Chicken Little’s Rebuttal

In my last missive (The “Dog Whistles of Math Reform”) I referred to Tom Loveless’ characterization of the Common Core (CC) math standards containing code words embedded within the standards.  Such words serve as cues for reform-minded/ progressivist educators in interpreting what are purported to be pedagogically neutral standards.

One reader, subtly inferring I was a Chicken Little, raised the question that if the CC standards lend themselves to such reform-oriented interpretations, wouldn’t that simply mean that people would continue to focus on the reform-oriented teaching they were inclined to practice? And if so what is changed? Wouldn’t things just stay the same?

The short answer is no, things would not stay–and have not stayed–the same. With CC now being the vehicle of “national standards” (except for those few states that have not adopted them) such interpretations have essentially become law.  The CC standards have not changed the minds of reform minded educators, nor the education schools that promote such philosophies.  The narrative that CC is pedagogically neutral while making sly winks via the embedded “dog whistles” is the same as authorizing all schools to continue to use techniques that cause many of the problems in math–it institutionalizes the problems.

With such prevailing interpretation, schools that might not have done so in the past feel compelled to change instructional practices.  These changes are made in the name of “alignment” with CC. The philosophies and practices promoted by ed schools now have considerable more cachet. Such instruction now carries with it an implicit authorization: CC requires you to teach this way.

In the year before CC went into effect in California, I was working at a middle school where teachers were routinely told by school–and school district–administration that “Next year there will be no more teacher at the front of the room saying ‘Open your books to such and such page, and do these problems’. There would be less reliance on textbooks, and more reliance on group-work, student-centered and inquiry-based approaches. This, we were told, was in keeping with what CC required. [Shameless self promotion: I wrote about this transition year in the book “Confessions of a 21st Century Math Teacher“]

The reform ideology permeating CC’s implementation is nowhere more evident than in how the teaching of the standard algorithms are implemented. Many reformers believe that teaching standard algorithms first eclipses students’ understanding of why the procedure works. In the CC standards, the first mention of a standard algorithm is in the fourth grade—for multidigit addition and subtraction. A delay also happens with multiplication (delayed to fifth grade) and long division (delayed until sixth grade).  Prior to that, the standards refer to drawings/strategies based on place value–but not specifically to the standard algorithms.

In line with reform type thinking, there are schools that happily comply with what they think is a mandated delay in the teaching of these algorithms, and teach the alternatives. As a retired teacher of thirty years commented on my last missive regarding the alternatives taught during the delay:

“These new ‘strategies’ simply become new procedures, which small children attempt to learn and memorize because that is what many small children do. Of course these strategies are unworkable, mathematically incoherent and very confusing.”

According to two of the lead writers of the standards, Jason Zimba and Bill McCallum, however, the standard algorithms can be taught earlier than the year in which they appear. Zimba in fact says this in writing, and recommends teaching it in first grade. Specifically, he states

The Common Core requires the standard algorithm; additional algorithms aren’t named, and they aren’t required.”  

But word is not getting out.  Some teachers in the lower grades have been sending notes home to parents telling them not to teach students the standard algorithms at home.

I still maintain a guardedly optimistic outlook however, as summarized in my closing statements in the post I referenced at the beginning of this one. Read it, and send a link to your local school board.

The “Dog Whistles” of Reform, Dept.

Tom Loveless of Brookings Institution coined the phrase “dog whistles of reform math” when referring to the Common Core math standards.  He was referring to code words embedded within the standards that serve as cues for reform-minded/progressivist educators in interpreting what are purported to be pedagogically neutral standards.

A recent article discusses this back-and-forth and contrasts traditionalists against progressives in the usual manner.  Representing the reform side of the arguments inthe article, are Alan Schonfield, a mathematician from UC Berkeley, and Steven Leinwand, a lead research analyst at the American Institutes for Research (AIR):

“The idea of practicing and practicing and regurgitating a procedure flies in the face of everything we know about how to take this body of knowledge called mathematics and have it work for everyone,” rather than just those at the top of the class, said Leinwand.

He argues that despite the noise, the Common Core, and the similar math standards states have held onto, are working for parents, students, and teachers. He rejects the arguments made by the traditionalists, whom he said are only animated about protecting their supply of high achieving math majors, rather than ensuring that each student has a solid foundation in the subject.

“Why the hell, after 100 years of that kind of math teaching” said Leinwand of the traditionalists, “was the U.S. doing so poorly, until recently?”

Aside from the usual mischaracterizations of traditionally taught math, I was particularly interested in this last statement of Leinwand’s and wondered what data he was using to bolster his argument given the rise of the remediation courses in math that college students have been taking over the past two decades.  (Some of these issues have been discussed previously here.)

I was also interested in a statement attributed to Schonfield on the debate overthe interpretation of the standards vs the standards themselves:

In an email exchange with InsideSources, Schoenfeld said that the Common Core had become a “Rorschach test” that people use to project their feelings about education. So while the debate over math practices is real, and the Common Core aligned math standards actually are more amenable to progressive teaching styles, the math standards themselves do not mandate how the subject should be taught.

It is true that on the Common Core website, we are assured that “These Standards do not dictate curriculum or teaching methods”.  But his statement that the standards are “more amenable” to the ideas of reformers/progressivists aligns with Tom Loveless’ “dog whistle” theory which would explain why many of the Common Core inspired assignments shown on television or the internet, bear the reform/progressivist imprints: student-centered and discovery-driven assignments; group-based and real-life-relevant; touted as fostering ‘critical-thinking.

The dog whistles that are picked up by reformers/progressivists are embedded in the standards:

  • “…explain the reasoning used.” (2nd year/1st grade)
  • Explain why addition and subtraction strategies work, using place value and the properties of operations. (3rd year/2nd grade)
  • Understand the relationship between numbers and quantities… (K)
  • Understand a fraction as a number on the number line…” (4th year/3rd grade)

The dog whistles of “understand” and “explain” feed into a key component of Common Core: A set of 8 standards called “Standards for Mathematical Practice” or SMP’s. The SMPs are eight practices that 1) supposedly embody the work habits and general mode of thought of mathematicians, 2) were defined largely by non-mathematicians. They also were based on what were called “process standards” in NCTM’s standards and were one of the main mechanisms for putting into action the reform math practices.

The SMPs are these:

1.Make sense of problem solving and persevere in solving them

2.Reason abstractly and quantitatively

3.Construct viable arguments and critique the reasoning of others

4.Model with mathematics

5.Use appropriate tools strategically

6.Attend to precision

7.Look for and make use of structure

8.Look for and express regularity in repeated reasoning

Taken at face value, they are seemingly benign. For example, there’s nothing wrong with the first point: “Make sense of problem solving and persevering in solving them.” Who wouldn’t want this? But the words “problem solving” are a signal for convoluted “real world” problems and “just in time” learning. The SMPs are largely interpreted as a means to teach “habits of mind” often outside of the context of the math courses from which they would arise naturally.

They don’t have to be interpreted this way.  But the reformers/progressivists’ view of CC in general and the SMP’s in particular is that they require inquiry activities and collaborative group work, students solving problems in more than one way and most importantly for “explaining their answers” rather than simply showing their work.

For this last, the assumption is if students cannot explain their reasoning, in writing or otherwise, students lack understanding.  But since students in lower and middle grades are novices and lack skill in articulating ideas, such explanations often have little mathematical value. It often amounts to students engaging in the exercise of guessing (or learning) what the teacher wants to hear. More “rote understanding” via “rote explaining”

It is my hope that we come to some agreement on how best to teach math, and in that vein, in closing,  I offer these guardedly optimistic statements:

  • Whether understanding or procedure comes first ought to be driven by subject matter and student need — not by educational ideology
  • Prior learning and knowledge is the greatest determinant of what children can learn, regardless of their physical age.
  • Curricula should be both mathematically coherent and logically sequenced for learning from novice to expert.
  • “Discovery” should not be conflated with “teaching understanding” as if they are one and the same
  • Mistakes in educational practices should not be clung to just because of the time spent making them


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?