Yesterday’s *New York Times *had a story regarding the results of the 2009 Mathematics portion of the National Assessment of Educational Progress. The results indicated that, at the fourth and eighth grade levels, American math skills have plateaued, with about 35-40% of students showing proficiency.

The *Times* also posted commentary from five individuals expressing their opinions about math education.

- Bruce Fuller, professor of education and public policy
- Lance T. Izumi, Pacific Research Institute
- Holly Tsakiris Horrigan, parent of public school children
- Richard Bisk, math professor
- Barry Garelick, U.S. Coalition for World Class Math

The first one of these commentaries to catch my attention was from Holly Tsakiris Horrigan, a parent in Massachusetts. She pinpoints what I think is an important problem in math education – the distinctly un-mathematical nature of many math textbooks.

If we want to improve mathematics education, we should banish nonsensical curricula like Trail Blazers, Everyday Math and Investigations and make sure that our teachers are properly educated and proficient in math content. These curricula substitute writing, drawing and calculator usage for solid math content, leaving children unprepared for more advanced math topics. The precious few children who go on to succeed in higher math after being subjected to one of these programs are those that had the benefit of substantial reteaching outside of the classroom.

Professor Richard Bisk, chair of the Mathematics Department at Worcester State College also points out the role that bad textbooks have played in the sorry state of America’s mathematical pedagogy.

To give students a firm foundation in math, we must start in the elementary grades by providing three things: a substantial improvement in elementary teachers’ knowledge of mathematics; a more focused curriculum that emphasizes core concepts and skills; and more challenging textbooks that teach for mastery and not just exposure….

Finally, many math textbooks in this country use the “spiral” approach, in which topics are not taught for mastery. Instead they are repeated year after year. I strongly agree with the report of the National Mathematics Advisory Panel, which said, “in elementary school textbooks in the United States, easier arithmetic problems are presented far more frequently than harder problems. The opposite is the case in countries with higher mathematics achievement, such as Singapore.”

It can be maddening to try to teach a “spiral” curriculum if you love the interlocking nature of mathematical skills and concepts. I was given a spiral curriculum textbook to use when I taught high school and couldn’t make sense of it because it jumped all over the place without ever coming to any kind of resolution for any of the topics.

Now, it is possible to design a proper spiral curriculum in which ideas are revisited at higher levels in later courses, but only after a certain amount of mastery has been achieved for the topic.

For instance, in Elementary or Intermediate Algebra, students may study the graph of the parabola simply as the graph of a quadratic function. They can see the solution(s) of the related quadratic equation by finding the zeros of the function. Application problems involving projectiles can also be explored.

These ideas can then be built on in a Pre-Calculus course with a deeper analysis of the parabola as a conic section defined by its focus and directrix. Completing the square and determining the focus and directrix of a parabola are more advanced skills that can be covered. Further applications involving the 3 dimensional paraboloid of rotation for headlights, microphones and satellite dishes (as well as solar powered electical generators) can also be explored.

The same can be done with other topics. But many of these textbooks change topics from day-to-day. In order to give students the foundations they need to build future knowledge on, each unit should ideally contain three or four ideas that are related to each other. Students can then spend several days on each topic, building a unit of study 10-14 days long (or whatever time period works for a particular situation).

Barry Garelick, co-founder of a parental advocacy group also stressed the importance of a cohesive curriculum:

The education establishment needs to understand that even process is based on skills and exercises, and a logical sequence of topics whose mastery builds upon itself. We need solid math curricula and textbooks that are based on the premise that procedural fluency leads to conceptual understanding.

The reader comments at the *Times* website for this story also discussed to this issue:

I have tutored elementary and Jr high school student for the past 10 years. I am appalled at the lack of math skills. When I tutored a program 4 days a week , one of the things I discovered was the lack of repetition and the lack of cohesiveness of the math lessons. One day, the kids did areas of a figure, the next, percentages, the next , something else. If they did not get the process in class, Oh well. I had 6th graders who could not do know the basic times tables and could not multiply 2 di[git] numbers.

Another commenter pointed out the differences between the American and Japanese curricula:

My son completed American (Main Line Philadelphia) and Japanese first grade curricula side-by-side. In the first grade, the topical coverage was similar but the Japanese curriculum presented the skills in a more logical order; later topics clearly built on previous topics. In the American curriculum, the topics were in a more scattered sequence. It was like building a brick wall with a brick here and a brick there with the expectation that a complete wall would result.

Though he also points out that the Japanese system isn’t perfect:

To be fair, the Japanese curriculum suffers from a belief that there is only one right answer.

In many cases there is only one right answer, but there are many different ways to get there!

Here’s another commenter discussing an important idea in math education:

You can’t teach something you don’t understand yourself. And “understand” means knowing not only how, but why.

Either all elementary school teachers need to learn the “why” of K-6 mathematics, or we need mathematics specialists in all elementary schools.

Li-Ping Ma published an excellent book in 1999 called *Knowing and Teaching Elementary Mathematics* that explores the mathematical knowledge of American and Chinese elementary school teachers.

For one thing, the Chinese teachers were generally math specialists. As a result, they often had a more profound understanding of the topics. This allowed them to understand WHY algorithms work the way they do and use this knowledge to effectively answer their students’ questions.

Ma points out that many of the Chinese teachers told her that they expect the students to “know how, but also know why.”

I first became aware of the problems in math education by doing an internet search on “Jaime Escalante” back in 1997 (I think I used Infoseek as Google didn’t exist yet!). What I found was the story of Escalante’s ostracism from the Garfield High School math department he had made famous in “Stand and Deliver.” The story was posted at a website called “Mathematically Correct.”

I spent a lot of time reading the content at that site in the late 1990s. A good summary of the state of math education at that time (which apparently hasn’t changed appreciably) can be found in the two-part article published in the Notices of the American Mathematical Society (Part I ; Part II) in 1997.

Professor H. Wu of UCal Berkeley also has many good articles about teaching mathematics and training math teachers as well. A favorite of mine is *Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education*.

There is one commenter at the *NY Times* site with whom I disagree. He is a science teacher who is concerned with his student’s math skills. He feels that mathematics should only cover “real life” applications:

Math courses do not give kids the skills they need to do REAL math…the kind of math that can be applied beyond theoretical math situations. The curriculum is so disconnected from reality as to be useless….

We need to rethink math. It has to go from being a bunch of theoretical ideas to actual practical skills. I’m not talking about the old “Farmer John has a fence 300 yards long…”

The Farmer and the Fencing is actually a particularly rich application problem that involves many different areas of mathematics. I have used this problem in classes ranging from 7th grade to Beginning Algebra to Differential Calculus, and, if a rectangular pen is split along the diagonal, the solution requires multi-variable Lagrangian methods. Creating the pen of largest area regardless of shape leads to the Isoperimetric Inequality.

If mathematics is to be centered around problems, then they should be problems chosen by each student individually and the problems should either be sufficiently rich in their own right so that the connections to many other ideas become clear or they should be embedded as a piece of a larger research project.

Physicists and other researchers in the hard sciences often spend time studying the mathematics they need to know in order to understand particular areas of their research. There’s no reason why students couldn’t approach mathematics in this way as well.

But, in a situation in which there are many different students whose different needs must be addressed in the same classroom, I believe that broad, generally applicable problems best serve the needs of such a diverse group.

on October 20, 2009 at 3:00 AM |Barry GarelickThis is a very nice blog about math; one of the best I’ve seen. Appreciate your comments about spiral curriculum and Everyday Math. You may be interested in my account of how I tutored my daughter with Singapore Math in order to stay ahead of the “train wreck” of Everyday Math. It’s at:

http://www.ednews.org/articles/one-step-ahead-of-the-train-wreck.html and is called “One Step Ahead of the Train Wreck.

on October 20, 2009 at 12:58 PM |richbeveridgeThanks very much for your comment. I liked your “One Step Ahead of the Trainwreck” article, as well as the “How Not to Teach Math” (http://www.city-journal.org/html/eon_3_7_03mc.html) you linked to.

I’ve read some about the Singapore Math (but never got around to buying copies) and it looks like an excellent curriculum.

on December 11, 2013 at 12:46 PM |The Farmer and the Fencing | Where the Arts Meet the Sciences[…] that mentioned the problem of “The Farmer and the Fencing.” The link for the post is here and a sample assignment from an Elementary Algebra (Algebra I) course is here. I mentioned in […]