(This is a modified version of an article that appeared in Education News on January 28, 2013. )
The idea of whether algebraic thinking can be taught outside of the context of algebra has attracted much attention over the past two decades. Interestingly, the idea was raised as a question and a subject for further research in an article appearing in American Mathematical Society Notices which asks, “Is there evidence that teaching sense making without algebra is more or less effective than teaching the same concepts with algebra?” I sincerely hope someone follows up on this question.
The term “habits of mind” comes up repeatedly in discussions about education — and math education in particular. Teaching algebraic habits of mind outside of and in advance of a proper algebra course has been tried in various incarnations in classrooms across the U.S.
Habits of mind are important and necessary to instill in students. They make sense when the habits taught arise naturally out of the context of the material being learned. Thus, a habit such as “Say in your head what you are doing whenever you are doing math” will have different forms depending on what is being taught. In elementary math it might be “One third of six is two”; in algebra “Combining like terms 3x and 4x gives me 7x”; in geometry “Linear pairs add to 180, therefore 2x + (x +30) = 180”; in calculus “Composite function, chain rule, derivative of outside function times derivative of inside function”.
Similarly, in fifth or sixth grade, students can learn to use the distributive property to multiply 57 x 3 as 3 x (50 + 7). In algebra, that is extended to a more formal expression: a(b + c) = ab + ac.
But what I see being promoted as “habits of mind” in math are all too often the teaching of particular thinking skills without the content to support it. For example, a friend of mine who lives in Spokane directed me to the website of the Spokane school district, where they posted a math problem at a meeting for teachers regarding best practices for teaching math.
The teachers were shown the following problem which was given to fifth graders. They were to discuss the problem and assess what different levels of “understanding” were demonstrated by student answers to the problem:
The problem is more of an IQ test than an exercise in math ability. Where’s the math? The “habit of mind” is apparently to see a pattern and then to represent it mathematically. Another drawback is that very few if any students in fifth grade have learned how to represent equations using algebra.
Presenting problems like the button problem above prior to a pre-algebra or algebra course will likely result in clumsy attempts at solutions that may or may not lead to algebraic thinking. For example, one student answered the problem as 1 x (11 x 3) + 1, which would be taken as evidence by some that the child is learning the “habit” of identifying patterns and expressing them algebraically. Another student answered it as 4 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 34.
Rather than establishing an algebraic habit of mind, such problems may result in bad habits. An unintended habit of mind from such inductive type reasoning is that students learn the habit of jumping to conclusions. This develops the habit of mind in which a person thinks that discovering a pattern is the solution and nothing further needs to be done. Such thinking becomes a problem later when working on more complex problems.
The purveyors of providing students problems that require algebraic solutions outside of algebra courses occasionally justify such techniques by stating that the methods follow the recommendations of Polya’s problem solving techniques. Polya, in his classic book “How to Solve It”, advises students to “work backwards” or “solve a similar and simpler problem”.
But Polya was not addressing students in lower grades who are on the novice end of the novice-expert spectrum of learning. He was addressing students who are well on their way to developing problem solving expertise by virtue of having an extensive problem solving repertoire. For lower grade students, Polya’s advice is not self-executing. Telling younger students to “find a simpler version of the problem” has about the same effect as providing advice on safe bicycle riding by telling a child to “be careful”.
As an example, consider a student who stares blankly at a problem requiring them to calculate how many 2/15 mile intervals there are in a stretch of highway that is 7/10 of a mile long. The teacher can provide the student with a simpler problem such as “How many 2 mile intervals are there in a stretch of highway that is 10 miles long?” The student should readily see this is solved by division: 10 divided by 2. The teacher then asks the student to apply that to the original problem. The student will likely say in a hesitant voice: “Uhh, 7/10 divided by 2/15?”, and the student will be on his way. Note that in this example, the problem is set in the context of what the student has learned — not based on skills or concepts to be learned later.
It would indeed be amazing if we could teach students algebraic thinking skills devoid of the content that allows such thinking to occur. I tend to believe, however, that a proper study of this will show what many have known since the time of Euclid: there is no such royal road.