Teachers are born not made, Dept.

I get fan mail from time to time and invitations to speak that most of the time never come to fruition.  One such invitation came from the treasurer of a Catholic school in the Los Angeles area.  He had read my book “Math Education in the US: Still Crazy After All These Years” and liked it so much that he ordered ten copies for various teachers in the school.

About two years ago he asked me if I would speak in late August at his school.  I was just starting my teaching job at my current school and had to report the week before school started–which coincided with when he wanted me to speak.  I said I could not given the circumstances, but maybe we could look at doing it in April since I got two weeks off, and surely one of those weeks his school would be in session.  He demonstrated a great amount of inflexibility and said August was the only time.  He then suggested the same time in August a year from then.

I said I couldn’t think that far ahead and let the subject drop. We continued in a back and forth conversation in which he was constantly buttering me up and saying things like “You are a national treasure but you probably don’t realize it.”  He would ask for my opinion on various things and I would give it to him.

One time, however, he said that he thought teachers are born not made, and wondered what my opinion was on the matter.  I said I disagreed and that I had learned a lot about teaching techniques from articles I’ve read from reliable sources, (and including talks I heard at a researchED conference that I attended). One can always improve one’s teaching if one has the inclination–there is always something to be learned.  He apparently didn’t like this, and I never heard from him again.

I’ve thought about this from time to time because I hear others saying it also.  Teachers are no more born than virtuoso musicians are born, or award winning writers or actors. Aside from the few prodigies who may exist (the mathematician Ramanujan comes to mind) in general it takes hard work and much practice and learning. (Even Ramanujan had to learn how to do proofs for what he felt were obvious statements that needed none.)

But the myth prevails, and there is a sub-culture of teachers who look at teaching as a journey. In their world, teachers are ninjas and superheroes in a world of unicorns They attend ed camps not to learn new things but to reinforce their misguided notions about ineffective practices being effective and to be among those who speak the same group-think. The slightest indication of going against the group-think will cast such person to wander in the desert–even those who are national treasures who may not realize who they are.


Clarification and Amplification, Dept.

In my last missive (a pretentious word, I admit, but I dislike the word “post” and I absolutely detest the phrase “smart and thoughtful post”, so please tell people who use such phrase to shut the hell up) I said the following:

“It’s a brave new educational world we live in. I want no part of it, nor any of the damned PD that comes with it either.”

Someone applied an interpretation I didn’t intend and tweeted: “These are the teachers who are retiring in droves.”

What I meant was that I want no part of the cheap rhetoric that passes as educational wisdom. Let me assure my faithful readers and followers that I have no intention of retiring. I will continue to teach despite any allegations, accusations or allusions that I’m doing it all wrong.

Of importance, however, is the fact that there are teachers who are retiring because 1) they can and/or 2) they’ve had it with being told, e.g.,  to not stand at the front of the room and to not teach but facilitate, and many other injurious and ineffective practices that are accepted (and mandated) by the edu-status quo.

Professional Development, Dept.

Teachers are routinely expected to attend professional development (PD) to supposedly help them in their teaching.  I recall one such PD I was forced to attend when I started teaching at my current school.  It was held at the school during the week before school started.  It was called “How to write rock star lesson plans” and it seemed to be all about collaboration, with the general message that writing lesson plans was a waste of time. Teaching should be organic and student-centered.

His ice-breaker was to go around the room asking everyone to name their “super power”. This is typical at PD sessions that seem to abound in references to unicorns, super heroes, Ninjas, rock stars, and other like-minded crap as if teachers are a special breed who must be spoken down to. The teachers at the session complied and he always had some witty comeback or conversation talking about the super power they named, except when he came to me and I said that my super power was “card magic”.  This left him speechless probably because I didn’t say anything about unicorns or something clever, so he went on to the next person.

The day proceeded along those lines.  His basic premise was that “constructivism” was the way to go in teaching.  He  started with a quote from Gary Stager, a known constructivist who publishes and gives talks and is generally well known by fellow constructivists.  I forget what the quote was and it doesn’t matter. It told me I really didn’t want to be in this room for 6 hours. (Yes, 6 hours.)

I thought he was on the right track when he talked about how in California schools, students in early grades, when learning about the California Missions, construct a mission out of sugar cubes.  “That only teaches you how to build a mission out of sugar cubes. Does it teach you anything about the history of missions?”  OK, so far.  But then he talked about a better alternative (wait for it):  Minecraft!

Again, I realized I had 6 more hours of this crap.  In retrospect I could have left and no one would have noticed. But I stayed til the bitter end which included him showing two pictures, one of a class where the seats were in rows and the teacher was at the front, and (one was supposed to assume) the students were bored and disengaged. The second picture was one with whiteboards all around the room with students up and about looking at the various math problems, supposedly engaging in meaningful dialogue about each problem, a la Jo Boaler.  “Now in which class do you think the students are more engaged?” he asked.  Reminds me of a question on a true-false test I had in Social Studies in second grade in which the question was “The fireman is my friend.”  How do I know? Depends on the fireman. And in the case of the classrooms, maybe the students in the first class were engaged. And if they weren’t, maybe the teacher wasn’t that great.

Finally, at the end, he had us write a program in some programming language. Of course, he called it “coding”–no one calls it programming anymore.  “Coding” is an important part of education, he said, though what it had to do with writing “rock star” lesson plans is anyone’s guess, other than one doesn’t have to write a lesson plan to have kids code. Just tell them “Find a way to draw a square using this coding program.”  The coding was the typical “Logo-like” program that allows one to draw line segments, skip around, rotate, etc. In short, it teaches nothing about programming other than how to get things to move on the screen to create shapes–the program has already been written.  I didn’t know how to proceed so I asked the teacher next to me how to do it and he showed me–he had worked with the program before.

Similarly the PE teacher a few seats away was expressing frustration and got mad at the PD leader for not providing instruction.  “You can figure it out,” he told her.  She too received instruction from the teacher next to me, and then she proceeded to tell other teachers how to work with it. The moderator was delighted with this and said “Look; a few minutes ago she didn’t know how to write code, and now she’s telling others”. As if this was proof that students learn better from each other than from a teacher.

I related this tale to a hard-fast educationist some months later and she responded: “Well, everyone knows that students learn more from hearing it from a classmate than from a teacher.”

It’s a brave new educational world we live in.  I want no part of it, nor any of the damned PD that comes with it either.


SteveH on problem solving transference and “understanding”

SteveH, a frequent commenter at this blog, has made some cogent observations about the transferability of problem solving skills, and about “understanding” in general.  Like me and others I know, he feels that far too much importance is placed on understanding in math in K-6 than is necessary. He also posits that mastery of procedural skills in the early grades and even high school, doth not a “math zombie” maketh.  I’ve taken two of his recent comments and placed them here for general interest.

I. Problem solving and transference

I want to say more about problem solving and transference. Polya is worthless, so what problem solving skills are transferable? You can draw pictures, label variables, and write equations to look for m=n. Everybody does that, but still, it’s difficult. You can study governing equations and their variations. This is classic homework p-set work. However, does D=RT problems and variations easily transfer to work problems? Yes and no. Both are amount = rate * time sorts of things, but there are lots of odd variations. Look at AMC test problems. You could know a lot about logarithms, but the testers are really good at finding odd problem variations. I look at some of their questions and feel really stupid. Success on the AMC is not really about general transference, but very specific and detailed preparation of problem classes for a timed test. You don’t study general problem solving skills. You study in detail every past test question you can get your hands on. Only that reduces transference distance.

How about general problem solving transference? First, it’s not a timed test issue. The question is whether you can do the problem eventually. I’ve seen cases where the smart kids in a class take longer to figure out the trick or angle. Knowing and working with lots of governing equation variations is probably the best foundation. How about this problem?

You are sitting in a row boat in a big tank of water that has a scale showing the level of the water in the tank. You take out the anchor and drop it into the water. Does the level of water in the tank go up, down, or stay the same?

What general problem solving skills help with this if you don’t know the governing equation? Do you have to do a JIT (thanks Barry) discovery of Archimedes? There really is no such thing as general magical transference and problem solving outside of m=n, and that’s not a big help even with a LOT of p-set practice. If you figured out this problem quickly, does that mean that you will do the same for any other problem? I doubt it. Even Feynman used to study up on trick problems so that he could impress people when he pretended to figure them out off the top of his head. Colleagues called him a faker.

Feynman’s true understanding brilliance was based on not only on knowing many governing equations in physics, but making sure he had a real physical sense and connection of those equations to reality. However, even that ability has limited trasference. You have to develop that sense for every different governing equation. Feynman struggled a lot. I still struggle a lot with problem solving. Epiphanies of understanding come to me, but only after I struggle and work on problems for a long time. Then again, there is a huge gap of hard work and discovery between insight and any sort of final solution.


II. Understanding and challenging assignments

This is from my son’s old Glencoe Algebra 1 textbook on factoring differences of squares – Exercises. (page 451)

Section 1 – Basic skill problems like this – Factor or prime?
256g^4 – 1

Section 2 – Factor and solve
9y^2 = 64

Section 3-6 various word problem applications

Section 7 – Open Ended
“Create a binomial that is the difference of two squares. Then factor your binomial”

Section 8 – Challenge
Section 9 – Find the error
Section 10 – Reasoning – find the “flaw”
a = b
a^2 = ab
a^2 – b^2 = ab – b^2
(a-b)(a+b) = b(a-b)
a+b = b
a+a = a (remember that a=b)
2a = a
2 = 1

Section 11 – Writing in Math

Section 12 – Standardized Test Practice

Section 13 – Spiral REview questions

Section 14 – Reading Math
Learn to use a two-column proof for algebraic manipulation

So what’s the problem here? Is it just that teachers only assign the problems in the first two sections? Clearly, those are the most important sections and they are not dumb, rote, busy work just for speed. You have to understand the basic concepts and see the variations. However, there are many layers of understanding and nobody can fully understand the implications with just a few problem variations. That’s why it’s common for many to not really “understand” algebra until sometime during Algebra 2.

You can get good grades in math, but still feel like you are struggling. It’s not a “zombie” issue. It’s normal. You can get poor grades and not understand, but that’s another issue. If you get good grades and really don’t understand, then probably your grades are OK, but dropping. Some honors (proper math) classes say that you can only enter if your grade is 80+ or some such thing. Gaps and weaknesses in understanding will eventually cause you to fall off the cliff. Math is tough in high school and college. However, it’s NOT that in K-6, but too many kids struggle. It’s NOT an understanding issue.

And just what are you after exactly, Dept.

Sorry to keep harping on the Education Trust “study” that finds middle school assignments lacking in the “conceptual understanding” department.  In the Education Week article on the “study” (and no mention whatsoever in the study of what mathematicians, engineers and/or scientists were consulted in writing it) a commenter agreed wholeheartedly with it and said:

And I COMPLETELY agree that my experience in MS (and HS) classrooms focuses way too much on procedural over conceptual understanding. 

Which caused me to wonder: Just what conceptual understanding do people think is missing from middle school math?  It isn’t that students are just given problems to solve without explaining what the concepts are.  Percentages are explained, as are decimals, as are fractions, and why one uses common denominators, and even the why and how of multiplication and division.  Anyone who teachers middle schoolers or even high school students, knows that students gravitate to the “how” rather than the why.

SteveH who comments here frequently notes the following about procedural vs conceptual understanding:

Kids LOVE being good at facts and skills. They are easier to ensure and test, and that success drives engagement and much deeper learning and understanding. Skills come before understanding and engagement, not the other way around. The best musicians are the ones with the most individual private lesson skill instruction. They didn’t get those skills top-down by playing in band or orchestra only. The process is difficult and many don’t like it and drop out. This is true for any real life competitive learning – it is not natural.

Such disputes are usually settled in the same manner as the one about whether inquiry or direct instruction is better, and someone says “You need a ‘balanced’ approach” without defining what that balance is.  In the argument about understanding vs procedure, the usual bromide is “they work in tandem”.  This tells us absolutely nothing.

Sometimes the conceptual understanding is part and parcel to the procedure like place value and carrying and borrowing (two terms for which I make no apology for using).  Other times it is not.

Having understanding is only part of the process. If I may talk about calculus for a moment. In upper level math courses in college, one learns the concepts behind why calculus works–including the delta-epsilon definitions of limits and continuity.  A student may be able to recite the definition of continuity and tell you what needs to be done to show continuity at a specific point  in a function.

(I.e., a function f from R to R is continuous at a point p ∈ R if given ε > 0 there exists δ > 0 such that if |p – x| < δ then |f (p) – f (x)| < ε.)  But having a student prove that the function y = x^2 is continuous at the point x=2 involves a procedural knowledge of how to go about doing that.  Just knowing the theorem is not enough.)

So I’d like to know.  Do these people who claim they focus too much on procedure think that the majority of middle school students are just operating blindly as “math zombies” as some bloggers like to call it, without any knowledge of what it is they’re actually doing?  Really?


More just in, Dept.

Looks like Education Dive isn’t the only one to write about how middle school math assignments lack “high cognitive demand”; Education Week is reporting on the “study” as well.

They give an example from the study of two assignments. Assignment A is considered purely procedural, while Assignment B is considered to require more cognitive demand:

A: Factor completely, and state for each stem what type of factoring you are using.   

x4 + x3 – 6x2

B: Create expressions that can be factored according to the following criteria. Explain the process you used to create your expression.

A quadratic trinomial with a leading coefficient of 1 that can first be factored using greatest common factoring. The greatest common factor should be 2x.

I find the wording of Assignment B a bit confusing but that’s besides the point. It appears that the people who did this “study” (and yes I will keep using quotes around that word, however offensive that may be to some) are not happy with a focus on procedural type problems. We are not given a complete view of the homework problem sets, so we don’t know if the problems scaffolded in difficulty. The “study” also did not examine the textbooks/assignments in K-6, which from what I have seen take an “understanding first, standard algorithmic procedures later, and only when understanding is attained” attitude. (See here for a clarification of what I mean).

Since I teach in a middle school, I see directly the casualty cases from what passes as mathematical education in K-6.  Middle school teachers in general realize they have to prepare students for high school math.  Given that burden, and having to teach students who are still counting on their fingers to add and subtract, don’t know their times tables and are flummoxed by the simplest of problems, it doesn’t take a brain surgeon to figure out what’s going on.  And what is going on is that middle school teachers are having to focus on the basics rather than the “critical thinking, depth of knowledge problems” held so dear by those who believe Common Core’s content standards are only there to support the platitudes known as the Standards of Mathematical Practice (SMPs).

I’ll also make a distinction here.  We are trying to teach students to solve problems, not “problem-solve”.  The latter is a term generally used to describe the process of solving one-off problems with little or no instruction in how to even approach them.

As for students taking algebra in 8th grade, I mentioned I use a 1962 textbook by Dolciani.  Here are two problems taken from the book. The first is about factoring:

“In the following problem, the given binomial is a factor of the trinomial over the set of polynomials with integral coefficients.  Determine “c” .    2𝑥−3; 10𝑥2−3𝑥+𝑐  “

And this is a word problem students are expected to solve:

“The plowed area of a field is a rectangle 80 feet by 120 feet. The owner plans to plow an extra strip of uniform width on each of the four sides of the field, in order to double the plowed area.  How many feet should he add to each dimension of the field?”

My students are not concerned with the “relevance” of the problem or whether it meets “real world” criteria. They want to solve such problems and draw on solid, explicit instruction and mastery of procedures in order to do so.

This just in, Dept.

From “Education Dive” (as in “deep dive”, “deep understanding” and other ridiculous jargon which unfortunately permeates the edu-world), a summary of a shocking new study:

Less than 10% of math assignments in the middle grades require “high levels of cognitive demand,” and only about a third of tasks expect students to show their thinking when providing their answers, according to a new analysis of more than 1,800 assignments, released today by The Education Trust.

Oh dear! Say it ain’t so.  What kind of high level cognitive demand do you want from a homework assignment?  What’s wrong with practicing procedures or solving word problems that escalate in difficulty–even if they aren’t the “open ended” variety? (Open ended, as in “The area of a rectangle is 24? What are the dimensions of the rectangle?”  Things like that, which supposedly get at “depth of knowledge” rather than the dreaded procedural “plug and chug”, which supposedly never scaffolds to higher difficulty problems.)

And what do they mean by showing their thinking?  A written paragraph? Showing work is not enough, I guess.  Can’t the teachers assess students’ reasoning by asking questions in class, like “Why did you subtract those two numbers? How did you come up with that approach?”  No, students now have “some ‘splainin’ to do!” Assignments that are “answer-focused” to use the jargon of the study, do not allow students to communicate their thinking.

And from the report itself:

Unfortunately, our analysis revealed that although roughly three-fourths of all assignments at least partially aligned to the grade- or course-appropriate math content, they also tended to:

  • Have low cognitive demand
  • Over-emphasize procedural skills and fluency
  • Provide little opportunity for students to communicate their mathematical thinking

And this tendency was often worse in higher poverty schools.

Which concludes with:

This analysis of middle-grades math assignments show that
schools and districts across the country are falling short when it
comes to providing their students with high-quality math tasks
that meet the demands of college- and career-ready standards.
The high percentage of aligned assignments demonstrates
that teachers are adjusting from the “mile-wide” philosophy
of previous standards movements and embracing the focused
prioritization of content that the math standards provide. These
high rates of alignment should be celebrated and strengthened.
However, alignment on its own is not enough to meet the high
bar set by rigorous college- and career-ready math standards.

And another “conclusion”:


Wow, it has all the right words doesn’t it?  And how does data show that we need to engage students with “opportunities for choice and relevance in their math content”? It might show that there is not much opportunity for such choice, but does it show that we need to provide such opportunities?  There are teachers (not just me) who will tell you that if students know enough to be able to tackle the problems given, they won’t care if it’s relevant or not.  OK, don’t believe me.

Look, I use a 1962 Dolciani algebra textbook to teach my algebra class. The word problems are plenty challenging for my students, though I’m fairly certain that the authors of said study would find such problems lacking in “real world relevancy” (as if my students care) and low cognitive demand.  Yes, I hear you saying “But they’re not from poverty and they would do well anywhere.”  Really? Got proof of that?

For my 7th grade class, I use JUMP Math, which uses micro-scaffolded approaches, but doesn’t skimp on the conceptual understanding behind the procedures either.  It has been given bad reviews by those who hole math reform ideologies in high regard as being “too procedural”.

Which brings me to one final question. Did the study in question look at how the students are doing on standardized tests?  And, oh yes, what types of approaches are used at Learning Centers, by tutors and by parents at home.  What is it that successful students are doing? Do they explain their work? Spend time on open-ended problems? Are do the stuff that’s held in disdain?  Any data on that anyone?