Following the lead of others, Dept.

A blogger who calls herself Quirky Teacher announced that she was finished with Twitter. She gave good reasons, and the more I thought about it, the more I realized that I’m just as tired of Twitter as she is.

Therefore, I too will be closing my account. I find I spend too much time trying to be right, being snarky when others say something that I deem to be 1) wrong, 2) idiotic, or 3) both, and trying to cajole others into ganging up on those whose opinions I find irritating.

I am also tired of the word “nuance” which is the usual rejoinder by those who do not agree with someone’s argument and criticize it by saying it lacks nuance.  I am tired of tweets promoting “smart and thoughtful posts” by edu-pundits and/or journalists who think they have the ultimate scoop on education.

It does have good attributes, but it is one of those precious “conversations” that rarely reaches any resolution and just seeks to further infuriate those who are furiating.

In ending my account, I am resisting the urge to tweet about this particular post.

How much deeper understanding do students really need, Dept.

In a recent op-ed in the LA Times, Dan Willingham, a professor in the department of psychology at the University of Virginia, addresses a particular aspect of math education in the U.S.  Blaming poor math performance on bad curricula, he argues, overlooks that elementary school teachers may not have the deep understanding of math that is required to teach it. In fact they may actually fear math.

Students without deep understanding, Willingham argues, may be limited to inflexible thinking. That is, their math knowledge is limited to performing specific operations for certain types of problems but they may falter when presented with problems in new settings or with slightly different wording. The result is an increasing number of high school students floundering in math “because the groundwork of understanding was never laid in elementary grades”.

Willingham suggests that the solution might then be to find and hire those teachers who have “deep math knowledge” and who know how to convey it. I have no problem with hiring teachers who have a thorough understanding of math. What troubles me are the premises that students are doing fine with math facts and standard algorithms. Also I question the disturbingly prevalent belief that student outcomes in math would improve if only they had a deeper understanding of math.

Students are not doing fine with basic math

While high school students may indeed be floundering, I disagree that it’s because it was not taught with understanding in earlier grades. In my opinion the contrary is true; there has been too much of an emphasis and an obsession with understanding in math in elementary schools.

Over the past three decades—in large part propelled by NCTM’s standards that came out in 1989—the preoccupation with understanding has manifested itself with a de-emphasis on learning math facts. Also, standard algorithms for the basic operations are delayed while students are presented with alternate strategies that require making drawings or using convoluted methods. Such methods are nothing new; they were taught in the past, but after students had learned and mastered the standard algorithms. Now, however, they are taught first in the name of providing the conceptual understanding behind why standard algorithms work as they do. Simple concepts are made more complex under what passes as “deeper understanding.” Students I have seen entering high schools do not know their math facts, and use alternate inefficient strategies for simple operations such as multiplication.

 The codification of reform math ideology

The Common Core standards have effectively cemented in the math reform ideology that is increasingly incorporated in today’s elementary school textbooks. Adding to that are the bevy of ineffective teaching methods (inquiry- and problem-based learning, group work, so called differentiated instruction) pushed upon teachers in ed school and in professional development seminars.

Furthermore, they are told in ed schools, in professional development seminars and other sources that memorization is bad and that teaching standard algorithms “too early” eclipses understanding. Teachers who elect to teach standard algorithms and teach in traditional manners are sometimes told to teach their lessons with “fidelity” to textbooks they are required to use. Young teachers who fear for their jobs will do so. Older teachers who may have the understanding that Willingham would like to see are sometimes told the same. Unlike the younger teachers, however, the older ones can simply retire.  And unlike the older teachers, the younger ones are likely the products of the ineffective math teaching and are probably just as confused about math as many of the students we are seeing today.

Given all that, I would agree that having more teachers with a better understanding of math might help the situation. The improvement would come from not only from a larger knowledge base. Specifically, in greater numbers such teachers are more likely to be able to reject the nonsensical approaches foisted on them and use the resources and methods shown to effectively teach students math.

What is “understanding” in math?

Willingham admits that hiring teachers with better understanding might be difficult because state tests have been shown to be inaccurate predictors of who teaches well. Adding to this, is confusion about what constitutes “understanding”. What educationists believe is understanding is in most cases visualization—drawing diagrams that demonstrate what two-thirds divided by three-fourths looks like.  That is not at all what a mathematician means by understanding.  Also, being made to use formulaic “explanations” and dragging work out far longer than necessary with multiple procedures and awkward, bulky explanations is not a sign of understanding. Forcing students to continually stop and explain becomes nothing more than “rote understanding” in the end.

There are levels of understanding that vary depending on where a student is on the novice/expert spectrum.  Novices do not learn like experts; it takes time for knowledge to accumulate with procedures and understanding working in an iterative fashion to support each other. Insistence on understanding at every point where students should be learning procedures while working effectively with their beginning levels of understanding lacks educational value.  With the prevalence of misunderstandings about what understanding is, the criteria for hiring teachers who possess understanding might well result in hiring more teachers with the same misguided views.

Another misconception of understanding is the notion that students who know why a procedure works are in a better position to solve problems via enhanced flexible thinking. In an article by Greg Ashman, he describes a study which suggests that it is a mistake to assume that students in possession of conceptual understanding will use it.

While I disagree with Willingham’s point on understanding in his recent op-ed, he provides, in my opinion, an  earlier article on Inflexible Knowledge that he also authored presents a more apt characterization of the interaction between understanding and domain knowledge, in a  that he authored.  In it, he says:

“Understanding the deep structure of a large domain defines expertise, and that is an important goal of education. But if students fall short of this, it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots. There is a broad middle-ground of understanding between rote knowledge and expertise. It is this middle-ground that most students will initially reach and they will reach it in ever larger domains of knowledge.”

Simply put, no one leaps directly from novice to expert. For sure, teach math with understanding, but don’t obsess over it. Teach the math students need to know.



Advice on the Teaching of Standard Algorithms Before Common Core Says it is Safe to Do So Dept. is an organization that rates textbooks/curricula with respect to how well they align with the Common Core standards. There are no ratings on the effectiveness of a curriculum or textbook–just whether it adheres/aligns to the standards.

They published a guideline for how to use EdReports’ reviews of texts.  Of interest is that under “Focus” for K-8, the key criterion to be assessed via their gradated ratings is: “Major work of the grade and no concepts assessed before appropriate grade level”

What captures my attention about this is the “no concepts assessed before appropriate grade level”.  Sounds similar to “no wine before its time” but it has more sinister implications in my opinion.

In my investigations and writing about Common Core standards, I have heard from both Jason Zimba and Bill McCallum, the two lead writers of the math standards. They have assured me that a standard that appears in a particular grade level may be taught in earlier grades. Jason Zimba also wrote an article to that effect.  So for example, the standard algorithm for multidigit addition and subtraction appears in the fourth grade standards. This does not prohibit the teaching of the standard algorithm in, say, first or second grade. A logical take-away from this would be that students need not be saddled, therefore, with inefficient “strategies” for multidigit addition and subtraction that entail drawing pictures or extended methods that have been known to confuse rather than enlighten.

Nevertheless, the interpretation of the “focus” of Common Core is used as a means to judge whether a textbook is aligned.  The term “focus” is discussed on the Common Core web site in a description of instructional “shifts” expected from implementation of CC. The instructional shifts do not appear as a standard anywhere within the Common Core content or the 8 Standards of Math Practice.

This means that if a publisher wishes their math book to sell, they will make sure that their assessment packages (otherwise known as tests and quizzes) do not include standards for later grades than for which the test/quiz is intended.  The result is that students will be tested on the inefficient convoluted and pictorial methods that are used in lieu of the standard algorithms.

While there are teachers who can overlook this, and accept a student’s use of a standard algorithm, there may be teachers new to the profession who adhere to the pre-packaged assessments. Furthermore, such practices may be reinforced by Professional Development vendors who specialize in Common Core.

The Common Core website insists that pedagogy is not dictated by Common Core:

Teachers know best about what works in the classroom. That is why these standards establish what students need to learn, but do not dictate how teachers should teach. Instead, schools and teachers decide how best to help students reach the standards.

In view of all this, my advice is to allow students to use the standard algorithms on a test even if it appears in a later grade. As far as alternate strategies, be aware questions on state assessments may include them. While scores on the state tests are not used in figuring the grade a student receives in a class, they may be used to qualify students for gifted and talented programs.

Bottom line advice: Teach the alternate strategies. Just don’t obsess over them.


Out on Good Behavior, Dept.

I am currently writing a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder”  When the series is complete, it  will be published in book form by John Catt Educational, Ltd.”

The chapters are being published in serial form at the Truth in American Education website.  If you are curious, the first seven chapters are available for your reading pleasure and can be found here:

Chapter 1 , Chapter 2 , Chapter 3 , Chapter 4 , Chapter 5 , Chapter 6 and Chapter 7

Rich Problems, Dept.

If you hang around long enough in the world of math education you’ll hear people refer to “rich problems”. What exactly are rich problems?
One definition is: “A problem that has multiple entry points and has various levels of cognitive demands. Every student can be successful on at least part of it.”
My definition differs a bit: “One-off, open-ended, ill-posed problems that supposedly lead students to apply/transfer prior knowledge to new or novel problems that don’t generalize.” 
For example: “What are the dimensions of a rectangle with a perimeter of 24 units?”
 A student who may know how to find the perimeter of a rectangle but cannot provide some of the infinitely many possibilities is viewed as not having “deep understanding”.
Rather, the student’s understanding is viewed as “inauthentic” and “algorithmic” because the practice, repetition and imitation of procedures is merely “imitation of thinking”.
I teach 8th grade algebra and use a 1962 textbook by Dolciani. Here’s a problem from that book that I gave to my students:
“If a+1 = b, then which is true? a>b, a<2b, or a<b?”
This is my idea of a “rich problem” but don’t tell anyone. I might get fired.

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.





More from the annals of Ed School, Dept.

In my Educational Psychology class, I gave a presentation on constructivism, showing the difference between minimal guidance, and guided instruction, and evidence that inquiry-based approaches are ineffective. The professor lauded me with praise afterward and said it really got her thinking, plus she really was intrigued with Singapore Math (which I used as examples of explicit and guided instruction).

A few minutes later, I overheard her saying to a student in the class:

“Direct instruction works well in the short term but there’s research that shows that over the long term, students who were taught with discovery learning retained more. They also did better on standardized tests over the years than students taught with direct instruction.”

But I got an A on the presentation.


UPDATE:  I contacted the professor to ask what particular research she was referring to.  Her response follows:

There isn’t just one study that shows this – there are several studies in different contexts. You can read about them synthesized here in the National Academies Press book, How People Learn  –

This book can be downloaded, which I have just done. Will let you know if I find anything of interest; stay tuned.

From the Annals of Ed School, Dept.

A new series that comprises a collection of things heard and overheard in ed school, uttered by students and professors alike. Ed school is the place where discarded and discredited psychological theories go to thrive.

Student:  Is the horse in George Orwell’s “Animal Farm” an example of cognitive motivation (i.e., motivated to achieve mastery) vs social cognitive
(performance appearance).

Professor: Yes, the horse exhibited cognitive mastery.

For those of you who haven’t read Animal Farm, or have forgotten it, the horse was forever criticized by the “collective” and responded to the criticism with “I will do better.” Not quite sure if I agree this is cognitive motivation or milieu control but as I’ve said before, there are no wrong answers in ed school.

Unintended Consequences of Teaching “Habits of Mind” for Algebraic Thinking

(This is a modified version of an article that appeared in Education News on January 28, 2013. )


The idea of whether algebraic thinking can be taught outside of the context of algebra has attracted much attention over the past two decades.  Interestingly, the idea was  raised as a question and a subject for further research in an article appearing in American Mathematical Society Notices which asks,  “Is there evidence that teaching sense making without algebra is more or less effective than teaching the same concepts with algebra?” I sincerely hope someone follows up on this question.

The term “habits of mind” comes up repeatedly in discussions about education — and math education in particular.  Teaching algebraic habits of mind outside of and in advance of a proper algebra course has been tried in various incarnations in classrooms across the U.S.

Habits of mind are important and necessary to instill in students.  They make sense when the habits taught arise naturally out of the context of the material being learned. Thus, a habit such as “Say in your head what you are doing whenever you are doing math” will have different forms depending on what is being taught.  In elementary math it might be  “One third of six is two”; in  algebra “Combining like terms 3x and 4x gives me 7x”; in geometry “Linear pairs add to 180, therefore 2x + (x +30) = 180”; in calculus “Composite function, chain rule, derivative of outside function times derivative of inside function”.

Similarly, in fifth or sixth grade, students can learn to use the distributive property to multiply 57 x 3 as 3 x (50 + 7).  In algebra, that is extended to a more formal expression: a(b + c) = ab + ac.

But what I see being promoted as “habits of mind” in math are all too often the teaching of particular thinking skills without the content to support it. For example, a friend of mine who lives in Spokane directed me to the website of the Spokane school district, where they posted a math problem at a meeting for teachers regarding best practices for teaching math.

The teachers were shown the following problem which was given to fifth graders.  They were to discuss the problem and assess what different levels of “understanding” were demonstrated by student answers to the problem:

The problem is more of an IQ test than an exercise in math ability.  Where’s the math?  The “habit of mind” is apparently to see a pattern and then to represent it mathematically. Another drawback is that very few if any students in fifth grade have   learned how to represent equations using algebra.

Presenting problems like the button problem above prior to a pre-algebra or algebra course will likely result in clumsy attempts at solutions that may or may not lead to algebraic thinking. For example, one student answered the problem as 1 x (11 x 3) + 1, which would be taken as evidence by some that the child is learning the “habit” of identifying patterns and expressing them algebraically.  Another student answered it as 4 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 34.

Rather than establishing an algebraic habit of mind, such problems may result in bad habits.  An unintended habit of mind from such inductive type reasoning is that students learn the habit of jumping to conclusions.  This develops the habit of mind in which a person thinks that discovering a pattern is the solution and nothing further needs to be done.  Such thinking becomes a problem later when working on more complex problems.

The purveyors of providing students problems that require algebraic solutions outside of algebra courses occasionally justify such techniques by stating that the methods follow the recommendations of Polya’s problem solving techniques.  Polya, in his classic book “How to Solve It”, advises students to “work backwards” or “solve a similar and simpler problem”.

But Polya was not addressing students in lower grades who are on the novice end of the novice-expert spectrum of learning.  He was addressing students who are well on their way to developing problem solving expertise by virtue of having an extensive problem solving repertoire.  For lower grade students, Polya’s advice is not self-executing. Telling younger students to “find a simpler version of the problem” has about the same effect as providing advice on safe bicycle riding by telling a child to “be careful”.

As an example, consider a student who stares blankly at a problem requiring them to calculate how many 2/15 mile intervals there are in a stretch of highway that is 7/10 of a mile long.  The teacher can provide the student with a simpler problem such as “How many 2 mile intervals are there in a stretch of highway that is 10 miles long?” The student should readily see this is solved by division: 10 divided by 2. The teacher then asks the student to apply that to the original problem. The student will likely say in a hesitant voice: “Uhh, 7/10 divided by 2/15?”, and the student will be on his way.  Note that in this example, the problem is set in the context of what the student has learned — not based on skills or concepts to be learned later.

It would indeed be amazing if we could teach students algebraic thinking skills devoid of the content that allows such thinking to occur. I tend to believe, however, that a proper study of this will show what many have known since the time of Euclid: there is no such royal road.