by Barry Garelick
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 has recently been raised as a question and a subject for further research in a recent 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 this request is followed up on.
The term "habits of mind" comes up repeatedly in discussions about education — and math education in particular. The idea that teaching the "habits of mind" that make up algebraic thinking in advance of learning algebra has attracted its share of followers. Teaching algebraic habits of mind 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:
Not only have students in fifth grade not yet learned how to represent equations using algebra, 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.
Such problems are reliant on intuition — i.e., the student must be able to recognize a mathematical pattern — and ignore the deductive nature of mathematics. An unintended habit of mind from such inductive type reasoning is that students learn the habit of inductively jumping to conclusions. This develops a habit of mind in which once a person thinks they have the pattern, then there is nothing further to be done. Such thinking becomes a problem later when working on more complex problems.
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. Since the students do not have the experience or mathematical maturity to express mathematical ideas algebraically, algebraic thinking is not inherent at such a stage.
Specifically, 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. It is not unusual, for example, to see students in algebra classes making charts for problems similar to the one above, even though they may be working on identifying linear relationships, and making connections to algebraic equations. By making algebraic habits of mind part of the 5th-grade curriculum in advance of any algebra, students are being told "You are now doing algebra." By the time they get to an actual algebra class, they revert back to their 5th grade understanding of what algebra is.
In addition, the above type of problem (no matter when it is given) is better presented so as to allow deductive rather than inductive reasoning to occur.
"Gita makes a sequence of patterns with her grandmother's buttons. For each pattern she uses one black button and several white buttons as follows: For the first pattern she takes 1 black button and places 1 white button on three sides of the black button as shown. For the second pattern she places 2 white buttons on each of three sides of one black button; for the third 3 white buttons, and continues this pattern. Write an expression that tells how many buttons will be in the nth pattern."
The purveyors of providing students problems that require algebraic solutions outside of algebra courses sometimes 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; he was addressing students who are well on their way to developing problem solving expertise by virtue of having an extensive problem solving repertoire — something that students in lower grades lack. For lower grade students, Polya's advice is not self-executing and has about the same effect as providing advice on safe bicycle riding by telling a child to "be careful". For younger students to find simpler problems, they must receive explicit guidance from a teacher.
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.
Giving students problems to solve for which they have little or no prior knowledge or mastery of algebraic skills is not likely to develop the habit of mind of algebraic thinking. But the purveyors of this practice believe that continually exposing children to unfamiliar and confusing problems will result in a problem-solving "schema" and that students are being trained to adapt in this way. In my opinion, it is the wrong assumption. A more accurate assumption is that after the necessary math is learned, one is equipped with the prerequisites to solve problems that may be unfamiliar but which rely on what has been learned and mastered. 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.