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Tuesday, November 4, 2008

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
“Student-centered” Learning (or “Constructivism”)

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.

This is how Spokane Public Schools defines constructivism (“Parent’s,” n.d.):

“This is an historically used method where skillful teachers personalize teaching to support learning for all students. A productive classroom is learner-centered and includes active instruction. Teachers provide students with experiences that allow them to hypothesize, predict, manipulate objects, pose questions, research, investigate, imagine, and invent. The teacher's role is to structure and lead this process with questions, ongoing assessment and attention to students’ progress toward defined learning objectives.”

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 which 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.”

Please note: The information in this post is copyrighted. The proper citation is:
Rogers, L. (November, 2008). "Constructivism and lack of practice." Retrieved (date) from the Betrayed Web site:

This article was also posted Nov. 9 on at


Anonymous said...

One myth I will gladly point out is that 'reform' math is as 'constructivist' as any other curriculum. It is a social theory about learning and has nothing to do with the reform movement, other than a few mathematicians thought it would be good advertising to use as an explanation for their supposed authority on curriculum, since Vygotsky was a new phenomena in US education circles at the time.
There are sparingly few teachers who could even begin to explain constructivism. References used in my thesis were Papert, Wilensky, Turkle, Tudge, Lave, Shaw, Harel, and DiSessa. Reading and foreign language teachers know more about constructivism than math teachers.

How could you use constructivism to provide practice for students is the appropriate way to frame the question of what good is constructivist theory? These are called activity structures and that has nothing to do with curriculum. I prefer working with pairs of students and I frequently use task cards from Marcy Cook, since they are self-correcting.

There are three very important criteria when working with potential dropouts.
1. content is accessible
2. student uses mental math in order to get many potentially correct answers before arriving at a final answer (adjusting).
3. activity is self-correcting so students can proceed at their own pace if they are so inclined.

An example of an actitivity structure is to pair two students together with a 'worksheet' and let one student work while the other student checks for mistakes. After half the worksheet is completed, have the students reverse roles. Students do all their thinking on lined paper. And often I will show a visual model to go with the algebraic explanation.

3 over 1/2 is a good example where structural approach fails. Since most teachers explanations would have students multiply by the reciprocal, so 3 x 2 would arrive at the correct answer with very little thought.

But provide a context like how many halves in $3? And the majority of my students would not be able to process their thinking correctly. They would give answers that made no sense like 3, or 5, or 7. They are somehow at a loss for words and that is an excellent starting point for one to begin learning about constructivism or how people learn to think or think to learn.

If the curriculum or textbook is bad of course well then of course constructivism isn't going to work, but then neither will any other method of teaching.

Unlike most teachers who use cooperative groups (e.g. Kagan), to impose structure on groups of students, I look for ways that takes advantage of children's tendencies to build group cohesion within the classroom.

My classrooms are frequently a last resort for most teenagers. I'll give you one example - I am teaching a 2 year algebra class of 40+ students, ages 14-18 of mostly marginalized Latinos who for a variety of reasons did not learn math from 4th grade on.

Neither Lappen, Treisman, or Isaacs have a Master's in Constructivism, while I do. It takes a decade of study and dedication. While they sell their curriculum, I study it.

The history of reform math is far more interesting. Most of it begins in Michigan and the Netherlands and it is based on a theory that was promoted in the 1950's called 'embedded learning'. On the surface (if you are comfortable reading) these textbooks look like they provide a 'natural' way for students to learn. The principal difference is that the curriculum is coded in text and therefore first requires students that are literate. The evaluations done with reform curriculum did not include children with low reading levels for good reasons.

Much has been said of fidelity of implementation - I believe these critics should have first applied this rule to the reform curriculum that were tested and evaluated.

Algorithms (a repeating set of sequential instructions) are also taught in Everyday math - only these are non-standard algorithms and are not counted as valid explanations by test correctors.

The secondary reform textbooks do not teach any standard algorithms, since mistakenly the authors assumed that everything that needed to be taught was done before the end of eighth grade.

This is not so. A careful examination of traditional structural textbooks like Algebra and Algebra 2 will show at least 10 standard methods of solving two polynomials with two unknowns.

Most reform textbooks treat the factoring of a trinomial as an advanced level subject - this is not the case in Asia and Europe where factoring with polynomials can be practiced as early as seventh grade.

The majority of Americans do not learn how to factor, much less use the quadratic equation.

My primary point is that constructivism can be combined with practice - this is not a tradeoff. And don't confuse constructivism with concepts. I teach my students how to solve problems with algebra. I prefer using a visual model, since it helps my language learners understand - and when they are confident, they can practice. A great math teacher is also an excellent reading teacher.

neal said...

Our research points out that the principles of constructivism are difficult to adhere to within the research design. The advocates of constructivism agree that it is the individual's spaced practice can also be implemented within a constructivistic approach.


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Anonymous said...

That's baloney, read the references first before you judge the research.

Most research in education isn't really research because it doesn't measure up to serious scrutiny - Milgram has always been first to point out obvious flaws in published work and why its always been argued that evaluation and curriculum writing should be kept separated due to obvious bias.

While its true education draws on multiple disciplines. Its quite a stretch to be making some of the outragious claims that I'm reading.

In particular, standardized advocates have a penchant for exageration and telling tall stories. The whole purpose of including an ethnographic study is to get a better understanding of what in blazes is being observed, because most of it would be bs if a person were actually seated in the classroom observing - eg. Didactique calculas and Core Plus.

Fred Strine, retired teacher said...

This latest article gets at the heart of why a major shift is needed away from current methods and philosophies in public education. The key to future success will be total rejection of constructivism. It's too utopian, socialistic, unreliable, and ineffective. It's positively antithetical to the traditional goal of educating students to be independent thinkers and self-reliant. How are those goals ever accomplished within the constructivist’s facilitated group discovery?

I'm afraid I'm beginning to sound like a conspiracy nut, but consider the wide acceptance of the constructivist, student-centered learning environment coupled with the abandonment of traditional history classes. The net result is an education system that has each group of students in every subject wasting precious time re-inventing every milestone in the history of human learning. It's as if post-modern educators have willfully set a match to the 21st century equivalent of the Library of Alexandria. I can't explain why anyone in his right mind would want to rediscover what is already known instead of using that as a solid foundation for NEW learning. I'm so old-fashioned.

I'm sorry to report I see a frightening connection between anti-individual, group think activities plus rejection of history and tactics used by 20th century totalitarians seeking control of the masses. It's precisely the tack taken by Hitler, Stalin, Pol Pot, et al. I'm not suggesting it's being actively pursued by conspirators, but rather negligently progressing under the aegis of education theorists who would love to dictate how the rest of us live.

I really think it's a power quest. The more we "train" the minds of our youth to look toward groups for solutions, the less likely we are to have adults willing to buck systems. It's the old bandwagon propaganda technique (logical fallacy, if you prefer) run amok. It's why traditional teachers are now scarce as hen's teeth. Constructivists want facilitators, not traditional subject experts. Since most young teachers have been indoctrinated by college ed. courses promoting the "student-centered" classroom, teaching has become a youth movement. I know 1st hand it doesn't include traditional geezers. In short, the whole constructivist movement is positively Un-American.

Your article is, as usual, brilliant, right on target, and gutsy. Same, too, for your stand with the school board. I really don't think proponents of discovery ed. are evil, but they are sheep, and sheep aren't exactly known for wise decisions. When pressed, they panic as a flock. They rarely stand their ground individually. I recall the anecdote about a particularly crafty border collie seeking revenge on his abusive owner. He herds the owner's entire flock of sheep over a cliff then runs away—supposed to be a true story. Sheep are easily MISLED.

Food for thought: Traditional Americans keep the Minute-Man spirit alive. We believe in the old "Don't Tread On Me" motto. If one were of a mind to convert Minute-Men Americans to a more easily manipulated, passive herd mentality, how would one proceed?

Anonymous said...

"whole constructivist movement"

theory about learning or a political movement?

lets put this in the proper perspective. Constructivism exists in all classrooms all the time. Either teachers and students are socializing and communicating or they are not.

Standardized movment is a political group which aimed to change education with top-down reform. What lawmakers have done in the process is helped the curriculum industry by writing minimum-achievable standards. Its the standards of education research which have been discarded that allows anyone with financial support to print a textbook that is basically rubbish. Look no further than What Works Clearinghouse, NSF, and the NCTM. That is Socialism.

Anonymous said...

The wishy washy constructivist approach to teaching is based on [assumed] psychological cognitive learning models and should never have been developed into a method of teaching in the first place [Sweller], especially not for novice leaners in elementary school. Its new age epistemology, pedagogy and psychology rolled into one, a psychological framework for teachers to use when they don't want to teach a subject properly. Which other framework would ignore final results, not bother to correct wrong answers or even pass students without achieving minumum standards? Only a framework whose aims were purely non-academic would accept such things.

The whole enquiry/constructivist theory is horrendous, leaves children confused, huge gaps in their knowledge and basically doesn’t work. Yes, it keeps teachers and children happily ignorant of what they don’t know, so from the outside it appears fun. Whatever problem solving skills or higher order thinking these people think is going on, it simply is not happening in those children. It’s chaos, cloaked in buzzwords and ignorance, and topped off with hegellian dialect.

The countries on top of the standardised world rankings are those still using traditional methods, the proof is there and will never be disproven by educational psychologist, you can ignore it and shout about non-academic skills such as critical thinking, but thats a hollow cry from those with sub-standard knowledge of maths or science due to constructivism. Effective maths professors and science lecturers all learnt from traditional direct methods, then constructed the knowledge their own way, they didn't rely on a teacher to 'help them build it in their minds using scaffolding'. That is a ridiculous concept and most parents unlucky enough to be burdened with constructivist teachers will probably agree that it's time for that fad to be dropped from national boards.