Dr. R. James Milgram, professor of mathematics at Stanford University, received his B.S. and M.S. from the University of Chicago, and his Ph.D. from the University of Minnesota. He has given lectures around the world and has been a member of numerous boards and committees, including the National Board of Education Sciences, a body created by the Education Sciences Reform Act of 2002 “to advise and consult with the Director of the Institute of Education Sciences (IES) on agency policies.” Dr. Milgram is the author of An Evaluation of CMP, A Preliminary Analysis of SAT-I Mathematics Data for IMP Schools in California, and Outcomes Analysis for Core Plus Students at Andover High School: One Year Later. This latter paper is based on a statistical survey undertaken by Gregory Bachelis, professor of mathematics at Wayne State University. Each of these papers identifies serious shortcomings in the mathematics programs: CMP, Core-Plus, and IMP. Below are his comments about the SC new math standards which are supposed to replace Common Core... THESE ARE MY COMMENTS ON A FEW OF THE STANDARDS IN THE SOUTH CAROLINA DRAFT STANDARDS THAT I WAS SENT BY JANE ROBBINS. OVERALL, THE OUTLINE OF THESE STANDARDS IS SIGNIFICANTLY SUPERIOR TO COMMON CORE. BUT THE DETAILS ARE FILLED WITH PROBLEMS, INCLUDING OUTRIGHT ERRORS, MISUNDERSTANDINGS, AND IMPROPER PREPARATION FOR THE INDIVIDUAL STANDARDS. ALSO IN TOO MANY PLACES THERE ARE FAR TOO MANY STANDARDS, AND MORE THAN A RELATIVELY SMALL PERCENTAGE CANNOT BE COVERED IN THE TIME ALLOTTED. Here are a very few of the eighth grade standards that caught my eye as I quickly went through the document. Realistically, virtually all of the entire document needs to be gone through with a fine toothed comb by a real mathematician. Ive just sampled a few points, and they are far from the only points where there are either outright errors or stupidities. 8.F.4 (a) (a) does NOT explain what slope is. What is a constant rate of change? The standard definition of slope is rise over run, and then provide examples to interpret this. 8.F.4 (c) (c) is so vague as to be meaningless. This is another point where examples are essential. Actually, one should insist on these changes as SLOPE is one of the absolutely most basic things one learns about in Algebra 1. (If the authors of this document are not capable of filling in the examples and focusing in on the key properties and applications of slope, then I would strongly suggest that you form new committee to fix things. 8.F.4 (d) For (d) see my comments on (c) directly above. 8.F.5 Again, I have no idea what this standard is meant to mean. Frankly, I have my doubts about the knowledge of the subject shown by the authors, but, at a minimum, there have to be EXAMPLES to illustrate what this word salad is supposed to mean. 8.EE1.2 (b) (c) Both (b) and (c) are essentially tautological. However, there is an issue here that is worth focusing on, which, of course, is not even mentioned in these standards. There are TWO real numbers that are the square roots of any squares, but there is only ONE real number that is the cube root of any real number! This distinction is important, and will later help support the learning of more advanced material in calculus and beyond. A1.F.1 (a) A1.F.1(a) is very sloppy. The critical property of a function is that a set of pairs (a, b), where a runs over points of the domain and b runs over points of the range is a function if and only if both (a, b) and (a,c) are in the set then b=c. Students need to understand what this means and how to recognize when a relation is a function. A1.F.1 (b) All these terms need to be defined, and there should be some discussion of what their definitions are in the standards. Perhaps you will find that doing this right is going to take too much time from the more standard Alg. 1 topics, and you will narrow things down. A1.F.1 (d) You are kidding, right? What equation? There is no reason to assume there is any equation involved in the creation of a function. Of course, if you want to restrict the notion of a function to one that is defined by an equation (whatever that means), then you should say so. Some comments on a few of the Algebra 1 standards. Virtually all of these standards need to be rewritten by people that actually know the subject, but here are some samples. A1.F.1 (f) This is the kind of standard that can mean almost anything, and without limiting examples, it is essentially meaningless. So I HOPE THE AUTHORS ARE CAPABLE OF PRODUCING APPROPRIATE EXAMPLES. (Not too many of the typical people involved in the specification of Standards are!) A1.F.2 What do scales have to do with anything? If you have something in mind here, it is ESSENTIAL that you produce sample problems to show what you mean. A1.F.2 (a) As things stand, this standard makes absolutely no sense. What in the world is a qualitative analysis of the graph? When I read something like this I am tempted to believe that the author is simply stringing words together without any real idea of what that want to do. Some comments on the Algebra 2 standards. Again, virtually all of the Algebra 2 standards need to be rewritten by people that actually know the subject. A2.NQ.1 (a) This standard is about as unclear as they come. understand that quantities are numbers with units. Quantities are amounts of something, to be sure, but its unclear how units come in. Usually, on talks numbers with units attached, such as miles per hour, or cubic feet per second, and thats already hard enough for students. Why make their lives even worse? A2.NQ.1 (d) talks about level of accuracy for the given context. This is very legitimate, but Algebra 2 is not the place to be doing it. Rather it should be done in a course such as chemistry or physics where the context is very relevant. A2.NQ.2 (a) In (a), perhaps the most crucial thing is the existence and properties of the conjugate of a complex number (if the number is a + bi then the conjugate is a - bi, and we have that (a + bi)(a - bi) = a^2 + b^2, so it is always non-negative. Moreover, it is non-zero unless both a and b are zero.) A2.F.1 What on earth is the average rate of change over a specified interval and why should one care? If you are going to have a standard like this you need to fill in such details. In general, one talks about the slope of a line segment joining two points on the graph that are very close to a specified point, and discusses what it might mean to take the limit of such slopes as the two points get closer and closer to the fixed point. One then studies explicit examples (trivial for linear functions, but not nearly so trivial for quadratic functions). But one has to think very carefully about whether this is such a good idea in Algebra 2. I would think it would be much more sensible in a pre-calculus course. A2.F.2 A2.F.2 is another example of word salad. I cant make any sense of it as its written. So Id have to know what the authors actually had in mind. Then maybe someone could translate into standard English. A2.F.3 (a) This is completely incorrect as stated. The key issue as that most functions dont have an inverse function that is actually A FUNCTION. What these inverse functions really are is RELATIONS, not functions, and this cannot be lost sight of as ignoring it will just make an already confusing situation much worse. Frankly, I think this entire area is not really appropriate for high school math. First, it is too formal and too general. Its something that someone who has lost sight of THE UTILITY of mathematics might force on students, but hardly anything that helps students to learn the things they really need to know. A2.F.3 (c) See my previous objections. If the authors really think this is something that students need to learn, then they should, at a minimum, focus on the key issue of recognizing when a function actually has an inverse function, and this isnt even mentioned. A2.F.3 (d) Finally, here we come to the heart of this topic, and it occurs at the very end as an afterthought. It should have been the emphasis topic. Then one could finally do what is actually important, constructing specific inverse functions such as arctan and arccos, determining the domain in which they exist and understanding why they cant be extended to a large domain. Equally important would be the log function as (sort of) inverse function to the exponential function. These are the topics that should be focused on here, not the excessively general discussion that, in fact, is present. A2.F.4 Here we go again. Note that the three parts of this standard are all entirely formal. There is no indication of any examples or any reason why one should know anything about composing two functions. Moreover, the most natural place for this topic is in studying transformations of the plane in geometry, where, for example, the composition of two reflections with their fixed lines non-parallel, is a rotation, not in Algebra 2. A2.P.3 This is a very important standard, but it should really be done in an earlier algebra course, as was the case for at least the hundred years till about 1980. So here is an explicit example of the degree to which the course has been dumbed down in recent years. How can we possibly think about having our students compete on an equal playing field with students coming from the high achieving countries A2.R.1 The key thing students need to learn about rational functions at this point is that a function like 1/((x-2)(x-3)) can be written in the form a/(x-2) + b/(x-3) (partial fraction decomposition) as this, first allows students to graph such functions and understand what the details of the graph signify, and second, prepares them for a crucial part of calculus and differential equations that is absolutely essential for any student wanting to major in any technical area in college. But, as usual, the only things that are mentioned here are purely formal properties, with the usual expected result that students will ask why they are supposed to learn it and forget about them as soon as possible. Geometry. These standards are all over the place. For example, immediately after what was, traditionally, the beginning of the geometry course -- the study of ruler and compass constructions, there is a huge and very complex set of standards talking about transformations, of widely differing levels. For example G.CTC.2(e) is a standard that virtually no high school math teacher can handle. And G.CTC.2(f) is even worse. On the other hand G.CTC.2 (a) is typically the content of seventh grade geometry, (see the standards in Common Core), and (b) is simply a waste of time, having very little to do with mathematics. In G.CTC.3, out of the blue, with no preparation, the students are asked to use slopes to determine whether lines are parallel or perpendicular. But as stated there is absolutely no requirement that the conditions for parallelism or perpendicularity are not expected to be demonstrated. (Indeed, the proofs are pretty advanced, and probably need to be prepared for in Algebra 2.) But if you dont include proofs, then students will simply treat the conditions as things to be memorized for the test, and then forgotten as soon as possible. On the other hand, the process of proof opens the doors for real understanding of the subject. Then we get to G.RP.1. This starts with the grandiose statement Understand the axiomatic structure of geometry. Good luck with that. It took well over 200 years for mathematicians to fully understand the strengths and limitations of this method, concluding with Paul Cohens analysis of the continuum hypothesis in the 1960s. Moreover, in point of fact, I doubt that there is more than one high school math teacher out of 500 who actually knows enough about this topic to do a decent job of teaching it, at a level that high school students would actually understand. Then, after this not so auspicious start, the standards just go on and on and on, including far more than could be handled in any one year course. This sequence must be entirely rethought and revised.
Posted on: Tue, 18 Nov 2014 20:54:38 +0000
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