When I work with students one-on-one, I get a unique window into their thinking. Everyone has a test this week, including several students who are taking the AP Calculus exam. As we are preparing, I’ve noticed the constant push-pull of the conceptual vs. procedural debate, because students need to finish by the end of the hour with me, and go away knowing “how to do it.” I know that giving them conceptual background will help make their learning more durable, but some of them resist going beyond the procedure and don’t welcome the “why does it work that way” explanation.
I’ve found myself with students referring to “that voice in your head” in order to get them to communicate their mathematical thinking, to connect the new knowledge to past related topics, to think about the underlying concepts for each process and help them build structures to support their understanding.
Here’s what I want that voice to be saying:
1. What Does It Look Like?
Knowing what the graphs of various function families look like allows for easy transformations using parameters. This week we transformed graphs of log functions and rational functions:
In addition, students found limits of functions without technology, based on what they knew of the nature of the graphs.
To find this limit
it is helpful to know these graphs:
To find this limit, think about the end behavior and how to determine horizontal asymptotes for rational functions:
2. What Am I Looking For?
A colleague noted recently that math is all about the verbs: solve, simplify, evaluate, and so on. When students pay attention to what they are being asked to do, the process follows easily.
For example, while solving equations, students are looking for the variable, which is located in different places in linear, quadratic, exponential, logarithmic and rational equations. Finding where the variable is now can guide students to a process for solving: inverse operations, factoring and zero product property, converting between exponential and log forms, condensing to a single log, finding a common denominator to clear fractions, etc.
This is an example of an exponential function in a quadratic format; relating prior knowledge of quadratics and “looking for x” enabled the student to solve successfully.
3. What Are The Tools In My Toolbox?
When faced with a problem, think about what tools are available. With rational expressions and equations, students begin by factoring, and then often (but not always) find a common denominator.
One of my students could easily add these fractions with an LCD:
But had trouble solving this equation:
She had learned one strategy for the expression, and then there was a “new” strategy for the equation that involved multiplying through by the LCD to “clear fractions”. She couldn’t keep track of which factors remained and was suceptible to errors.
Instead, we built on the strategy of creating common denominator fractions; once all the fractions have the same denominators, she can work with only the numerators and solve successfully:
We also used this strategy to simplify a complex fraction; we created a single fraction in both the numerator and denominator, then remembered that a fraction means DIVIDE:
The AP Calculus students also think about their toolbox when faced with an integral problem: what are the integration strategies they can use as tools?
- Do I know an antiderivative? (Can I simplify algebraically to make one)?
- Is there a known geometric area I can find?
- Is part of the integrand the derivative of another part? (U-substitution)
- Can a trig relationship help me rewrite the integrand?
- Does the integrand contain a product? (Integration by parts)
- Is this a definite integral on the calculator section? Use the calculator!
4. Where Are The Trouble Spots?
When finding the domain of a function, focus on the numerical values that “cause trouble.” Where should we look for trouble in these?
Finding limits and derivatives of piecewise functions also puts students on the hunt for trouble, and certain integrals need special treatment due to discontinuities:
5. What Should I Write Down?
Write enough to show your mathematical thinking to a teacher/reader who doesn’t know you. Write enough to be clear and get it right. No bonus for doing it in your head on multiple-choice, and there is definitely a penalty for doing too much in your head and getting it wrong. And on free-response questions on the AP (and most questions on teacher-created tests), you need to show work that supports your conclusion.
6. Does My Answer Make Sense?
Even if calculators aren’t available, students can estimate square roots, logs, and other results. For example,
In word problem situations, does the answer make sense? If Amy can do the job in 4 hours and Josh can do it in 6 hours, together they should take less time than either of them working alone. And don’t forget appropriate units if a problem is situated in a real context.
7. How Are These The Same/Different?
Analyzing the small differences between examples helps students home in on important features.
What intercepts and asymptotes will these functions have in common?
(Calculus) What are the different requirements and results of the Intermediate Value Theorem and the Mean Value Theorem? What is the difference between average rate of change and average value of a function?
8. How Do I Feel About This?
Finally, I ask my students how they are feeling about the material: Are you finding this unit easy or hard? What parts are more difficult for you? If you find this to be challenging, you need to put on your thinking cap. Saying “I can’t do it” gets in the way of your understanding; instead, say “I can’t do it YET, I’m learning” and focus on the MANY things that you do know. Don’t overthink the easy things or overlook the tough details. Being confident is an important ingredient for your success.
You’ve got this.
NOTES & RESOURCES:
Two Geogebra applets for transformation of functions are found here (multiple parent functions) and here (rational functions).
More on “easy” and “hard” labels and their impact on students in these blog posts: “Things Not To Say” and “The Little Phrase That Causes Big Problems”.