Algebra: What's the Point?

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The NY Times ran an op-ed piece by Andrew Hacker titled “Is Algebra Necessary?” Since I had such a memorable high school experience in Ms. Collins’ Algebra II class (see last post), I’ll weigh into this fray. 

Hacker makes at least two main points: (1) The current algebra curriculum has little practical relevance for students, except for the 5% entering the work-force in careers like producing animated movies, developing investment strategies or setting airline ticket prices. (2) Proficiency in algebra is a major stumbling block to high school graduation and success in college.

Hacker’s proposed revision of the math curriculum would involve better teaching of practical math for our personal and professional lives.  Instead of pure algebra and calculus, students would be taught the history of math and how math is applied to professions.  The standard math achievement (for the 95% who will never directly apply algebra) would be mastery of basic math needed for personal finance, measuring, estimation, and “citizen statistics” (understanding the reliability of numbers and how we use numbers to describe important issues).

Many who oppose Hacker’s view argue that algebra plays an important role in developing reasoning and problem solving skills. They suggest it is a form of mental training. Like an athlete who lifts weights to improve physical performance, students study advanced forms of intellectual disciplines to improve thinking and reasoning. I’m not sure about the accuracy of this analogy.  The brain is not a muscle. But, if we’re going to have schools, we should aim to develop advanced thinking skills.  We don’t just need advanced thinking skills for work, we need them to solve complex problems at home and in our personal lives.  I doubt we can develop those skills without putting students through a rigorous math curriculum.

The debate should not be about whether or not to teach algebra, or to whom.  We can teach math to young children in ways that lead to success in algebra. Children manipulate objects to solve problems. They sort, group, count and rearrange things when adding and subtracting and learning geometry. Algebra is the use of symbols to represent unknowns in math problems. In the equation x + 2 = 7,  “x” represents the unknown. Algebraic reasoning allows us to find the numeric value of “x”. Algebra is thinking symbolically about the physical realities of the universe.  Sal Kahn provides a more thorough and inspiring explanation in his introduction to algebra.

Here is a relevant story from my recent observation of kindergarten mathematics:

A teacher began a math unit with a pretest that required her 20 kindergarteners to solve two types of problems: (1) If I have 3 apples and you have 4 apples, how many apples do we have together? (2) If together we have 5 apples, and you have 3 apples, how many apples do I have?  She read the problems to the students and gave them paper apples to help work the problems. Five students solved the first problem, but only 1 of 20 solved the second problem.

After several days of practicing similar problems, always using objects to arrange, group, add and subtract, the teacher gave another test. Eighteen of 19 students solved the first problem and 11 of the 19 students solved the second problem.

The teacher was disappointed in the results, but I was surprised by how many students solved both problems.  Some of the students were only 5 years old. Previously, kindergartener math instruction taught children only to recognize numbers and count objects.  Addition and subtraction were first grade math. In 2012, good kindergarten teachers show children how to add and subtract by thinking “algebraically”.  The children do it surprisingly well if the teacher demonstrates the process and provides plenty of practice.

We should rethink the context and method of teaching algebra.  Without the manipulation of objects and without the application of algebra to the real world, many students will miss its purpose - to think logically about how the universe works. 

Algebra is not the only tool for abstract reasoning. Grammar is an abstraction of language.  “Subject + predicate = sentence” is an abstraction of a simple sentence structure. Obviously, we can learn and use language rather well without knowing its grammatical structures and principles; but, learning grammar and using it to fine-tune our use language is relatively easy as long students are capable of thinking abstractly about language. Failing grades in grammar results from the same flawed teaching methods that makes algebra difficult to learn – instruction that relies too much on the students’ memory of “rules” and not enough on students’ understanding of the relationships among things.

The highest level of schooling, whether it occurs in high school, college, university or graduate school, should produce learners who can think and reason abstractly.  Algebra and grammar are tools of abstract reasoning and we’ve got to teach those disciplines at the times and in the manner that nurture thinking, reasoning, and abstracting about the natural and human world.  Teaching algebra in the context of real objects and real-world problems must be added to drilling students in proper mathematical procedures.  Teaching grammar in the context of meaningful and interesting language is the only effective way to teach grammar.  But we cannot set a precise deadline by which all students show mastery of these abstract disciplines.  We must allow for the natural variations in human ability. Believing that all students can apply abstract reasoning (without the use of real objects or meaningful language) by a particular age is the flawed belief of people who never mastered algebra or grammar.