One recent semester I permitted my Algebra students to use apps on their cell phones and a program on my computer to solve every numeric or algebraic problem on the tests. My tests contained all the usual questions for an Algebra test and then on another portion of the test I asked questions that addressed the use of abstraction, generalization, deductive reasoning, creativity, and critical thinking.

A popular app was called Photo Math. With this app all one needs do is point the phone camera at an equation, system of equations, trig questions, calculus question, etc. and the answer appears immediately on the cell phone. The point here is that technology in the 21st century removes the need for humans to perform mundane computations.

Matrix computations can be performed by any number of computer programs (apps). The need to manually perform those tedious computations can be avoided. The saved time can be spent using the elementary row operations together with deductive reasoning, creativity, and critical thinking to solve previously unencountered problems. Students will leave the course with an understanding of the relation of elementary row operations and equivalent systems of equations.

If we remove all the training involved with computational skills from an algebra course we will have lots of time 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. These problem-solving skills will be used long after the course has ended and far beyond “math” problems. These problem-solving skills have always been at the heart of mathematics, but in schools we have systematically conflated mathematics and computation.

To accomplish the above goals requires the content of algebra classes be changed.

# Teach Less Computation

Share Cognitive Science Implications for Teaching Mathematics

Picking up the conversation we started on your previous post: it is still unclear to me what difference you see here between computation and deductive thinking. You say that technology removes the need for humans to perform mundane computations. That is true, but I suppose that most things you call deductive reasoning in (say linear) algebra can be done by computers. For example elementary row operations, the Gauss algorithm, determining the space of solutions of a system of linear equations, computing dimensions of subspaces, finding eigen values and vecotrs etc. Could you provide an example of a problem/task that should be taught in algebra, whose solution cannot be automated by a computer?

I think all of the tasks we do in elementary coursess are automated. We should teach students how to use those tools, how to find new tools, how to evaluate tools, etc. Most importantly we should spend more time teaching how to use abstraction, generalization, deductive reasoning, and critical thinking to convert a real-world word problem into a mathematical model which can then be solved with the computer tools.In this case selection of the tool might become an issue for some students.

The questions and issues you bring up push me to rethink what I am writing for some of the future blog posts.

BTW I have started working on the “Books” page on the blog. Take a look. Suggestions and comments are welcome.