My First Educational Treatise and What I Learned

My mother was my first critic of my first educational treatise, written when I was a senior in high school. During my junior year, I had tried unsuccessfully to get transferred out of a geometry class taught by a teacher whose reputation had preceded her for years. In retrospect, she was a very bright woman who had emotional problems and spent the class talking about this, that and anything other than about geometry and left it to us to read the book, do the problems, and make presentations on the board.  At that time, however, I wanted someone who taught.  The math department head was more than familiar with her problems but when I asked for a transfer had told me that 1) I wasn’t a teacher and 2) I had only had her for three days; thus: how could I judge?

My father (who was not familiar at all with how school politics worked) tried to intercede on my behalf but was outmaneuvered by the high school bureaucracy/double talk. I managed to survive the semester with her, and the next fall when I was a senior, I decided to do the student body of the high school a favor and wrote a little pamphlet called “A Manual for Personal Objectors”.

The title was a take on the “Manual for Conscientious Objectors” which was required reading in the mid-60’s for those who were trying to escape the draft (and being shipped to Viet Nam) on the basis of religious and other grounds. I felt that trying to get out of a bad teacher’s class was as hard as getting out of the draft.  I typed my treatise up on ditto masters (which was like a mimeograph that used that awful smelling solvent, and the print came out purple) that my mother had, since she was a teacher).  I typed it in “landscape” mode so I could fold it over and bind it like a pamphlet. My mother ran off the pages at her school and because two-sided pages were unheard of for dittos, I then had to paste pages together to get a back-to-back, real pamphlet look. My mother helped me to glue them and we put the pamphlets together.

I started selling the pamphlets at school for 25 cents a piece which got me in a little bit of trouble at school, but that’s another story. One day my mother got mad at me for something I said to her (probably complaining about something I didn’t like about something she did) and she let me have it. It was one of those “After all I’ve done for you” type rants, and once she got going there was no stopping her.  Among the things she mentioned was running off the dittos and helping me glue them together.

I knew at that point that I had lost this battle, but she wasn’t done by a long shot.  “And another thing,” she said.  “How do you think I felt when I read what you wrote about how teachers’ grading policies are unfair and you said ‘This is especially true of English teachers.’ ”  She was an English teacher.

There was only one thing I could say, and I said it through tears: “I didn’t mean you, Mom.”

I often think of this when I hear complaints from students about a teacher as happened in a school where I taught. I was one of two math teachers for the seventh and eighth grades at the school—we had both been hired at the same time. I was friendly with the other math teacher and she would often tell me of the frustrations she was facing in her classes.  I knew that we didn’t see eye to eye on math education, given that she was a fan of Jo Boaler, and she believed that memorization eclipsed understanding.

In that particular school, I was available for tutoring students prior to the first period, and some of her students would occasionally come in for help. In one case, a boy was having difficulty with proportion problems, such as 3/x = 2/5. I showed him how it is solved using cross multiplication (without explaining why cross multiplication works—I was after just getting him through the assignment). I ran into the boy’s father during the summer, and he made a point to thank me for helping his son.  I found this odd since I had only tutored him maybe one other time.  “He was getting D’s and F’s on his test and since you helped him, he started getting A’s and he ended up with an A- in the class.”

From what I knew about the teacher, I suspected that she offered little instruction, expected students to collaborate and discover, and make connections. As it turned out, there were many complaints to the school from parents and she was fired that year. I recall her telling me that she was not going to be rehired, that parents were complaining about her, and that she felt as if she were the victim of a witch hunt.

Since she had read one of the books I wrote, I’m fairly certain she knew we were on opposite sides of how math should be taught. But she never said anything about my methods, nor I about hers. I enjoyed the autonomy the school gave me to teach as I wished so I kept my thoughts to myself. I was friendly with her, and my wife and I had her over a few times.

There are those who might say it is my duty to speak up about how a teacher teaches. But in a school, we are in a fragile situation. There are vast differences in teaching philosophies within the teaching profession, but you have to work and get along with fellow teachers as well as the people in power. The tightrope I walk is remaining loyal to how I believe math should be taught, while finding the common bond with the other teachers and the administration. And more importantly, realizing that many teachers are victims of the indoctrination of ed school group-think which has dominated the education profession for many decades. In the meantime, I’ll continue to write my criticisms of math education—being careful to call out the group-think and its perpetrators.




Misunderstandings about Understanding, Dept.

What do we mean by “understanding” in math? I gave a talk about this at the researchED conference in Vancouver. I have included an excerpt from my talk, and added some commentary at the very end which is designed  1) to further elucidate the issues and 2) to infuriate those who disagree with my conclusions.

Understanding Procedures

One doesn’t need to ‘deeply understand’ a procedure to do it and do it well. Just as football players and athletes do numerous drills that look nothing like playing a game of football or running a marathon, the building blocks of final academic or creative performance are small, painstaking and deliberate.

Many of us math teachers do in fact teach the conceptual understanding that goes along with an algorithm or problem solving procedure. But there is a difference in how novices learn compared to how experts do. Requiring novices to retrieve understanding can cause cognitive overload. Anyone who has worked with children knows that they are anxious to be able to solve the problem, and despite all the explanations one provides, they grab on to the procedure. The common retort is that such behavior comes about because math is taught as “answer getting”. But as students acquire expertise and progress from novice to expert levels, they have more stored knowledge upon which to draw. Experts bundle knowledge around important concepts called “neural links” which one develops in part through “deliberate practice”.

Furthermore, understanding and procedure work in tandem. And along the pathway from novice to expert, there are times when the conceptual understanding is helpful. But there are also times when it is not.

It’s helpful when it is part and parcel to the procedure. For example, in algebra, understanding the derivation of the rule of adding exponents when multiplying powers can help students know when to add exponents and when to multiply.

When the concept or derivation is not as closely attached such as with fractional multiplication and division, understanding the derivation does not provide an obvious benefit.

When the Concept is Not Part and Parcel to the Procedure

One common misunderstanding is that not understanding the derivation of a procedure renders it a “trick”, with no connection of what is actually going on mathematically. This misunderstanding has led to making students “drill understanding”. Let’s see how this works with fraction multiplication.

Multiplying the fractions  is done by multiplying across and obtaining But some textbooks require students to draw diagrams before they are allowed to use the algorithm.

For example, a problem like  is demonstrated by dividing a rectangle into three columns and shading two of them, thus representing  of the area of the rectangle.

Fig 1

The shaded part of the rectangle is divided into five rows with four shaded.  This is 4/5 of (or times) the 2/3 shaded area.  The fraction multiplication represents the shaded intersection, giving us 4 x 2 or eight little boxes shaded out of a total of 5 x 3 or 15 little boxes: 8/15 of the whole rectangle.

Fig 2

Now this method is not new by any means. Such diagrams have been used in many textbooks—including mine from the 60’s—to demonstrate why we multiply numerators and denominators when multiplying fractions. But in the book that I used when I was in school, the area model was used for, at most, two fraction multiplication problems. Then students solved problems using the algorithm.

Some textbooks now require students to draw these diagrams for a variety of problems, not just the fractional operations, before they are allowed to use the more efficient algorithms.

While the goal is to reinforce concepts, the exercises in understanding generally lead to what I call “rote understanding”.  The exercises become new procedures to be memorized, forcing students to dwell for long periods of time on each problem and can hold up students’ development when they are ready to move forward.

On the other hand, there are levels of conceptual understanding that are essential—foundational levels. In the case of fraction multiplication and division, students should know what each of these operations represent and what kind of problems can be solved with it.

For example: Mrs. Green used 3/4 of 3/5  pounds of sugar to make a cake. How much sugar did she use?  Given two students, one who knows the derivation of the fraction multiplication rule, and one who doesn’t, if both see that the solution to the problem is  3/4 x 3/5, and do the operation correctly, I cannot tell which student knows the derivation, and which one does not.

Measuring Understanding

 Given these various levels of understanding, how is understanding measured, if at all?  One method is by proxies involving procedural fluency and factual mastery but which involve some degree of mathematical reasoning.

Here’s an example. On a multiple choice placement test for entering freshmen at California State University, a problem was to simplify the following expression.


In case you’re curious, here’s the answer:   (y+x)/(y-x)

This item correlated extremely well with passing the exam and subsequent success in non-remedial college math. Without explaining one’s answer, simply recalling the arithmetic properties of fractions along with being fluent in factoring was enough for a reasonable promise of mathematics success at any CSU campus.

In short, the proxies of procedural fluency demonstrate the main mark of understanding: being able to solve all sorts of variations of problems. Not everyone needs to know the derivation to understand something at a useful—and problem solving—level.

Nevertheless, those who push for conceptual understanding, lest students become “math zombies”, take “understanding” to mean something that they feel is “deeper”. In a discussion I had recently with an “understanding uber alles” type, I brought up the above example of fraction multiplication and the student who knows what the fraction multiplication represents. He said “But can he relate it back to what multiplication is?”  Well, that’s what the area model does—is it necessary to make students draw the area model each and every time to ensure that students are “relating it back” to what multiplication is?

They would probably say it’s necessary to get a “deeper” understanding. My understanding tells me that what is considered “deeper” is for the most part 1) not relevant, and 2) shallower.


IN CASE YOU’RE INTERESTED: The entire talk can be obtained here: It is the PowerPoint slides, which if viewed in notes format contain the script associated with each slide.