There are a number of points in the Summary of Administrative Response to MMSD Mathematics Task Force Recommendations which should be made. As a mathematician, let me just comment on comments on Recommendation 11. There are other comments which could be made, but I have a limited amount of time at present.
The first question I have is in the first paragraph. “One aspect of the balanced approach is represented in the four block approach to structuring mathematics lessons. The four blocks include Problem Solving, Number Work, Fluency and Maintenance and Inspecting Equations.” There is a missing comma, since it is not clear whether Maintenance goes with the previous word or the last two. However, in either case, “Inspecting Equations” is a strange phrase to use. I am not sure what it means, and when a mathematician who has read extensively in school mathematics does not understand a phrase, something is wrong. You might ask Brian Sniff, who seems to have written this report based on one comment he made at the Monday meeting, what he means by this.
In the next paragraph, there are the following statements about the math program used in MMSD. “The new edition [of Connected Math Project] includes a greater emphasis on practice problems similar to those in traditional middle and high school textbooks. The new edition still remains focused on problem-centered instruction that promotes deep conceptual understanding.” First, I dislike inflated language. It usually is an illustration of a lack of knowledge. We cannot hope for “deep conceptual understanding”, in school mathematics, and Connected Math falls far short of what we want students to learn and understand in many ways. There are many examples which could be given and a few are mentioned in a letter I sent to the chair of a committee which gave an award to two of the developers of Connected Mathematics Project. Much of my letter to Phil Daro is given below.
The final paragraph for Recommendation 11 deals with high school mathematics. When asked about the state standards, Brian Sniff remarked that they were being rewritten, but that the changes seem to be minimal. He is on the high school rewrite committee, and I hope he is incorrect about the changes since significant changes should be made. We now have a serious report from the National Mathematics Advisory Panel which was asked to report on algebra. In addition to comments on what is needed to prepare students for algebra, which should have an impact on both elementary and middle school mathematics, there is a good description of what algebra in high school should contain. Some of the books used in MMSD do not have the needed algebra. In addition, the National Council of Teachers of Mathematics has published Curriculum Focal Points for grades PK-8 which should be used for further details in these grades. Neither of these reports was mentioned in the response you were sent.
I have pointed out errors and omissions in Connected Mathematics and Discovering Advanced Algebra to Sniff, and suggested that teachers be informed about these problems and given suggestions for how to work around them. You might ask him what has been sent to teachers about rational numbers and repeating decimals in Connected Math and the geometric series in Discovering Advanced Algebra. I wrote the principal author of Connected Math about their treatment of repeating decimals in the first edition, in 2000 and 2002. Nothing was changed in the second version. It is still a very poor treatment. I will send separately a paper I gave at a meeting in Lisbon last November. It deals with the help teachers should be given, and how inadequate it frequently is.
The National Mathematics Advisory Panel recommended that the geometric series should be done in first year algebra, since it is not hard to derive the sum of a finite geometric series and it has many interesting applications. In Discovering Advanced Geometry, the sum of this series is stated but not derived. What understanding is this giving students?
There never has been a serious public discussion about the direction of mathematics education in the Madison Schools. There should be. There was a committee set up to report and the part which surprised me most was the survey of elementary school teachers, who reported that most of them did not use a textbook as a primary resource. Decades ago my daughter went through a year at Cherokee with a teacher developed program in math. It was a disaster. I wonder about the results mentioned in a Capital Times article on the charter school Nuestro Mundo. Here are the result on WKCE Third Grade tests.
Percentage scoring proficient or advanced in reading
|Madison School District||72||88||47|
Percentage scoring proficient or advanced in math
|Madison School District||72||87||52|
Both the reading and math tests were given in English. In every other study I have seen about schools like Nuestro Mundo, the math score relative to the district score is much closer than the reading score is to the district average. Does the math staff at MMSD have an explanation for this dramatic difference?
Here is most of my letter to Phil Daro mentioned above. If you have any questions about what I have written, please feel free to contact me. My phone number is 233-7900.
Recently I read the announcement of the prizes awarded by ISDDE. The Connected Math award singled out two of their books. The 8th grade book, “Say It With Symbols”, had the following written about it:
Say It With Symbols tackles the development of robust fluency in symbolic manipulation (always a high priority) by focusing on “making sense with symbols” at every stage. Work on interpreting symbolic expressions leads on to creating equivalent expressions and thus to sense-making solution of linear and quadratic equations, and to modeling.
Let us look at a little of this book. There is some work on factoring quadratics, but clearly not enough for students to become fluent with it. The quadratic formula is stated but not proven, nor is there a proof (much less a motivated one) in the Teacher’s Guide. Completing the square is never mentioned. There are a couple of problems like the following: Page 51 in Second Edition. [I can give comments on the First Edition if that is what you used, but I am giving them a break and using the Second. It has been through even more use than the first, but still has a lot of flaws.]
44. You can write quadratic expressions in factored and expanded forms. Which form would you use for each of the following? Explain. c. To find the line of symmetry for a quadratic relationship Answer: The line of symmetry is a vertical line perpendicular to the x-axis through a point with an x-coordinate half way between the x-intercepts. The factored form can be used to find this point. How about the case when the factors are not real? y=x^2+2x+2. There is still a line of symmetry, but without complex numbers, which few will treat in eighth grade, factoring does not work. Of course one can make it work by subtracting a constant, but this is a book for students who are just learning algebra. Whenever the word “Explain” is used in a question, I look to see what the explanation is. There is no reason given for why half way between the intercepts gives the line of symmetry. A explanation can be given using either form, but the authors do not do this. I can give you many examples where the “Explain” answer in the Teacher’s Guide is far from an explanation, and sometimes is wrong.
Part d asks how to find the coordinates of the maximum or minimum point for a quadratic relationship. Here completing the square is clearly the better method at this stage, if one is aiming for the very important goal of fluency in symbolic manipulation, but that is not their goal. They seemingly never make the vital step of changing variables in an expression. There were many places where this could have been introduced and then used to give mathematical closure at the level they deal with, but it is not there.
Let us skip to the end of this book. There is an introduction to tests for divisibility in problem 9 on page 77 and problem 10 on the same page for divisibility by 2 and 4. The answers in the Teacher’s Guide are reasonable. Then in problem 41 the problem of divisibility by 3 is considered. The answer pulls out the idea of changing 100a + 10b + c to 99a + a + 9b + b + c and then writes this to get the usual criteria. What is missing is an explanation for why one does this. One looks for the closest numbers to 100 and to 10 which can be divided by 3, which mimics the argument in divisibility by 2 and 4. The teachers will not know this, nor know that this can be extended to divisibility by 11 by a similar argument, although unlike the case of 3 and 9, the step from 11 to 99 to 1001 is only easy for 11 and 99. Before seeing how this extends one cannot just divide 1001 by 11, but write 1001 as 990 + 11. This extends. This is what should be in the Teacher’s Guide. One recommendation from the National Mathematics Advisory Panel is that instruction should not be either entirely “student-centered” or “teacher-directed”. The problem should have been given with some explanation about how divisibility by 2, 4, and 5 works, and then after remarking that divisibility by 3 cannot come from just looking at the last digit, ask the students to figure out what the closest number to 10 is which is divisible by 3, and then the closest number to 100 which is divisible by 3, and to use this information to try to find a simple test for divisibility by 3.
Let us consider the last problem. Judy thinks she knows a quick way to square any number whose last digit is 5. (Example 25) Look at the digit to the left of 5. Multiply it by the number that is one greater than this number. (example 2*3=6) Write the product followed by 25. This is the square of the number. Try this squaring method on two other numbers that end in 5. Explain why this method works. [Explanation: Students may find it easiest to explain why this method works by forming an equation [sic] to represent the value of any number ending in five, such as (10x+5), where x can be any whole number. Then a student taking the square of this value they [sic] will get (10x+5)(10x+5)=100x^2+100x+25)=100. [The 100 is only part of what should be there. It should be 100x(x+1) + 25.] This equation represents Judy’s method of finding the square. [The word “equation” is wrong. They mean “expression”.]
If they are going to let x be any whole number, then Judy’s method is wrong, since she said to look at the digit to the left of 5, and multiply it by the number that is one greater than this number. So 125^2 would be the same as 25^2, or with careless reading, the same as 2*13 with 25 appended. This is not symbolic fluency in the textbook.
The next to the last problem dealt with divisibility by 6, and the correct statement is given in the Teacher’s Guide, but the argument pulls out heavy machinery in the form of the Fundamental Theorem of Algebra when it is not needed. However, the related problem of assuming that a number is divisible by 2 and by 4 (rather than 2 and 3) does not imply it is divisible by 8 is missing. That is a mistake since students at this age will often not see the difference.
I have yet to talk to a high school teacher who thought that students who have had Connected Mathematics Project are better at symbolic calculations than those they had had earlier before CMP was introduced. Some, but not all, say the students have better conceptual understanding. Thus I find it strange that fluency in symbolic skills is singled out as a strength of CMP. Have you read the books which were mentioned?
In other areas, such as geometry, CMP has few if any of the problems which are common in East Asian countries, to help students learn how to solve multistep problems, including quite a few nice problems where auxiliary lines need to be drawn. I have books from Nigeria which have better geometry problems than CMP does. You should know this if what I found on the web is correct, that you are helping develop a middle school program based on Japanese models. Instead of giving CMP an award, it would have been much better to have read the first edition carefully and made constructive suggestions about how to improve it. It needs a lot of improvement.