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Tuesday, April 22, 2014

Professional development in math should focus on math, not on pedagogy or materials

By Laurie H. Rogers
 Creativity springs unsolicited from a well prepared mind.”
“Fundamental knowledge is the basis of creativity.”
-- John Saxon, co-author of the Saxon Math textbook series

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.
Chief Academic Officer Steven Gering said the district plans to teach fractions to teachers, and I’m glad to hear it. But those skills should always have been required. Fractions are a small "fraction" of what’s missing in the skill set of many K-8 teachers.

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. (Professors of education are largely in charge of educating the country's prospective teachers.) This assistant professor said substantial evidence supporting constructivism 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:
I am interested in this country you speak of that has been in the grip of constructivism for 30 years. Most studies indicate that American classrooms incorporate few (if any) constructivist practices espoused by schools and colleges of education. ... What we do have, I would argue, is a fairly widespread attempt at ham-handed implementations of constructivist-oriented reforms. .... The mathematical and pedagogical knowledge needed to run a constructivist mathematics classroom is not possessed by most teachers now, or at any point in the last half-century ... (W)e graduate and hire most anyone with a pulse and a clean-ish criminal record. ... (A)s a whole our system in its current form couldn’t teach constructively if it wanted to. And it doesn’t even want to because it hasn’t seen much (if any) constructivist teaching.
This stance is typical. As our conversation continued, this man refused to acknowledge the problem and ducked most of what I told him. It wasn't long before he began to call me names. He is courteous in the way of many reformers: Ostensibly civil, yet still calling me a conspiracy theorist, "closed" to the conversation, "dogmatic" and even "indulging in intellectual dishonesty." I'm sure he sees himself as polite and restrained. His entire defense boils down to this: Constructivism works; they just aren't doing it right.

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)
Below are processes that tend to result in increased errors and misunderstandings. (Proponents of fuzzy math and excessive constructivism typically emphasize these):
  • 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
The success and clarity achieved with direct instruction are motivating and empowering. The confusion, struggle and failure found in constructivist classrooms are spirit killers, producing students who hate and fear math. Many develop depression, panic and performance anxiety. Their constant hunt for consensus leads to distraction, dependence, insecurity and groupthink.

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.)

Please note: This information is copyrighted. The proper citation is:
Rogers, L. (April 2014). "Professional development in math should focus on math, not on pedagogy or materials." Retrieved (date) from the Betrayed Web site:


Bruce Price said...

Good news here. Manitoba leads fight against bad math programs. This article discusses most of the same issues as in this blog. What's really surprising is that there's a newspaper up there which reports the whole thing. If the US had newspapers like this, we wouldn't be having all the problems we are having.

Anonymous said...

It's been a while since I've posted here, but I read your writing regularly. For the record I am flat opposed to CCSS and the millions that we unload to standardized testing, I am 100% pro-union because I believe that union membership protects good teachers who piss off incompetent administrators, and I am part of a coalition of teachers that wants local district control of curriculum and materials, as opposed to state and national mandates by the ineffective OSPI and US DoEd.

I teach upper level mathematics (AP Calc, AP Stat, PreCalc, Alg 2) at college/honors levels, and I agree 100% with your article here. (We have met, I came to Olympia to help you lobby against CCSS a couple of years ago.)

Under the new TPEP evaluation, my principal and evaluator ranks me as Distinguished in terms of my content knowledge. He comes to my classroom to get his "math fix" as he is a former math teacher (which is nice, because most admin know jack squat about math either in content or pedagogy.) I say that only to establish myself as a REAL math teacher. I did 2 years in middle school using Connected Math and it was HORRIBLE, I literally left middle school because I refused to use it any more - I started gathering my own materials to teach my Advanced 7th grade classes actual Algebra, and I was castigated by the district office as "not having the right to use materials other than those approved by the school board". And yet, my 7th grade class went on to score some of the highest scores on the 7th grade math MSP that the district has seen (to date, and this was 5 years ago now), even to the extent that the middle school's improvement measurement on the state matrix was a 7, the highest possible, and 95% of those students ended up passing the Algebra 1 EoC in their 8th grade year, a feat that the school or district has not even come close to since I left.

Back to the TPEP evaluation - where I get marked down is "student engagement". Because the preparation in fundamental skills in other lower level classes is so poor (fractions, basic equation solving techniques) and students get socially promoted instead if skills promoted (and truly I blame OSPI for this because their bar for passing the Algebra and Geometry EoC exams is so low), I have lots of students in my classes (Algebra 2 and above) who are simply not equipped to handle the level of math (either in concept or in work ethic), and thus when I am "direct teaching" my students, yes some of them are bored and disengaged. But there isn't anything I could do for them except to go back and teach them Algebra 1 all over again - at which point it is an Algebra 1 class, not an Algebra 2 class. I cannot do justice to the class unless I have students who are prepared for the class (a concept that administrators resist.) I get marked down for not engaging the students enough in the limited amount of time I have them. The engagement needs to come from the student - taking notes, asking questions, paying attention to the teacher/work (as opposed to checking Facebook or Twitter on their phones). Students have been trained poorly as to what their role is in the classroom. If I'm not giving them group project work, if I'm not calling kids up to the board to work out problems, if I'm not giving them an exit task or some other work that I can assess right there on the spot while I am being observed, I get marked down on my student engagement. It really is quite ridiculous, because when students follow the basic plan for being a student (do your homework, pay attention, take notes, ask questions) they are quite successful with me in math, because the focus is on the math and not on some meaningless fluff that doesn't contribute to them improving their skills.

Anonymous said...


Anyway, all of this is to point out that there are some real math teachers out here who are trying to fight the dumbing down of the system by administrators (the US DoEd, OSPI, district admin). But I'd also like to point out that the partners in the dumbing down of the system are those who intend to profit from public education - those who want to privatize and corporatize so they can sell standardized tests and standardized test prep materials and standardized textbooks, they are part of the enemy. If we are really about improving math instruction and math understanding in our students, these profiteers need to go. Education is not somewhere any corporate entity should be seeking to make a buck - the minute that creeps in, the system is corrupted.

mike miller said...

Dear Anonymous,
Kudos to you for doing what's necessary to improve the education of your students, consequences be damned. I was right with you, agreeing with you 100%right up to your last paragraph.
The subject matter of this article is important, and I do not wish to commandeer or distract from its message. But I had to comment on your last paragraph which in essence condemns or belittles a, the, or any profit motive.
You should be complimented for "gathering your own materials" to better help your students and improve your instruction. But please tell us sir, which materials did you gather that were not produced and/or marketed by a "for profit" entity? For that matter, look around your classroom. What, if anything, exists that was not produced "for profit?" From the linoleum tiles on the floor to the paint on the walls, and every chair, desk, whiteboard, and even the clock on the wall is a product of a "for profit" entity. Then there's all the technology... from the light-switches on the walls to all the digital tools within those walls. Indeed, in the theoretical absence of "profit", public education could not exist. Where do you think the tax revenue comes from to keep the schoolhouse doors open (which were also made for profit), and to pay your inarguably meager salary?
The freedom of self-interest (which translates into the "profit-motive") is not, to use your words, "the enemy." It's only become problematic in education because those whom are supposed to be representing our (the children's) best interest have paved the way for monopolistic influence. Broad, mandatory standardization serves the interests of both our elected-appointed representatives-leaders and a handful of "corporate interests." This collusion is the problem, and this collusion was/is enabled and facilitated by our government officials, be they bureaucrats or representatives.

Anonymous said...

mike miller -

I understand what you are saying, and I am not fundamentally anti-profit or anti-profit motive. (For what it's worth, I created a lot of my own materials for my 7th graders and could do so for many of my other classes also, but yes I also used published material.)

I'll give you an example of where things go wrong and need regulating. Consider that many colleges and universities use textbooks that are exorbitantly costly (as are HS texts - my PreCalculus texts costs $75, a ridiculous amount, and a cost that is ultimately paid by the taxpaying citizen.) A college may use one edition of a text for Fall Quarter, then an "updated" edition for Winter Quarter. It's essentially the exact same text, whatever the adjustment for the "update" - yet students are then not allowed to work into a buyback program at their local university bookstore and are forced into buying brand new textbooks when essentially an older version would suffice. This is a corrupted relationship between universities and publishers (and for that matter, professors who want to sell more of their texts.)

That's just a small piece of the unnecessarily high cost of higher education.

Standardized tests in their current manifestation are an absolutely useless product. They do not serve to "improve instruction", they exist purely for the profit of publishers (and as a tool to wield against teachers.) And the companies that own the publishing houses are huge conglomerates with lots of "speech" in state and national governments. Have you seen the extent of McGraw-Hill's holdings, for example? Graft, plain and simple.

Again, I am absolutely not anti-profit. However, if an entity is going to profit from supplying materials to the public education system, those materials had better be effective. I also believe that constructivist theorists are complicit in this because they want to sell their "ideas" that are not effective, which is as bad as selling a product that is not effective. And, no doubt, they are getting paid well by Pearson (for example) to keep propping up those faulty ideas.

I have no problem with a construction company being paid market price to build a good school, teachers being paid well to teach a good class, or materials companies being paid to provide good and necessary materials. My point is when the squeeze is put on for the sole purpose of profit.

Anonymous said...

I completely agree with the article. I teach high school math in Washington state and I feel that most math professional development I have taken has taught ineffective math teaching strategies and/or content. I also feel pressure to agree with the new ideas rather than criticize them.