Is the NCTM Opposed to Mathematics Education?

The National Council of Teachers of Mathematics (NCTM) is the nation's largest professional organization for K-12 teachers of mathematics. The idea that the agenda of the NCTM would in fact be opposed to teaching mathematics seems, on the face of it, absurd. And yet that is the thesis of David Kline's paper, "A Brief History of K-12 Mathematics Education in the 20th Century". If Kline were an isolated crank, this idea would not matter much. But he is a professor of mathematics at California State University Northridge and according to Google Scholar his paper has been cited by 51 researchers since its publication as a chapter of Mathematical Cognition in 2003. Furthermore, the paper was described as must reading for all interested in mathematics education by John Mighton, author of the influential education best-seller, The End of Ignorance: Multiplying Our Human Potential, which I reviewed earlier.

Klein is firmly in the traditionalist camp of mathematics education, and in fact presents himself as a member of Mathematically Correct, the most famous of the traditionalist groups. To boil down a detailed argument of over 40 pages to a few sentences, Klein feels that the mathematical reform movement which came to prominence in the 1990s under the auspices of the NCTM and the National Science Foundation (NSF) disenfranchised students by offering mathematics instruction grounded in constructivist theory and based on "textbooks with radically diminished content and a dearth of basic skills". He traces the reform movement to the progressive education movement beginning in the early 20th century, whose leaders had a documented hostility to mathematics. For one example of many, he quotes the influential progressive educator William Heard Kilpatrick as saying that mathematics is "harmful rather than helpful to the kind of thinking necessary for ordinary living". He states that the NCTM was created by the MAA (Mathematical Association of America) in part to counter these progressivist ideas, though later the NCTM embraced these same ideas.

As is well-known, the reform movement in mathematics is characterized by educational constructivism, the theory that "only constructed knowledge – knowledge that one finds out for oneself – is truly integrated and understood". Constructivism is originally a psychological term, and Klein quotes several psychologists who claim that educators misapplied the concept, so that claims by constructivists they support "brain-based learning" ring hollow. In any case, educational constructivism is now connected with child-centered, cooperative, self-paced, problem-based discovery learning.

Education can be viewed as a wedding of pedagogy and content. In theory, the two are separate. However, constructivist learning takes longer since the student spends time exploring blind alleys on the way to getting a correct answer. For example, if the student is discovering their own algorithm for multi-digit addition it will take longer than if they are simply told how to do it, and moreover the algorithm they derive may be inefficient, making subsequent work take longer. So when traditionalists say they are designing a pedagogy-neutral curriculum, they are being somewhat disingenuous. By proposing a lengthy list of content to be covered, they insure that strict constructivist approaches will not work.

In essence, Klein feels that the NCTM has been taken over by professional educators, not teachers, and that these educators are pursuing a progressivist agenda with little regard for actually teaching mathematics.

While this critique seems to make sense, it does not fit with my personal observations of many practitioners of constructivist mathematics education. Most of these people have a deep love of mathematics and of teaching. After all, anyone who has done original mathematics has practiced discovery learning. Indeed, the "Moore Method" pioneered by topologist R. L. Moore at the University of Texas was an extreme form of discovery learning for graduate mathematics majors, Moore is regarded as one of the most successful teachers of graduate mathematics in American history, based on the number and quality of his Ph.D. students.

I think most mathematics educators would agree that students should receive some direct instruction in standard algorithms and basic theory and some opportunity to explore mathematics on their own. As always, balance is important.

I also think that most mathematics educators would agree that being able to teach well using a constructivist approach is more difficult than using a traditional approach. Part of the reason why constructivist approaches have not been more successful therefore has to do with inadequate training of teachers, and of a failure to recruit the best students into a difficult and underpaid profession.

For some eloquent defense of constructivist, problem-based learning from people who clearly love mathematics see:
A Mathematicians Lament by Paul Lockhart
In Math You Have to Remember, In Other Subjects You Can Think About It by Keith Devlin

Fractal Video from Teamfresh

I found this on Steven Strogatz's NY Times Math Blog

Classic newton fractal from teamfresh on Vimeo.

From a Spreadsheet Problem to the Umbral Calculus: A Mathematical Odyssey

I'm planning to write a paper where I describe how a colleague's challenge to come up with an Excel formula to compute a weighted average of grades led me to make a couple of mathematical conjectures, and how I was able to prove the conjectures and solve the problem. Along the way, I got a lot of help from many people and I discovered a lot of combinatorial mathematics that I had not known, including the Binomial Inversion Formula and the Umbral calculus. In describing this odyssey I will explore the social nature of mathematics and the different ways that people from different disciplines approach mathematical problems. Also, I hope to show that experiences of this sort can be replicated in the classroom through a problem-based method of learning.

My Problem Published

My problem about finding the kth largest element of a set has been published (with a very slight misprint) in the American Mathematical Monthly:
Monthly Problem 11520  

E18. A locus related to a rectangle

This problem is related to my earlier E16.

Let ABCD be a rectangle. Find the locus of all points P such at PA + PC = PB + PD.

The End of Ignorance: Multiplying our Human Potential

I've just finished reading The End of Ignorance: Multiplying our Human Potential, by John Mighton who has developed a mathematics education program called JUMP (Junior Unrecognized Mathematics Prodigies). His system has met with amazing success with a very wide range of elementary school students and considerable hostility from the Mathematics Education establishment in his native Ontario.

He challenges the NCTM orthodoxy, and the tenets of constructivist math education. I feared that this might be another "Mathematically Correct" screed, but it is far from that. Mighton has an enviable record of success in reaching the most "hopeless" students, and an admirable humility in recognizing that his system is not the only way to improve math education.

Mighton has a Ph.D. in mathematics, a career as a playwright, and a firm grasp of philosophy. He and a large cadre of volunteers have developed the program over a number of years, and refined it by trial-and-error. The major ideas are:

1. Learning takes place with a balance of concrete and symbolic, guided and independent, and procedural and conceptual.
2. Compared with constructivist methods, the teacher is expected to be a very active guide. Concepts are broken into small units, gaps in student understanding are detected and filled, lessons are carefully designed, sequential, and scaffolded. Weaker students are motivated by carefully graduated challenges, and stronger students are given extra challenges.
3. Whole-class lessons allow students to experience the thrill of discovery collectively.
4. Teachers give frequent and specific encouragement to all students.
5. Formative assessments are given continuously, and used to modify instruction. Students who don't know the material necessary to begin the lesson are given additional instruction before learning the new topic.
6. There is a strong emphasis of the development of  procedural knowledge through use of workbooks and individual work.

I think any educator who reads Mighton's calmly recollected stories of the hostility and closed-mindedness that his ideas have generated among certain math curriculum consultants (some of whom are subject to conflicts of interests due to their relationships with textbook publishers) is bound to feel a sense of embarrassment for our profession.

Mighton's book has caused me to rethink some of my pro-constructivist positions. I also will be following up on reading some of the work on cognitive psychology that he cites as having been seriously misinterpreted by the mathematics education establishment as supporting constructivist and situated learning approaches.

NES/MAA Meeting

The NES/MAA meeting I mentioned in my last post was held at Salve Regina University, which is located on the grounds of an opulent mansion on the ocean at Newport Rhode Island. There were a number of interesting invited presentations. I particularly enjoyed the talk by David Abrahamson and Rebecca Sparks on Baseball Statistics and Keith Conrad's talk on Check Digits (in credit card numbers, etc.). Both dealt with fairly elementary mathematics, but related the results to the everyday world in a compelling way.


Ed Burger's Battles Lecture on p-adic norms was a bit more technical, but Ed's high-energy and humorous style of presentation made the medicine go down very well.

My presentation on The Pythagorean Theorem (Revisited) was well received. I was a bit surprised and gratified that none of the mathematics professors or students previously knew the main theorem I was presenting, the Pythagorean Theorem for right tetrahedrons.