Thursday, August 13, 2015

Dr. James Milgram Has Harsh Words For Oklahoma's New Math Standards Draft



James Milgram is a professor emeritus of mathematics at Stanford University. He served on the validation committee for the Common Core mathematics where he did not sign off on the standards. ROPE asked the Standards Writing Committee to include Dr. Milgram in the standards writing process in some way, but his expertise went unused.

The following information was sent to us by Dr. Milgram after asking him to comment on the drafts made available online. 


1) From what I can tell, no one on the Oklahoma Mathematics standards writing committee has a college degree in Mathematics, in particular, no one appears to have a PhD in Mathematics; the committee members all appear to have degrees in Education (some in Math Education). I believe that without degrees in the pure subject of Mathematics, the members of the committee are not fully qualified to write Mathematics standards for the state of Oklahoma.

Agreed. I went somewhat further and checked the qualifications of the 4 members identified as university faculty members. One is a specialist in remediation, something worth having but all three of the others show no evidence of any knowledge of actual mathematics beyond the most elementary – and I mean elementary. Their expertise appears to be focused on no more than the first 2 to 3 grades. Also, nobody on the committee appears to even be qualified to handle things at the level of any high school material past a weak Algebra II course. In fact this draft standards cuts off well before the Common Core math standards, which were, as numerous people have pointed out, themselves far below the expected level of high school mathematics necessary for students wishing to major in technical areas including STEM in a solid four year college or university.

2) In the early grades, the Standards do not specify that students learn, practice, memorize, and be able to demonstrate instant, accurate recall of the basic math facts of addition, subtraction, multiplication, and division up to at least 10s, preferably 12’s or 15’s. This is certainly true. But there are far more problems than just this (as serious as it is). Here is a brief rundown of the first grade standards that were so imprecise - or more likely actually incorrect mathematically - that I couldn’t make sense of them as written.
1.A.1.1, 1.A.1.2: Imprecise to the degree that I can’t understand what they mean. Create and extend repeating or growing patterns etc. On what basis? are the students given rules for how to extend or not. If not, then this has nothing to do with mathematics, and will, later, very negatively impact the liklihood of students being successful in the area.
1.N.1.1 Does this have anything to do with actual mathematics? If so show me the research. Here’s the normal way this goes. One “recognizes” and names the Symbol (e.g., ten frames, arrays, etc) and one might or might not count the number of each, but one does not see the number that these objects are expected to represent.
1.N.1.2 Where did the “in terms of tens and ones” suddenly come from? Students should still be counting, it is almost certainly too early to introduce base ten place value notation.
1.N.1.3 Represent? Come on! Would this committee like to define “represent” and send me a recording of the session. My suspicion is that, at least in mathematical terms, virtually nobody on that committee understands what they are talking about. To be specific, base 2 place value notation is a very natural way of representing numbers, as is base 3 or base 5. Is this what you mean. I hope not! Likewise what do you mean by read and write? I think what you mean is make the English sound for the symbol “5” or the symbol “7”, but this is virtually anti-mathematics. After all, the committee should know that the sounds in Spanish or German for these same symbols are completely different, but mathematically, they represent the same number.
1.N.1.8 Do you really mean “equivalence.” I think not. I think equivalence is far too sophisticated to even talk about before, at earliest, sixth or seventh grade. I think what you really want to say is “equality.” It does not speak well for the competence of this committee to make mistakes like this.
1.N.2.1 – 1.N.2.3: These three standards together are virtually identical to the single most criticized and ridiculed first grade standard in the Common Core. Please look at both John Stewart’s and Steven Colbert’s discussion of the Common Core mathematics standards for clarification.
1.N.3.1 I don’t know what you can mean here. Using physical models it seems very, very difficult to partition. But it is fairly easy to use physical models to construct polygons from equal pieces. There is a vast difference between partitioning and combining.
1.N.4 This is actually a very good section. The failure to handle coinage is one of the most glaring issues with the Common Core in the early grades.
1.GM.1.1: Probably too advanced for first grade. See my comments above.
1.GM.2.2: As written, total nonsense. Length is absolute, it is the measurement of length that depends on the size of the unit. Get it? There is a distinction between the mathematical concept length, and the particular ways in which one might assign a number to that length.
3) Grade 2.
2.A1.1, 2.A1.2 Same issues as with the corresponding first grade standards.
2.A.2.2 What on earth is “number sense?” What does it mean to say “Introduction to properties, but not mastery of vocabulary.” After all, vocabulary is not mathematics. Properties probably are. The test is whether things change in different languages.
2.N.1.1 “Read, write, discuss, and represent.” Read, write and discuss do not have any mathematical significance. And I think that instead of represent a number you mean the reverse, determine the number that represents the cardinality of a small finite set.
2.N.1.5 “emphasis on understanding how to round instead of memorizing the rules for rounding.” How to round is a sequence of (somewhat arbitrary) rules so this completely confuses me.
2.N.2.1 Amazingly like 1.N.2.1 – 1.N.2.3, and it will and should be soundly criticized.
2.N.2.2 is way too low level for second grade. It is more appropriate for Kindergarten or first grade.
2.N.2.4 seems to be a way of suggestion practice with the standard algorithms for addition and subtraction without mentioning the terms. It would really be simpler to just say “Use the standard algorithms to add and subtract two and three digit numbers.”
2.N.3.1 It might be too soon to introduce fractions. Also, as usual, the committee has reversed the meaning of represent.
Much of grade three appears to be too advanced, comprising material that us usually done in the fourth grade, even in the high achieving countries.

4) In the middle grades, the standards do not emphasize the teaching and learning of standard formulas for the computation of area of various geometric shapes, etc. Instead, the standards seem to emphasize students inventing their own formulas. (Inventing formulas may have value for advanced students after they have learned thoroughly the standard formulas).

This does not bother me so much, though it would be helpful if you were to give me examples of specific standards. Then I could comment more accurately. Overall, I felt that the fifth sixth and seventh grade standards were better than average for the states, though the coverage of ratios and rates (particularly rates) was somewhat skimpy. There should have been much more attention paid to problems involving motion at constant speed and flow at constant rates. This is what is done in the high achieving countries. Moreover, these kinds of problems occur much earlier there than in this country and by sixth grade are far more involved than anything our kids will see at least until pre-calculus.

Of course there are still problems in these grades. For example, look at 7.N.2.3. It has already been pointed out that the rational numbers include integers, fractions, terminating decimals, and, moreover, that every rational number expands as an ultimately periodic infinite decimal. Moreover, every ultimately periodic infinite decimal is a rational number, though this is NOT CLEARLY STATED. But assume this early standard has been fixed so students are expected to understand this characterization of rational numbers.

For fractions ONE HAS THE DEFINITION OF ADDITION (SUBTRACTION) 
a/b ± c/d = (ad ± bc)/bd. And the only known “algorithm” is to apply this definition directly. It is also known that for arbitrary infinite decimals there is not and cannot be any efficient and finite algorithm for any of the 4 basic operations. Thus, we must conclude that 7.N.2.3 is simply incorrect and must be entirely revised.

Also, consider 7.GM.3.1. This purports to give definitions of area and volume. But both are incorrect since it is usually impossible to even exactly fill even a reasonably nice three dimensional region such as a triangular prism with cubes without gaps or overlaps.

Indeed, one is virtually forced to talk about limits when discussing volume and area, but limits are far beyond the expectations for seventh grade, even in the high achieving countries. 7.GM.3.1 needs to be completely rethought and revised.

When we get to Algebra, things start to go south again. But most of the issues I’ve found are also present in the Common Core which appears to be very (probably too) similar. For example, look at PA.N.1.3.

It is also important to note that key topics that are even present in Common Core (which is famously devoid of any discussion of pre-calculus, calculus, as well as a lot of standard material in Algebra II, trigonometry and even geometry) are not present at all in these draft standards. Thus, except for one mention of right triangle trigonometry in the introduction to the Oklahoma Standards for Geometry, there is no discussion of trigonometry at all, and there are only two mentions of rational functions (A2,F.1.8 and A2.F.1.9)

Likewise the only mention of matrices occurs in A2.N.3.1 - A2.N.3.3, and even the weak discussion of matrices in Common Core is far better than this.

6) To answer the argument that not all students desire to enter STEM majors or elite universities, I suggest that Oklahoma high schools offer three diploma tracks for mathematics:
1) Work-Ready Diploma (students not going to college: minimum of Algebra II).
2) General College-Ready Diploma (students going to community or state colleges: minimum of Pre-Calculus/Trigonometry).
3) Elite College/STEM-Ready Diploma (students desiring to attend elite universities or to enter STEM majors: minimum of Calculus I).
Agreed, but where does this argument appear?