# Great Thinking!

One of the bonuses of working with students one-on-one is that I can get a window into their mathematical thinking by asking questions and having them “narrate” their work as they proceed.  Several of my recent conversations demonstrate unique and flexible thinking that helped my students work through computations effectively.

How might we enable this type of visible thinking in our classrooms?  I tried these prompts:

• Tell me what you were thinking.
• What would you do to compute this if you didn’t have a calculator?
• What math operation did you do first?
• Can you explain the steps you did in your mind?

What follows are some observations of “great thinking” by my students, who used flexible number sense for success.

A. Using number sense for easier multiplying or dividing

B. Fantastic Fraction Flexibility

Here, noticing the multiplier is easier than solving the proportion directly:

In this question about slope, the student noticed that the first denominator must be 2 if the fractions are to be equal.

It is hard to find perfect square factors of large numbers; this strategy employs factors and the student never found the original product.

Here, the student used an alternative to the traditional process of rationalizing the denominator.

D. Factors and Multiples

Rather than set up proportions, this student noticed multiples

I’ve avoided the term “cancel” for some time, so here we talked about “dividing away common factors”.

E. Other interesting choices for computation and order of operations

The formula for area of a trapezoid can be thought of as “height * average of the bases”.  This student also applied binomial multiplication to compute without a calculator.

Here the typical order of operations would do the distributive property first.  Instead, the student saved a step.

Also on my mind as I write this are three things I have read this week.  First, James Tanton addresses a multiplication mistake made by an education official in this blog post, forgetting what 7 x 8 is.  Tanton suggests that the “best answer” in this case would have been “This is a tricky one. But I do have in my head that 7×7 is 49, so 7×8 is seven more than this: 56” because it highlights the thought process and downplays the memorization aspect of number facts.  His discussion is thought provoking (and he applies the same idea to trig identities too!).

The second item I was contemplating was in Sarah Carter’s Monday Must Reads post; she and I both liked David Sladkey’s “No eraser allowed” technique of insisting students leave their mathematical thinking available for the teacher to see.

The third concept informing my thinking is the book study I’ve been doing with colleagues from Teachers Teaching with Technology (#T3Learns) on “Visible Learning for Mathematics” (2017) by Hattie et al.  Hattie describes a spiral relationship between surface learning, deep learning, and transfer learning that enables students to achieve.  He notes that surface learning is NOT shallow learning (p 29) but is instead “made up of both conceptual exploration and learning vocabulary and procedural skills that give structure to ideas” (p 104) that “sets the necessary foundation for the deepening knowledge” (P 131) on the path to understanding. Techniques such as number talks, guided questioning, worked examples (including accurate work and work with mistakes), and highlighting metacognitive strategies all enhance the process of surface learning for students, according to Hattie.

I constantly remind students to “Show Your Mathematical Thinking”—which is the updated version of “Show Your Work.” By asking them to “narrate” their thinking, I am focusing on my students’ surface learning to build their foundation of skills and tools on their learning journey.

# That Voice In Your Head

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?

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”.

The students with whom I work are heading into midterm exams.  For some of them, the mathematics concepts and procedures come easily, and others have to work harder to feel confident in their understanding.  All of them can benefit from diligent preparation, and although a few still resist, here is the advice I am giving:

1. Do the whole review packet and check your work against the solution key.  It isn’t optional. Show your mathematical thinking so you can analyze your process.
1. Review past tests and quizzes, looking at both the questions you got wrong and the correct ones.  Even if you got it right in earlier in the semester, make sure you remember how to do it now.  Re-do the questions, don’t just “read it over”.
1. The more practice and review you can do BEFORE you get to the review session, the more productive the time will be.  Mark the ones you get wrong and/or don’t know how to do so you have a list of questions ready.  Don’t wait til the last minute to start studying.
1. Keep track of important formulas, graphs, examples & concepts on a self-created study guide.  Do it as you go through the review packet.  If you need to memorize something, write it out each time you use it until you know it. If you need to be able to solve something without a calculator, practice it that way.
1. Make use of other resources: if your teacher has a website, go back to unit review sheets and solution guides from the semester. Check another teacher’s website from your school if your teacher doesn’t have one.  Work with a classmate, but don’t merely divide up the work: make sure you both can complete the problems.  Utilize Khan Academy, YouTube & Google.
1. Cumulative exams are challenging; scores are often somewhat lower than your typical quiz/test scores have been.  However, remember to be confident in the things you know—yes it is a big job to prepare, but you can do it!
1.  Take care of yourself physically over these weeks:
• eat right and choose healthy snacks (think protein/fiber not sugar)
• stay hydrated (more water less soda)
• stay active because it helps relieve stress and is good for your brain: go to sports practice, work out or run or shoot hoops, or even just take the dog for a brisk walk during a study break
• wash your hands frequently, etc. so you don’t get the bugs that will inevitably be going around
• get enough sleep: it is far more important for your brainpower to sleep an extra hour than cram an extra hour.

Good luck!