“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 
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 
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.
Chris Hunter 
I agree that long division is overemphasized, but polynomial division is useful in calculus. It is used for computing the partial fraction decomposition of a rational expression. It is also used to find the zeros of a polynomial of degree three or higher, which often arises when finding the intervals of increase and concavity of a function.
As a HS teacher, I taught students how to find the zeros of polynomials, but we used synthetic division. The thinking was that it wasn’t about polynomial division but about the characteristics of the function.
Oh, I always loved synthetic division. I don’t know why; certainly I liked doing it more than I found stuff to do with it.
You asked the question “how could she know?” — did you ask her? If not, then … who is pointing the fingers at whom? I know my folks here in college *do* trip flat over “I don’t remember any long division” a few times.
That said, “They’ll need to endure this later” is an entirely bogus argument to me. Teaching things that are developmentally appropriate is what enables students learn from, not just endure, the lectures later. Walking in with those big ideas is critical. I’m interested in part 2 🙂 When somebody says “they’ll get hit with this later” I say “if I see somebody walking down the middle of the road, do I run them down because somebody else is bound to do it?”
Fair question. No. This comment was part of a more important conversation that I didn’t want to derail. Next time.
(They trip over it when they’re learning to factor. That’s also when that not knowing the times tables smacks ’em. Unfortunately for a nastily high percentage of them, they honestly don’t *really* understand multiplication and they definitely don’t understand division.)
The fundamental meaning of multiplication and division… definitely part of Part 2.
I can’t even describe how much I love this! I teach middle school and I frequently hear teachers use that as an excuse about why they don’t use manipulatives in their classroom. I don’t care if high school teachers never get out an algebra tile that doesn’t mean we shouldn’t be doing it in middle school. I can’t wait to read Part 2!
Thanks Brooke. I’ve heard that too. In my experience, use of manipulatives is similar in both elementary and secondary — hit and miss.