Jump Math

Just completed our annual Teachers’ Convention. It was not particularly eventful for me, but as always, I walked away with something to think about.


1403816949849On Friday, I caught two sessions with playwright-mathematician John Mighton. Mighton is an interesting character. After an undergraduate degree which, in his words, left him with the impression that he wasn’t much good at anything, he put his energy into learning to write. He developed a modestly successful reputation as a playwright in Toronto, but still found by his early 30s that this was a tough way to earn a living. He made ends meet by tutoring kids and discovered that he was able to learn (re-learn) mathematics if he managed to break the learning into small, concrete steps. And he found that his pupils learned the same way. This led Mighton to graduate school and a career as a mathematician.

Somehow, Mighton maintained his interest in student learning, and put together the basic structure of Jump Math, which has grown to a not-for-profit organization championing Mighton’s vision and cooperating with university researchers to provide a research base to support, refine and change the program.

I’ve linked to the organization above, so I won’t repeat their basic information. Rather, I’ll recall and reflect on Mighton’s sessions.

Mighton hits on some themes that I’ve reflected on in the past—mainly the virtues of practice and mastery—but he takes it further. Jump Math is predicated on the principles that

·       Scaffolding is essential to learning.

·       The teacher is responsible for making the scaffolding increments small enough so that every student can make every move.

·       Every child must climb every rung of the scaffold before the class can move forward.

·       Success is its own reward.

·       Every child (with enough cognitive capacity for language) can make significant progress in elementary mathematics.

·       Competence precedes understanding.

On the one hand, this is a pretty unsurprising list of principles (do remember that this is my reading of the talk, not necessarily the voice of John or of the organization). On the other, it’s about as un-trendy as you can be in contemporary education.

Before I go further, a note about the Jump Math materials. You can buy stuff from them, but the teachers’ guides (Grades 1-8) are available for free on the website. Mighton emphasized that the student materials are quite uninteresting, as they are nothing more than overly-large practice sets. They’ll save you time and effort, but they are the least important part of the program. The teacher’s guides articulate the recommended scaffolding for the classroom lesson. If you only have one thing from Jump Math, Mighton says, make sure it’s the teacher guide. This seems eminently sensible to me.

So how might a teacher prepare a lesson utilizing the Jump ideas?

First you need to know where on the continuum of background knowledge and skills each child in the classroom lies. This is relatively straightforward. Mighton used the miniature whiteboards that are common in classrooms today. (Actually, it was a sheet of white paper inside a plastic folder; participants wrote in dry-erase marker on the folder.) The Do Now activity is simple and relevant to the activity. If today’s lesson, for example is the addition of simple rational expressions, then I need to know if each student is comfortable adding fractions. So I might put a simple fraction addition on the board, and ask each student to perform the addition on the white boards and hold it up. We’re talking simple 10-second stuff here. When each student holds their board up (no exceptions; no one is allowed to opt out) then you can see if you’re ready to move forward. If anyone requires attention, do it now. Again, you move together in a group.

Side note: it’s easy to see Mighton’s background in theatre. When an audience is unified, there is an energy in the room, that far exceeds the energy of individuals working separately.

The key part of the lesson is “what’s next?” When the simple addition is successfully performed, add a SMALL bit of extra complexity to the problem. The idea is that even fairly simple mathematics requires a surprisingly large number of small pieces for it to be sensible. If I am going from 2/3 + 4/5 to (x+1)/(2x-3)+5/(x-6), I should do it in baby steps. Make sure that every idea of common denominator, gathering of like terms, reducing fractions, etc. is in place. If I give all at once, I can be sure that some will get it (eventually) and some will not. The belief behind Jump is that everyone can and should get each step before moving on.

I’ll not belabour the lesson further, as I think the point of the lesson is reasonably clear. In a sense, I think every teacher is onside with much of the above, but there are still moments of uncertainty. Am I going to slow? Am I going to bore my brighter students? What if someone doesn’t catch on? Do I have time for all of this?

Of course, the answers to these questions are found in practice, not theory.

The other family of concerns is with the “richness” of the problems. There is a powerful movement in mathematics education that asserts that openness, richness and exploration are the keys to mathematical learning. I’ve looked at this in the past, with my reflections on a video by Dan Meyer. Mighton is even more skeptical than I am about the virtue of open problems for most students most of the time.

Ultimately, the sessions have provoked me to look harder at my scaffolding and to be more precise in my progression. I have always taught in a similar way, but I have not been as scrupulous as I might have been about always taking small steps and always ensuring 100% understanding before moving on. In my defense, I teach high school academic mathematics, and students are capable of storing anomalies for now, and resolving them later. Or have I been assuming too much? How big do these steps need to be? I will report back.

Finally, Mighton offers a view of inclusion that is curiously out of sync with most established views. In most of the literature on inclusion (see my earlier entries on UDL, for example) we plan for multiple entry points for students with multiple means of participating in activities at their own level. Jump suggests that this is excessive. Jump Math claims that we can harness the group dynamic (Weber’s “collective effervescence”) and differentiate by working together at all times. If this is so, then much can change in education—well, mathematics education at least. It is likely that this structured approach is well suited to mathematics because of its rigid internal structure; learning to read is likely a different sort of cognitive experience. But that’s a talk for another day.



Search Engines and Post-Truth

I had a rather mindless question this morning: In Harry Potter, what is the first name of  the namesake of Hufflepuff house? So I went to Google, and typed simply hufflepuff. Here’s what I got.


“Ah,” I thought, “Helga.” But then I looked at the google page. Stylistically, it looked awkwardly familiar. The layout, the references, the “similar links” appeared to be no different from what Google provides for living souls.

Uncomfortably, I typed “Haile Selassie”. Google quickly responded.


My queasiness has not subsided much. Most people know that Helga Hufflepuff is fictional and that Haile Selassie was an Ethiopian emperor. Well, sadly, far more people know Hufflepuff than Selassie. But that’s another matter.

The point is that truth and fiction have precisely the same online frame. If you come into the frame with knowledge, you are able to understand the picture. But how is someone who does not enter with knowledge make sense of all this?

Honestly, I don’t know.



Postman: “Teaching as an Amusing Activity”

As I noted previously, Postman’s 1985 book Amusing Ourselves to Death: Public Discourse in the Age of Show Business seems oddly appropriate in the connected world of the early 21st century.
Chapter 10 of this book, “Teaching as an Amusing Activity” looks at the role of television in shaping 1980s educational thinking and practice. My goal today is to briefly review what Postman had to say, and think about today’s connected classrooms in light of this.
As I wrote in my previous discussion of Postman, TV and prose have very different structures, giving them very different relationships to knowledge. (Postman says that different media have different epistemologies; this, I think, goes too far.) Postman begins with a discussion of “Sesame Street”, then reflects back to the classroom.

As for educators, they generally approved of “Sesame Street,” too. Contrary to common opinion, they are apt to find new methods congenial, especially if they are told that education can be accomplished more efficiently by means of the new techniques. (That is why such ideas as “teacher-proof” textbooks, standardized tests, and, now, micro-computers have been welcomed into the classroom.) “Sesame Street” appeared to be an imaginative aid in solving the growing problem of teaching Americans how to read, while, at the same time, encouraging children to love school.
We now know that “Sesame Street” encourages children to love school only if school is like “Sesame Street.” Which is to say, we now know that “Sesame Street” undermines what the traditional idea of schooling represents. Whereas a classroom is a place of social interaction, the space in front of a television set [is not.] (p. 143)

There are two big ideas here. First, we have the displacement of responsibility from human teachers to an entertainment medium. Second, we have the displacement of a social setting—the classroom—to a private setting—in front of a screen. 30+ years later, the issues have not changed.
After some reflection, Postman continues:

This does not mean that “Sesame Street” is not educational. It is, in fact, nothing but educational—in the sense that every television show is educational. Just as reading a book—any kind of book —promotes a particular orientation toward learning, watching a television show does the same. “The Little House on the Prairie,” “Cheers” and “The Tonight Show” are as effective as “Sesame Street” in promoting what might be called the television style of learning. And this style of learning is, by its nature, hostile to what has been called book-learning or its handmaiden, school-learning. (p. 144)

Now we’re getting to the main issue: by changing the media of education, we change not only the content of education; we also change our notions of what is educationally valuable. Whatever it is that students get out of television learning, we have to be prepared to understand that it is different from what they get out of reading, writing and socially interacting. Watching the movie is not the same as reading the book. Not only do we need to understand it, we need to be prepared to challenge our values. What is important in education?
In the connected world, we often hear bland generalizations about the modern “knowledge economy” and “21st century skills” and “21st century literacies” and so on. But are we really aware of the price we are paying? Postman lays it out with what he calls the three commandments of television-based education.

Thou shalt have no prerequisites
Every television program must be a complete package in itself. No previous knowledge is to be required. There must not be even a hint that learning is hierarchical, that it is an edifice constructed on a foundation. The learner must be allowed to enter at any point without prejudice. This is why you shall never hear or see a television program begin with the caution that if the viewer has not seen the previous programs, this one will be meaningless. Television is a nongraded curriculum and excludes no viewer for any reason, at any time. In other words, in doing away with the idea of sequence and continuity in education, television undermines the idea that sequence and continuity have anything to do with thought itself.
Thou shalt induce no perplexity
In television teaching, perplexity is a superhighway to low ratings. A perplexed learner is a learner who will turn to another station. This means that there must be nothing that has to be remembered, studied, applied or, worst of all, endured. It is assumed that any information, story or idea can be made immediately accessible, since the contentment, not the growth, of the learner is paramount.
Thou shalt avoid exposition like the ten plagues visited upon Egypt
Of all the enemies of television-teaching, including continuity and perplexity, none is more formidable than exposition. Arguments, hypotheses, discussions, reasons, refutations or any of the traditional instruments of reasoned discourse turn television into radio or, worse, third-rate printed matter. Thus, television-teaching always takes the form of story-telling, conducted through dynamic images and supported by music. This is as characteristic of “Star Trek” as it is of “Cosmos,” of “Diff’rent Strokes” as of “Sesame Street,” of commercials as of “Nova.” Nothing will be taught on television that cannot be both visualized and placed in a theatrical context.
The name we may properly give to an education without prerequisites, perplexity and exposition is entertainment.

Whew! This is strong stuff. I can’t help but agree. And, again, the issues have not changed. Indeed, many educators have taken these three commandments to heart, merely by being part of the 21st century.

I hope to return to this later.


Alternative Facts

I post in utter amazement.


What in the world are “alternative facts”? Sometimes the facts are unclear. Sometimes the facts are in principle or in practice unknowable. But alternative facts?

As Chuck Todd noted, “alternative facts are not facts; they are untruths”.

No matter how many times gurus tell us that appearance is reality, it isn’t.

Is this what post-truth looks and sounds like? First claim: he gave facts. Second claim: he gave “alternative facts”. Third claim: nobody really knows the facts.

So far as I can see, all political thinking for years past has been vitiated in the same way. People can foresee the future only when it coincides with their own wishes, and the most grossly obvious facts can be ignored when they are unwelcome. . . To appreciate the danger of Fascism the Left would have had to admit its own shortcomings, which was too painful; so the whole phenomenon was ignored or misinterpreted, with disastrous results.

The most one can say is that people can be fairly good prophets when their wishes are realizable. But a truly objective approach is almost impossible, because in one form or another almost everyone is a nationalist… The most intelligent people seem capable of holding schizophrenic beliefs, or disregarding plain facts, of evading serious questions with debating-society repartees, or swallowing baseless rumours and of looking on indifferently while history is falsified. All these mental vices spring ultimately from the nationalistic habit of mind, which is itself, I suppose, the product of fear and of the ghastly emptiness of machine civilization….

I believe that it is possible to be more objective than most of us are, but that it involves a moral effort. One cannot get away from one’s own subjective feelings, but at least one can know what they are and make allowance for them.

–George Orwell, London Letter in: Partisan Review (Winter, 1945)



Back to the 80’s # 2: The Keegstra Affair

In 1981, the village of Eckville, population about 700, became home to the most notorious incident in the history of Alberta education.

Eckville, Alberta

Susan Maddox, a relative newcomer to town, read over her son’s high school social studies notes with alarm. His notes were riddled with anti-Semitism, holocaust denial, fear of the “Jewish-controlled international banking system” and supposed threats to Christianity posed by the new world order.

Her son’s teacher, James Keegstra, was not only a veteran teacher, well-liked by staff and students, he was also Eckville’s mayor. Maddox’s complaint finally led to some investigation into Keegstra’s classroom practice. The Minister of Education removed Keegstra’s teaching certification (for failing to teach the approved curriculum, interestingly), and Keegstra became a test case for Canada’s hate speech laws, taking up court time into the 1990s.)

Keegstra’s initial conviction for willing promoting hatred, 1985. Photo from The Calgary Herald

(I do not wish to give air to Keegstra’s views at this time. If you can get a hold of Robert Mason Lee’s article “Keegstra’s Children” from the May 1985 issue of Saturday Night magazine, you’ll get a fuller sense of what Keegstra taught, and what his students learned.)

My interest today is in one narrow but crucial part of Keegstra’s teaching: that the mainstream press cannot be trusted to tell the truth.

But it was far more insidious than that.

Suppose you found a book that said that the Holocaust were, indeed, an actual historical event. Keegstra had a ready-made reply: the book is fake news. Photographs: they can be faked easily. So the inquisitive mind asks the logical question: why would someone make this stuff up?

The answer, according to Keegstra (and his ilk) is simple: it’s part of a conspiracy to promote the Zionist Agenda.

You see, on this view, the texts that agree with Keegstra’s point of view are the work of honest, decent people working to bring the truth to light. The texts that disagree are the work of the conspiracy and its well-intentioned dupes. The more evidence you find to support the reality of the Holocaust or the falsity of the Conspiracy, the stronger becomes your belief that the forces of darkness are powerful. Once in, you’re in deep.

Once you buy into this conspiratorial worldview, there is no escape. All of your information is second-hand, and you have to decide who to trust. And there is no guidance for who to trust. You have two opposing views and no means of adjudication. Well, almost.

As a student, you have your teacher to trust. Who can blame the children of Eckville for trusting their teacher, a well-known Christian, and mayor of the town to boot?

Of course, this is why we have approved textbooks, controlled curricula and curated libraries for school. Right?

Ask yourself any question about anything that you are not an expert. Then turn to the internet for answers. Are humans causing global climate change? I don’t know about you, but I do not understand nearly enough about the science to have an informed opinion. I do know (thanks to my education) something about science in general, about scientific credibility, and about the simple strategy that if I have to trust, I should trust the most qualified person in the room. It’s not infallible (what is?) but it’s better than just trusting anybody.

But this is more than a bit unnerving. Think about recent accusations that the media are biased—about anything you care to consider. There is a trivial sense where this is true. Media tend to report on things that fit their business model and that fit their market research. They tend to trust authority figures, including government. And, I don’t like where this is going.

It’s not that we’re trapped in a relativistic spiral. It’s just that it feels like it.

And students are trapped even more deeply than adults are.

But you don’t have to look far to find adults—pre-internet adults, mind you—who believe the most outlandish things because they are online. During the recent US election, I thought that nobody would believe the silly fake news articles about the Clinton’s having all their enemies murdered and getting away with it. Nope. I was wrong. Once a slice of the media (and the politicians it favours) are able to cast doubt on all disagreeing parties, we’re in precisely the same spots as Keegstra’s children.

This, in my mind, is the most crucial issue facing modern education.

Everybody says we should teach “media literacy”.

Teachers think they’re teaching “media literacy”.

We are an increasingly media illiterate society.

Where next?

I’ve got a couple more stops in the 80s before I can start to build my case for an appropriate response. See you in a couple of days.


Back to the 80’s # 1: 1984

Back in 1983-1984 everybody and his dog read or reread George Orwell’s 1984 and made a pronouncement of some sort.

cache_27680941No, we didn’t tweet it or post about it on on Facebook. No one could have imagined such inanities back in the 80s. But we were aware of Orwell’s dystopian vision and we were aware that the anointed year had come. So we spoke.

And spoke.

And wrote.

And spoke.

The general consensus was that Orwell had missed the mark. Sure, we said, the Soviet Union and China were awful dictatorships, but we in the lovely west were seeing nothing of the horrors of Orwell’s vision.

That is to say, we missed the point of the book.

our_northrop_frye2Orwell’s central thesis, neatly summarized by Northrop Frye (I intend a Frye “Back to the 80s” soon), “is that there is only one way to create a hell on earth that we and our children can never escape from, and that is to smash language.”[i]

To smash language. To remove the possibility of articulating non-trivial thought.





067466020xAs the philosopher Charles Taylor tirelessly argues, language has both a constitutive as well as a designative function. Language designates insofar as it provides expression to thoughts, and provides a vehicle for interpreting others’ thoughts. But language does much more than this. Language not only allows us to name our world; it also is integral to the creation of much of it as well. By naming the world, we come to understand it on our terms, and we create possibilities of understanding and application that could not be present otherwise.[ii] To live in language is to live in a world of meaningful creation and participation. Without language, we would be much less than we are now.

If, as Orwell feared, language were to be destroyed, then the possibility of meaningful human creation would die with it.

Our world looks nothing like Orwell’s dictatorial dystopia; aren’t we safe?

I don’t think so. Twitter—the medium of choice for the US President-elect—restricts communication to 140 characters. This has had a fascinating effect on communication. First, a number of tweeters have revived the long-dead art of the epigram. It is possible to be insightful and hilarious in 140 characters. George Takei is a master of the art. But neither can you provide a detailed or nuanced argument. Twitter is a way of telling people what they already believe. If you have Twitter talent, it’s a fun and entertaining way of helping people to laugh at their own cleverness. Try tweeting an explanation of what’s at stake in the Trans Pacific Partnership.

But it’s not just tweets. It’s memes on Facebook and Instagram. Wonderfully entertaining, both, but utterly incapable of sustained argument or novel conception.

Perhaps I’m being a bit hasty here. Sure we have social media for exchanging our daily trivialities, but that doesn’t mean that all written communication is or will be trivialized. This is where there is hope.

Education is the place where language has a chance to make a stand.

71ew4kattblIf education fully embraces the “new technologies” and “21st century literacy” then the future is bleak. As Neil Postman (another likely topic for my “Back to the 80s” kick) argued, continuous prose is one of the greatest technologies created by humans for the purpose of making and communicating meaning. Nothing compares to the ability of prose to structure arguments, to demonstrate cause and effect, to offer thought experiments. But prose is not simple. It takes a lifetime to master clear, lucid prose. We have to decide if we, as a society, are up to the task. Do we care to give the gift of prose to future generations?

(As an aside, imaginative prose is one of our great cultural legacies. It is being displaced by video. This is a whole different discussion for another day.)

There is much more to say, and I will return to this topic.

Let me close with my favourite epigram–a pre-Twitter tweet–from an earlier time. In the 18th century, epigrammatic wit was highly valued (as were extended prose and poetical treatment of serious issues). Alexander Pope gave a dog to Frederick, the Prince of Wales in 1736. The dog had a collar, engraved with the following couplet.

I AM his Highness’ dog at Kew; 
Pray tell me, Sir, whose dog are you?

[i] Northrop Frye (1988) “The Authority of Learning.” A talk given to the Empire Club in Toronto on 19 January, 1984.

[ii] Charles Taylor (2016) The Language Animal: The Full Shape of the Human Linguistic Capacity. The Belknap Press of Harvard University Press.


What you do is not what you understand

A very short entry today.

I’ve previously mentioned the work by Perkins et. al., as part of Project Zero at the Harvard Graduate School of Education  (from quite a while ago) in which it was suggested that understanding is a performance. I’ve tried in vain to get a full-bodied theory from this work; it just doesn’t seem to be out there. If I had to guess, I’d think that the authors figured out that they didn’t have a defensible theory and abandoned the project. They have ceased academic publishing on the idea, but continue to sell books and PD materials to schools…

Anyway, here’s a quick thought experiment to dispel the notion that understanding can be reduced to performance. Imagine watching two people play chess. Can you tell something–it doesn’t need to be a complete picture–about what the players understand by watching their moves?

Well, first, you’ll need to understand something about chess yourself to make this judgment (careful: this is in danger of circularity!). So now you, using your chess knowledge and understanding, try to form a judgment about what the players understand about the position on the board. Are they beginners? Are they good coffee-house players? Masters?

But wait. What about context? Who are these people? How do you know that they aren’t actors playing through a script? You don’t. Unless you have access to their internal processes–their thinking–you are in a position of significant uncertainty. We don’t even need the fanciful possibility of acting or fraud. Perhaps the players have memorized some moves from master games, and can play them so long as the opponent doesn’t stray from the memorized lines. (The longer the game goes on, the less likely this becomes.)

This is a real problem in modern chess tournaments. From time to time accusations of cheating come forth–perhaps the player is getting electronic assistance during bathroom breaks. How do you decide if the player understands the game well enough to make those moves? The standard approach is laughably simple. You give the player some sample positions in test conditions and get him/her to explain what moves to make and why.

Natural language dialogue appears to be the most obvious and sensible approach to the question of student understanding. This is not a theory but is a starting point. More to follow.