Archive for the 'Math Education' Category

The Disconnect Between School and College Math

Wednesday, January 9th, 2013

Why are many college students unprepared for math?

This article is about what I call the “disconnect” between mathematics instruction in the schools and in our colleges and universities. In a nutshell, the teaching of math in the schools generally downplays theoretical aspects (such as proofs), whereas university math courses do stress theory. According to Dr. Kim Maltman of York University, “…we have found that our first-year science majors [are] 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….” (see Math Remediation…). This statement probably applies to almost every post-secondary institution in North America.

The leap from school math without theory to proof-based college math courses is usually abrupt, and takes many students by surprise. As one student said to me, “I can’t hope to understand math. My only chance of passing your course is to memorize the problem solving techniques needed on the exam.” She didn’t pass.

The best students eventually get through the disconnect, and learn to master theory. This works well for students in math, physics, and engineering, but not for many of those going into biology, economics, or psychology. Many students find their educational hopes dashed because of their failure in basic math courses.

Knowing Theory is Necessary for Understanding Mathematics

The trouble with studying mathematics without learning basic theory is that the student then fails to understand the math that he or she has studied. He or she may be able to solve routine exam problems, but even that ability begins to fade as memory falters later on. Such rote “learning” is all but useless in more advanced courses, and in later life generally. Among people I meet who are not scientists or engineers, well over half tell me that they never really understood math in school, and have now forgotten most of it. Why is so much time and effort spent teaching math in school if this is the result?

Teaching – and learning – mathematics with a secure understanding of each topic is much more difficult than just memorizing mysterious formulas and methods for solving routine problems. Understanding math as you learn it has the following important consequences:

  1. The effort of understanding the logic helps immensely in imprinting a given topic permanently in your memory.
  2. Understanding a given topic helps greatly later when learning more advanced topics.
  3. Understanding each topic fortifies your confidence in using that topic later to solve problems, both routine and novel.
  4. Learning basic theory (proofs) in mathematics places the learner in a long historical-cultural stream extending from the ancient mathematicians down to the present day. Our current knowledge of mathematics is entirely dependent on the existence of valid, tested proofs for every known result.

The lazy approach, whether in learning or teaching math, is to ignore the theoretical, logical foundation and simply concentrate on memorizing techniques for solving routine problems. This approach may get the student through exams, but the long-term consequence will be an increasing level of confusion and uncertainty about math.

Of course, even if you do understand things, you still need to memorize many techniques and formulas. You also need plenty of practice in problem solving, in order to firm up both memory and understanding.

But What Can be Done, Today?

Perhaps some day in the distant future the teaching of math in the schools will emphasize memorization and comprehension equally. Meanwhile what can a student do to re-learn school math with a proper level of understanding, as required for college studies? Read through the school textbooks again? – not. Search the web? Hire a private tutor? Take a remedial course?

A better solution – if you’re willing to work hard on your own – may be to obtain a copy of Math Overboard! (Basic Math for Adults). This recently published book reviews all of school mathematics, from kindergarten to Grade 12, with the aim of instilling detailed understanding of every topic. Nothing is left out; everything is included. Part 1 is now selling from the website for $24.00 net. Part 2 (expected in June 2013) completes the book, and will sell for about $20.00.

The Math Education Crisis

Friday, November 23rd, 2012

 Mathematical Mis-education

Mathematicians and other scientists are upset about the current state of math education in American schools. Here are a few nuggets taken from the website

“We support a balanced approach between understanding and skills. Unfortunately, in the shift towards ensuring that children understand math concepts, which we support, several important elements of mathematics have been neglected, or completely eliminated, from curricula and math classrooms.”

“The most recent version of the WNCP (Western and Northern Canadian Protocol) math curriculum omits all standard algorithms for addition, subtraction, multiplication and division.”

“Martin Scharlemann, while chairman of the Department of Mathematics at the University of California at Santa Barbara, wrote an open letter deeply critical of the K-6 curriculum MathLand, identified as “promising” by the U. S. Department of Education. In his letter, Professor Scharlemann explains that the standard multiplication algorithm for numbers is not explained in MathLand. Specifically he states, “Astonishing but true — MathLand does not even mention to its students the standard method of doing multiplication.” ”

“Post-secondary instructors are also frustrated by the weak math skills of many new graduates and are troubled by the fact that many math teachers are not receiving adequate training in math before entering classrooms in Canada.”

It is inconceivable to me that anyone would think that you can understand Arithmetic, let alone Algebra, without mastering the basic algorithms for addition, multiplication, subtraction and division. Yes, you can buy a $10 calculator that will “find the answer” to any given numerical calculation, but this does NOT imply that learning these algorithms is not necessary.

The 7-11 checkout clerk tells you your purchases add up to $8.63, so you hand him a $10 bill and he gives you $1.24 change. When you get outside you ask yourself how come a coke ($1.49) and some chips ($2.49) can add up to over $8.00. So you reach for your calculator – oops, it’s at home. So you return to the store to complain, but the clerk explains that there was tax of 87c. Now what? Shrug it off? Too bad you never learned how to add or multiply, and after that how to do quick approximate sums in your head.

Then you go to a political rally, where the candidate tells you that his opponent’s tax policies will cost taxpayers $750 billion. Is this realistic? And is it dollars per year, or over a 4-year period? And how much is that per average taxpayer? Would you dare take out your calculator there among all the screaming audience? And how do you enter the number 750 billion into the calculator, anyway?

Math Overboard!

If that was your experience in school, you might want to re-learn your basic math from scratch. I would like to recommend my recently published book Math Overboard! (Basic Math for Adults). Covering all of school math, from kindergarten to Grade 12, Math Overboard! stresses the importance of understanding math in detail, as you learn it, or in this case, re-learn it. Frequent Problems test your understanding as well as your skills. It’s not an easy book, but it’s your best hope to really learn what math is all about.

For further information, please visit Math Overboard!