All posts by Richard Askey

Articles and Books on Mathematics Education

The winter 2009-2010 issue of “American Educator”, has a number of interesting articles. Here are two of interest for people interested in mathematics education.
Daniel Willingham “Is It True That Some People Just Can’t Do Math”
Patsy Wang-Iverson, Perla Myers, and Edmund Lim W.K. “Beyond Singapore’s Mathematics Textbooks – Focused and Flexible Supports for Teaching and
Learning”
The first has a number of useful references as well as comments. Here is one. There have been many papers written in Madison on student’s lack of understanding of the equal sign. I once asked Liping Ma if this was a problem in China. She said that as far as she knew it was not. There is confirmation of this in one of the references.
Four questions asked of sixth grade students in the U.S. and China.

The paper which includes this is “Sources of Differences in Children’s Understanding of Mathematical Equality: Comparative Analysis of Teacher Guides and Student Texts in China and the United States”, by Xiaobao Li, Meixia Ding, Mary Margaret Capraro, Robert M. Capraro. It appeared in Cognition and Instruction, vol. 26, no. 2, pages 195-217, in 2008.
The second article in American Educator has comments on curriculum, teacher induction and education and support while teaching. There is also a one page supplemental article on teacher professional development and evaluation by Susan Sclafani and Edmund Lim W.K.
In addition there have been two very interesting books on school mathematics education written by mathematicians. The first is “Arithmetic for Parents: A Book for Grownups about Children’s Mathematics” by Ron Aharoni, Sumizdat, 2007. An article by Aharoni about his experience teaching mathematics in an elementary school in Israel can be read here. This is a good introduction to his book, and more useful details are in the
book.
The second is “And All the Children Are Above Average: A Review of The End of Ignorance: Multiplying Our Human Potential” by John Mighton, a Canadian mathematician and playwright. The paperback version of this book was published by Vintage Canada. You can read about Mighton here. and there is also information about his math program JUMP here. This program was developed after Mighton learned a number of things while tutoring students who had significant problems in learning elementary mathematics. A review of this book by David Kirshner appeared in the Journal for Research in Mathematics Education in the January, 2010 issue.

Deja vu: Report of the 1965 Madison School District Math 9 Textbook Committee

1.7MB PDF by Robert D. Gilberts, Superintendent Madison School District, Ted Losby and the Math 9 Textbook Committee:

The mathematics committee of the junior high schools of Madison has been meeting regularly for four rears with one intention in mind — to improve the mathematics program of the junior high school. After experimenting with three programs in the 7th grade, the Seeing Through Mathematics series, Books 1 and 2, were recommended for adoption and approved in May of 1963.
The committee continued its leadership role in implementing the new program and began evaluation of the 9th grade textbooks available. The committee recommended the adoption of Seeing Through Mathematics, Book 3, published by Scott, Foresman and Company, and Algebra: Its Element and Structure, Book 1, published by Webster Division, McGraw-Hill Book Company, and the Board of Education adopted them on May 3, 1965.
A number of objections to the Seeing Through Mathematics textbooks were made by various University of Wisconsin professors. Dr. R. C. Buck, chairman of the University of Wisconsin Mathematics Department strongly criticized the series. A public objection to the adoption was made at the Board of Education meeting by Dr. Richard Askey of the University Mathematics Department. Later, a formal petition of protest against the adoption of Seeing Through Mathematics, Book 3, was sent to committee members. [related: 2006 Open Letter from 35 UW-Madison Math Professors about the Madison School District’s Math Coordinator position]
The sincerity of the eminently qualified professional mathematicians under Dr. Buck’s chairmanship was recognized by both the administration and the committee as calling for reconsideration of the committee’s decisions over the past three years relative to the choice of Seeing Through Mathematics 1, 2 and 3.
Conversely, the support of the Scott, Foresman and. Company mathematics program and its instruction philosophy, as evidenced by numerous adoptions throughout the country and the pilot studies carried out in the Madison Public Schoolsvindicated that equitable treatment of those holding diametric viewpoints should be given. It was decided that the interests of the students to be taught would be best served through a hearing of both sides before reconsideration.
A special meeting of the Junior High School. Mathematics committee was held on June 10, 1965.
Meeting 1. Presentations were made by Dr. R. C. Buck, Dr. Richard Askey, and Dr. Walter Rudin of the University of Wisconsin Mathematics Department, and Dr. J. B. Rosen, chairman-elect of the University of Wisconsin Computer Sciences Department.
The presentations emphasized the speakers’ major criticism of the Seeing Through Mathematics series — “that these books completely distort the ideas and spirit of modern mathematics, and do not give students a good preparation for future mathematics courses. Examples were used to show that from the speakers’ points of view the emphasis in Seeing Through Mathematics is wrong. They indicated they felt the language overly pedantic, and the mathematics of the textbooks was described as pseudo-mathematics. However, it was pointed out that the choice of topics was good the content was acceptable (except for individual instances), and the treatment was consistent. A question and answer session tollowed the presentations.
……….
After careful consideration of all points of view, the committee unanimously recommended:

  1. that the University of Wisconsin Mathematics and Education Departments be invited to participate with our Curriculum Department in developing end carrying out a program to evaluate the effectiveness of the Seeing Through Mathematics series and, if possible, other “modern” mathematics series in Madison and other school districts in Wisconsin;
  2. that the committee reaffirm its decision to recommend the use of Seeing Through Mathematics, Book 3, and Algebra: Its Elements and structure, Book 1, in grade nine with Seeing Through Mathematics, Book 1 and 2 in grades seven and eight, and that the Department of Curriculum Developnent of the Madison Public Schools continue its study, its evaluation, and its revision of the mathematics curriculum; and
  3. that en in-service program be requested for all junior high school mathematics teachers. (Details to follow in a later bulletin).

Related: The recent Madison School District Math Task Force.
Britannica on deja vu.

Response to the Madison School District’s Math Task Force Recommendations

To: comment@madison.k12.wi.us
Cc: askey@math.wisc.edu
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.

Continue reading Response to the Madison School District’s Math Task Force Recommendations

Letters: ‘A’ Is for Achievement, ‘E’ Is for Effort

Letters to the Editor: NY Times:

Student Expectations Seen as Causing Grade Disputes” (news article, Feb. 18) indicates a rather recent phenomenon among college students.
Students from the earliest grades are encouraged to work hard and told that the rewards will follow. Students must realize that a grade is earned for achievement and not for the effort expended.
Yes, some students can achieve at higher levels with far less effort than others.
This mirrors the world beyond college as well.
In my experience as dean, when students complain about a professor’s grading, they seem to focus more on their “creative” justifications (excuses) rather than on remedies. Most faculty members stress the remedy that leads to achievement of instructional goals.
The time-honored mastery of the material should remain paramount. After all, this is what our society expects!
Alfred S. Posamentier
Dean, School of Education
City College of New York, CUNY
New York, Feb. 18, 2009

To the Editor:
As someone who recently went through the ordeal of contesting a grade, I was quite impassioned on reading your article. I have done this only once in four years, so not all of us take the matter lightly.
I resent the suggestion that students feel “entitled” to “get/receive” good grades.
What is so irrational about believing that hard work should warrant a high grade? I would argue that the very core of the American dream is the sentiment that one can achieve any greatness that he or she aspires to if he or she works hard enough.
When one puts one’s all into a class, it’s not shameful to hope that grades reflect that. The same applies to professionals and their salaries. Instead of psychoanalyzing their students, perhaps these professors should ask themselves this question: If your students are all really this despicable, why are you teaching?
Aimee La Fountain
New York, Feb. 18, 2009
The writer is a senior at Marymount Manhattan College.

Singapore Math Bill Approved in Utah

Lisa Schencker:

Some lawmakers want Utah to follow the lead of a tiny Asian country when it comes to teaching math.
A senate committee Friday morning approved a bill, SB 159, that would allow districts and charter schools to apply for grants to use the Singapore method to teach math. Singapore is one of the highest scoring countries on international math tests.
In Singapore, math students are encouraged to think visually and develop mental strategies to solve problems. They’re discouraged from using paper to compute math problems.
“We seek to create a school system that will produce a significant percentage of the scientists and engineers needed by our country,” said Sen. Howard Stephenson, R-Draper, who is sponsoring the bill.
SB 159 would offer competitive grants to districts that come up with plans for teaching Singapore math in kindergarten through sixth grade and some secondary school classes. The bill would also require districts to train teachers in Singapore math and offer grants to colleges and other groups to train mathematicians to be teachers.
“I believe this will raise the math abilities of everyone in the state,” said Aaron Bertram, chairman of the University of Utah mathematics department.

10 Lessons of an MIT Education

Gian-Carlo Rota:

Lesson One: You can and will work at a desk for seven hours straight, routinely. For several years, I have been teaching 18.30, differential equation, the largest mathematics course at MIT, with more than 300 students. The lectures have been good training in dealing with mass behavior. Every sentence must be perfectly enunciated, preferably twice. Examples on the board must be relevant, if not downright fascinating. Every 15 minutes or so, the lecturer is expected to come up with an interesting aside, joke, historical anecdote, or unusual application of the concept at hand. When a lecturer fails to conform to these inexorable requirements, the students will signify their displeasure by picking by their books and leaving the classroom.
Despite the lecturer’s best efforts, however, it becomes more difficult to hold the attention of the students as the term wears on, and they start falling asleep in class under those circumstances should be a source of satisfaction for a teacher, since it confirms that they have been doing their jobs. There students have been up half the night-maybe all night-finishing problem sets and preparing for their midterm exams.
Four courses in science and engineering each term is a heavy workload for anyone; very few students fail to learn, first and foremost, the discipline of intensive and constant work.
Lesson Two: You learn what you don’t know you are learning. The second lesson is demonstrated, among other places, in 18.313, a course I teach in advanced probability theory. It is a difficult course, one that compresses the material typically taught in a year into one term, and it includes weekly problem sets that are hard, even by the standards of professional mathematicians. (How hard is that? Well, every few years a student taking the course discovers a new solution to a probability problem that merits publication as a research paper in a refereed journal.)
Students join forces on the problem sets, and some students benefit more than others from these weekly collective efforts. The most brilliant students will invariably work out all the problems and let other students copy, and I pretend to be annoyed when I learn that this has happened. But I know that by making the effort to understand the solution of a truly difficult problem discovered by one of their peers, students learn more than they would by working out some less demanding exercise.

A Letter to Jay Matthews

To Jay Matthews:
Let me suggest that Gerald Bracey is not an appropriate person to quote when dealing with mathematics education. First, it was TIMSS in 1995 rather than 1999 when students in the last year of high school were tested. Second, while some of our students who took the advanced math test had only had precalculus, all of them had studied geometry and we did worse in geometry than we did in calculus. Bracey never mentions this. Check the figures yourself to see the disastrous results in geometry.
We had 14% of our students take this test so the fact that some other countries did not test students in vocational tracts is irrelevant since they have a much larger fraction of their students in academic programs than 14%, as we do. About the ETS restudy, while they claim that the original sample was not comparable with other countries, their population was also not comparable with that of other countries. When you take the top say 7% of our students, judged by the courses they take which is not a perfect match but
not bad, and compare them with the top say 20% of the students in another country, that is not the same as comparing them with the top 7% in another country. ETS never mentions this in their press releases on this study.
Richard Askey

SAT Comparison: Wisconsin, Minnesota, Illinois, Michigan and Iowa

The average SAT scores for Wisconsin and the neighboring states are summarized below. The higher the percentage of students who take the test, the lower
the average score is likely to be.

State % Taking Test Critical Reading Math Writing
Minnesota 8 596 609 579
Illinois 7 583 601 578
Michigan 6 581 598 572
Wisconsin 5 587 604 577
Iowa 3 603 612 582
The Wisconsin Department of Public Instruction Press Release [255K PDF] only compared Wisconsin to the National Average, below.
National Average 45 502 516 494

College Board 2008 SAT information.

NAEP Math Results: Ohio and Wisconsin Comparison

The 2007 NAEP results have just been released. There are many interesting results one can learn by looking at this data. In addition to the very serious racial gap in Wisconsin which has been commented on by The Educational Trust [Grade 4 Math NAEP Analysis | 80K PDF ] [Grade 8 Math NAEP Analysis | 80K PDF] and the Wisconsin Department of Public Instruction [172K PDF], there are strong indications of other problems in mathematics education in Wisconsin. Consider the following data comparing results for whites and blacks in Ohio and Wisconsin from the first year NAEP results were given by states and the 2007 results. As background, 12 points on NAEP is generally thought to be about the change from one year to the next on a given test. This is not a good estimate when looking over 15 to 17 years, since part of the rise in the test score likely came from changes made in textbooks and in what teachers teach because of the change in the NAEP Framework in the early 1990s.


For example, in Trends in Mathematics and Science Study, TIMSS, fourth grade math was tested in 1995 and 2003, and the results were flat while the NAEP results went up enough to allow statisticians to conclude the increase was statistically significant.


I assume that some of the rise in NAEP over this period is because students are learning more about the topics covered in NAEP, but that this is not the only
reason for the rise in NAEP scores.


The data below is comparison data between the results in two states at two different years, so the point estimate for a year of schooling seems to be a reasonable guideline. If so, Wisconsin has lost about a year to Ohio. Something needs to be done about this.

NAEP Fourth Grade Mathematics
Whites 1992 2007
Wisconsin 233 250
Ohio 222 250 Ohio gained 11 points on Wisconsin
Blacks 1992 2007
Wisconsin 195 212
Ohio 194 225 Ohio gained 14 points on Wisconsin
NAEP Eighth Grade Mathematics
Whites 1990 2007
Wisconsin 279 292
Ohio 268 291 Ohio gained 10 points on Wisconsin
Blacks 1990 2007
Wisconsin 236 247
Ohio 233 258 Ohio gained 14 points on Wisconsin

Madison Literary Club Talk: Examinations for Teachers Past and Present

2.1MB PDF
First, a disclaimer. I am far from an expert on most of the topics which will be illustrated by questions. One of my aims in giving this talk is to let others know about a serious problem which exists beyond the problem of mathematical knowledge of teachers.
I have written about the problem in mathematics and hope that some others will use the resouces which exist to write about similar problems in other areas.
In his American Educational Research Association Presidential Address, which was published in Educational Researcher in 1986, Lee Shulman introduced the phrase “pedagogical content knowledge”. This is a mixture of content and knowing how to teach this content and is the one thing from his speech which has been picked up by the education community. However, there are a number of other points which he made which are important. Here is an early paragraph from this speech:

We begin our inquiry into conceptions of teacher knowledge with the tests for teachers that were used in this country during the last century [the 19th] at state and county levels. Some people may believe that this idea of testing teacher competence in subject matter and pedagogical skill is a new idea, an innovation spawned in the excitement of this era of educational reform, and encouraged by such committeed and motivated national leaders as Albert Shanker, President, American Federation of Teachers, Bill Honig, State Superintendent of Schools, California, and Bill Clinton, Governor of Arkansas. Like most good ideas, however, it’s roots are much older.

It took Wisconsin almost 20 years to adopt this “good idea”.

1989-2006 Math Comparison: Are Students Better Now?

math8906.jpg
W. Stephen Wilson [75K PDF]:

Professors are constantly asked if their students are better or worse today than in the past. This paper answers that question for one group of students.
For my fall 2006 Calculus I for the Biological and Social Sciences course I administered the same final exam used for the course in the fall of 1989. The SAT mathematics (SATM) scores of the two classes were nearly identical and the classes were approximately the same percentage of the Arts and Sciences freshmen. The 2006 class had significantly lower exam scores.
This is not a traditional research study in mathematics education. The value of this study is probably in the rarity of the data, which compares one generation to another.
….
Nineteen eighty-nine is, in mathematics education, indelibly tied to the National Council of Teachers of Mathematics’ publication, Curriculum and Evaluation Standards for School Mathematics (1989), which downplayed pencil and paper computations and strongly suggested that calculators play an important role in K-12 mathematics education. My 2006 students would have been about two years old at the time of this very influential publication, and it could easily have affected the mathematical education many of them received. Certainly, one possibility is that mathematics preparation is down across the country, thus limiting the pool of well prepared college applicants.

Wilson is a Professor of Mathematics at Johns Hopkins University.

WGN Program on School Reform & “Lets Put Parents Back in Charge”

Milton Rosenberg is a retired social psychologist from the Univ. of Chicago. He has a radio show on WGN, 720 on AM. Next Tuesday, March 7, the topic is “School Reform”. The two guests, Joseph L. Bast and Herbert Walberg, have written a new book: “Let’s Put Parents Back in Charge: A Guide for School Reformers“. The show starts at 9 PM and ends at 11.

Madison and Wisconsin Math Data, 8th Grade

At a meeting on February 22 (audio / video), representatives of the Madison Metropolitan School District presented some data [820K pdf | html (click the slide to advance to the next screen)] which they claimed showed that their middle school math series, Connected Mathematics Project, was responsible for some dramatic gains in student learning. There was data on the percent of students passing algebra by the end of ninth grade and data from the state eighth grade math test for eight years. Let us look at the test data in a bit more detail.

All that was presented was data from MMSD and there was a very sharp rise in the percent of students scoring at the advanced and proficient level in the last three years. To see if something was responsible for this other than an actual rise in scores consider not only the the Madison data but the corresponding data for the State of Wisconsin.

The numbers will be the percent of students who scored advanced or proficient by the criteria used that year. The numbers for Madison are slightly different than those presented since the total number of students who took the test was used to find the percent in the MMSD presented data, and what is given here is the percent of all students who reached these two levels. Since this is a comparative study, either way could have been used. I think it is unlikely that those not tested would have had the same overall results that those tested had, which is why I did not figure out the State results using this modification. When we get to scores by racial groups, the data presented by MMSD did not use the correction they did with all students ( All 8th grade students in both cases)

MMSD Wisconsin
Oct 97 40 30
Feb 99 45 42
Feb 00 47 42
Feb 01 44 39
Feb 02 48 44
Nov 02 72 73
Nov 03 60 65
Nov 04 71 72

This is not a picture of a program which is remarkably successful. We went from a district which was above the State average to one with scores at best at the State average. The State Test was changed from a nationally normed test to one written just for Wisconsin, and the different levels were set without a national norm. That is what caused the dramatic rise from February 2002 to November 2002. It was not that all of the Middle Schools were now using Connected Mathematics Project, which was the reason given at the meeting for these increases.

It is worth looking at a breakdown by racial groups to see if there is something going on there. The formats will be the same as above.

Hispanics
MMSD Wisconsin
Oct 97 19 11
Feb 99 25 17
Feb 00 29 18
Feb 01 21 15
Feb 02 25 17
Nov 02 48 46
Nov 03 37 38
Nov 04 50 49
Black (Not of Hispanic Origin)
MMSD Wisconsin
Oct 97 8 5
Feb 99 10 7
Feb 00 11 7
Feb 01 8 6
Feb 02 13 7
Nov 02 44 30
Nov 03 29 24
Nov 04 39 29
Asian
MMSD Wisconsin
Oct 97 25 22
Feb 99 36 31
Feb 00 35 33
Feb 01 36 29
Feb 02 41 31
Nov 02 65 68
Nov 03 55 53
Nov 04 73 77
White
MMSD Wisconsin
Oct 97 54 35
Feb 99 59 48
Feb 00 60 47
Feb 01 58 48
Feb 02 62 51
Nov 02 86 81
Nov 03 78 73
Nov 04 88 81

I see nothing in the demography by race which supports the claim that Connected Mathematics Project has been responsible for remarkable gains. I do see a lack of knowledge in how to read, understand and present data which should concern everyone in Madison who cares about public education. The School Board is owed an explanation for this misleading presentation. I wonder about the presentations to the School Board. Have they been as misleading as those given at this public meeting?

Richard Askey

Why is the MMSD Afraid to Have a General Discussion of Their Mathematics Program?

A year ago the Jefferson PTO planned to have a mathematics night, with a discussion of their math program. I was asked if I would appear and said yes. The Madison Metropolitan School District was asked and they refused to send anyone, saying that they did not want to do this school by school. but district wide. When Mary Ramberg was asked when this would be done, she said they had no plans to do this.
Here is part of the report from 1882 from the State Superintendent about textbooks. At this time changes in textbooks had to be approved by the State Superintendent. The following should be done:

  • 3d. That regard shall be had to the merits of the books, and that if the change is sought to be made in the interests of better books, the superior merits of the books proposed to be introduced shall be stated.
  • 4th. That the change shall not be against the pronounced public opinion of the locality interested.

Why is the MMSD afraid to have a general discussion of their mathematics program?