Constructivism and lack of practice

Here are two of the clues to America's current mathematics problem:

1."Student-centered" learning (or "constructivism")

2.Insufficient practice of basic skills

In an October email, Spokane's secondary mathematics coordinator reaffirmed this district's commitment to a "student-centered" approach to teaching (also sometimes called "discovery learning" or "constructivism"). In this approach, students often work as partners or in groups, and teachers act as "facilitators" rather than as "instructors." Students are encouraged to come up with their own multiple solutions to problems and to ask fellow students for help before asking the teacher.

Reform math curricula are typically built around a constructivist approach, probably because the 1989 Standards document from the National Council of Teachers of Mathematics calls for it (Stiff, 2001c; "Curriculum," 2004). Proponents say the approach leads to "deeper understanding," helpful collaboration and better student enjoyment of the process. Others say a dependence on it can hinder the learning process and frustrate students.

A local parent told me this story about when his daughter took a math class that used reform math curriculum Connected Mathematics: Students were told that "Juan" was mowing a lawn in a right-angle triangle. He wanted to figure out the length of the diagonal. The term "Pythagorean Theorem" (a2 + b2 = c2) wasn't presented. The students were to work in groups and figure out a way to get the answer. Finally, one student who knew the theorem provided it to her group. (Her group was the only one to get the right answer.) Incredibly, the teacher "chastised" the student for using the formula.

"A lot of parents don't believe it at first," the parent said to me. "Like, their kids are younger, they don't know, and they feel that parents are exaggerating, but it is the honest-to-God truth, and these stories get worse."

In small doses, constructivism can provide flavor to classrooms, but some math professors have told me the approach seems to work better in subjects other than math. That sounds reasonable. The learning of mathematics depends on a logical progression of basic skills. Sixth-graders are not Pythagorus, nor are they math teachers.

Meanwhile, anti-reform advocacy group Mathematically Correct provides an amusing take on constructivism ("What Is," 1996): "This notion holds that students will learn math better if they are left to discover the rules and methods of mathematics for themselves, rather than being taught by teachers or textbooks. This is not unlike the Socratic method, minus Socrates."

Insufficient Practice of Basic Skills

Another problem in math classrooms is the lack of practice. Instead of insisting that students practice math skills until they're second nature, educators have labeled this practice "drill and kill" and thrown it under a bus. I wish I had a dollar for every time I heard that phrase. It's a strange, flippant way to dismiss a logical process for learning. Drilling is how anyone learns a skill. Removing drilling from the learning process is like saying, "We'll just remove this gravity. Now stay put." Everyone drills - athletes, pianists, soldiers, plumbers and doctors. Drilling is necessary. It isn't good or bad - it's simply what must be done.

Imagine if I told chess players they had to figure out the rules of chess on their own, in fits and starts, by trial and error and by asking their fellow players. Imagine if I expected them to win games when they hadn't had a chance to practice.

In American education, the "worm" is not yet turning, but it might be looking over its shoulder. In its March 2008 report, the National Mathematics Advisory Panel reintroduced the notion of practicing the basics: "Practice allows students to achieve automaticity of basic skills - the fast, accurate, and effortless processing of content information - which frees up working memory for more complex aspects of problem solving" ("Foundations," 2008, p. 30).

But children in the system now are stuck with a process that asks them to work in diverse groups to reinvent thousands of years of math procedures that they then don't get to practice.

Some people enjoy puzzles on logic and process, where things might not be what they seem and where they've got to figure out subtle differences and new ways of thinking. But this esoteric, conceptual approach to math, with a constant struggle to understand the process, doesn't seem like a logical approach for children. Children are concrete thinkers who tend to appreciate concrete ideas. Children want instructions, direction and things that make sense. Many don't appreciate the daily grind of writing about math, of having to figure out what they're doing, of having to count on classmates for guidance, of trying to remember things they've done just once or twice and several weeks ago.

It's ironic that proponents of reform math criticize traditionalists for supposedly not knowing "how to teach math to children." The reform method seems completely oppositional to how children learn best.

I asked a Spokane student if she prefers the Connected Mathematics she gets in school over the Singapore Math she gets at home. She said, "In a way, Connected Mathematics is easier because you don't have to know as much math, but in a way, it's harder because you have to know more. You have to know exactly what they want."

She gave me an example of the classroom approach: Students are to gather in groups to discuss a problem. The problem might be a complicated twist on simplistic math, or it might be a concept they've never seen before. As the groups muddle around, they don't always agree on what's required. Sometimes, they don't have the necessary underlying skills. Some students become frustrated or bored. Trying to help each other, some confuse the others. They might come up with the right answer, or they might not, but - without practicing the new concepts - the class moves on to something new. Singapore Math, on the other hand, "might be harder as far as the math goes," she said, "but at least you know what they want."

I told her I thought her answer was articulate and enlightening. "I've spoken to a lot of people now," I said, "and you explained things very well." "That's because they teach it," she replied, "but I'm the one who has to learn it."

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British head teacher suspends a quarter of her pupils in a year... and exam results soar

A head teacher has transformed academic achievement at her school by adopting a zero tolerance approach to bad behaviour. Caroline Haynes, 49, has handed out 478 exclusions at Tendring Technology College over the past year - an astonishing one in 20 of all those issued across the county. The crackdown has seen the number of pupils getting A* to C grades at GCSE soar from 48 per cent in 2004, when she joined the school, to 74 per cent this year.

Mrs Haynes attacked political pressure on schools to reduce exclusions in order to improve their Ofsted behaviour ratings. 'Statistics paint a false picture,' she said. 'Because we refuse to buckle under the pressure we had to work very hard to convince Ofsted inspectors that pupil behaviour is good, despite the figures. 'I could reduce exclusion rates tomorrow by not suspending pupils, but it would have a detrimental effect on the quality of teaching and unruly behaviour.'

Academic results at the college in Clacton-on-Sea, Essex, were rated as being below average in a 2003 Ofsted report. Mrs Haynes joined the following year and now issues more than two suspensions each day on average to the 1,880 pupils. The total of 478 over the year is the equivalent of one in four of the school's pupils. However, a recent report found the school to be 'good' or better in every category.

Under her regime, there is an escalating level of sanctions, including extra work, detentions and being placed 'on report'. One-day exclusions are dished out for offences such as failing to attend two after-school detentions. Longer exclusions are given for offences such as bullying, stealing, disruption, abuse of staff or fellow students, vandalism and racism. During exclusions pupils are set work and cannot return to classes until it is completed. They and their parents must also meet the head to discuss how to improve their behaviour.

Mrs Haynes said it was important pupils knew when they had 'crossed the line'. She added: 'Our pupils learn to deal with the consequences of their actions and our teachers are allowed to concentrate on their job rather than battle bad behaviour. Exam results have soared. I'm very proud.'

The school permanently excludes pupils for possessing an illegal substance or offensive weapon. Two were expelled last year. The Department for Children, Schools and Families said it trusted heads' judgment 'to decide what sanctions will work best'.

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