# Developing Math “Spidey Sense”

A recent post in the math blog Divisible by 3 made me reflect upon the role of importance of estimates and initial gut feelings in math class.  In the blog post, Mr. Stadel shared some estimation anecdotes from his middle school classroom, and a great Ignite video on quality instruction from Steve Leinwand (do yourself a favor and watch Steve’s 5 minute rant on instruction….you’ll be glad you did!).  How often do we challenge our students to communicate initial guesses or predictions of what will happen next?

Recently, I tutored a young man named Kevin in AP Statistics.  Like many AP Stats students, Kevin was quite comfortable with using his calculator, to the point where I often grabbed the calculator from the table as he was reading a problem.  Consider the following problem:

In a recent survey, adults were randomly selected to provide their opinion on presidential campaign spending.  200 adults were randomly selected from Pennsylvania, and 200 were randomly selected in California.  In PA, 130 of the adults supported campaign spending limits, while only 122 in California supported limits.  Do the data show a significant difference in the opinions of all adults in the two states?

Before diving into the computations, I found it helpful to ask Kevin what his “spidey sense” told him about the problem?  Does the result “feel” significant?  Could he predict the p-value?  Does the student have a feel for what the numbers might bear out after we crunch them?  These “spidey sense” discussions were fruitful, in that the conversation would focus on important concepts like the effect of sample size on sampling distributions.  Have your students make initial predictions before performing any computations, and see how understanding and misconceptions are revealed.

How can we develop “spidey sense” in other high school courses?  Consider the following scenario, which is often used to introduce systems of equations:

• Dave and Julie are each saving money in a bank account for a new television for their room.  Dave started his savings with \$80, and adds \$10 each week to the account.  Julie started with no savings, but adds \$15 to the account each week.

I have used problems similar to this one, and often the lesson requires students to create a data table for each week, graph their data, and answer a series of questions which lead to Julie having more money than Dave.  Eventually, algebraic expressions are introduced and we can solve systems!  Ta-dah!

But since this is a money problem, there’s a great opportunity here to communicate and develop initial opinions.  Ask your class, “Who will be able to buy the TV first?”, “What does your spidey-sense tell you?” and see how many of the concepts develop organically.  Or, if you want to compare the two students, ask which of the following is true?

1. Dave will always have more money than Julie.
2. Julie’s savings will pass Dave’s soon.
3. Julie’s savings will eventually pass Dave’s, but it will take a while.

Let the discussions drive the instruction.  Let students tap into their built-in intuitions and share ideas.  And, as Steve Leinwand exclaims, “value and celebrate alternative approaches”.

How do you challenge students to tap into their math-spidey-sense?