“Fundamental knowledge is the basis of creativity.”
Recently, I asked Spokane Public Schools about the new professional development (PD) in math for teachers. I was sent a link to a district page where upcoming courses focus on the implementation of a new and unproved math curriculum, not on mathematics.
Where else but in our public schools are employees persistently deficient in necessary skills? Where else are they taught that pedagogy is infinitely more important than expertise in the subject? Where else are billions of our dollars used to train employees in skills they should have had before they were hired? Who in the private sector knowingly hires doctors who don’t understand medicine, contractors who don’t know how to build, or rocket scientists who don’t understand rocket science? Who would knowingly hire an endodontist who doesn’t know how to do a root canal?
It’s true that many teachers don’t understand enough math. I don’t blame them. They learned what they were taught.
I’ve been conversing with an assistant professor of education who maintains that constructivism is the best way to teach math. He said substantial evidence is easy to find, so I asked him for some. He sent me a few articles containing older, questionable methodology or anecdotal criticisms of a teacher. I pointed out to him the last 30 years of the failure of excessive constructivism. He responded:
Avid proponents of constructivism typically seem certain that they’re correct, that they don’t have to prove anything, and that all problems are due to incompetent implementation. After 30 years and trillions of taxpayer dollars spent pushing fuzzy math and constructivism on public schools, they don’t see today’s nationwide math problem as due to fuzzy math and constructivism.
Many maintain their faith by denying the problem. Obvious academic failure is explained away or deemed to be irrelevant; they focus on an undefined notion of “deeper conceptual understanding.” (They ignore the fact that this “understanding” can’t be achieved without acquisition of skills.) When confronted with irrefutable evidence, they blame it on teachers, parents, students or society.
I don’t blame teachers. Most received garbage for math instruction – in K-12, in college and for years after they were hired. How could they teach math properly? They were taught that math is hard and that they don’t have to know math in order to teach it. (They must be so tired of being lied to.) Their training has intellectually disarmed them, their students and this country. These are unforgiveable sins.
If proponents of fuzzy programs and constructivism had to use math in the “real world,” and were held accountable for the results, they would have to modify their views. In the "real world," math is a tool, used to get a job done. What matters are clarity (understandable by others); efficiency (done relatively quickly); and accuracy (the result is correct). Math is a tool – like a hammer or drill. One doesn't come to consensus on the philosophy of a drill; one learns to use the drill and then one uses it.
We use math to help us cut the wood, build the bridge, fill the ditch, fire the rocket, heal the sick, fire the bullet, cook the food, calculate the pay, run the business, combine the chemicals, fly the plane, build the software, measure the floor, balance the checkbook, project the earnings, and balance the budget.
Math is critically necessary to the functioning of the country. A mathematically illiterate populace puts America’s future in jeopardy. K-12 math is inherently understandable and doable, but proponents of fuzzy math and excessive constructivism have made it incomprehensible.
Luckily, I was taught properly, and I refuse to be “disarmed” now. I’m engaging in some PD of my own. I recently picked up Saxon Algebra I and read it cover to cover. That was instantly helpful. I’m now doing the problems in Saxon Algebra II, chapter by chapter. I wondered if this PD would change my views, but it’s reinforcing everything I’ve been thinking about how math should be taught and learned.
“Deeper conceptual understanding" in K-12 math comes with knowledge and practice to mastery, not with pointless struggle and reinventing of the wheel. Efficiency on paper is critical; the calculator tends to get in the way of learning. Each day, as I work through another chapter, I think, "Oh, yes. Right. I see that now." Proper process is being reinforced for me; each time I cut a corner, I pay for it with an error. As I practice, I’m becoming faster, more efficient and more accurate. Recently I tweaked an algorithm to make it more efficient; this would not have come to me without skills and understanding.
Constructivists claim that the materials don’t matter (as they insist on fuzzy materials), and that it’s the teacher who matters. (This is how they duck criticism of their materials and blame everything on teachers.) But proficiency is gained via solid instruction, such as from textbooks that provide sufficient explanation and practice, examples, structure, and an incremental and logical progression of skills.
Below are some processes that are conducive to the development of solid math skills. (Proponents of fuzzy math and excessive constructivism typically refuse to implement these):
- Direct instruction of sufficient material, emphasizing the most-efficient, most-effective processes (including long division; vertical multiplication; arithmetic; exponents; negatives; the number line; polynomials; fractions, decimals and percentages; the clock and the calendar; and proficiency with paper and pencil).
- Practicing concepts to mastery, with constant refreshers of previously learned skills
- Using good process:
- Working vertically
- Writing down the equation, filling in what’s known, solving for the variable, checking the work, making sure the question is answered
- Writing clearly, separating equations from calculations
- Going from simple skills to complex, working forward in a logical, linear fashion. (Classes should NOT begin in the middle of a math textbook)
- Excessive use of mental math
- Prioritizing methods and processes that are inefficient, confusing, nonstandard, not useful long-term, and complicated for children
- Constant distractions through group work, discussion and premature “real-world application”
- Dependence on calculators, classmates and achieving consensus, rather than emphasizing individual understanding and proficiency
- Forcing children to “construct” their own methods, manage their own classmates, explain things to themselves, and understand concepts at a level that is wildly inappropriate for their age
- Dependence on teachers who don’t understand math, refuse to correct or explain work, and don’t provide students with answers so that students can check their own work
- Dependence on administrators who refuse to give textbooks to children, destroy solid materials by inserting loopy processes and philosophy, and force teachers to begin in the middle
I’ve come to see fuzzy math and excessive constructivism as abusive. Indeed, the assistant professor described his own reeducation in math as “painful,” “brutal” and “ego-crushing.” Why would he want that for children? I can’t think of a reason to demand that children suffer. He insists his efforts are to benefit children, but he appears to be too far removed from classrooms, math, children and outcomes to understand how fuzzy math and excessive constructivism destroy skills, self-esteem and futures.
I would never do that to a student. Math should be enjoyable. Individual understanding and proficiency should be the goals. Direct instruction can easily incorporate laughter, play, practice and application. Students can work through the material, eating the elephant one incremental bite at a time. Over and over I see students relax as they realize I'm not going to make them reinvent it. They smile. They gain confidence. They tend to say, "I get it. I can do this." It's the nature of direct instruction.
Doing the math I’ve done has reinforced what I knew. The truth is evident in the children. Thirty years of fuzzy math programs and constructivism have led us here, to a nation that can't do much math.
It's no wonder that Eastern Washington University decided in 2011 to disband its masters in math. After two decades of fuzzy math and excessive constructivism in surrounding school districts, it's likely that few high school graduates were able to get through the EWU program. Sadly, the EWU situation reflects just the tip of the national math iceberg. We are clinging to the edge of a grim precipice that teeters over complete national mathematical illiteracy.
All K-12 teachers and parents should have received at a minimum the instruction I did, but it isn’t too late. Take a placement test so you can assess your level and start teaching yourself. Buy a textbook online or at a secondhand bookstore. You don’t need highlighter pens, sticky notes, butcher paper or group work. All you need is about $15 and some time. (Teachers will have to do it on their own because their district is not likely to give them this PD, nor is Bill Gates, Pearson Education, the Broad Foundation, the Common Core, Texas Instruments, EngageNY, Arne Duncan, or the teachers union).
Get yourself some math skills, and pass them on. Watch as the children soar.
(P.S.: You might want to buy a complete set of Saxon Math now, before some well-meaning and ostensibly polite person wants to make them illegal.)
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