Teach 180: Conceptual Equity and Access in Calculus (Day 67)

Algebra is a gateway to higher level math.  If a student can't do algebra, they are bound to have difficulty in higher level mathematics.  Students that typically have problems in calculus have problems, because they have issues with their algebra.  However, we can teach conceptual understanding and allow students access to calculus even though they struggle with getting all the details of the algebra right.

NCTM's Position Statement on Access and Equity in Mathematics Education states "creating, supporting, and sustaining a culture of access and equity require being responsive to students'...knowledge when designing and implementing a mathematics program and assessing its effectiveness."  In addition, we need to be sure we are "acknowledging and addressing factors that contribute to differential outcomes among groups of students".  So, equity and access doesn't just mean addressing cultural, gender or socioeconomic differences, but addressing differences in the levels of mathematical understanding that students bring with them to class.

If learning calculus is built on algebraic manipulation, students who are deficit in algebra skills won't be able to do calculus. So, how can we get those students to build a conceptual understanding of calculus?  Do we allow algebra to be a barrier to building conceptual understanding?

Here is where technology can help to bridge the gap. Consider the function: f(x) = (2x - 5)^1/3 + 1 and identify its extrema, inflection points, intervals where the function is increasing, intervals where the function is decreasing, intervals where the function is concave up and intervals where the function is concave down.  We could do all of the work of calculating the derivative and second derivative by hand and consider the sign of the first derivative and the second derivative.  Lots of algebra.

Let's look at the graph of the first derivative instead.

What does this graph tell us about the function?  Are there any places where the derivative is undefined?  Is the location where the derivative is undefined a maximum, a minimum or neither?  Describe how you know this from the graph of the first derivative.

Now let's look at a graph of the second derivative.

What does this graph tell us about the function?  Do we have any places where there is an inflection point or change in concavity?  Is the function ever concave up?  Is it ever concave down?

Now that students have analyzed the graphs of the first and second derivative.  Have them make a sketch of the graph.

This analysis allows all students to gain conceptual understanding without being held back by their errors in transposing numbers, their errors in simple arithmetic calculations, or their errors in algebra.  I am not proposing that we ignore the weaknesses in student's algebra skills. What I am proposing is one way to build conceptual equity and access in calculus.