In arithmetic we use and manipulate numbers to create concrete answers to specific numeric questions.

Algebra is the theory of arithmetic. The two main tools used to create this theory are abstraction and generalization. First, we try to describe abstractly, without reference to specific numbers, the manipulations and processes that are used in arithmetic. The goal of such abstraction is to arrive at a general rule which will be applicable to many questions. Secondly, we attempt to generalize the abstractions. The hope here is to arrive at an even higher level mathematical structure which may be useful in describing a wider array of questions.

Algebra is one small part of a much larger discipline called mathematics.

According to The North Central Regional Education Library: “Mathematics is a study of patterns and relationships; a science and a way of thinking; an art, characterized by order and internal consistency; a language, using carefully defined terms and symbols; and a tool.” ^{(1)} Clearly this has very little to do with numbers and computation.

Dr. Robert H. Lewis, Professor of Mathematics at Fordham University correctly observes: “The great misconception about mathematics — and it stifles and thwarts more students than any other single thing — is the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student’s duty is to memorize all this stuff.”^{ (2)}

According to the National Council of Teachers of Mathematics, “One of the primary goals of mathematics education is to enhance students’ ability to reason deductively. The capability to think logically is needed in every discipline, and it is particularly important in mathematics.”^{ (3)}

According to Illana Weintraub, “Algebra is a very unique discipline. It is very abstract. The abstract-ness of algebra causes the brain to think in totally new patterns.”^{ (4)}

Smith, Eggen, and St. Andre voice a basic tenet of all mathematics when they state: “The characteristic thinking of the mathematician, however, is **deductive reasoning**, in which one uses logic to draw conclusions based on statements accepted as true.”^{ (5)}

The purpose of early college level algebra courses is therefore to introduce the student to the use of abstraction, generalization, deductive reasoning, creativity, and critical thinking while exploring the patterns and relationships of a variety of algebraic entities including, but not limited to, equations, inequalities, algebraic fractions, polynomials, and functions.

Please observe the emphasis should be on abstraction, generalization, deductive reasoning, creativity, and critical thinking not on the mind-numbing robotic-like practice of computational activities with the sole purpose of “getting the answer” to insipid and inane textbook exercises.

- http://www.ncrel.org/sdrs/areas/issues/content/cntareas/math/ma3ques1.htm
- http://www.fordham.edu/mathematics/whatmath.html
- http://illuminations.nctm.org/LessonDetail.aspx?ID=L384
- http://www.mathmedia.com/whystudal.html
- Smith, Eggen, St. Andre.
__A Transition to Advanced Mathematics__. Monterey: Brooks/Cole (1986). p.1.

I won’t directly object to what is said here (since in parts I sympathise), but I have always found it difficult to make sense of some of these claims. For example: what precisely is the difference between computation and deductive reasoning? Say we have a system of two linear equations with two unknowns, we rewrite those equations to arrive at other equations (or the solutions). Is this activity dumb computation or deductive reasoning (in which one uses logic to draw conclusions based on statements accepted as true)? Or even simpler: when you plug in values for x in an algebraic expression involving x and you simplify until you get a number. I see no a priori reason for calling this dumb computation as opposed to deductive reasoning? Do you see one? Also: is there any scientific evidence to such lofty and popular claims like “The abstract-ness of algebra causes the brain to think in totally new patterns.”?

Consider the problem of solving a linear equation in one variable.

Computation: The student can simply memorize some steps that lead to what is called the solution.

The disadvantage to this method is that a procedure must be memorized for each equation/problem.

Deductive Reasoning: The student knows certain general procedures which generate equivalent equations. The student uses those procedures to generate a sequence of equivalent equations until a simplest equation is obtained. The solution set is easily determined for a simplest equation.

The advantage of the deductive reasoning approach is that those procedures can be, and are, used for any equation. Keep reading the blog. Later this year I will address this issue in some detail.

Re “The abstract-ness of algebra causes the brain to think in totally new patterns.”

It seems tome that the recent neuroimaging shows that practice retrieving information strengthens those retrieval paths. Thus those abstract concepts are easy to recall when needed in an unfamiliar problem situation.

Thanks for your questions and your interest.

Have your read the book “Make It Stick” available at Amazon.

I hope to put together a reading list on the blog site in the next few days.

Thanks for your answer. I’ll keep reading and look forward to when you address this in more detail. In the mean time I’ll leave some questions, as I don’t fully understand your examples to distinguish between computation vs. deductive reasoning. (Sorry if what follows seems pedantic, I’m just genuinely interested in understanding the distinction.)

Let’s stick with your setting of solving one equation that is linear in one variable for that variable.

Regarding computation: Are you assuming that the steps memorised by this “computational student”

onlyallow him to solve the task when the equation is presented in some very specific standard form? Like say ax+b=0, with a,b explicit numbers in decimal notation and the unknown named x?Or are you assuming the steps he memorised

alwaysallow him to solve the task, even when it is presented in an “arbitrarily” complex way as: Solve (a+b)/4- c^2 R = 27(R/b-2c)/5 – 34.2 for R, with a,b,c unknown constants?Regarding deductive reasoning: so this kind of student knows all the rules that he can apply to transform an equation into an equivalent one, and he was told what it means for an equation to be solved for one variable, but he wasn’t told with which strategy he can arrive from a linear equation to an equivalent one which is solved? In other words, he knows the rules of the game, but he wasn’t told that there is always a winning strategy, and he is expected to figure out a strategy by himself? Is that correct?

Michael;

Most of the students coming into my college algebra class could solve equations of the form ax+b=c if a, b,and c were integers. If any of a, b, or c were irrational numbers, about 80% of the students could not solve the equation. If any of a, b, or c were rational numbers many more (maybe 60%) could solve the equation. However they used a completely different procedure than the procedure use if a, b, and c were all integers. When I explored their approach, it turned out that they had memorized two different procedures and saw no relation between the two.

Students learn methods of generating equivalent equations. They learn that generating a sequence of equivalent equations terminating with a simplest equation is the preferred method for solving equations or systems equations. They learn when the approach always works. They learn how to look for modifications when that procedure fails.

The student also learns that there are many unsolvable equations and how to recognize some of these.