“They’ll Need It for High School” (Part 4)

In Part 1, I catalogued and critiqued common uses of “They’ll need it for high school.” Samuel Otten’s take on “When am I ever going to use this?” was an obvious influence. “Poor Pedagogy Preparation” was one of my categories: “I want them to get used to it” as an indefensible defence for the mad minute. Since that post, I’ve wondered about putting a positive spin on “I want them to get used to it.”

As a K-12 Numeracy Helping Teacher, I have the opportunity to teach in elementary and secondary classrooms alike. Often, I’m struck by how pedagogical similarities overwhelm any differences. Malcolm Swan’s ordering decimals lesson — a current favourite of mine — and my sinusoidal sort illustrate this.

In Grade 5, students are asked to put the decimal cards in order of size, smallest to largest.

Card Set A - Decimals

The most common response is 0.4, 0.8, 0.04, 0.25, 0.75, 0.125, 0.375. This can be explained by how students compare whole numbers. Misconceptions are revealed, but not corrected. (The rest of the activity will take care of that.)

In Grade 12, students are asked to put the equation cards in piles.

Sinusoidal Sort - Equations

The most common response is to make two piles: sine and cosine. This can be explained by how students were introduced to y = sin x and y = cos x. This isn’t a misconception; there is no “right” sort. But it is unsophisticated. And it goes unchallenged. (The rest of the activity will take care of that.)

In Grade 5, students match hundred grids and number lines to their decimals. They explain how they know that the cards make a set, building connections between decimals and their understanding of fractions and place value. They argue. When there’s consensus, students are again asked to put the cards in order. I note their strategies that emerge (benchmarks, place value, equivalent decimals) for a class discussion.

In Grade 12, students match graphs and characteristics to their equations. They make conjectures. They explain how they know that the cards make a set, building connections between transformations of trigonometric functions and their understanding of transformations of other functions. When there’s consensus, students are again asked to put the cards in piles. I note their sorting rules that emerge (amplitude, period, phase shift, vertical displacement, range, maximum/minimum values) for a class discussion.

Elementary and secondary lessons need not be as closely aligned as above. It’s not about matching card matching activities. Or parallel “Which one doesn’t belong?” prompts. Or three-act math tasks for K-5. More generally, exploring and discussing ideas, working collaboratively in pairs/small groups, problem solving (and posing!), … you could make a case for students needing these experiences in elementary school because… wait for it… they’ll need it for high school. We want them to get used to it. I hope.

A nagging thought… I’m not convinced that pedagogy preparation is ever a satisfactory response to “So, what will they need?” Even if that pedagogy is positive. If we’re going to follow that pathway, then this question demands an answer not in terms of teaching methods but in terms of the mathematical habits of mind (or processes or practices or competencies) that these teaching methods aim to promote. (See my “Affective Domain” concerns in Part 1.)

That’s not to say that I wasted my time with this post. For me, it’s always helpful to think about and present examples of secondary mathematics education as something other than passive. Maybe that — reframing those pedagogical “TNIFHS” conversations with colleagues into something more promising — is the real value here.

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Sinusoidal Sort

On Monday, I was invited to Sandra Crawford’s Pre-Calculus 12 classes to try out an activity we created together. Thanks, Sandra!

Sandra’s students were familiar with how transformations of functions affect graphs and their related equations. They’ve stretched & shrunk (vertically & horizontally), flipped (in the x-axis & in the y-axis), & slid (up, down, left, & right) linear (& piecewise linear), quadratic, absolute value, reciprocal, & radical functions. These were topics in prior units. In this unit, students were previously introduced to radian measure, the unit circle, the six trig ratios, & the functions y = sin x, y = cos x, & y = tan x. Next up: determining how varying the values of a, b, c, & d affect the graphs of y = a sin b(x – c) + d & y = a cos b(x – c) + d.

Such was the case when I last taught trig functions (in Principles of Math 12). Back then, my approach was to provide clear and concise explanations, connecting these transformations to those transformations (or, better, transformations of these to transformations of those). But was this necessary? Shouldn’t students be able to make this connection? On. Their. Own.

In small groups, students were handed a set of equation cards to sort and were asked to explain their sorting rule. We designed the equations so that there were plenty of similarities and differences in terms of whether or not there were leading coefficients, coefficients of x, brackets, etc., as well as in terms of the values of a, b, c, & d themselves. After all that, most groups just sorted the equations into sine and cosine functions — to be expected, I guess, given the focus of the prior lesson.

sort1

Next, students were handed graph cards and were asked to match each to the corresponding equation card. We encouraged students to make predictions, then test these predictions using technology. Interestingly, few reached for their graphing calculators or phones. We asked students if, having seen the equations and their graphs together, they wanted to re-sort.

sort2

This process was repeated with characteristic cards. Note: The terms amplitude and period were introduced the lesson before; phase shift and vertical displacement were not. Hence, horizontal translational and vertical translation at this stage of the lesson.

For the most part, students were communicating and reasoning mathematically, making connections, and problem solving. They were engaged with mathematics. A minority probably would have preferred to be engaged with taking notes.

Groups shared their sorts the following day. In the end, the functions were sorted in a variety of ways, which allowed Sandra to highlight each transformation.

sort3

A few groups struggled with matching all of the cards. Therefore, I reduced the number of functions. If finished, some students could be given two additional functions. Each of these is actually a phase shift of one of the initial eight (e.g., y = cos x + 2 ↔ y = sin (x + 90°) + 2). I wonder what they’d do with that.

Sinusoidal Sort (doc)
Sinusoidal Sort (pdf)

(Note: I’ve triple-checked these. Still, no guarantees.)

Revisiting GeoGebra

Four years ago I learned about GeoGebra and made some applets to be used in my classroom. I started by creating applets that demonstrated the effect changing slider values had on the graphs of trigonometric functions. I’d change a value and then ask the class to describe what happened to the graph. These constructions made excellent demonstrations. But that was the problem. The spectator experience was improved, but students remained spectators. (SMART Board fans take note: having one student at a time come to the front of the class does not change this.)

I also posted these applets on my class website. I thought students would try them at home to reinforce learning and check for understanding. They didn’t.

I wanted to move more towards having students themselves do the investigating. I constructed dynamic worksheets to explore slope and circle geometry in Math 10 and 11. Twice, I threw in the towel halfway through the period because of technical difficulties. The 15 laptops had to remain plugged in because their batteries no longer held a charge. The wireless network couldn’t handle having 15 laptops on it. The files were copied from my flash drive to desktops but only worked on some of the computers.

So, we went back to pencil and paper. Each student drew and then measured his or her own angles. Some students immediately observed the relationship. Others observed it after seeing the results of each group member. They asked “What if we move the inscribed angle off to the side more?” and “What if the central angle is larger?” Then, they set off to find the answers. Listening to these conversations, I wondered what this would have looked like had I been able to carry out my lesson plan.

In the four years since then, I’ve seen several GeoGebra/Sketchpad constructions created by other math teachers but very little that really excites me. A new tool to use while I stand and deliver? An e-version of an investigation that my students do using pencil and paper? Okay. I guess. Just don’t try to sell it to me as being more than what it is.

I want to incorporate technology into my teaching in meaningful ways. Here’s something from David Cox that could get me back on the GeoGebra bandwagon. It starts with a great problem that is enhanced because it is posed using GeoGebra. Students continue to interact with the applet as they attempt to solve the problem.

I uploaded some of my dynamic worksheets to GeoGebraTube and was very pleased to see that they worked on my iPad. I’d love some feedback on them. Was my assessment of them correct or are they salvageable? Also, I’m willing to give GeoGebra another try. Can you point me to exemplars?

“Under the M… the square root of 12”

On this blog, sometimes I share my thoughts about transforming math education. This is not one of those times.

Here, I’m using my blog as a digital filing cabinet.

One activity that my students enjoyed was MATHO (and its variations FACTO and TRIGO).

Have students select and place answers from the bottom of each column to fill up their MATHO cards. In some versions, I pulled prepared questions from a hat. In other versions, I translated answers to questions on the fly. For example, if I grabbed 2√3, I called out “Under the M… the square root of 12”. After a student shouts “MATHO!” ask potential winners to read aloud their numbers. (Remember to keep track of answers you have called.)

Nothing revolutionary here – just a fun way to review content.

Squares & Square Roots
Exponent Laws
Simplifying Radicals
Rational Exponents
Factoring Trinomials x^2+bx+c
Factoring Special Products
Trig Functions

By the way, if you are looking to read about changing things, please check out Sam Shah’s recent post, The Messiness of Trying Something New.