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.



Critical Thinking: UFOs

The press is having fun with the discovery that the Pentagon has spent a few million dollars checking out UFO claims.

Here’s a little video of Neil deGrasse Tyson talking about the program. In his typically witty way Tyson points out that the presence of unidentified flying objects does not imply that they are evidence of alien visitation.

It’s ok not to know. It’s just fine to suspend judgment. Admit that you don’t know things. But there’s a huge gap between something defying explanation, and its being explained by guesses like “aliens” or “gods” or “ESP” or “magic” or what have you.

Not knowing is ok. It’s what pushes us to look for evidence. Recognizing your limitations, recognizing that there is work to do, and actually doing the investigation is at the heart of critical thinking. When you have decent evidence, be prepared to revise your position. Easy to say. Sometimes not so easy to do.

The Gift of Sloth

Cartoonists Jerry Scott and Jim Borgman remember high school all too well.

zits december 15 2017


Veritas Omnia Vincit, indeed


Canada watched, torn between the comedy and the horror of good intentions turned to viciousness at Wilfrid Laurier University last month.


Lindsay Shepherd, a 22-year-old M.A. student was at teaching assistant for an undergraduate Communications class. She prepared a seminar on the politics of gendered language/pronoun usage/binary concepts. In what appears to be an act of unexceptional diligence, she showed 3 minutes of a “debate” that had recently aired on public television (TVO). I’m not sure which 3 minutes Shepherd showed, but the full 9-minute clip looks like this.

In the clip we see (in)famous psychology professor Jordan Peterson squeeze in a few words about his views on gendered pronouns. Taking the offensive is Nicholas Matte, from the University of Toronto’s Mark S. Bonham Centre for Sexual Diversity studies. There isn’t much of a debate here. But perhaps there is enough material to stimulate some interesting discussion. What are the issues? Where are the divisions?

Apparently, a student in the class took exception to the clip, complaining that giving Peterson air time is tantamount to giving legitimacy to his position.

So far, so what? Another boring day of right vs left, conservative vs reformer, freedom of speech vs security of psyche. But of course, things are never so simple. Shepherd was called into a meeting with her supervisor, the program chair, the manager of the university’s Gendered and Sexual Violence Prevention and Support Office. Not a very psychologically safe place, if you ask me. The power is entirely on the side of the university, and it’s 3-against-1. Further, the three have tenure, and Shepherd is a 22-year-old graduate student with very little infrastructure to support her.

Predictably, Shepherd was told that she should not have shown the clip and—believe it or not—was told that playing Peterson’s views was little different from giving a platform to Nazism.

Fortunately, Shepherd was wise enough to audiotape the entire ordeal. To her credit, she went to the press with her story, but said that she did not wish to release the recording unless it became absolutely necessary. Of course it became necessary.

CBC article with audio

So what is all this doing on a blog about publicly-funded K-12 education? First, any high school teacher could play that panel discussion with impunity, at least in my jurisdiction. Nobody has any doubt that 15-18 year olds can separate the signal from the noise, sort through the issues and makes some sense of what’s going on. It is patently absurd for a university to assume that undergraduates cannot do the same. And if they can’t, shame on the university and the high schools that feed it.

Second, it’s crucial that we remember why the post-Enlightenment world values freedom of speech so highly. There are many well-known arguments, but let me just focus on two ideas.

  1. Human beings are fallible. We struggle with the contingencies of our experiences, with the historical realities of ourselves, our cultures and our institutions. There are damned few things about which we are so certain that there is no point in hearing alternatives. This is not to say that a reasonable person should treat all alternatives as equally worthy of our time. But it is to say that we should be humble and recognize that the potentates of old were simply wrong to silence the other.
  2. People need to have the right to learn. I do not have all the same values, beliefs and knowledge that I had 10, 20, 30 or more years ago. I have changed. And I have changed for many reasons. I understand some things more deeply now than in the past. I have new information, new theories, new means of analysis. I have new experiences. The world has changed around me, and I have had to negotiate my way through this change. And this is true of everyone. The deep questions is: how can I learn if I am held to pre-ordained “truths” that I cannot question? Whatever I believe about gender expression today, it certainly is not what I believed in 1980, and it is highly unlikely to be what I believe in 2030. Why should a single point along this journey—say, 2017—be privileged above every other point of change? Why would we say, “this year, we have the TRUTH so you can stop thinking now”? To even suggest such a thing is to deny a person the right to learn.

As a teacher I must protect each student’s right to learn. Whatever they believe today, it must be open to reflection, to challenge, to reform. As a teacher, it is my obligation to question, to encourage the student to recognize both the justifications but also the implications of each belief. And perhaps most important of all, to recognize that intelligent, sincere people can disagree with us. The only way forward is through openness, not through bludgeoning our beliefs into others.

Wilfrid Laurier University’s motto is Veritas Omnia Vincit: Truth Conquers All. The irony is still hot.

Dr. Nathan Rambukkana, the supervisor who initiated the meeting (and who made the absolutely absurd comparison to Nazism) has issued a lukewarm apology in the form of an open letter. He does himself no credit.

Open letter from Nathan Rambukkana to Lindsay Shepherd

The President of Wilfrid Laurier has also apologized.

Apology from Laurier President and Vice-Chancellor Deborah MacLatchy

The university has shamed itself. A young woman’s career has been put in jeopardy. It’s a truly pathetic tale.

If we don’t believe in freedom of expression for people we despise, we don’t believe in it at all.–Noam Chomsky

Let’s give Lindsay Shepherd the final word.


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.




Constricting the modern mind

I will be brief and rambling today.

As I was wasting a few minutes on “social media” I noticed post after post where a meme–often a very clever one–was offered as evidence for a political opinion. You know what I mean “X destroys opinion Y with one example” blah blah blah. I find it both irritating and a bit frightening.

I find it frightening because more and more it appears that memes are displacing newspapers and news broadcasts as the fundamental information for voters. I see memes blaming Prime Minister Justin Trudeau for Trans Canada Pipeline’s corporate decision not to further pursue the Energy East pipeline. I see local mayors tarred and feathered over I’m not sure what. I see that “the left” all agree on everything, and it’s all stupid. I see the “the right” all agree on everything, and that it’s all racist.

And I fear for the future of democracy.

One of the strongest responses could come from education. But I’m not sure that the will is there.

As I sadly looked at my Facebook page, a line from JS Mill’s On Liberty came to mind.

“He who knows only his own side of the case knows little of that. His reasons may be good, and no one may have been able to refute them. But if he is equally unable to refute the reasons on the opposite side, if he does not so much as know what they are, he has no ground for preferring either opinion… Nor is it enough that he should hear the opinions of adversaries from his own teachers, presented as they state them, and accompanied by what they offer as refutations. He must be able to hear them from persons who actually believe them…he must know them in their most plausible and persuasive form.”—John Stuart Mill, On Liberty (1859).

Mill knew better.

People who disagree with us are not all fools, nor are they all morally defective. (Some may be, of course.) For virtually any position worth fighting over, reasonable people can disagree on the details, and sometimes the fundamentals.

But do we take other people’s disagreements with us seriously enough?

What are the real issues behind athletes’ kneeling during the national anthem? They are not all fools; they are not all anarchists hell-bent on destroying a nation. You don’t have to agree with anyone to take him seriously. But you do have to have some intellectual and moral courage.

Can we teach students to take others seriously? I think we can. I don’t think we do it well enough. Classroom debates rarely get to the heart of the matter. Debates tend to quickly degenerate into glib contests of verbal cleverness.

Student writing (or oral, or visual representation, or film, or whatever) should always consider the strongest opposition to the point being argued. In fact, this should be one of the main points of assessment in the scoring rubric (that’s scoring guide, or rules, for non-teachers). Perhaps as much as 30-40% of the student’s grade should be contingent on whether she takes her opponents seriously and gives them fair and honest voice.

I’ll stop here. I’ve given this much thought over the years, but I’ve never tried to articulate it until now. Hopefully, you’ve noticed that I haven’t given fair voice to someone who disagrees with taking opposing views seriously. It’s a dilemma.


Gilded Monuments and History

In Halifax, Nova Scotia, controversy over the proposed removal of a statue of Edward Cornwallis has been quietly brewing for nearly four decades.

300px-CornwallisStatueHalifaxNovaScotia[1]Cornwallis was a British military man, who was given the task of establishing the city of Halifax, and  was Governor of Nova Scotia from 1749-1752, after which, he retired back in England. At the very least, the statue is a tribute to colonialism, to the “conquering” of the new world and its first inhabitants. In short, the statue symbolizes the beginning of the modern Canadian nation-state, as well as the beginning of the decimation–some would say genocide–of indigenous Canadians.

We are hearing similar debates in the USA this year, with discussions about public commemorations of Civil War leaders. As in Canada, one side wants the statues to celebrate “glorious history” and the other wants to put an end to the public display of the vile politics of an earlier era.

There really isn’t much to say about the issue in general, apart from the observation that every public monument is different. Staying within Canada, consider Mt. Stalin. After the end of the second World War, the Canadian government honoured our great allies by renaming three peaks in the Rockies after Churchill, Eisenhower and Stalin. Since the war, we haven’t learned much to make us regret the first two, but as the realities of Stalin’s rule of terror became known, Mt. Stalin became a public embarrassment. But it took a fair bit of debate before the mountain was renamed Mt. Peck  in 1987. The fact is that Stalin was a major contributor to victory in the War. Yet, this wasn’t enough to outweigh his atrocities. Canada chose to stop honouring Josef Stalin. And rightly so.

Cornwallis will be a more difficult case. He was, undoubtedly, a man of his time, and we can’t hold that against him. On the other hand, we are acutely aware of the pains of colonialism and its legacy. The people of Halifax will have to weigh three things:

  1. Cornwallis was founder of the city.
  2. The statue celebrates the colonial imbalance of power that was instrumental in the decimation of First Nations.
  3. The statue itself has been a part of the civic culture for almost a century.

I have no pony in this race, and I leave it to the good people of Halifax to figure this out.


But what about history? What are we to make of the claim that the removal of statues is the destruction or denial of history.

It’s nonsense.

These statues are not history. They are public markers of admiration. When they were installed, the admiration was undoubtedly real. But it’s our world now, and we have both the right and the duty to carefully consider and reconsider who we choose to display publicly. Do the people of Halifax value the British presence and dominance in Nova Scotia more than they decry the destruction of indigenous peoples? Does the civic pride in looking at a depiction of a dead general outweigh the personal anguish of the victims of colonialism and their descendants?

Weigh carefully, my friends.

Finally, let’s think about historical precedents. Often after revolutions or even after minor insurrections, zealous crowds have toppled statues. I’m talking about something different. Let people use the wisdom of public assembly to guide decisions of public policy.

In the end, though, time will be the judge. All these dead guys displayed in bronze remain in the historical record. And the judgment of historians–and history students, will continue to evolve as do our collective beliefs of justice and goodness change. Time gets the final word.

I met a traveller from an antique land,
Who said—“Two vast and trunkless legs of stone
Stand in the desert. . . . Near them, on the sand,
Half sunk a shattered visage lies, whose frown,
And wrinkled lip, and sneer of cold command,
Tell that its sculptor well those passions read
Which yet survive, stamped on these lifeless things,
The hand that mocked them, and the heart that fed;
And on the pedestal, these words appear:
My name is Ozymandias, King of Kings;
Look on my Works, ye Mighty, and despair!
Nothing beside remains. Round the decay
Of that colossal Wreck, boundless and bare
The lone and level sands stretch far away.”

–Percy Bysshe Shelley, 1818