“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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“They’ll Need It for High School” (Part 3)

I’m picking “TNIFHS” back up. At the end of Part 1, I promised Part 2 would answer “What are the big ideas in elementary school mathematics that students will need for high school?” Instead, I talked times tables.

In this third half, I’ll refocus on these big ideas. Or one of them — one that came up in that initial “they’ll need long division for high school” conversation.

More than the standard algorithm, what students will need is an understanding of division as sharing (finding the number in each group) and measuring (finding the number of groups). More generally, what students will need is an understanding of the fundamental meanings of all four operations.

Here’s part of a task I presented to my secondary math colleagues:

Evaluate, or simplify, each set of expressions. Make as many connections as you can conceptually & procedurally, pictorially & symbolically.

fundamental meanings of all four operations

Sticking with division, in this task (−6) ÷ (+3) was chosen to bring to mind sharing (3 groups, −2 in each group) whereas 6/5 ÷ 3/5 was chosen to evoke measuring (3/5 in each group, 2 groups). (This often leads teachers themselves to revisit 6 ÷ 3.)

division

(−6) ÷ (+3) as sharing (top) and 6/5 ÷ 3/5 as measuring (bottom)

Flexibility is key. Consider (−6) ÷ (−3), 6/5 ÷ 3, 6 ÷ 0.3, 0.6 ÷ 3, 6x ÷ 3, 6x ÷ 3x, etc. (Note: division of fractions & integers are high school topics in Western Canada.)

(I’m not saying that dividing by a fraction — or decimal fraction — always means measuring. You can think sharing, which can be challenging. Andrew Stadel’s estimation jams are my favourite examples of this. How long is “All Along the Watchtower”?

4082414_orig

Did you see 2/3 in the picture? Did you divide by two, then multiply by three? In other words, did you invert and multiply? What’s the meaning of 2:40 ÷ 2/3 in this context?)

The subtraction set above is interesting. Teachers pick up that the expressions are variations on a theme: five “take away” two. Their pictorial representations tend to show subtraction as removal: “if you have five apples/quarters/x‘s/square root of two’s and I take away two…”

282px-Subtraction01.svg

Pictorial representations that show subtraction as comparison (the “difference”!) are less frequent, but maybe more helpful.

subtraction

5/4 − 2/4 as removal (top number line) and comparison (bottom number line)

Consider (+5) − (−2). To “take away” negative two from positive five means introducing zero pairs whereas the “difference” between positive five and negative two means understanding that positive five is seven greater than negative two.

This second meaning is probably more meaningful in high school. For example, subtraction as removal reduces (1.89t + 15) − (1.49t + 12) to an exercise in collecting like terms whereas subtraction as comparison has students contrasting rates of change (e.g., cost per additional pizza topping) and initial amounts (e.g., the price of a plain cheese pizza). Similarly, if F_1(C) = 9/5C + 32 and F_2(C) = 2C + 30, then (F_2 − F_1)(C) compares conversions given by an estimate and the formula. When solving |x − 5| = 2, it’s more helpful to ask “What numbers differ from five by two?” than think missing minuends in take away problems.

Addition and multiplication — as well as other big ideas needed for high school such as proportional reasoning (or “multiplicative thinking”?) — will be addressed in future posts.

“They’ll Need It for High School” (Part 2)

So Part 2 was supposed to be about the big ideas in K-7 mathematics that students will need for high school. But that’ll have to wait for Part 3. Instead, more on times tables.

Three oft-used arguments for the importance of memorizing times tables:

  1. When learning higher levels of math, there just isn’t time to use calculators or strategies to determine basic facts.
  2. Besides, thinking taxes working memory which means by the time you’ve worked out the first part of the question, you will have forgotten the… Where am I?
  3. Because factoring.

1 & 2 are gospel. Well, so is 3; nevertheless, it’s the focus of this post. I have a couple of thoughts on times tables and factoring trinomials.

The reason some students struggle with factoring trinomials is not because they haven’t memorized products to 10 × 10. I can get away with this if we’re talkin’ Pythagoras. But factoring?! I mean, that’s all it is, right? To factor x² + 7x + 10, you just have to ask yourself, “What two numbers multiply to 10 and add to 7?”

HS math teachers, try this: give your students a quiz on factoring. Include both x² + 10x + 24 and x² + 25x + 24. Get back to me. For extra credit (yours, not theirs), throw x² + 6x + 5 in there. If your students are anything like mine, I bet x² + 25x + 24 gives them at least as much difficulty as x² + 10x + 24. What does this mean for these students? More practice multiplying by one?!

Of course, 1 × 24 falls outside most times tables. Recall of products to 10 × 10 gets us the factors of x² + bx + 60 – if b = 16. But x² + 17x + 60, x² + 19x + 60, x² + 23x + 60, and x² + 32x + 60 are fair game, right? Try c = 48. Or 72. Or 96. Or 100. What role does memorizing times tables play? What role does being flexible with numbers play?

My point, I think, is that these are different, albeit related, skills. In other words, the “it” they’ll need for factoring (trinomials) is factoring (numbers). And number sense. This has some implications for K-7: not necessarily more “What’s 4 × 6?” but more “A rectangle has an area of about 24 square units. What could its length and width be?” or even “The answer is 24. What’s the question?”; not thinking digits/standard algorithm but thinking – and talking! – factors/mental math strategies, e.g. 16 × 25 = (4 × 4) × 25 = 4 × (4 × 25) = 4 × 100 = 400 (via Sherry Parrish).

origami by @Mythagon nothing to do with post @k8nowak says put pictures in posts

origami by @Mythagon
nothing to do with post but @k8nowak says put pictures in posts

Say you’re still asking, “How am I supposed to teach them factoring when they don’t even know their multiplication facts?” When I introduced polynomial division in Math 10, some of my high school students didn’t even know long division. So I taught division of numbers and polynomials side-by-side, highlighting connections. Can the same miiindset (channeling my inner Leinwand) be applied to factoring trinomials and times tables?

And what about something like x² − 2x − 24? If that – asking yourself, “What two numbers multiply to -24 and add to -2?” – is all it is, why not factoring trinomials to teach multiplication (and addition) of integers?

Part One

“They’ll Need It for High School” (Part 1)

“They’ll need it for high school.” I hear that. A lot. From elementary and secondary alike. I’ve been doing the K-12 Numeracy Helping Teacher thing (think “Math Coach”) for four years now. Previously, I taught Math 8 to 12. Twelve years. In Part 1, I’m going to look at math topics, teaching practices, and other things related to readiness where this phrase is used.

The Chestnuts

Long division and times tables.

Teaching long division may be the greatest time suck in all of elementary mathematics education. When I was new to this gig, I asked an intermediate teacher “Why the em⋅PHA⋅sis on long division?” “TNIFHS,” she answered. Having taught HS, her answer surprised me. A HS student will spend 5 years × 90 classes/year = 450 classes, give or take, in math. She will not need long division in 449 of them. HS math teachers, back me up here — one lesson: polynomial division. That’s it. Her turn to be surprised. But don’t blame her: this idea gets a lot of play in the media.

Over lunch at a recent pro-d workshop — the tortelloni was lovely — a mathematics professor from a local university complained that her Calculus students struggled with long division. How could she know? What’s long division got to do with Calculus? Finger Pointing 101.

This is not a call for scrapping the standard long division algorithm in K-7. We need more history of mathematics in math class, not less. Wanna argue dividing multi-digit dividends by multi-digit divisors without using technology is an important life skill? Fine. But don’t point to HS math.

“How can I teach them when they haven’t even memorized their times tables?” is my Groundhog Day conversation. Granted, recall of the multiplication facts is important. And overblown; it’s no silver bullet.

Worse still is “they need to quickly recall the basic facts for high school.” How fast? Faster. But “faster equals smarter” is not a productive belief for learning mathematics at any level. And we know Mad Minutes cause math anxiety. This bleeds into the next category…

Poor Pedagogy Preparation

“They’ll be lectured to at high school.” Often, this is an assumption, one many HS teachers I know take issue with. And, even if it is true, “I want to get them used to it” is not much of a defence. The same holds true of assessment and homework. Future poor practice should never be the reason for current poor practice. High school math teachers are guilty of making assumptions and justifications looking ahead, too.

They’ll Need High School Math for High School Math

Michael Pershan posted a few calculus readiness tests on his blog. One question jumped out at me:

Let f\left( x\right) =2x^{2}-2x. Simplify \dfrac {f\left( x+h\right)-f\left( x\right) } {h}.

If this isn’t calculus, it’s damn close. I can’t think of a conceptual context outside of calculus in which there’s a need for the difference quotient. (Compare this with what they’ll really need for calculus from Christopher Danielson’s NCTM session from a year ago.)

I wonder what this looks like at HS. Maybe SWBAT simplify \dfrac {-\left( -7\right) \pm \sqrt {\left( -7\right) ^{2}-4\left( 2\right) \left( 4\right) }} {2\left( 2\right) } as readiness for quadratics? I should stop, lest my HS brethren get any ideas.

This is silly, but it does illustrate one problem I have with TNIFHS: we meet students where they’re at, not where we want them to be.

The Affective Domain

So, what will they need? “Give me a student with a positive attitude towards mathematics, who’s persistent, who’s curious, etc. and she will be successful in high school,” I’ve answered in the past. I stand by this.

But there’s a problem with this answer. Implied in “they’ll need it for high school” is “they’ll need it before high school” (see times tables). I’ve met HS math teachers waiting for these curious, persistent students to one day show up at their classroom doors.

Another problem: there are big ideas, or enduring understandings, or key concepts, or whatever you want to call them, in mathematics that students will need for high school and this answer gives them short shrift. These will be discussed in Part 2.