All I want for Christmas is... maybe not your two front teeth. Perhaps a slow morning to enjoy your cup of coffee? Or a new pack of Papermate Flair pens? This year we channeled our wishlist energy into pinning down what big ideas we reeeaaaalllllly want our students to grasp from the courses leading up to AP Calc. Here are the top 5 things we wish all students knew before they entered our class. We’ll be distributing these at the next department meeting ;)
1. Slope as a Rate of Change
When I say “slope”, you say _____. How would your students complete this blank? If they’re anything like ours they would say “rise over run.” This is the most common conception of slope in beginning Algebra courses and while it does present a strategy for calculating the slope of a line on a coordinate plane, it ignores an even bigger truth about slope. Slope is a rate of change, a measure of how fast a function is increasing or decreasing. Long before students calculate the slope of a curve with a derivative, they can learn to interpret slope as an indicator of how fast something is changing. They can compare the slope at two points and interpret what this means about the quantity being graphed. They can interpret positive and negative slopes. They can learn about functions in terms of their rate of change. Linear relationships are a huge focus of an Algebra 1 course, and a solid understanding of a constant rate of change would set students up well for later coursework. Without getting too formal, even students in middle school and early high school can distinguish between linear and nonlinear rates of change from a graph. They can estimate the slope of a curve or compare the relative steepness of two curves or at two points of a single curve. Instead of just memorizing a formula for slope, students would be well served by understanding slope as a rate and be able to attach proper units to a slope in a contextual problem. Students should be able to explain why a slope is represented by a change in the dependent variable over a change in the independent variable.
2. Interpreting Answers in Context
Interpreting derivative and definite integral values is a big component of AP Calculus. We teach our students “the big 4” to help them remember what to include in a good interpretation. Many of these components apply not just to derivatives and integrals but all interpretations! A greater focus on interpreting a value in context in earlier courses (like an x-intercept, a maximum, a horizontal asymptote, or even a point on a graph) would greatly help students as they transition into Calculus. We want students making sense of information, not just reciting it. The ability to calculate a certain value is important, but this skill needs to be coupled with an understanding of what this value means! Why do we care about this answer? What does it tell us about the context? Understanding the meaning of a value is also an invaluable tool for determining if the answer makes sense and the ability to catch mistakes. A focus on units, including context, and using precise language (rather than vague phrases like “it”, “the graph”, or “the function”) will help students better understand the meaning of their answer and prepare them for future coursework.
3. Log and Exponent Properties
All Calc teachers know that when students struggle in our class, they are most often struggling with algebra, not calculus. Calculus introduces a few big conceptual ideas but also requires a great flexibility and fluency with algebraic manipulation to be able to apply those ideas to problems. Using derivative shortcuts is easy, if students can easily and quickly rewrite an expression with a root as a power. Extensive simplification of expressions is often not needed in Calc class (because all equivalent answers are accepted), but the ability to recognize equivalent expressions is paramount. Additionally, the ability to evaluate simple log and trig expressions can make a problem much more manageable because expressions can be simplified and allow students to determine if an answer is reasonable. If students can identify that ln (0.8) is a negative value or that 20^(⅓) is between 2 and 3, they will be able to quickly tell if a given solution is in the ballpark of reasonable answers. Additionally, knowing that ln (x^2) is equivalent to 2 ln x can make a multi-step derivative problem into a one-step problem.
We do not recommend spending days and days drilling exponent and log properties into students’ heads. Instead, spend time building up rationale for why those properties are true, and help students build a toolkit for manipulating a wide variety of algebraic expressions, with an emphasis on equivalence. We also suggest using spiral review to keep these fundamental ideas fresh in students’ minds. For example, in a trig unit, why not ask students to evaluate sin (ln 1)? Or in a unit on polynomials, why not present the equation as a product of power functions, giving students the opportunity to review their exponent rules when determining the degree of the polynomial?
4. Multiple Representations
We know students are adept with solving equations, but have they ever been asked to solve that equation as represented on a graph? Or to estimate a solution for that equation from a table of points? Using and recognizing multiple representations of functions is so important in the AP Calculus world that it represents one of our Mathematical Practices! But if students never see problems like this throughout their math experience, we are setting them up for struggle when they finally do encounter a function given only by a graph or table of values. Even more important, they miss important connections between concepts, like that the solution they find has a visual meaning on a graph. How cool is that?
5. Math in the Wild--Applications that Interest Students!
Finally, we know that students absorb concepts so much better when they see the relevance or real life use for that skill. Asking when two trains will meet or how many oranges you end up with may address the concept but how much more interesting would it be if those trains were football players running a play or if the oranges were apps on their phones? We believe that giving students authentic opportunities to see how math plays out in the "real world" will keep them not only engaged, but willing to continue taking math courses right up to AP Calculus, one of the most applicable math subjects there is! This can be as simple as giving a task that requires decision making and genuine problem solving (no clearly defined path to a solution), or can extend all the way to enacting a project based unit.
So, what’s on YOUR Christmas wish list? Share your ideas in the comments below!