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.
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.
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.
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.
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[sup]S[/sup] (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}.
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.
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… :eek:
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.
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.