Engaging your way through math


Learning math through engagement: fun until the very end.


Over-Corrections Revisited

In a piece I wrote earlier I talked about how Dweck’s and Duckworth’s respective theories of growth mindsets and grit were being exaggerated and misinterpreted, exhibiting the tendency to “overcorrect”.

I also referenced Dan Willingham’s famous quote: “Sometimes I think that we as teachers are so eager to get to the answers that we do not devote sufficient time to develop the questions.”

I admitted I hadn’t read the book and made accusations of people running with that quote and using computer graphics/animations as approaches to “engage” students. I was met with a comment criticizing me for misinterpreting what the person was doing based on a book I never read. Fair criticism, so I read the entire context of Willingham’s quote. Having put the quote in context, what he was talking about was in fact, putting topics “in context” for students, rather than teaching a topic with no connections whatsoever, in a “do this, do that, answer this, answer that” mode. (Traditionally taught math is often mischaracterized this way: as students being taught skills and facts “by rote” with no connections to how such procedures were arrived or what they mean.)

The school where I teach has recently subscribed to a service called Pear Deck, which allows teachers to construct presentations in which students can interact, by answering questions on their computer.  What caught my eye was this particular promotion of a Pear Deck presentation. I tweeted something about the presentation. I was met with various replies such as:

“Some tedium part of the deal. Not all things worth learning can be presented as fun, games.”

“Love algebra, no interest in video games. Not wondering anything about Super Mario.”

“I’d rather try to calculate when two trains going different speeds are going to meet up.”

I also received this:

“We believe that you can’t teach students before they have a question.”

I’m not sure if the “we” refers to people at Pear Deck and that the person who tweeted it works for Pear Deck. But it’s evident that “they” are running with the Willingham quote.

Context is important, for sure. And while there’s nothing wrong with a video game or graphic approach, per se, it also is not the only way to engage students. Sometimes, just asking the question in an interesting manner is enough to stimulate curiosity. One method I have used is in warm-up questions before class begins. Some of the questions are based on what they may have learned the day before. We might have talked about the equation of a line in the form y = mx + b, let’s say. But a warm up question might then ask “What is the y-intercept (b) of the line y = 5x + b if it contains the point (2,3)?”Which was not discussed. In fact, it is the topic of the day’s lesson.  Some students may be able to stretch and use prior knowledge to actually answer it, others may be stuck. But curiousity has been aroused: “What’s the answer? How do you do it?” And the day’s lesson begins.

Activities purported to be engaging, such as finding the relationship between maximum area for a given perimeter of a rectangle, using graphics technology  tend to be distractions. They are distractions if the math necessary to solve it in a more straightforward manner is not provided. As such, it may be “engaging” but does not provide mathematical information that can actually be transferred in solving other problems.  (For example, instead of trying combinations of widths and lengths, why not graph the parabola that describes area as a function of various lengths and widths with a constant perimeter, and find the maximum point? Such an approach is a link to preparing students for the optimization approach that they will eventually learn in calculus)

Robert Craigen, a math professor at University of Manitoba, wrote about this in an earlier post about this topic, and I reproduce his comments here for you to think about:

“Or even more to the point, it may represent a perfectly fine instructional piece all by itself in isolation, but not contribute meaningfully to coverage of the curriculum for a given class. That is what is wrong with this stress on digressive activities of the sort I collect under the rubric “candy” above. In a piece we wrote some time ago, Barry and I made a point about the distinction between “main dish” and “side dish” instruction. Side dishes can indeed make the experience of classes interesting and engaging, but they may or may not actually represent progress toward the main dishes in the course, as laid out in the curriculum. The problem with these miscellaneous “bright ideas” for one-off lessons you find on the internet for free or offered by these mercenary educational software groups is that they tend to be side dishes. Too much of that and your students may not get to through the main dish, and all these sparkly bright ideas will turn out to be a poor idea in the big picture.”


Common Core Against the World, Dept.

On a website promoting Common Core tests being superior to other tests,  ExcelinEd cites some examples of questions. In all fairness, some of the questions are indeed better than those which can result in correct answers by guessing or other means. But in other cases, they seem to mince words.
For a high school question they cite an example of the “Previous math question”, in which “previous” means pre-CC:
If 3(y-1) = 8, then what is y?
They feel that this question “is an example of solving equations as a series of mechanical steps.”
The CC type question however is in their minds much better: “What are two different equations with the same solution as 3(y-1) = 8?”
They state: “This question is an example of solving equations as a process of reasoning.”
Well, I guess, but they still both, at the end of the day, involve a series of mechanical steps. And both require that the initial equation be solved.
Seems like a distinction without a difference. And there are some tests that have been used for years that have done a nice job of distinguishing the abilities of students, despite the “mechanical’ nature of the questions.
It’s good to bear in mind what Zalman Usiskin, a noted anti-traditional math person had to say about testing:
“Let us drop this overstated rhetoric about all the old tests being bad. Those tests were used because they were quite effective in fitting a particular mathematical model of performance – a single number that has some value to predict future performance. Until it can be shown that the alternate assessment techniques do a better job of prediction, let us not knock what is there. The mathematics education community has forgotten that it is poor performance on the old tests that rallied the public behind our desire to change. We cannot pick up the banner but then say the tests are no measure of performance. We cannot have it both ways.”
Zalman Usiskin What Changes Should Be Made for the Second Edition of NCTM Standards. UCSMP Newsletter, n12 pp. 10 (Winter 1993)

You’re NOT the Exception

I will be starting a teaching assignment in a middle school this August.  A few weeks ago, I met with the woman who will be mentoring me for two years.  (This is part of California’s licensing requirements for new teachers–sort of like being out on parole from ed school.)  As part of some initial advice, she emphasized that students should do math not only in the classroom, but outside–i.e., give examples of real world problems. For example, she said, take fractions–and then added in a hushed confidential tone, “And I wish they would just do away with them, but there they are so we have to deal with them”.  Readers of my book “Letters from John Dewey/Letters from Huck Finn” know that my experience in ed school and student teaching has taught me excellent verbal and vomit suppression skills, so I kept quiet.  She then went on and described how students could measure a table, say, with fractions of inches and then find the area, or perimeter, etc.  She remarked how it is common that many adults say “What on earth did I learn that algebra for?”

I responded that in my experience with word problems or any kind of problems, the relevance to real-life never mattered to me. What was important was whether I could do the problem, and that I noticed with students, if they can’t do something they get frustrated and that’s when they ask “When will I ever use this stuff?”

She stated very matter-of-factly, like a doctor who has heard the complaints of a particular symptom from many patients over many years, that I probably liked math and therefore had an inclination to learn it, but there are some kids who, for whatever reasons, hate it, and have a hard time with it.

What she was saying was a variation of the “you’re the exception” argument. It is offered as a counterpoint to anyone who defends learning math in a traditionally taught manner.  Usually, such conversations focus on misrepresenting traditionally taught math as “rote memorization of facts and procedures” with no understanding, and no connection to previously covered ideas–it is all tricks and mnemonics with students “doing” but not “knowing” math.  Such mischaracterization is picked up by supposedly objective education reporters who do not question their source’s assertions, but take it on faith.

Those who did well in math taught in the manner so derided and mischaracterized are thus put in a category of “You’re smart; you would have learned it any way it was taught.”

In light of such discussions, I find it interesting therefore that Jo Boaler who has become the cause celebre of math education of late with her latest book “Mathematical Mindsets” (incorporating the ideas of Carol Dweck’s “growth mindset” thesis) says the following at her You Cubed web site:

“When mathematics is taught with an attitude of elitism and is held up as being harder than other subjects and suitable only for the gifted few, a tiny subset of those who could achieve in mathematics—and the scientific subjects, which require mathematics—do so.”

So, using Boaler’s own logic, the idea of “giftedness” or exceptionalism is harmful, since everyone is capable of learning math.  But if the idea of giftedness is so overused and abused, then those people who benefited from traditionally taught math should not be viewed as “exceptions”.  Doing well with the old style of math does not necessarily signify giftedness.

She (and others) then do the usual mischaracterizations of traditional math.  Her underlying premise is that math be taught the right way.  And “the right way” means “not the traditional or conventional way”.

She concludes: “It is imperative for our society that we move to a more equitable and informed view of mathematics learning.”

All you need is the right “mindset”, I suppose.  That and “full inclusion” classrooms” so there are no special classes for students who may in fact be gifted.   A positive “I can do it attitude” is great, so nothing against Carol Dweck’s theories.  But such mindsets need to be coupled with good instruction, study habits, initial worked examples and problem sets with well-scaffolded and increasingly difficult problems that stretch thinking beyond the practice problems.  It also takes hard work.

What I’m finding is that the “exceptional” students in various classes I’ve taught are those students who benefitted from solid instruction–acquired in school, but often via traditionally taught math by parents, tutors, or learning centers.





Shut the Hell Up, Dept.

Big shock; NY state’s new math standards look a lot like Common Core’s.

The excuse offered is, well of course they do, because CC standards weren’t that bad to begin with.

Here is where I stopped reading:

“So if a new set of learning standards is so similar to the unpopular Common Core, what’s the point? “We want students to be college and career ready. And so, if you look at what colleges expect and what employers respect, it’s probably not too dissimilar with what’s expected in Common Core. So there probably wasn’t a lot of wiggle room so to speak there. And what defines college and career ready in one state is probably very similar to what would define college and career readiness in another state. The other component of this is that the SAT and ACT exams, which are college obviously entrance exams, are aligned to Common Core standards. And so if students really do want to score well on those exams and get into the college of their choice, then chances are they’re going to have to know or be familiar with the kinds of material that Common Core expects them to be familiar with.”

Sorry, but I’ve seen the deficiencies in CC’s high school math standards. I’ve also seen how the “code words” of reform that are embedded in the standards have resulted in a tsunami of reform math practices and philosophies in many classrooms. The idea of “open ended” questions and problems, is but one of many examples of this type of thinking that has become more popular than Pokemon Go.

I’ve also seen parents who complain vilified as “old fashioned” and standing in the way of real progress. In the meantime, Sylvan, Huntington and Kumon aren’t exactly adopting constructivist tutoring services. You figure it out.

The Idiocy of Idiots, Dept.

North Carolina would have allowed students to choose between the traditional path of math (i.e., separate courses of algebra 1, geom, alg 2, pre-calc, etc) vs the integrated approach. But the NC House rejected the Senate version of a bill that would have allowed such choice.

“The Senate passed its version of the bill last week despite objections from opponents who said a dual curriculum could burden smaller schools that have limited teachers and resources. Lawmakers also said they worried students and parents would make knee-jerk decisions at the beginning of the year without fully understanding the positive effects Common Core can have on testing outcomes and career readiness.” 

Right. Parents’ decisions are always “knee-jerk” aren’t they, and Common Core was not a rush job. Is that the narrative being pushed here? Believe me, even with a traditional path, there would have been plenty of Common Core crap left in the courses, from what I’ve seen of CC-aligned algebra and geometry text books, so what was everyone worried about?

ResearchED in Wash. DC on October 29

I participated in the ResearchED conference in June at Oxford. ResearchED is a great UK-based, grass-roots organization consisting mostly of teachers. The goal of the organization is to bring more scientifically based research on educational practices into the education arena. Currently such arena is dominated by questionable studies that reference everyone else’s questionable studies and which are used to bolster and proliferate so-called “evidence” for bad and destructive teaching practices.

So you don’t want to miss ResearchED’s conference to be held in Washington DC on October 29, 2016.  Here is the list of speakers so far:

Seth Andrews: Founder, Democracy Prep Public Schools, and Senior Advisor at the White House

Giselle Kniep: Giselle O. Martin-Kniep is an educator, researcher, program evaluator, and writer

Ulrich Boser: Senior Fellow at American Progress

Tom Bennett: Founder of researchED International, advisor to UK Government on behaviour in schools, and author.

Robert Craigen: Associate Professor of Math, University of Manitoba. Cofounder of WISE Math

Pedro de Bruyckere: Educational scientist at Arteveldehogeschool, Belgium

David Didau: writer, speaker, trainer, educational maverick, UK

Steve Dykstra: adolescent psychologist and a founding member of the Wisconsin Reading Coalition

Lisa Hansel: Director of Knowledge Matters campaign & author of Loving Language Baby Books

Dr Gary Jones: educational consultant, UK

Eric Kalenze: author of Education Is Upside-Down: Reframing Reform to Focus on the Right Problems

Annie Murphy Paul: book author, magazine journalist, consultant and speaker

Professor Bryan Penfound: Professor in the Mathematics department at University of Winnipeg, Winnipeg, MB

Robert Pondiscio: Senior fellow and vice president for external affairs at the Thomas B. Fordham Institute

Ben Riley: Founder of Deans for Impact

Assistant Professor Yana Weinstein: Department of Psychology, University of Massachusetts – Lowell

Professor Dylan Wiliam: Emeritus Professor of Educational Assessment at University College London

Paul W. Bennett, Adjunct Professor of Education, Saint Mary’s University

Glenn Whitman, director of the Center for Transformative Teaching and Learning

John Mighton, founder, JUMP Math

Beth Greville-Giddings, Research Lead and a Teaching Assistant at Westbury School


Why, What or How, Dept.

Michigan State Senator Colbeck has introduced a bill that would replace the Common Core Math Standards, with the 2009, pre-CC math standards that were in place in Massachusetts. This would be a first, given that other states that repealed CC math standards replaced them with “rebranded” math standards–tweaked just a bit, but were essentially the same.

“Colbeck says looking more closely at the standards made him realize the Common Core methodology is flawed.

” “You have de facto national control over curriculum and what gets taught in each of the classrooms,” Colbeck said. “That for me was concerning, especially when you start looking at the content of some of the resource kits they provide to help schools with their curriculum and course materials. That’s where the real concerns came in—in particular, on math. As an engineer, I thought there was too much of an emphasis on the ‘how’ versus the ‘what’ on several of the standards. I don’t think the ‘how’ should be anywhere in the standards. You should focus on the ‘what.’” ”

I think he meant “the ‘why’ versus the ‘what’ ” given that CC math standards have resulted in an interpretation that essentially extends the reform math ideas of the past 25 years. Those ideas center on “understanding” versus the false dichotomy between understanding and procedures. Detractors of traditionally taught math think of procedures as “rote memorization” that obscure the underlying concepts.


Then again, he might have meant that the “how” was dictating how teachers were to teach–i.e., what pedagogical approach to use, despite the background material on CC insisting that the standards do not do that.  They do when they insist on using specific examples to get a concept across, or in their famous “instructional shifts” that they say CC calls for. What shifts are those?  Well, for example, on the CC web site, the powers that be state “Mathematics is not a list of disconnected topics, tricks, or mnemonics; it is a coherent body of knowledge made up of interconnected concepts. Therefore, the standards are designed around coherent progressions from grade to grade.”  There is a presumption that up until now math has been taught as a list of disconnected topics and tricks and that no effort at connection with previous concepts was ever made.  Thus, the mischaracterization of traditionally taught math is built in the Common Core narrative provided in their background material.  (For more on mischaracterizations of traditionally taught math, see the presentation I gave at ResearchED at Oxford. )

Edu-Entrepreneurialism and Overcorrections

I am grateful to Eric Kalenze, author of “Education is Upside-Down” for pointing me to the concept of over-corrections in education. There is a tendency for solutions to problems to go overboard. So if there are a few teachers who give praise in unhelpful ways, (e.g., saying “Good job” for someone sharpening their pencil, say), then a whole movement develops to teach how to give praise, and not over-praise. The result can be the extreme of not giving praise at all, or saying ridiculous things like “Seth is not being disruptive”

We are seeing this most often with Dweck’s “growth mindsets” and Duckworth’s “grit”. Both Dweck and Duckworth did research and carefully qualified their statements and theories about both concepts in their books. But the edu-entrepeneurs have risen to fill the perceived void such theories left in their wake, and we have seminars on how to teach growth mindsets and grit, that border on –no, cross the border of–idiocy disguised as lunacy.

The latest edu-entrepreneur movement that I’m detecting seems to stem from something Dan Willingham said in his book “Why Don’t Students Like School?“:

Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question.” 


I admit I have to read Willingham’s book, but knowing how he is a careful writer and thinker, I suspect he probably goes into more detail on this.  What has happened, however, is similar to what’s happened with concepts like “grit” and “growth mindsets”.  Now the Willingham quote joins the party.  Case in point: Dan Meyer’s talk at the annual NCTM conference earlier this year keyed into Willingham’s quote and almost overnight, a multitude of teaching blogs seem to be running with it. The idea is being used via computer graphics and other approaches to “engage” students, (an example is talked about here) because “engaged” students are supposedly stimulated to ask questions, seemingly satisfying Willingham’s goal to “develop the question” for which teachers can then supply an answer.  Except that from what I see, when the students actually do ask the questions that the adherents believe need to be asked, those questions may remain unanswered until the students discover the answers.