Love Notes of the Past, Dept.

One of many comments received on the first article I wrote criticizing Common Core 

In general, I think there is a problem with Garelick’s body of work in that it tends to limit the conversation about math to arithmetic alone – and not only to arithmetic, but to its operations – and not only to its operations, but to its whole-number operations – and not only to its whole-number operations, but to the algorithms for those operations – and not only to the algorithms for those operations, but to the results of those algorithms. Garelick does discuss fractions in this piece, but it’s still concerned solely with results of algorithms for operations – again, as if that’s what math equals. I also think he is off base in the fraction critique itself. Many mathematicians (which Garelick isn’t) would say the relationship between multiplication and division is an important aspect of learning about rational number arithmetic.

Pretty flat criticism considering that the article in question as well as my “body of work” at the time focused on the foundational portion of math; aka arithmetic.


On “inflexible knowledge”

The concept of inflexible knowledge has made the rounds on the edu-circuit for years now.  In short, math education isn’t working, because if it were, then people would be able to apply prior knowledge to solve problems in new situations. That is, knowledge would just transfer like a hot knife through butter.

Robert Craigen, math professor at U. of Manitoba has this take on the idea of “inflexible knowledge”:

It isn’t knowledge that is inflexible — it is people. One should learn the knowledge, and then have practice and exposure applying it in flexible ways. The error of progressive ed is the notion that if one doesn’t acquire knowledge in particular ways then it isn’t flexible. 

Somebody please kill me, Dept.

I am required to attend six (6) all day professional development sessions over the course of this school year.  The sessions encourage “collaboration” amongst the math teachers in the county where I teach. It will be “facilitated” (as in “moderated” and other soft words that mean the word that shall not be mentioned: “led”)  by the person who wrote this blog entry.

The basic assumption of the blog is that students who are not at grade level can be brought to the appropriate level through “just in time” learning.

Like a magician who is adept at slight of hand, we start off with a definition that many readers swallow hook, line and sinker: I.e., that “gaps in learning” are somehow essentially different from “unfinished learning”. The author then posits that “We can ensure access to grade level mathematics even if a student has unfinished learning by intentionally planning just in time formative learning process.”

Skipping over the buzzword of “formative” which used to mean “teaching”, and just in case there were any misgivings over the assumption that the author wants us all to swallow, there’s this:

“I’ve never believed a student comes to us with holes or gaps in understanding, as in my own mind that is deficit thinking. It assumes that students can’t or aren’t able to attain grade level or mastery. How many of us have had students that just weren’t quite there yet, and given a bit more time or a different approach got it!”

Well, to tell you the truth, different approaches are nice, but if someone has difficulty multiplying or adding/subtracting, does not understand how 4 can be expressed as 3 and 4/4, continually balks at finding a common denominator to add fractions, and by grade 9, say, has no proficiency with fractions, decimals or percents, then in my experience (as with many others who I’ve met through the years) different approaches don’t make much difference. But the author contends that the “unfinished learning” can be addressed via “just in time” planning:

“If we are able to anticipate conceptual misconceptions, procedural disconnects or skill-based errors, we can prepare an activity or a few questions, for a just in time intervention, that will support learners in completing their learning.”

I’ve talked before how use of the word “learners” instead of “students” is more than a bit annoying, as if we have upgraded our approach to education so much that to refer to students as students means we are going back to “the old ways” which everyone knows didn’t work. So no need for me to go there. Let me focus therefore on the author’s method for doing this “just in time” intervention:

“How do we begin just in time planning? Understand The Shifts : Focus, Coherence, and Rigor. ”

OK, let’s stop there a moment. “The Shifts” (which reminds me of an ad campaign in the mid 80-s that was supposed to promote the State of California by calling the state “The Californias”) refers to a discussion on the Common Core website on how Common Core results in shifts in instructional strategy. Note that the discussion of “the shifts” is commentary on the website, and not part of the standards. They are the authors’ view of the consequences of Common Core standards and represent what the authors believe would and should happen upon implementation of the standards.  In other words, they are not part of the standards themselves. Nevertheless, even before the ink dried on the Common Core standards, proponents of Common Core talked about “The Shifts” as if they were/are enforceable parts of the standards themselves.

One of the shifts is “rigor,” which the website translates as: “Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity.” The site also mentions that students should attain fluency with core functions such as multiplication (and by extension, multiplication of fractions): “Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures.”

I learned of the connection between these “instructional shifts” and the current practice of drilling understanding in a conversation I had with one of the key writers and designers of the EngageNY/Eureka Math program. EngageNY started in New York state to fulfill Common Core and is now being used in many school districts across the United States. I noted that on the EngageNY website, the “key shifts” in math instruction went from the three on the original Common Core website (focus, coherence, and rigor) to six. The last one of these six is called “dual intensity.” According to my contact at EngageNY, it’s an interpretation of Common Core’s definition of “rigor.” It states:

Dual Intensity: Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year.

He told me the original definition of rigor at the Common Core website was a stroke of genius that made it hard for anti-intellectuals to speak of “rigorous” in loosey-goosey ways. He was able to justify EngageNY/Eureka’s emphasis on fluency. So while his intentions were good—to use the definition of “rigor” to increase the emphasis on procedural fluency—it appears he was co-opted to make sure that “understanding” took precedence.

In our discussion, I pointed to EngageNY’s insistence on students drawing diagrams to show place value in adding and subtracting numbers that required regrouping (a.k.a. “carrying” and “borrowing”—words now anathema in this new age of math understanding). I asked if students were barred from using the standard algorithm until they acquired “mastery” of the pictorial procedure.

His answer was evasive, along the lines of “Of course we want students to use numbers and not be dependent on diagrams, but it’s important that they understand how the algorithms work.” He eventually stated that Eureka “doesn’t do standard algorithms until students know the prerequisites needed to do them.”

Thus, despite Common Core’s proclamations that the standards do not prescribe pedagogical approaches, it appears their definition of “rigor” leaves room for interpretations that conclude understanding must come before procedure.

And the author of this blog seems to think that all is a matter of understanding and that a “just in time” exercise will fit in the missing pieces.

This is all part of a misguided mish-mash that passes for what Common Core is all about and what math education should be all about. There are those of us who see the results of these ideas.  Many of us have had to tutor our children, or pay for tutoring.  And some of us are forced to take six (6) PD sessions led by the author of the blog in question. In my case and others like me, we are told to try the things we know are not working well, despite good results among our students using methods held in disdain by those in power. Those of us in this situation seem to know better but are relegated to the sidelines of a never ending mutual admiration circus that passes as “evidence-based, research-based” education.

This despite much research and evidence to the contrary.

Love Notes from my Fans, Dept

Some love notes from my fans:

“According to your bio, teaching is not your primary profession. It is something you are just doing post-retirement. This does not mean you do not have useful insights. It does mean that you need to know your limitations for formulating educational policy.”

“Ph.D. in applied math here. The traditional form stinks. By focusing on the “algorithms”, the traditional method gives no intuitive insight into the relationships or what you’re actually doing. It doesn’t build on first principles. That’s why so many kids get an A in, say, Calculus but really don’t understand it at all- they just know how to plug & chug. There’s a reason why 20% of freshmen at the top universities in the country (Stanford, Berkeley, Harvard, etc.) have to be remediated even after acing their AP courses.”

Yes, my fans dearly love me. And with plaudits like these, who can resist reading more of what I have to say. You can start with “Math Education in the US: Still Crazy After All These Years”. Makes an ideal gift for people whose idea of viable educational policy is formed by what they learned in ed school and reading articles in Phi Delta Kappan.

Available at Amazon

Shameless Promotion, Dept.

This just in from Sinal Singh, author of “Pi of Life: The Hidden Happiness of Mathematics” and a math consultant at Scolab:

“Barry Garelick is a surly sycophant for all things anachronistic.”

What more reason do you need to buy my book; available now at Amazon. Buy 10 copies and give them to your enemies!

Math Education in the U.S.: Still Crazy After All These Years “Hell hath no fury like a mathematician whose child has been scorned by an education system that refuses to know better.” 


And while you’re at it, you may as well buy the whole set.  Your enemies will love you for the gifts!

Confessions of a 21st Century Math Teacher  (For anyone concerned with what Common Core is bringing about in the name of 21st century math education, STEM education, and “21st century skills, this book is a must-read. )

And the book that started it all: Letters from John Dewey/Letters from Huck Finn  “Few refuges exist from the multicolored tomes posing as math textbooks. No one is safe from this modern day invasion of the body snatchers. And just like in the movie, those with the power to do something have already been taken over by the seed pods of education school dogma.”

Canadian Tropes, Dept.

Interesting to see that what passes as sound educational policies in Canada are based on the same nonsense that US educational policies are based on. To wit and for example in this latest Globe and Mail article about Ontario’s math ed crisis, is this quote from Annie Kidder, executive director for the advocacy group People for Education who is welcoming a reform of all aspects of the math curriculum in Ontario:

“I think that this is a very important move, that we recognize the importance of what are sometimes called … global competencies to student learning in all areas. It’s important that we recognize that there’s more to life than the three Rs,” Ms. Kidder said. She added: “It’s obviously important that kids read, write and do math. But it’s also important that they know how to collaborate … and they’re able to be successful in a knowledge economy and in jobs that don’t exist as yet.”

I don’t understand the hang-up with “collaboration”. If math were taught properly there would be less of a need for students to collaborate since they would be better able to figure things out on their own. The assumption that the working world is now based on collaboration more than ever before is one of those characterizations that comes from people who don’t work in the real world on a day to day basis. Yes, people work on projects collaboratively but usually each person brings their own particular expertise to the table. There is a difference between experts and novices. The collaboration one sees with students is usually the riding on the backs of the students who can do the work–because perhaps they learned it via parents, tutors or learning centers where they were taught what wasn’t being taught at school.

The math wars in Canada continue

Anna Stokke tells it like it is in Canada in a no holds barred piece in the Globe and Mail.

“After six years of advocating for better math education in Canada, I have noticed a frustrating cycle that ministries have done little to break. Expensive consultants are hired to provide teacher professional development on unproven fads. Resources are then purchased to support these ineffective methods in the classroom, which produces more struggling students who need extra support. After a round of testing shows that students are doing more poorly in math, the same people who created the problem decide that teachers need more support using the ineffective methods. More PD and resources are then purchased, and the cycle continues. Parents are often left with no other option but to hire tutors to cover the gaps and those who can’t afford tutors watch helplessly as their children get further behind.”

Canada’s approach to math education differs from the US in that some of the provincial Ministries of Education (MoE’s) are mandating via the standards the inquiry-based approaches and bad practices. In the US such practices are not mandated, but strongly hinted at through the “dog whistles” of reform math embedded in the Common Core standards. Previous to Common Core, the NCTM’s standards paved the way.

Canada’s MoE’s are hanging tight, but at least there is a press that represents the parents much more strongly than one sees in the US as evidenced by Anna’s piece.

Tell it, Anna!

Articles I didn’t finish reading, Dept.

This article in Business North was titled “Teachers Learn to Make Math Fun, Engaging”, which alone would have caused me not to read any further, but I was curious as to how many tropes I would run into. The article doesn’t disappoint.

It starts off with an opening as classic (and nausea inducing) as Denny’s bacon and egg eye-opener:

“When will I ever use this in my real life?” is a plaintive question students often pose to their math teachers. For over 20 area elementary and middle school teachers, the “Engage, Learn, and Connect Math Topics” summer workshop at The College of St. Scholastica (CSS) provided teachers some solid ways to respond to that perennial question.

I usually find that students ask this when they are frustrated and/or having difficulty with a particular type of procedure or problem. When they are capable of doing the procedure, they tend to be just as engaged as they would with any activity. Not to mention that the prevalence of this question is helped along by TV sitcoms that may feature such a situation. The question is met with the predictable laugh track as the camera zooms in to a close up of teacher’s frustrated expression.  

“Engagement is key. There is no one size fits all approach to teaching. Workshops like ours help to bridge some gaps and help teachers dispel some misconceptions of what it means to study math.”

Of course, what is ignored is what I pointed out above; that proper instruction that allows students to be able to do problems and stretch beyond the initial worked example breeds success.  And success in turn breeds motivation. But the pervasive group think is that engagement comes first, and then success and motivation follow.  

And part and parcel to engagement, is this gem of a trope without which no article would be complete:

“It is important for our society to prepare kids for the jobs of the future. There are new jobs that don’t even exist yet that will come as a result of the new ideas in the math and science fields. We want to engage students so they know they can succeed in those fields and to prepare them to be ready for the new opportunities yet to come.”

Oops, I guess I did finish the article, didn’t I?


Oh, Really, Dept.

There is a very long article in Quanta about various schools in the New York City area and how teachers go about teaching science and math.  The context of the article is the usual “The US has not done well in science and math”, and the not so subtle sub-text of “That’s because we’re teaching it wrong”. Per the article, “Wrong” means teaching by rote, by cookbook, by telling students formulas instead of having them derive it, by having them do experiments with predictable results. In short, the traditional model of science and math treats students as novices. The collective wisdom of the teachers featured in this article, however, treats students as if they are experts.

The emphasis is project-based/inquiry-based structures in which students are given an assignment or problem with some prior knowledge, and expected to make up what they don’t know on a just-in-time basis.  Guidance is kept to a minimum, with the student expected to discover what they are supposed to learn–or, in the case of science experiments gone awry, to discover what was the cause, and correct it.

The article profiles several teachers who have used these various methods. If a teacher can have success using such techniques, that’s great. But in reading the article, I do have to wonder about such methods and their basic assumptions and mischaracterization of the conventional/traditional techniques which they hold as “not working”.

For example:

Too often, said Vice Principal Elizabeth Leebens, students “get the history of science rather than getting an opportunity to do it for themselves.” Still, Leebens was surprised when, after a group of science club students asked Comer for help with their ice-cream-making experiment, which had already failed seven sticky times, Comer told them, “Go back and check your process.” Off they went to the bathroom to dump the latest batch and start over.

“I call it productive struggle,” Comer said. “That’s where the growth happens.”

In meetings, Leebens said, Comer has “challenged me to stop doing the cognitive work for kids: ‘Let them do it themselves. They can do it; they can do it.’” For Comer, she added, it’s about “life lessons and also high expectations for kids in letting them see what they can do before the adult decides what they can do.”

The “struggle is good” philosophy prevails at these schools and among these teachers. I am guessing that because such approach has come under criticism in the past, that it is now called “productive struggle”.  In any event, there are good ways and poor ways to make students struggle.  One hears about “scaffolding” problems so that students stretch beyond initial worked examples or experiments but have enough prior knowledge to make the leap.  But there are those who consider “scaffolding” as handing it to the student, giving the student too much information so they don’t get the “deep understanding” nor learn how to “problem solve” (to use the edu-trendy parlance that has gained popularity over the past few years).

The article talks about a math teacher and how she came to her particular philosophy of productive struggle.

Midha has always loved math — it came easily. She suffered some doubts about her abilities in 10th grade, thanks to an algebra teacher who did the standard “chalk and talk” at the blackboard. But during freshman year at Wesleyan University, she took calculus with a professor who considered mathematics to be an art. “Everything he said was so profound,” she recalled. “I was like, ‘Oh, my god. This is the coolest thing ever.’” And that was that. She became a math major — and also a music major; she plays classical guitar.

With this quote, the article then lays the groundwork  that the standard “chalk and talk” doesn’t work (and making the general assumption that all teachers using traditional methods never ask questions of students, but just lectures in a boring and uninspiring manner).  Rather, teachers are required to be inspiring and profound and math is to be taught as an art. Or something like that.  I don’t know much else about her calculus professor or what he did in class, but I suspect that he relied just a little bit on “chalk and talk”. 

The article goes on to describe a lesson the Midha gives her students in which they have to determine the heights of buildings outside their classroom using an iPhone app that served as an inclinometer, and using what they know about trigonometry.  She calls this a “struggle problem”:

Using two iPhone apps, they measured the angle of elevation from point A to the top of the building and the distance the walker traveled from point A to point B, at the base of the building. These two measurements gave them crucial pieces of information about a right triangle, and from there — using what they’d learned so far in class about trigonometry — they are now charged with the task of calculating their building’s height. But first there are ponderous stares, frowns, diagrams drawn and redrawn amid plumes of eraser dust, and a collective buzz of puzzlement:

“I dunno, man. I really don’t know.”

“Help! Help!”

But they won’t get much help from their teacher.

“I’m the teacher who stopped giving them the answer,” said the 30-something Midha. “In every unit that we do, I warn them: ‘I’m going to give you the tools that you need, but I’m never going to tell you how to do something. You have to figure out how to do it, you have to figure out the answer, and you have to prove to me why you think that answer is what it is.’” She also offers reassurance through an oft-repeated mantra: “The only way that you can fail is if you give up. If you continue to persevere, if you continue to try, if you continue to work through this, you will get this. But if you give up, you will fail.”

But after a few days of this, with students not getting anywhere she decides to give them some hints:

“Remember: Where was the angle of elevation measured from? The eye. So when you are drawing and calculating, remember that. Your job is to calculate the total height of the building. … Remember: There is a reason we measured the eye height. There is a reason we measured the eye height.” Repetition, and more repetition, is key for penetrating the adolescent brain.

Midha also provides a bonus hint of sorts, pointing out to her students that they measured the eye height in inches and the building distance in miles, and that the worksheet asks for the height of the building in feet. With that, she leaves her students to their own devices —“Good luck!”

She circles the room, surveying the progress, asking simply: “Does it make any sense? … Why doesn’t it make any sense?” Despite her hints, the relevance of the eye height is proving elusive, and the inch-feet-miles conversions are confusing. She reiterates her tips one-on-one with the groups, and then lets them loose again, declaring: “I’m going to walk away now … ”

Which raises the question of whether it would do any harm to just tell students what they need to know when after some grappling with the problem they are clearly in a receptive mode to receive and process such information.  No, I guess not.  Best to just let them struggle.

I will stick with the more controlled variety of struggle, and provide the scaffolding needed and when it appears fruitful, tell them what they are now ready to hear.  And in so doing, it appears that many will brand me “old school”. I recall a professor I had in ed school saying to a student that such direct instruction may be good in the short term, with students scoring higher on tests than those using inquiry-based techniques. “But in the long term, those using inquiry have a deeper understanding.”

This appears to be the premise that guides the teachers described in the article, as well as an ever-growing body of teachers indoctrinated into this type of teaching.  

I’ll risk being called old school–and other names.






Count the Tropes, Dept.

Yet another in an unending series of articles about math education and what parents should (and should not) be doing to help.  This particular one is from Chicago Parent and contains the usual tropes/mischaracterization about how math used to be taught and why the new ways are so much better.  We start with the usual ever-popular one:

STOP TEACHING THE TRICKS: A large amount of research has gone into the progression of teaching students mathematical concepts. The shift has moved away from teaching students to blindly follow rules and toward making sure they understand the larger mathematical ideas and reasoning behind the processes.

I have written extensively about how math was taught in the past, citing and providing examples from textbooks from various eras. It might be interesting to look at some of the books used in previous eras that have been described as teaching students to “blindly follow rules”.  Many, if not most, of the math books from the 30’s through part of the 60’s were written by the math reformers of those times.  It makes the most sense to start with the series I had in elementary school: Arithmetic We Need.  The reason is because not only is it from the 50’s, but also one of the authors was William A. Brownell, considered a leader of the math reform movement from the 30’s through the early 60’s. Today’s reformers also hold Brownell in high regard, including the prolific education critic Alfie Kohn, who talks about him in his book The Schools Our Children Deserve .

In arguing why traditional math is ineffective, Kohn states “students may memorize the fact that 0.4 = 4/10, or successfully follow a recipe to solve for x, but the traditional approach leaves them clueless about the significance of what they’re doing.  Without any feel for the bigger picture, they tend to plug in numbers mechanically as they follow the technique they’ve learned.”  He then turns to Brownell to bolster his argument that students under traditional math were not successful in quantitative thinking: “[For that] one needs a fund of meanings, not a myriad of ‘automatic responses’. . . . Drill does not develop meanings.  Repetition does not lead to understandings.”

Brownell, however, requires students to do the practice and exercises held in disdain by those who believe traditionally taught math did not work.  The series contained many exercises and drills including mental math exercises. Such drills might appear to run counter to Brownell’s arguments for math being more than computation and “meaningless drills,” but their inclusion ensured that mastery of math facts and basic procedures was not lost. Also, the books contained many word problems that demonstrated how the various math concepts and procedures are used to solve a variety of problem types. Other books from previous eras were also similarly written—most authors were the math reformers of their day—and provide many counter-examples to the mischaracterization that traditional math consisted only of disconnected ideas, rote memorization, and no understanding.

But let’s move on. It gets worse:

…By teaching the trick before a child has this foundation, you may be inadvertently doing more harm than good. Students become reliant on tricks and fail to master the conceptual understanding needed to use the tricks appropriately.  Remember, kids will be more successful in the future as a problem-solver than as a memorizer.

What this article and many look-alikes caricature as “tricks” are actually mathematically sound algorithms. The idea that teaching standard algorithms “too early” eclipses the underlying conceptual understanding most likely stems from Constance Kamii’s infamous study “The Harmful Effect of Teaching Algorithms to Young Children” which was published by the National Council of Teachers of Mathematics (NCTM) in an annual review.  It has become the rallying cry that has garnered more believers than the idea that the substance called “laetrile”, extracted from apricot pits, is a cure for cancer.

With respect to the math books of earlier eras, they started with teaching of the standard algorithm first.  Alternatives to the standards using drawings or other techniques were given afterwards to provide further information on how and why the algorithm worked.  This is opposite of how reformers are advising it be done now.  What happens, typically, is the first way a child is taught to do something becomes their anchor, with everything else being supplemental.  By teaching the supplements first, there is a mix-up of main course versus side dish, with many students unable to tell the difference.  The popular theory is that students now have a choice and can pick the method that works for them.  I have tutored students showing profound confusion, asking me what method they should use for particular problems, feeling that various problems demand different versions of the same algorithm.

Note also the ever-popular warning against memorization: Memorizers are not problem solvers apparently. Well, sure, there may have been teachers who taught math poorly and had students memorize day in and day out with no conceptual context. To listen to the people who write these articles, it seems that the nation was plagued with such teaching, as if poor teaching was/is an inherent quality of traditionally taught math.  I would argue otherwise and go so far as to say that many of the “understanding-based”, student-centered, collaborative techniques that dominate many of today’s classrooms are inherently ineffective and damaging.

Memorization is the seat of knowledge. Eventually students just have to know certain facts and procedures and do them automatically. The idea that memorizing eclipses the understanding of what, say, multiplication is presumes that students are taught the times tables with no connection to what multiplication is, and what types of problems are solved using it.

Other than that, the article is pretty good, I suppose.