I was at a session at a small math conference last fall, and the presenter was going through their list of things students must do to learn mathematics. When the presenter said that students had to do things according to the instructor’s method (“They have to do it MY way!”), I am sure my jaw fell to the floor. In my experience you have to let students think and experiment in order for them to learn and understand mathematics.

As the years have passed I have tried to allow students time to think in class, especially how they think they should solve the problem.

When I tell them how to do it I rob them of a chance to think mathematically.
When I tell them that this is the only way to do it I am telling them that they no longer have to think.

For students who have struggled with mathematics this approach takes away motivation and inspiration.

Last semester, in an intermediate algebra class, we were reviewing for a final exam and were discussing solving absolute value inequalities. This is a topic where I only teach one method: convert the inequality to a compound linear inequality and solve. As I reminded students of the strategy we used in the past, one student asked if we could solve this type of inequality in the same way that we solve quadratic inequalities and rational inequalities. (Find critical values and test intervals.) I was so proud of that out-of-the-box thinking and praised the student for being able to tie the concepts together. Even better – several students used this approach on their exam.

I observed a colleague last semester and he was having students come to the board to factor polynomials. One of the polynomials to factor was x^2 – 4x. Many students quickly confuse this with the difference of squares x^2 – 4, missing the fact that there is a factor of x that is common to both terms. The student who went to the board surprised me by adding on “+ 0”, rewriting the expression as a trinomial. He then proceeded to factor it as (x + 0)(x – 4), and ended up writing the factored form as x(x – 4). I loved that the student thought of such a unique way to factor the expression instead of just working their way down a checklist.

When you tell students to find the maximum/minimum value of a quadratic function that they must complete the square, you rob them of trying to find the most efficient way to find the vertex for that function.
When you tell students to graph lines by first rewriting the equation in slope-intercept form, you rob them of perhaps finding a more efficient way to graph the line, such as finding intercepts.
When you tell students to solve a system using the elimination method, you rob them of the chance to think whether the elimination or substitution method would be more efficient.

Let your students try to solve new problems before you tell them how to do it. Thinking leads to understanding and growth. Don’t restrict them to only one approach to a particular type of problem. Don’t tell them the technique to use on an exam. Try changing MY way to YOUR way, and watch your students flourish!