were learning and using math concepts for years before set theory.

When I took math in grade school I don't remember that there was any discussion about how set theory related to basic arithmetic.

[Tully Mars]

Originally Posted by gazpacho

When I took math in grade school I don't remember that there was any discussion about how set theory related to basic arithmetic.

My memory is fuzzy (pun intended), but I seem to remember that part of the resistance to Set Theory and New Math in general was that it wasn't being presented as being related to basic arithmetic or as an addendum, it was being taught instead of basic arithmetic.

[ultrafilter]

Originally Posted by gazpacho

Every simulator I have use have modeled it as a driver state. If a node only has drivers driving Z it will be shown as Z of X to indicate the node is not driven because that is important to know in simulation but the physical gates will intemperate this as either high or low. Perhaps not consistently either high or low but the gates will not use three levels of logic. The only places I have seen that use multi level things if flash memory. There is not really any multi level logic. There are digital to analog converters storing multiple levels in the cells and analog to digital converters after or as part of the sense amps. Nobody does any thing like logic gates with anything but two levels. I would be interested in seeing other than two level logic gates.

I could've sworn that at some point in my digital logic class the professor showed us a truth table for a tri-state device of some kind, but that was many years ago, so I might be misremembering.

Anyway, it's easy to think up many-valued gates (although implementing them might be a bit more difficult). For any set S, the power set 2^S (i.e., the set of all subsets of S) is a boolean algebra. Define 0 as the null set, 1 as S, X AND Y as the intersection of X and Y, X OR Y as the union of X and Y, and NOT X as the set containing all elements of S that are not in X. Logically, this behaves exactly like the two-element boolean algebra you're used to. In fact, the two-valued boolean algebra is constructed this way, with S = {1}.

[gazpacho]

I am certain that you are remembering correctly truth tables for driver gates that included an output state of Z or high impedance they are common. My point ,and it is probably a nit picking kind of point, is that hi impedence outputs are used to share more than one driver on a node not as part of a multi valued logic.

Thinking up many valued gates is easy to do I would be interested in seeing a physical implementation of them.

[DSYoungEsq]

Originally Posted by Tully Mars

My memory is fuzzy (pun intended), but I seem to remember that part of the resistance to Set Theory and New Math in general was that it wasn't being presented as being related to basic arithmetic or as an addendum, it was being taught instead of basic arithmetic.

That is doing it a disservice. Like I said, I was working under New Math texts, but I most certainly had to fill all the squares in first grade with stars to show I knew my add by 1, add by 2, add by 3, etc. facts. Same in third grade with multiplication. So it wasn't that basic arithmetic was ignored, so much as it was that time previously devoted to basic math concepts was devoted instead to more abstruse basic math concepts. And, because many who were teaching math at the elementary and Jr. High levels were not mathematicians, they weren't easily able to make the connections between the set theory stuff and the arithmetic stuff. It just kinda sat out there on its own.

Brahier talks about this in the textbook Teaching Secondary and Middle School Mathematics (Allyn & Bacon) 2005 2d ed. He notes the basis for the movement (a sense that we needed to be able to compete and a recognition that the teaching of math hadn't changed in 300 years), and the basis for the opposition that grew to the movement (see, e.g.: Why Johnny Can't Add (1973) by Morris Kline). In the end, the New Math movement catered to the top students with better mathematical skills, was applied by teachers who didn't know what they were doing or why, and was underappreciated by a public that didn't understand it. Thus, it died a relatively swift death.

Contrast to the approach of the New Math the approach advocated by the National Council of Teachers of Mathematics in 1980, which said we needed to focus on problem solving skills, regardless of how basic math facts are taught. This is eminently sensible, and yet, 27 years after they said this, in your average classroom, the only nod made to this concept is to include "story problems" in tests, as if that is what is meant by "problem solving." The TIMSS reports from 1995 and 1999 showed that the United States was significantly behind several countries in diverse areas of the world when it came to solving math problems. A look at the typical classroom in the US, when compared to a classroom in, say Singapore, or Japan, shows why. In the former (and the one I'm observing now for my "methods" classes is no different), the focus is on providing the students with a mathematical fact, then having them drill the use of that fact with a series of problems that vary only in the numbers used. By comparison, the Asian classroom involves the students in discovering the "fact" in question (say, for example, the relationship of the two shorter sides of a right triangle to the hypotenuse), and does not bother to drill them on it much at all. Students who feel the need for drill obtain those drills outside of the classroom (Japan, for example, has supplementary programs done privately which offer what we in the US would consider normal seatwork/homework). End result: when a US student is provided a problem (that is, not an equation to solve, but a fact set that requires deciding on a method of resolution and successfully carrying that method out to an accurate conclusion), the US student has no classroom experience in doing that.

In the face of that, the fact that Susie can't add 4 and 5 without punching buttons on her calculator seems almost immaterial...

[Mathochist]

Originally Posted by Stranger On A Train

Set theory really isn't that complicated, and the concepts behind it should be accessible to any eight year old of average intelligence.

The basics we're discussing (a.k.a. "naïve set theory") isn't that complicated. The real thing gets quite a bit more complicated, and the pure foundationalist approach characteristic to the field is definitely something we never even tried to teach to an 8-year-old.

[Measure for Measure]

Anybody adept at using search engines probably also has a good grasp of boolean logic. Admittedly most users of google use their implicit "And" exclusively. Perhaps one day there will be a search engine that permits intricate uses of AND, OR, NOT and parenthesis, as Nexus and certain other indexes do.