View more books by Colin W. Clark: Math Overboard! - Part 1, Math Overboard! - Part 2

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Archive for December, 2012

Math Remediation and Tutoring

Monday, December 31st, 2012

Math Remediation

The following Abstract refers to a talk presented recently at my home university (British Columbia) by York University math professor Dr. Kim Maltman:

Abstract. At York, we have found first-year science majors coming to us from the
Ontario high school system in general rather poorly prepared for first-year
university mathematics. The result is very high drop-plus-fail rates in
our first-year math courses and a resulting high attrition rate in the
early years of our degree programs. A major source of the problem appears
to be the widespread use in the schools of an approach heavily emphasizing the
memorization of solution problem templates, an approach which leaves a
majority of our incoming science majors with deficiencies in very basic
algebra, trigonometry, and, even more problematic, their intuitive
understanding of the basic operations of arithmetic. In this discussion,
I will outline an approach I have developed involving 4-day, 4-hour-per-day
intensive remediation sessions focused on changing the way such students
approach mathematics. The program was begun in 2005 and significantly
expanded in 2009, now handling between 15 and 20% of the incoming
class each year. I will present statistics outlining the significant impact
we have seen on student performance.

It seems that the phenomenon of “high drop-plus-fail rates” is almost universal in North American colleges and universities. I don’t know how other universities are dealing with the problem, but at UBC until recently little was done, other than to provide students with a math-help facility where they could obtain help with their calculus courses. Many students still had difficulties, as attested to by the large number of posters advertising Math Tutoring. Besides this, almost every shopping mall in the area has an office advertising Help classes in Math.

Why is the phenomenon of high dropout rates so common and persistent? I think the Abstract has got it right – “…heavily emphasizing the memorization of solution problem templates,…which leaves incoming science majors with deficiencies in … algebra, trigonometry, [and] their intuitive understanding of … basic operations….” Just so.

But how to deal with the problem? Two possibilities are:

  1. Re-design the school curriculum to place more emphasis on understanding math.
  2. Provide assistance for poorly prepared college entrants.

The first alternative probably cannot be achieved easily, if at all. Remediation sessions are one approach to the second possibility, and it’s certainly impressive if a total of 16 hours’ class time can reverse 12 years of “math-is-memorization” instruction. I don’t know how  many universities are providing this service at the moment.

A second approach is streaming, with poorly prepared students being forced to take a course such as “Precalculus.”  But unless this course includes remediation for weak training and poor habits in arithmetic, algebra and geometry, I doubt if it will succeed in rescuing many students.


Private tutoring is yet another approach, but whether tutoring is likely to result in the necessary changes in the way that students perceive math is doubtful. Commercial tutoring may be better, but expensive.

Rather than hiring a tutor, a struggling student might try to re-learn math by searching the web. But what should they re-learn? Proofs? Set theory? Long division? Inverse functions?

Math Overboard!

These were the issues that prompted me to write Math Overboard! (Basic Math for Adults), which is a complete review of school math, emphasizing understanding and algorithmic, problem-solving skills. Math Overboard! is a self-study resource that helps motivated students to upgrade their comprehension of basic math to a level suitable for college math courses.

Here is what Prof. Marc Mangel (UC Santa Cruz) says:

I have now had a chance to read through MATH OVERBOARD and like it very much; I look forward to Part II. I will recommend it to grad students in biology as a reference book and once I am back on campus … I will hawk it to my colleagues who teach pre-calc and calculus. I think that for the calculus classes it would be wonderful resource book, if the students would use it.


Understanding Mathematics

Monday, December 10th, 2012

Why Should You Try to Understand Math?


Many students in America graduate from high school with a limited understanding of the math that they studied in school. If they go on to college or university, these students often have difficulty with the math courses there, for three reasons:

  1. College math courses are founded on basic school-level math.
  2. A lack of understanding of basic math usually leads to faulty memory, confusion, and errors.
  3. A habit of studying math without trying hard to understand it backfires in college courses.

These considerations lead to the following questions:

  1. What constitutes “understanding math,” anyway?
  2. What is the proper emphasis on understanding basic math?
  3. How can students re-learn their school math so that they do understand it?


What does Understanding Math Mean?

The phrase “understanding math” refers to the understanding of each specific topic in math. Do you understand quadratic equations? Logarithms? Inverse functions? and so on.

Understanding a specific topic implies:

  1. Knowing exactly what the topic (result, theorem, algorithm) is, including knowing the precise definitions of all relevant terms.
  2. Knowing why the result, theorem, or algorithm is true or valid.
  3. Knowing how to use the topic.
  4. Knowing how the topic is related to other topics in math.

Understanding a certain topic in math is more difficult than just memorizing it. Failure to understand a topic may have been the result of laziness on the part of the student, or the result of the teacher’s lack of insistence on understanding. SAT tests don’t usually test understanding directly, so there may be little incentive to learn understanding.

The Distributive Law

Here’s an example, the distributive law:

a (b + c) = ab + ac

This equation is absolutely fundamental in all of mathematics, from algebra to trigonometry and beyond. It’s also one of the most frequent sources of error made by weak students.

Are you familiar with the above equation? What does it say in words? Can you give some examples? What do the letters a, b, and c represent? And especially, why is the equation true?

Does it matter why the equation is true, in general? If you don’t know why it’s true, how would you know whether the similar-looking equation a/(b + c) = a/b + a/c is true? (It isn’t.) It’s quite a feat of memory to remember that the first equation is true while the second is not, without understanding why in either case. But a student who remembers (even if vaguely) why the first equation is true, is also likely to realize that the same logic does not apply to the second equation, which is probably therefore incorrect. This student could also probably prove that the second equation is wrong, by supplying a counterexample.

Knowing why the formulas, theorems and algorithms of basic math are true is an important aspect of learning the subject.  Without such knowledge, math becomes just a jumble of unrelated, mysterious factoids.

Too Much Emphasis on Understanding?

Is it possible to over-emphasize understanding of basic math? Yes, if this means that problem-solving skills are downplayed. Students need lots of practice (“drills”) to perform routine operations quickly and accurately, without consciously thinking of the underlying theory. But the fact that you do subconsciously remember the basic logic is strongly confidence-building. A sound mathematical education combines theory (understanding) and practice in balanced proportions.

Math Overboard!

My recently published book Math Overboard! (Basic Math for Adults) reviews all of school mathematics, from kindergarten to Grade 12, emphasizing both understanding and skills. To my knowledge no other available book accomplishes this outcome. Please visit Math Overboard!