Sunday, April 19, 2015

...And to teach reading, study math instruction

In her recent Edweek commentary To Teach Math, Study Reading Instruction, Marilyn Burns asks:

How can we connect literacy and math, so that teachers bring the strengths they have with language arts instruction to their math teaching? How can teachers make links between mathematics and language arts pedagogy that will enable them to engage children with math in the same way they bring children to the wonder of reading?
One way, she proposes,
is for teachers to think about leading classroom discussions in mathematics as they often do when teaching language arts. Probing students' thinking during math lessons is valuable, so that the goal is not only getting correct answers, but also explaining why answers make sense.
Asking students why answers make sense: this idea is so novel that it apparently hasn’t occurred to most math teachers. Along these lines, Burns advises, it’s important
even when [students’] answers are correct to ask: "Why do you think that?" "How did you figure that out?" "Who has a different idea?" "How would you explain your answer to someone who disagreed?"
Naturally, Burns is also a fan of verbal explanations and peer tutoring:
It's useful to have students comment on their classmates' answers as well, asking them to explain what a peer said in their own words, or asking students if they have a different way to explain the answer. If students are stuck, it's sometimes useful to have them turn and discuss the problem with a partner and then return to a whole-class discussion.
Her takeaway?
There's much for us to think about to help teachers teach math more effectively. But I think we can make headway if we take the two most important areas of the curriculum—reading and math—and look at them side by side to analyze what's the same, what's different, and what we can learn from one to enhance the other.
I agree. And so here are some of my suggestions about how good math instruction (the kind done in countries that outperform us in math) can teach us about reading instruction (so as to make it more closely resemble that done in countries that outperform us in reading):

1. Basics first, learned to mastery: just as good math instruction teaches basic arithmetic facts and procedures to automaticity; reading instruction should teach phonics to automaticity. Since many students, even older ones, currently lack automatic symbol-to-sound decoding skills, this means much more time on phonics than is currently occurring.

2. Focus in depth on the one best method(s) rather than covering a bunch of less effective methods superficially. Just as the best math classes focus on standard arithmetic and algebraic algorithms rather than drawings of groups of objects, digit splitting, skip counting, number bonds, repeated addition, repeated subtraction, landmark numbers, and lattices, reading classes should focus on phonics, vocabulary acquisition, and close readings rather than on sight word recognition, context clues, text-to-world references, and text-to-self references.

3. Make sure students have sufficient background knowledge: just as good math instruction waits until students have mastered relevant concepts before having them do novel applications in novel problems, good reading instruction should provide relevant background knowledge to new books (e.g., for Pride and Prejudice, information about English hereditary law; for the Great Gatbsy, information about the Jazz Age).

4. Keep it content-focused: just as good math instruction doesn’t focus away from the actual math via verbose word problems, verbose explanations, and overly concrete, detailed, real life situations, reading instruction should focus on inferences within the text, rather than on inferences that take readers out of the world of the text--and, worse (in the case of “text-to-self” references), distract or annoy them with the task of having thoughts about themselves rather than about what they’re reading.

Friday, April 17, 2015

Math problems of the week: Common Core-inspired quadratic structure problems

A sample high school math problem from PARCC (The Partnership for Assessment and Readiness for College and Careers, a consortium of 23 states involved in developing Common Core tests):

PARCC's discussion of how this problem aligns with Common Core Standard MP.7:

A more challenging set of "seeing quadratic structure" problems from over a century ago (Wentworth's New School Algebra):

Extra Credit:

1. Are there other useful structures one could recognize the PARCC problem as having other than Q2 + 2Q = 0? For example, might recognizing it as having the form a*a = b*a be an alternative, non-brute-force way of seeing its solutions?

2. If "seeing structure in a quadratic equation" warrants a special Common Core standard (MP.7), why aren't students getting more challenging quadratic structure problems like those in Wentworth above?

3. Was "seeing structure in a quadratic equation" even more important a century ago, before we needed the Common Core authors to remind us of how important it is?

Wednesday, April 15, 2015

Teaching Math in the 21st Century

Barry Garelick's new book is out!

Also available on Amazon.

An insider's account of math education in the Brave New World of the Common Core standards. Fans of Barry's Conversations on the Rifle Range series here on this blog will very much enjoy this compilation. More broadly, it is a must-read for anyone concerned with 21st century math education, STEM education, and "21st century skills."

Monday, April 13, 2015

But is he really ready for college?

In my previous post (below), I wrote about all the efforts that got J where he is now. But I left out one key thing. None of the social skills training, none of the GrammarTraining, and none of the educational strategies and opportunities would have gotten him anywhere without external incentives and external consequences. Even now, J would be happy to spend the entire day doing ceiling fan stuff. When he got his first college acceptance letter (by email), he didn’t tell me about it for four days—not until I thought of asking him whether he’d heard anything. Excited though he was to talk about it, in the grand scheme of things it simply wasn’t that important to him.

The same goes for doing well in school: he cares somewhat, but not that much. And the main reason he cares are all the external incentives we’ve attached-most of which boil down to ceiling fans.

In elementary and middle school, his dependence on parent-administered incentives for scholastic success created constant tension between us and J’s teachers. We could only incentivize assignments we knew about; we could only reward him for remembering to turn in his assignments if teachers let us know whether he had managed to do so; we could only reward him for good grades on assignments and tests if teachers communicated these to us shortly after the fact. But parent-teacher communication and online gradebook tools were not (to put it mildly) the school’s strong suit, and so we were constantly finding out, too late for timely incentives, that things were going badly.

“He needs to learn to be more organized,” said the school, repeatedly.

“He needs external incentives to be more organized,” said the mom, repeatedly. “We can’t provide these unless we know about problems as they arise. And you can’t punish a kid whose lack of organization is part of his disability by giving him low grades.”

If I hadn’t kept repeating this, J’s grades would have been low enough that he wouldn’t have gotten into the math and science magnet he now attends. The alternatives—the one remaining dysfunctional neighborhood high school, or a vocational tech school that has since been shut down—wouldn’t have led anywhere good.

Even at his current school, J has his egregious moments of minimal effort, maximal disorganization, or failure to notice and follow important directions, and teachers remind us of how important effort, organization, and following directions are in college and the workplace. Some of them say that it’s reasonable for the consequences for these shortcomings to match what the consequences will be in college or the workplace. It’s high time, surely, for J to stop depending on external motivations for effort, etc.: after all, to function in the real world, independently of hovering parents, he should be prepared for built-in, real-world consequences.

But had his high school grades fully reflected all his moments of minimal effort and maximal disorganization and failure to notice and follow directions, he wouldn’t have gotten into college.

Perhaps, then, J isn’t ready for college. But what is the alternative? If he’s not mature enough for college, he’s certainly not mature enough for the work place. He could theoretically stay in high school until he’s 21, but he’s taken all there is in the way of math and computer science classes, and we don’t want those skills, his most promising ones, to stagnate. Does it really make sense for him to ride out the next few years of his life in a holding pattern until he (eventually, hopefully) crosses some threshold of emotional maturity, when what he needs is a structured setting in which to further develop his most promising skills?

So, yes, he is ready for college. He is ready, in particular, for the best college he got into: one with an autism support program and minimal distribution requirements. One that, equally importantly, is within walking distance of home--where he will continue to live, and where, yes, we will be monitoring him very closely and continuing to give him all the external incentives it takes for him to continue to succeed.

As for the real world, we will cross that bridge when we come to it.

Saturday, April 11, 2015

Autism diaries: between "glimmers" and "miracle cures"

Towards the end of last calendar year, I posted the following questions:

The SAT Critical Reading score of Applicant X is 3/8 of his SAT Math score. There is a 500 point difference between the two scores.
1. Assuming that Applicant X grew up among native English speakers, what is his likely diagnosis?
2. Should you admit him to your engineering school?
The schools in question have now given us their answers, but first, a bit of background.

When J was four years old, he was downgraded by a autism specialists at one of the top autism research centers, after an extensive evaluation using the then-new ADOS tool, from PDD/“mildly” autistic to “moderately” autistic. One of the evaluators suggested we focus on life skills.

“He already has life skills,” I said. I described how J had figured out, on his own, how to operate the bread machine, and how he had recently tracked down an apple slicer at the grocery store so he could slice his apples the way they did at school.

The evaluator looked momentarily surprised. But then, delicately and diplomatically, she explained that “There may be glimmers.” Glimmers of specific abilities. She didn’t need to elaborate: we’d all seen Rain Man. Clearly she was thinking of parents who observe their 2-year-olds writing phrases like “FBI warning” with sidewalk chalk and assume that all would be well; kids who turn into adults who can tell you, in an instant, the day of the week of your great-grandfather’s 40th birthday, but can’t make simple purchases or pass job interviews. Equally clearly, I was not the first parent whose hopes she’d had to check with this reality.

But rather than life skills, we opted for a language support classroom, a year of neurofeedback, years of classroom-based tss support, years of daily extracurricular social and academic tutoring, and about three years of training in all 109 lessons of GrammarTrainer. By 6 J was functioning OK in mainstream classrooms, and by 16 he was fully included, with no tss support, at the first school to welcome rather than fear him--which also happens to be the best math and science magnet in the city (and one of the city's 3 top schools over all, with some of the best teachers I've ever seen).

So what has our moderately autistic four-year-old turned into? Among other things, a high school senior with high scores in math and computer science and writing skills not far below the neurotypical average. With a great sense of direction and extensive knowledge of the city transit system; with extensive experience conducting independent transactions at stores and negotiating fan visits and decommissioned fan donations from neighbors; and with hundreds of dollars earned through snow shoveling, yard work, household repairs, and, (somehow!), Google Ads, he has life skills galore. Such are his “glimmers.” But this is no miracle cure. J’s diagnosis hasn’t changed: recent tests confirm that his autism is still, not “mild,” but “moderate”; he still lags far behind in language comprehension, social skills, and emotional maturity.

So is he ready for college? 15 years ago, no one, even his hopeful parents, would have predicted it, but the answer, according to three different admissions offices, is yes!

Thursday, April 9, 2015

Math problems of the week: Common Core-inspired 5th grade geometry

From PARCC (Partnership for Assessment and Readiness for College and Careers), a consortium of 23 states involved in developing Common Core tests:

Sample Mathematics Item: Grade 5 “Two Aquarium Tanks”:

PARCC's rationale for this assignment in terms of Common Core Standards:

Extra Credit:

1. Discuss the ratio of mathematical challenge to nonmathematical challenge in solving these two problems.

2. Discuss the ratio of the effort involved in coming up with problems like these to the effort involved in justifying them--i.e., explaining how effective they are in meeting the Common Core standards (from "Evidence Statement" all the way down to "Scoring Information").