Letter to Editor of Brandon Sun on Math Education

In response to a letter that was written to the Brandon Sun (Brandon is a city in Manitoba), Anna Stokke and I responded in the following letter that was published today (March 25, 2017) in the Brandon Sun:

The author of “A Message To Armchair Educators” (Sound Off, March 20) is nauseated by non-teachers who express concerns about Manitoba’s poor rankings in international tests.Negating opinions of anyone who is not involved in teaching or education is, at the very least, prejudicial.

Parents bear the burden for ineffective and injurious practices that pass as “research- and evidence-based.”Too many parents are forced to spend time teaching their children what isn’t being taught in school. Other parents, who can afford it, hire tutors or send their children to learning centres such as Sylvan and Kumon. Enrolment in tutoring centres has nearly tripled in the U.S., according to the census there, and a similar increase is occurring in Canada.

Parents who do not have sufficient income to seek help outside of school may, as the author of the Sound Off suggests, be victims of poverty. But unlike the author’s assertion, it isn’t poverty that is causing poor performance in school; it is the ineffective teaching practices that do harm, and poverty-stricken parents have no means by which to rectify it via tutors or, in many cases, by teaching their children themselves.

The author goes on to celebrate bogus practices like “multiple intelligences” and teaching to “preferred learning styles.” But fads like this do not work.

“Multiple intelligences” is one of those platitudes that if repeated enough times becomes taken as truth. We would be happy to provide a list of the research showing that there is no evidence that multiple intelligences and tailoring instruction to preferred learning styles are effective. On the contrary, research shows this to be ineffective.

The author of the Sound Off states: “memorizing math facts does not mean a student understands.” This is a mischaracterization that is often used to denigrate traditional math practices that have proved effective for years. The assumption is that math facts were traditionally taught in isolation without linking them to what they are used for, and that math was taught without “understanding.” A glance at textbooks used in previous eras shows that this is not the case. Explanations of procedures were provided in the past and students were given problems that, ironically, many students today would not be able to solve. Many of the people who claim that traditional methods do not work have benefited from the very techniques that they hold in disdain — yet they promote teaching methods that prevent today’s children from enjoying similar benefits.

Lest we be told that we have no voice in this argument because we “know nothing of the education system,” we are both teachers. One of us teaches math in a middle school and the other is a math professor. Both of us have been involved in the math education debates for many years and have written numerous articles on the subject.

There is a problem with education in Manitoba. International tests show that the percentage of students who struggle in math doubled over 10 years, while the percentage of students who excel in math was cut in half. Manitoba students once performed at the Canadian average — on par with Ontario — but on the most recent national assessment (PCAP), Manitoba was the lowest-performing province in Canada. Things are unlikely to get better if unproven fads dominate the teaching practices in Manitoba schools.

Unfortunately, the opinions expressed by the author of the Sound Off seem to prevail in North America despite much evidence to the contrary. The press and others would be doing the public a great favour by questioning the so-called “evidence” called up by the “experts.”

Barry Garelick is an education writer and middle school math teacher, based in California. Anna Stokke is a math professor at the University of Winnipeg, a Brandon University alumna and author of the C.D. Howe report “What To Do About Canada’s Declining Math Scores.”


Republished from the Brandon Sun print edition March 25, 2017

Lest we forget,Dept.

Let’s not forget where it all started: “Letters from John Dewey/Letters from Huck Finn” in a relatively new second edition, with new intro, bringing things relatively up to date. (We can never be fully up to date because the future is constantly changing; that’s why we don’t need facts, just Google. But I digress).

Order your copy now and send it to a school superintendent who needs to be enlightened. Changing the world one book at a time!

No, we’re really not, Dept.

I’m always wary of those who intercede in debates on (in my case) math education and say “Do you agree that apart from A, that you and he/she are on common ground on B?” This is a foot-in-the-door strategy to then get to “You’re really both saying the same thing.”

I assure you that in most if not all cases, we’re not saying the same things. But thanks for trying.

Shut the hell up, Dept.

An “expert” on Common Core is interviewed in this Ed Source piece. Excuse the quotes around the word expert, but it reminds me of a special I saw a while ago on PBS TV, focusing on the TV series called “The Prisoner” which starred Patrick McGoohan and was a big deal in the late 60’s.

They had an “expert” on The Prisoner series; some guy in his twenties who looked like he never worked a day in his life, whose expertise ranged from DC and Marvel comic book lore, to TV shows from the 50’s and 60’s. He gave a rather detailed analysis of the last episode of The Prisoner as if he were talking about one of Shakespeare’s later works.

That’s how this expert on Common Core struck me, particularly when I came to this gem of a statement, which caused me to stop reading:

“I like to tell people that I’m ‘classically trained’ in mathematics. I was brought up the traditional way, which is to never ask, ‘Why?’ – just be able to do the problem. You didn’t have to know the hows and the whys.”

Maybe you have the stomach for the rest of the interview. Let me know what you think.

Ed Speak, Dept.

Since the imposition of Common Core and ubiquitous propaganda that Common Core math standards require different methods of instruction (despite claims on the CC website that the standards do not prescribe how teachers are to teach), there has emerged an expression that causes me great anguish whenever I hear it. The expression is “to problem solve”, in which “problem solve” has become a new verb form.

It has caught on like wildfire and replaced older forms such as “solving problems”. The phrase has taken on specific meanings; it is a “dog whistle” of reform math (a term Tom Loveless of Brookings coined). It means a departure from the standard word problems that have been held in disdain by reformers as tedious, repetitious, “plug ‘n chug’, not relevant to students’ interest and not reflective of how math is used in the real world.

Case in point would be the type of word problems (which used to be called ‘story problems’ in simpler times) that one finds in great supply in older algebra books and diminishing to none in newer ones, about trains catching up with each other, people mowing lawns together at different rates, and so on. There has been universal agreement amongst reformers that these problems did not do anyone any good except the brighter gifted students who, as reform legend has it, would have learned math anyway by virtue of having been born that way.

The term “to problem solve” now means giving students so-called “rich” problems that cause them to “dig deep” into concepts, learn what is needed to solve the problems on a just-in-time, as needed basis, and other wonderful-sounding things that in the end are more often than not ineffective. Such problems are often open-ended, with multiple right answers, and students solving them in multiple ways. A classic example offered in this genre is “A rectangle has an area of 28 square inches; what are the possible dimensions of the rectangle?” Others are laborious one-off type problem which require intepretations of graphs that are then used to plug in to various formulae in order to answer various questions that the educators responsible think represent how people go about solving problems (collaboratively, of course) in the real world. These type generally fall into the category of problem-based, or project-based learning. There are those who make a distinction between problem- and project-based approaches. I am not one of them.

This is not to say that all such problem constructions are bad. And I recognize that there are teachers who are able to effect a good balance of different problem types, and hook in to prior knowledge and skills.  I offer, however, something that veteran middle school teacher Vern Williams has said:

I have always stated that if a reform minded teacher produces competent, intellectually passionate students, they will absolutely escape any criticism on my part. But the opposite seems never to occur. Regardless of stellar results, the traditional teacher will always be criticized for being a self centered sage on the stage, controlling student learning and running a draconian classroom. Their students may be the happiest most accomplished students of all time but the teacher will never be good and pure until they cross over to the reform side.

In that vein, what I often hear are arguments that these “rich” problems justify the rejection of traditional type problems used over the years to teach students fundamental problem-solving skills–skills that are generalizable and transferable to many types of problems.

I happen to use the old type of problems in teaching my students how to solve problems–not problem-solve. People at my school marvel at how my students are challenged, and who show improvement in solving problems.  I am often asked what I do to get such positive results. An answer I haven’t yet uttered but am tempted to do so is: “You know all those types of problems that they tell teachers we shouldn’t give students to solve? Well, that’s what I do.”

And Another Thing, Dept.

Back in November, 2015, online Atlantic published an article that Katharine Beals and I wrote on explaining your answer in math.  It generated some controversy in the comment section, as well as some discussion in math blogs.

As I’ve indicated in other posts, the article was selected for inclusion in the anthology “Best Writing on Mathematics, 2016” published by Princeton University Press.

Apparently, one particular physics professor got annoyed with it first time around, and annoyed again when it made the rounds a second time after publication in the anthology. So annoyed in fact that he ranted about it in an article he wrote for Forbes. He states:

There’s a lot in the Garelick and Beals piece that I intensely dislike, but their core argument boils down to “The real point of math is being able to get the right answer, so as long as students get the right number, nothing else should matter.”

Actually that’s not what we were saying. We were saying that in lower grades, requiring explanations of problems so simple that they defy explanation confuses rather than enlightens–“englightens” as in providing “deep understanding”.

The author claims that he “hated being required to ‘show work’ for math problems too…” Again, Katharine Beals and I have nothing against showing one’s work, and in fact the main premise of our article is that showing the math that one did to arrive at an answer provides an explanation in and of itself.

We also have nothing against teachers asking students questions about how they arrived at their answers–such questioning technique provides guidance to students in learning what their reasoning actually was, and how to verbalize it.

Our objection is the emphasis on written explanations, particularly in lower grades (K-6, although admittedly the article uses an example from middle school).

Using diagrams as a means of explaining concepts has its use, particularly in teaching place value, but the insistence on using it, and insistence on requiring an “understanding” before students are allowed to use standard algorithms acts as a “barrier to entry” that does more harm than good. In teaching students a new procedure you need to keep it as clean as possible. Some context is good to introduce why the procedure works as it does, but one needs to move beyond that quickly.  Some students pick up on the underlying concept, but most do not. Insisting on introducing visual representations and explanations into the mix will more than likely confuse most students, who now have to try and link mentally the skill of doing the procedure and linking that to the much harder task of identifying what it might mean in a physical way.

The people who propose these ideas that images and explanations lead to “deep understanding” do so because 1) they have forgotten that they themselves benefitted from the methods that they now hold in disdain, and 2) have viewed the world through an adult and expert lens for many years so that they implicitly understand the link between numbers and their representations in the real world. Young students largely do not. Anyone who teaches knows that problems are much harder when put into contexts rather than just left as math, yet they suggest that putting new concepts into contexts makes understanding easier.

Anna Stokke, a math professor at University of Winnipeg and is an advocate for better math education in Canada puts it this way:

The understanding piece is a lot more difficult for students. They generally don’t like it and it’s something that really comes with much experience and mathematical maturity. It won’t make students like math more if we spend more time on understanding…it will just confuse and frustrate them more. In my experience, I’ve found that students like step-by-step procedures and algorithms more than anything else.

The fetish towards understanding predates Common Core and has been going on for 28+ years with the advent of the NCTM standards which pushed these ideas. (Many of these ideas and ideals are embedded through what Tom Loveless of Brookings calls the “dog whistles” of math reform that appear in Common Core.) I strongly suspect that the reason that students arrive at high school profoundly confused is because far too much emphasis is put on “understanding” before the students are ready to do that.

Understanding, critical thinking, problem solving come when students can draw on a strong foundation of domain content relevant to the topic currently being learned. As students find their feet and establish a larger repertoire of mastered knowledge and methods, the more articulate they can become in explanations. Put in neuroscience terms … the pre-frontal cortex (where critical thinking takes place) is underdeveloped in early and middle school years. It undergoes rapid development through teen years (where self-concept is growing) and this is where students should be challenged to more sophisticated reasoning, explanation of meaning and so on. It is not fully developed until one reaches early adulthood, sometime in one’s 20s. When a small child is asked to engage in critical thinking about abstract ideas, they will produce a response that may look like independent reasoning to an untrained adult, but it will involve more of a limbic response. That is, they are responding emotionally and intuitively, not logically and with “understanding”. That may be behaviorally interesting, but it is not mathematical development and it leaves them behind in the development of their fundamental skills.

Articles I Never Finished Reading, Dept.

This one is on how it’s good when parents don’t understand their kids homework, and the usual folderol about how students used to learn via “tricks” and now it’s all about “understanding”.

They pulled out the old chestnut about the fractional division rule of invert and multiply.  It starts out the same as it always does:

“Back when you were in school and when I was in school, the way we learned mathematics — and I’ll talk about the division of fractions — we all learned the trick. You flip (the fraction) over, then you multiply and that’s how you come up with the answer,” said Principal Fernando Hernandez. “It worked, but that didn’t mean that you understand the concept.

I have my own ideas about that but what caught my interest (and stopped further reading) was what came after:

“So something we would ask the students to do now is we might actually give them the answer. ‘One divided by two-thirds is one-half. Please justify that, prove to me that that is true.'”

In case you’re wondering, 1 divided by 2/3 = 3/2 not 1/2, but aside from that, what they’re trying to say is that they want students to be able to show that just as in dividing whole numbers, you can reverse the process and multiply the quotient by the divisor to get the dividend.  Which doesn’t really explain why the invert and multiply rule works.  But as I’ve said many times, if you give me two students of whom one knows why the rule works and the other doesn’t, but both can solve a word problem that requires fractional division, I can’t tell which one knows why the rule works.

Articles I never finished reading, Dept.

Another in the never-ending series on how ed tech can be used in positive ways.

This section was as far as I got (which admittedly is pretty much near the end of the article.)  I have a strong stomach for this kind of stuff, but I have my limits:

“Exploration: The technology should provide opportunities for students to explore by conjecturing, testing out different ideas, and making mistakes. We should avoid digital learning programs that focus only on memorization or funnel students’ thinking.”

Nothing wrong with this idea per se but the disdain for memorization is quite apparent. Kids need to memorize their facts, period, and some programs actually help them do that. We should avoid such things unless it has the trappings of “conjecture” and other sound-good words?

“Multiple Solution Strategies. Identify technology applications that have more than one way to solve the problems. For example, rather than using digital flashcards such as 3+4 = ?, we can identify apps that ask students to find pairs of numbers that add to 7. The latter question has many solutions such as 1 & 6, 2 & 5, 0 & 7 and supports students to understand how one whole number (in this case 7) can be broken into parts in multiple ways.”

This is like those problems that reformers love to have TED talks about: “The number is 28; tell me everything about it.” Or “The rectangle has an area of 36; what are its dimensions.” Everything open-ended, nothing confined. You still have to know your math facts, no matter how you dress it up, and the open-ended approach serves as just another way to avoid that. In my opinion and no one else’s of course.

“Connections between concepts and procedures. Good educational technology supports students to focus on relationships, not discrete facts. Rather than choose a digital program that solely focuses on doing the same procedure over and over, identify a program that supports students to understand why the procedure works. For example, with regards to the earlier problem 3+4 = ?, a digital program that includes other representations, such as images of objects that students move around can better support to develop meaning of the procedure. Digital math games that focus solely on procedures should only be considered after students have strong understanding between concepts and procedures.”

Yes, and no article on education would be complete without the “procedures-bad, concepts-good” recitation. Reminds me of a boss I had who whenever he used the word “strength” when talking to the people working for him, as in “you have some very good strengths” he would be quick to add “but you have weaknesses too”, ostensibly to forestall any of us asking for a raise. In the above quote note the allegiance to “understanding must come first”. Then and only then can students do all the procedures you want them to do. How’s that been working out for the nation for the past 28+ years?

In Defense of Common Core Criticisms

In an article called “In Defense of the Common Core”, Scott F. Marion, president of the National Center for the Improvement of Educational Assessment, argues that the CC will stop the book publishers from dictating curriculum and how subjects will be taught:

Even those Jeffersonian critics of the Common Core would have to acknowledge that while they are fighting against one authority, they are allowing a more distant and surreptitious entity (textbook publishers) to dictate the curriculum in their schools. It is hard to blame publishers, because except in limited cases, they survive by appealing to the broadest market possible.  

The Common Core has helped change this in a couple of key ways. First, the textbook publishers, spurred in part by evaluations produced by organizations such as EdReports, have begun to produce materials aligned with the Common Core. Theoretically, this will lead to more focused, shorter, and relevant texts in many states.

What he fails to mention is that, with respect to math, before Common Core, math education was dominated by the reform-math approaches embedded in the standards of the National Council of Teachers of Mathematics (NCTM).  Interpretations of these standards are in part–some might even say in large part–responsible for why the textbooks evolved to the products they are today. He seems to think that the Common Core standards will result in changes in textbooks, assuming that alignment with Common Core will produce something different than what we have today. In fact, interpretation and implementation of Common Core is along the ideological lines of reform math and that for all intents and purposes, the Common Core math standards has become a reform math document, an opinion shared by not a few mathematicians, including Jim Milgram, Wayne Bishop, and Frank Quinn.

The CC math standards are promoted as ‘pedagogically neutral’, as ‘guidelines, not a curriculum’ and ‘Teachers can use whatever tools they want to help students meet the standards’. Why is it, then, that many Common Core inspired assignments bear the reform/progressivist imprints: student-centered and discovery-driven assignments; group-based and real-life-relevant; touted as fostering ‘critical-thinking’?  One clue comes from the language embedded in the standards–what Tom Loveless of Brookings Institution refers to as the “dog whistles” of math reform, picked up by reformers.

The words “explain” and “understand” are the prime examples of this, which are embedded in many of the standards. These are key signals to the reformers/progressivists. And given the reform background of some of the people on the math standards team, I do not believe this is entirely coincidental. The three lead writers were Phil Daro, Bill McCallum and Jason Zimba. Phil has a degree in English literature, and I’m told has a minor in math and taught high school algebra, briefly. He has been active for years in promoting reform math. Bill McCallum, a math professor who teaches at University of Arizona has been sympathetic with the reform approach. Jason Zimba who has a doctorate in Physics and taught at Bennington College is less sympathetic to reform math ideas than the other two; but public statements he has made indicate he is not averse to such ideas. The rest of the writing team as well as the team that reviewed and commented on the standards, were largely reform oriented.

Scott concludes his defense of CC by saying “If we reject the Common Core, painstakingly developed with input from educators and researchers, we are essentially ceding our standards back to textbook publishers.”

Ignoring the fact that few if any educators and researchers were sought in developing the standards, if anything is being ceded, it is its how CC is being interpreted. Being a neutral standard regarding pedagogy means that it has become law. Being neutral is the same as authorizing all schools to continue to use techniques that cause many of the problems in math. Alignment with the CC standards, far from taking us away from the status quo, means that there will only be more of the same.

Frank Quinn, a math professor at Virginia Polytehnic University, summarizes the state of affairs in a paper that examines how Common Core treats how fractions are to be taught.  He states:

Elementary education was largely insulated from the pressures driving the profession. Instead, for at least the last century, the main pressure on educational methodology comes from the need to sell it to administrators, legislators, and the general public. The current Reform movement represents a breakthrough in this direction, as powerful in its own way as any technical innovation in professional practice. They have made a reduction in skill levels sound exciting, and done it so well that they have become the dominant movement in just a few decades. The introduction to the Common Core document, and the “Standards For Mathematical Practice” that follow, offer visions and abstract goals that are much more compelling than any Traditional or Modern account. They have the practical effect of reducing skill expectations almost to zero, but this gives the Movement a further tactical advantage: modern methods are distinguished by their efficiency and power, but in the Common Core there is not much for them to do. A hammer does not look good when it is used to squash ants.

Using Quinn’s analogy, the only breakthroughs with respect to Common Core appear to be ways to squash ants more effectively with hammers.

Best Writing on Mathematics, 2016

Happy to report that the article that Katharine Beals and I wrote which was published in the online Atlantic in November, 2015, was selected in the anthology of “Best Writing on Mathematics, 2016”. 

The article generated a great deal of interest at the time and drew both hostile and supporting comments in the comment section and in several math education blogs at the time.

Katharine and I are honored to be included in this anthology.