Precal Blog

We reviewed the concept of principal branch today in terms of the sine function. Remember that sine is not one-to-one and so we need to restrict the domain in order to define its inverse. We use some common sense ideas to make the restriction. The branch should be a function (duh), centrally located, positive if possible, as continuous as possible, and must contain the entire range of the trig function we are interested in.

I recommended that you sit down and plot out by hand and from memory the parent graphs of all six trig functions. Clearly show the critical points, inflection points and asymptotes. When that is done we can think about how to graph the inverse. We talked about reflecting over the line y = x. It might be easier to think about switching the x and y coordinates and plotting the new points.

Once we have the inverse graph we can restrict it to the principal branch we defined earlier. For sine we would use negative pi/2 through pi/2 for the principal branch so this would be the range of inverse sine. The domain is obviously from -1 to 1. You can see this clearly on your grapher by graphing the inverse sine of x.

Tomorrow, we’ll take a look at cosine and tangent and then the reciprocal functions. Happy graphing.

If you didn’t hear it, Charlie went on the offensive about what he calls “fuzzy math” after hearing that the Waukesha school district adopted CMP for the middle school. He characterizes curriculums like CMP, Everyday Math, and IMP as dumed-down [sic], constructivist, no-math-in-them, ridiculous mistakes that for some reason edu-crats keep adopting. He asserts that the reason students in the US cannot (among other things) add, subtract, multiply, and divide whole numbers, as well as fractions, is because of “fuzzy math”. He expounds on this in the book he wrote in 1989.

He spoke first with Wisconsin House Representative Leah Vukmir who for a long time has fought against “fuzzy math” in Wisconsin. She for the most part repeated what Charlie was saying, but if you listen to what she says, and understand the philosophy behind “relational math” curriculums, it is interesting to hear her correct herself when she tries to define “fuzzy math”. She says, “The children are led, ah, the children construct meaning…so that they can discover the principles of mathematics that mathematics were founded on.”

Here, Charlie characterized “fuzzy math” by an analogy. He likened it to letting the kids sit down at a piano and play around with the keys until eventually they play Beethoven’s 5th Symphony. In other words he believes that “fuzzy math” asks kids to re-create, on their own, the mathematics that they need to learn. Leah interjects that what should happen is someone should “teach” them, they are drilled, and then they do it over and over again. She goes on to say that there is a lot to be said about the rigors of putting your mind through memorizing.

A colleague of mine corrected Charlie’s analogy. “Relational” curriculums would first let the kids hear the symphony. Then we would show them the notes (or maybe just the piano part so as not to overwhelm them). Then we would ask the kids “How can we get from what we know right now to playing this finished result?” We would then spend 7 or 8 weeks learning whatever we need to learn in order to play the symphony. I suppose we would have to start with notes – what they are and the notation used to write them. We would choose a particular passage (one with some interesting qualities) and use what we learned about the notes to play through that passage. At the same time we would recognize that playing a piece of music involves much more than just pressing keys. Using our interesting passage we would discuss timing, phrasing, dynamics, tempo, and anything else that will help the kids learn how to not only play this piece, but the general concepts that will allow them to play any piece of music they may encounter. We would then look for other passages that are similar to the one we just studied and learn to play those. Eventually we would approach other passages in the same way until we have learned to play the piece. Concepts of melody, harmony, counterpoint, arranging, and composing would also be woven in to the discussion.

Chalies’s approach mentioned earlier would look like this: Sit the kid down at the piano, play the first line for them and then ask the kid to repeat that – offering corrections and answering questions along the way. Send them home to memorize and practice that line and then review it tomorrow before repeating the process with the second line. Eventually, line by line, very sequentially, the student will be able to play the piece.

I hope I’m being clear about which method of instruction yields a richer and more meaningful approach. Given the conceptual basis we built, my students are now prepared to compare the symphony we just learned to other symphonies or even to study a unit in composing. The students should feel empowered to tackle other pieces of music based on the principles of music we learned. Charlie’s students are chained to the symphony they learned. They can play it – but aren’t prepared to speak intelligently about it nor use it to help them with future pieces.

He then took two phone calls even though the phone lines were jammed. The first lady used to be the head of a math department who eliminated CMP in her school. She bases her condemnation of CMP on the fact that her ninth grade classes at the time could not work with fractions. This should be interesting to anyone who has ever taught ninth grade mathematics. She comments that when she showed the teachers a traditional curriculum and asked them to compare it to CMP, not a single teacher was in favor of CMP. She also makes the point that when she hears about research in the teaching of mathematics and CMP, it really is irrelevant, because you have to be “in the trenches” and then you know what works. What works in her opinion is simply “give them the algorithms and make them do a little bit of drill.”

Then she goes on to say that she actually taught CMP her first year because she was forced to and didn’t know any better. In that experience, CMP left students confused, frustrated, and not knowing anything. She herself was frustrated and confused. (My assumption at this point was that she did not have any training.) It doesn’t surprise me that she was frustrated and that her kids were too. Teaching math for understanding actually requires understanding on the teacher’s part.

This is already too long. Charlie offers no evidence (research) to back up his claims besides a bunch of web-sites. I can find a bunch of web-sites too that say anything I want. How about having an informed discussion? How should kids learn mathematics? I would suggest as a starting place, Adding It Up by the National Research Council. This report, at the behest of Congress, asked proponents of both sides of the issue to review the research, engage in dialogue and provide suggestions and recommendations for the teaching and learning of mathematics. For research on specific curriculums, the What Works Clearinghouse, at the behest of the US Department of Education, provides solid research based ratings of various elementary and middle school curriculums.

I don’t blame Charlie. He is doing what he is best at and what he gets paid for. As an educator, I’ll do the same.

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