When teaching a new topic, we have to resist the temptation to just tell our students how to solve a problem and let them think for themselves. This helps our students to develop their intuition and their problem solving skills.We also have to keep in mind that there is more than one way to solve a problem.

### Example 1 – Intro Stats

The example that jumps to mind is from a statistics class when students need to find P(a < z < b)  using the normal distribution. In class we have already learned how to find P(z < a) using the z-table and P(z > a) using the complement of the area from the z-table. I draw a bell curve and ask my students to think about how they can find this shaded “middle” area. For me, the most efficient approach is to find P(z < b) and then subtract P(z < a).

When I ask my students for their ideas, someone will always say to

1. Find P(z < a) using the z-table.
2. Find P(z > b) using the complement of the value found in the z-table.
3. Add these two areas together, and subtract their sum from 1.

To me, this approach is not as efficient as the one I laid out, but I love it when my students suggest it. First, it shows me that my students have considered the work we have done so far and applied it to this new problem. Second, it allows me to explain why the other approach is more efficient.

### Example 2 – Series

Last night my son (HS Sophomore) was working on a midterm review. One problem asked him to find the sum 3 + 1/3 + 1/9 + 1/27 + … This sum is not quite in the form of a geometric series, but it’s close. The approach that jumped out to me was to rewrite 3 as 2+1, resulting in 2 + (1 + 1/3 + 1/9 + 1/27 + …). I could then use a formula to find the sum in parentheses and add 2 to this result.

My son had a different idea – why not add in a 1 after the 3 to make the series a geometric series, find the sum, and then subtract 1? What a great idea! There are many times in math where we use the add & subtract a number trick, it was nice to see him come up with that on his own.

I then asked him if he could find a third way of finding the sum. After a little thought, he realized that 1/3 + 1/9 + 1/27 + … was itself a geometric series and that he could find that sum and add 3 to the result.

(Side note: The formula for the limit of a geometric series with initial term a1 and common ratio r is given by r=a1/(1-r). The sum for the given problem is 3.5.)

I think my son gained a lot more than the solution to the one particular problem – he developed mathematical intuition, increased his understanding about geometric series, and further developed his critical thinking skills. Win – Win – Win!

### Summary

If we tell our students to simply do what we do, what will they do when they run into a problem they have never seen before and we are not there to tell them what to do? Encouraging students to think, and to develop strategies on their own, prepares them to be a lifelong problem solver. Isn’t that what we want? Let your students think.

-George

I am a math instructor at College of the Sequoias in Visalia, CA.  If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

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