For Wilbert, a True Friend and Mentor

“This can’t be true.” I looked again
but the words remained there, plain to see.
“Life’s setting sun is sinking low.”
I wondered, “how could this be?”

The very one who, so many times,
stood firm for what you knew was right
and who broke down walls to get things done,
with modesty, honesty and foresight.

And who, for friends, and home and family
stood strong, the worst you’d face
is now facing the end, but still my friend 
holds courage, strength and grace.

And for my part, stronger than the sadness
I know some time soon I must endure
from my memory springs the many things
I have yet to thank you for.

I’m reminded of the first time we met.
You, then the province’s consultant for Math Ed.
agreed to help this “young gaffer” (your words)  
with advice on a paper I just couldn’t put to bed.

Well, based on the insight and knowledge
you so freely gave me on that day
my mediocre work improved
and, thanks to you, the paper earned an “A”.

And then in the early nineties when
distance education physics was set to go
and I hinted “hire me” to who was in charge
my rejection came as a humbling blow.

But when my school board got a call
asking if they could second me to do that very chore
I knew right then there was one more thing
I had to thank you for.

And then, later on, when the axe fell,
budgets slashed and so much was let go
you found a way for me to stay,
even gave me more chances for to grow.

Like when you managed the Vista project
which reinvisioned online learning,
you found for me a space–and t’was through that grace
my career took its next turning.

For the skills I learned all through that time
and the responsibilities you helped me learn to bear
I did apply as we built CDLI,
so thanks, too, for all those years.

I’ve often wondered how best to thank you for
the opportunities, support and, of course, advice
but I came to see you just did it unselfishly.
There never was a price.

And so, now I work my time with those
on the same path that that we once chose.
And the journey they’re on is so very long.
It’s a less kind world these days, I do suppose.

Still it seems to me that the best way
to honour all the kindness you’ve bestowed
is to pass it on and try to help those
with whom I share this road.

So when I find a way to help another
through some thing I say or do
I’ll feel that same sense of gratitude
and smile, then and there,
as I think of you.

My favourite picture of you, taken back in ’99 on the occasion of your retirement. Back Row: Lloyd Gill. Wayne Oakley, Harvey Weir, Dave Dibbon, Rachel Handrigan, Wade Sheppard, Rene Wicks. Front Row: Wilbert Boone, Maureen Boone, Jean Brown, Maurice Barry

They Sure Can Slide Fingers on a Pane of Glass

Increasingly it seems to me that if, twenty years from now, we took the time to assess what the young folk of today got from their childhoods, the one thing we’ll be able to state with surety is that there was never a generation so adept at sliding their fingers along a small sheet of glass.

Thirty-five years ago I began (getting paid for) my teaching career. In those days I self-identified as a science – math teacher. I loved it, especially the lab activities. I was lucky because at the time–the early eighties–science curricula were designed to be very hands-on. It was great, but there was something else: I generally found that the activities jived very well with the students’ personal experiences. Students could, for example, relate to labs studying motion because the objects of study seemed so very familiar. For non-accelerated motion the students were used to gliding along ice, rolling along level ground on skateboards, bikes and rollerblades. For accelerated motion, they could similarly draw on tobogganing or biking downhill, playing ball and just throwing rocks in the ocean. For circular motion they had experience on playground merry-go-rounds, swings and even with twirling things on the end of string.

But then time passed. I noticed it first for circular motion, never an easy topic and one that you had to ensure that students had up-close-and-personal experience with before digging in through the lens of physics. Students could not relate anymore to any of the once-familiar events. Not even twirling stuff on strings! I just put it down to the increased time that the children were spending playing video games indoors, figured, “That’s sad, but I guess we’ll just have to redouble our efforts with the hands-on activities in school,” and thought no more of it.

…until the penny dropped.

Talking to the young people who attend the university at which I work it became increasingly obvious that, not only are the students not directly experiencing the physical world (aka playing outdoors) but neither are they doing that in school! Regardless of what happens in k-6, once they hit Intermediate and then High School, their days in science class are mostly spent with their bums in uncomfortable ancient school desks, all neatly arranged in rows, and listening to an adult talk, talk, talk about scientific knowledge or show off how well they can “solve a problem,” which, by the way, is not that at all but, rather, a boring run-through of some algorithm for dealing with some contrived situation or other.

And there’s shag all interaction with the physical world.

Once there was a thing called “core labs,” hands-on activities that HAD to be done. In the eighties they numbered 12 to 15 per course. These days the number is more like six and, guess what, less than that are actually done. Oh, they’re talked about and sometimes even simulated–you know, rubbing your fingers across the glass top of a tablet or whatever to simulate motion, or something equally banal–but rarely ever really done.

What a shame. It turns out that our remarkable, wonderful brains are ideally suited to experience the world in two different but complementary ways. One way is procedural, logical, even rules-based. It is dealt with mostly–but by NO MEANS EXCLUSIVELY–by the left side of the brain. Talking, reading and experiencing simulations feeds it nicely. The other was is more holistic, even probabilistic, and, similarly is mostly handled by the right side. It’s best fed through direct physical and / or sensory experience with the phenomenon in question. Two views, ideally nicely merged and coexisting, producing a complex and useful representation of whatever the senses encounter.

Too bad that the simulated and  talked about and PowerPointed-to-death world is mainly processed procedurally. It’s not real in the experiential sense and, as such, the processing that the (mostly) right brain is so good at never gets to happen. As a result, the young people can talk and diagram about the physical world, even “solve” paper-and-pencil problems but give ‘em something real like some electrical components or mechanical parts and they have no clue whatsoever what to do with them.

Because they’ve never had the chance to. Still, they have experienced glass displays and they are no doubt adept with that.

Homework and Questions in Math: a Lob Scouse

I was discussing homework in mathematics with my friend and colleague Susan Ryan just the other day. Initially the topic of conversation was homework. Soon we realized we were pretty much of the same mind about it—specifically that it’s a good practice when given in the appropriate amount, thus implying that teachers need to be judicious regarding exactly what tasks they assign.

While there are no doubt many homes in which creative effective learning can take lace it’s been my experience that quite a few are not, and, in the interest of fairness to all, it’s vital to ensure that homework tasks are things that all students have a fair opportunity to get done well at home. I am therefore of the mind that it’s generally a good idea to assign some of the more routine tasks such as review drill and practice as homework. For example back when I taught grade ten math, before covering the factoring of trinomials I found it useful to assign a review of math facts (I am not kidding. That was grade ten and I am thinking of the 1980s. The whole problem about remembering math facts is truly nothing new) before getting started as a significant number of the students were rusty, especially regarding the multiplication facts bigger than 7×7. Since it was not worth taking class time with I’d just pass out a boring review worksheet. Similarly, the following year, just before we started the unit on quadratics I’d also assign review on factoring polynomials as homework, again for the same reason. These days if I was still doing it I’d certainly make use of a flipped classroom approach and provide a modern version of essentially the same thing.


The talk then turned to things that do not work well for homework. Both of us, as parents, could easily recount numerous stories of especially frustrating experiences we’d had when our children asked for help with certain tasks that had been sent home. We then put on our cynical retired teachers’ hats and concluded that in many of these instances what had been sent home were items from the textbook that the teacher did not feel comfortable with handling in class.

My particular “favourites” were the ones that start with, “explain why…” such as “Explain why you chose the solution you did,” or “Explain why a polynomial function of degree n can have, at most, n  zeros.” I was particularly not fond of questions that started with “Write to explain…” Now, don’t get me wrong—I am very much in favour of students working on their mathematics communication skills. It’s just that sometimes the questions do not lend themselves to independent thought and are much better handled as a group or whole-class discussion when many ideas can be drawn out, thus forming something of a mosaic of shared understanding.

When tough questions like the ones above are posed to individuals, the students tend to find themselves totally stuck and most either totally give up or turn to the parents. In the absence of classroom context it’s not hard to see how those same parents would become equally confused, frustrated and absolutely angry at a mathematics program that would expect from young children to be able to respond to such open ended and difficult questions; questions that in the absence of context (after all they were not in class and can’t be expected to see how it fits in) seem pointless.

Me—I would never knowingly do that. I would have no problem taking up those “explain why…” questions but never for homework.


And that’s where things got interesting. Susan then more-or-less agreed, but took the conversation in a different direction. She encouraged me to think about those questions and how, so very often, they form the stuff of angry, frustrated Facebook posts. You know what I’m talking about, don’t you? Some samples that I have seen at one time or another follow.

Regarding those posts, as is often the case, the extreme frustration is generally due to two errors. First, the teacher likely made an error in judgement and either selected a problem that was too difficult and, second, the parent assumed that the child know much more than they did. For example, in the above letter back to ‘Jack” the parent clearly has no idea of how difficult place value is for young children, and that the idea of ‘borrowing’ to do computation makes absolutely no sense until the child does have that firm grasp. Simply put, children and not small adults. Sure, the suggested parent solution seems obvious to someone with a university education but for a grade school child—no! That response–the letter to Jack–while venting pent-up frustration, just suggests the enormity of the gulf that exists between the child’s grasp of math and the adult’s grasp. As Susan would point out, the parent’s solution amounts to no more than “squiggles on a page” for most young children until a LOT of development work is done on number sense, place value and subtraction itself.

We don’t always acknowledge the simple fact that things that we know very well seem obvious to us. The better we know them the more obvious they are and, in fact, the less perceived need for any external learning strategy.

Just because things are obvious to us, doesn’t mean that are equally discernible for others, though. That’s a fundamental problem we all have as parents—we think that young children are just smaller versions of their adult selves and totally forget the fact that as the body develops physically so, too, does the brain. The mental capacities we have as adults are not necessarily developed to the same extent for kids.


That, then led to Susan showing me one of the activities she has used with parents to help explain why teachers do what they do. She asked, “why is 2 + 3 = 5?”

I was caught unawares with the question and mumbled something. She laughed and said, “hang on. Let’s think what students might say instead of what you might say.” She got out a sheet of paper and wrote this:


“There will be some students in the class who just ‘get it’ intuitively and it will seem obvious to them. They see it as most adults do. It’s not that they can’t explain it—no it’s just that it’s so obvious to them they suspect you’re daft to even ask (just like most adults). They are where they need to be and are ready to move on.”

Then she wrote this:

3   4   5

“Some students know to start from 2 and just count up three more. They’ve also pretty much got it. They need a little help, perhaps, as they’re not thinking abstractly just yet, but it’s safe to say they probably understand what’s going on well enough to take it to the next level.”

Then she wrote:

1   2       3  4  5

“Some students need to write the whole thing out from the start. They are still pretty concrete as they have to have 5 ‘things’ before them in order to understand what’s happening. Perhaps they will even draw 2 dots and 3 dots. They have the essence of addition but are still heavily reliant on the concrete. Perhaps it’s a development thing, and you just have to allow them time for their brains to develop more, or perhaps they just need more help—a decent explanation of what addition is might be all that’s required for some. Either way, you know there’s a hump to be overcome before they have it nailed down.”

Then, finally she said, “and, finally there are the ones who have to look at what everyone else has done. You might call it cheating or copying, or whatever. At any rate they will be unable to complete the activity and are certainly not ready to do anything further unless more development is done. They can complete work and make it look good, but it’s not their own work. For these students it makes little sense to memorize addition facts as they have no idea what it’s all about anyway.”

“So people often wonder why teachers ask what seem to be silly, even stupidly obvious questions. The fact is, though, that they’re not necessarily silly. The answers the students give is tell an important part of the story regarding the degree to which they understand any given topic. In turn this informs our own practice.”


Funny how one thing leads to another, isn’t it? Homework led to a discussion of questions and how what seems to be nonsense on the surface can be something quite useful when you consider it in context. Most things are like that, aren’t they?

Oh, and since this rambling disquisition seemed to start with homework it might as well end there. Any way you look at it homework is something that should be carefully considered. After all, while students’ time in the classroom is precious and needs to be spent wisely, so too, is their  time out of it. Many other things besides school subjects need attention—sports, the arts, learning social interactions, volunteer time, and just having fun, to name just a few. If a school subject needs to occupy that time then it seems to be the respectful and wise thing to choose carefully the when, the how much and, most importantly, the what.

Final Note: perhaps you don’t know what a lob scouse is? Click the link to find out.

Zero’s Nothing, Right, so What’s Wrong with It?

There’s a lively conversation in my province around something called a “no-zero policy.” This was something several school districts implemented several years ago, however there seems to be no evidence of any such thing being currently in existence in the current school board configuration. At the moment it’s my understanding that new policy is being written and that in the meantime the existing regions that were folded into a larger board are expected to continue with whatever they previously had. The “no zero” policy applies variously, then, depending on where you are located. That, however, does not seem to matter  to anyone, especially now in pre-election times when cries of “end the no-zero policy” seem to be coming from several quarters, with the assumption that at the moment it applies to all.

Just what is a no-zero policy and why do some think it’s important? And, just as importantly why are others so bitterly opposed to it?

Let’s try and make it simple. It’s generally reported that “no zero policies” state that students cannot be given a grade of zero for late, un-submitted, or plagiarized work. The most often reported justification is that evaluation consultants (who are sometimes accused in the media of never having to step foot into a classroom) recommend this because young people should not be unduly punished for making the kind of stupid mistakes they have always been want to make. Presumably by offering second chances the students have the chance to learn by their mistakes and, hopefully, not suffer any long-term negative consequences as a result.

Sounds OK, right? So why is it that so many are bitterly opposed to it?

For an answer to that let’s briefly consider human nature. What if there were absolutely no consequences for not submitting work on time or not being honest, that is, plagiarizing work? To answer that, just ask this question: why do we insist that work get passed in on time and that it be the individual’s own efforts? Simple—if you don’t do that, many (perhaps even most) will not bother putting in the required effort and will just put it off for some other time. That’s why we have deadlines and that’s also why we attach consequences to them. If we didn’t most work would never get submitted in a timely fashion and the small trickle of always-late work would result in very poor learning and an impossible-to-manage situation for the classroom teachers. Small wonder that much of the resistance to “no zero” policies comes from practicing teachers who are tired of dealing with this.

Well, then. Fine—it seems that, in light of this, it makes perfect sense to totally ditch no-policies, right? After all it’s one thing to give a student a break, but the removal of consequences in the form of zero-grades will likely result in a situation that is far worse: (a) students will do worse because they don’t take the assigned work seriously, always figuring they can do it later or maybe even get a do over and (b) the steady, unregulated trickle of inbound work that happens in the absence of enforced deadlines results in an unmanageable situation for the teacher.

Just think about how this might play out. Suppose that you are the type of student who leaves things until the last possible moment. In all likelihood there will come a time that you will finally have to deal with a back-load of work. Perhaps it’s the few days before the first progress reports are due to be sent home. “Alright,” you’ll say, “I can’t get zero so I won’t submit. What’s the worst that can happen?” So, the teacher does the best they can. When scanning through your work they notice that several important things were never submitted and so, instead of giving you a grade—as they should be able to; after all the work should be there to be evaluated; it’s not the teacher’s fault—they instead have to write something like, “I am unable to evaluate your son/daughter because they did not submit any work.”

Stay with me.

The report eventually gets home. In all likelihood the parents would have to find out about it themselves. After all, what student would be stunned enough to bring home what is essentially a blank report card? Perhaps the parents get notified via email, or maybe from a friend. Whatever. At any rate the parents / guardians eventually see it.

And freak out.

You know what happens next: angry words are exchanged with the child and then frantic calls are made to the schools. The end result is that the parent swears that the late work will be submitted asap. Within a few days the student brings a pile of paper to the teacher and dumps it on the desk. “Here’s all my late work.”

The teacher groans. First of all there’s really no telling how much of the work was the student’s own. Perhaps it was, but under the extremely tense situation that would have unfolded at home in all likelihood other hands were involved in the production. Perhaps the parents “helped” or maybe a tutor was enlisted. Perhaps—heaven forbid—some of the stuff was even purchased online. It’s easy to do that. The teacher knows that too, and then is left in the unfair position in which they have to make an evaluation based on work that may, or may not, have been done by the student.

It gets worse, though. Recall that this work was done at the last minute. This, in turn, places great strain on the teacher. Evaluating student work is always time consuming and difficult. It’s also best done efficiently and well when the tasks are combined and grouped. Simply put, a teacher can do a better job in marking all of the work at once than in doing it in dribs and drabs. It will take much longer overall and will likely not be done with the same level of consistency. The end result is not good—much more work for the teacher along with the likelihood that evaluation is nowhere at the same level of quality and consistency.

So, with that in mind it seems to make perfect sense to ban all mention of “no-zero” policies, right?

No, it doesn’t.

Why? It still could still be about the fact that young people do dumb things and need to be given second chances (a thing I wholeheartedly agree on, by the way) but even if, in light of the previous argument, we decided that human nature will have to trump humanity, there still remains a tricky, insurmountable obstacle: grades are not “rewards.”

What is a grade? There are two answers:

  • (The informal one that seems to be prevalent in general use) It is a reward for “good work.” The better the work the better the grade. No work, therefore, translates to a grade of zero.
  • (the CORRECT one) it is a measurement of how well the student has achieved the curriculum outcomes.

Curriculum outcomes? Since 1995 the curriculum in this province, and for that matter, the rest of Canada, has been defined in terms of specific curriculum outcomes. These are statements that express what students must be able to do and are organized more-or-less hierarchically, and broken down by key-stage (grades k-3, 4-6, 7-9,a high can be key stages), by course, and then down to more specific statements that apply to a given course at a given grade level.

For example, one specific outcome from grade 6 mathematics is, “express improper fractions as mixed numbers.” (Note: improper fractions have a larger number on the top and mixed numbers are a combination of a whole number and a fraction. For example 9/2 is an improper fraction that, when expressed as a mixed number is 4 1/2)

Every course is defined this way and the Department of Education (DOE) has expended considerable resources in developing curriculum guides for teachers that, among other things, explain and describe the outcomes, offer teaching suggestions (contrary to popular opinion the DOE does not prescribe the method by which they are taught. It prescribes the what, not the how. Take note you people stuck saying “oh the Department imposes discovery learning” nonsense.) along with suggested methods by which achievement of the outcomes can be evaluated.

Here’s how it works. The DOE describes what is to be taught and the school district takes care of getting the job done—the how. Teachers are therefore expected to provide evaluations that provide an indication of the extent to which the outcomes have been met. It’s all about the outcomes. They–and nothing else–are what define the curriculum. It’s not about what individuals feel should be in the curriculum but, rather, what’s been agreed to by curriculum committees staffed by teachers and led by officials from the DOE.

Evaluation? Grades are not rewards; they are measurements. In the example above it comes down to this: to what extent can the student write improper fractions as mixed numbers? In general terms a grade of 80-100 says they do it with excellence, a grade between, say 65 and 80 means they do it very well, a grade between 50 and 65 mean they do it reasonably well but could do better as this will impact future work. Grades below 50 mean that in the teacher’s professional opinion they do not do it well enough.

What, then, does a grade of zero mean?

This: that the student knows NOTHING WHATSOEVER about converting improper fractions to mixed numbers.

When a teacher assigns a grade of zero to a particular assessment they are certifying, professionally, using everything they have learned through 5-8 years of university learning (and backed up by numerous years of professional practice) that this is the case. A grade of zero, in this case means the teacher is saying “The student knows nothing whatsoever about improper fractions and that’s my professional judgement.”

Seriously, how can you certify that? They must know something about the topic.

I know what you’re thinking. You’re thinking, “But, what else is the teacher to do? In all likelihood the student had loads of opportunity to show the teacher they could do this. There’s no way the teacher sprang the assessment on them at the last minute and surely the teacher would have been open to working something out if the student had to miss the assessment for a valid reason. It’s probably the case that the student threw away the chances they had and so, what else could the teacher have done? They deserved the zero and if we did this more often they’d probably pull up their socks and get the work done.”

There’s still the fundamental problem, though. The above argument just brought the whole thing back to the original—incorrect—definition of what a grade is. In the same way that a grade cannot be considered a reward for good work it also cannot be considered a punishment for un-submitted work. That’s not what assessment is about. Any way you look at it, as long as our curriculum is defined in terms of outcomes the grade has to be a measure of how well they have been achieved.

What then do you do in the case of un-submitted or plagiarized work?

That is the real question.

Let’s draw a box around the answer so before defining what one should do, let’s specify what you should NOT DO. You should not:

  • Assign an arbitrary zero as there’s no way the student knows nothing about the outcome.
  • Roll over and do whatever the student / parent wants you to do.

At this point in the essay the preachiness will come to an end. Clearly there are no simple answers but something needs to get worked out that is in everyone’s best interests. Perhaps this means a provisional “no-zero” that imposes practical limits to prevent abuse. Middle ground is the only workable solution but it’s very difficult to state the procedures in simple terms as by codifying the contingencies and responses, all you will do is (make a game of it and) construct something that starts to look as onerous as the criminal code of Canada! It might be best to express what is needed in the form of a framework, a more general set of intents and values that leaves the major decisions to the professional judgement of the teacher and school.

So what do you do when a student consistently fails to turn in work? You give them a reasonable opportunity to address the situation. Hopefully they will make good use of the chance given to them. Most will. What if they don’t? If there is evidence that the student has behaved in an unreasonable manner despite being given chances then an incomplete or failing grade will have to be justified and assigned. Recall that a zero grade doesn’t mean the student DID nothing, but rather that they KNOW nothing whatsoever. Frankly it`s difficult to see how anyone can score below, say, 20, so maybe that should be the arbitrary minimum.

But that’s not for me to say. It is, rather, an issue for the school district to continue to grapple with and hopefully it gets to do so without political interference. Know what? Right now, as the politicians rant and rave about the no-zero policy I’m willing to bet that teachers are busy behind the scenes trying to work through the complexities that have just been laid out. I wish them all the best.

Three Myths About Math Education: Part 3-The Advocated Methods Were Different

The two previous posts dealt with two of three myths that exist in math education: (1) that the current curriculum is a “mile wide and an inch deep” and (2) that the current curriculum is built around discovery learning. This post will address a third: that current advocated methods are radically different from the ones in vogue decades ago.

Before going ahead let’s try to make this clear: the current curriculum advocates a balance between memorization and understanding, Consider, as evidence, the snippet, just below, from the grade 4 mathematics curriculum guide. Read from the second column the two bullet points. Notice that the first one talks about strategies going as far as 9×9 and that the second one states clearly that by the end of grade 4 the students need to have committed up to 7×7 to memory.


So that’s what it is for today, but what about times past, the times when people claim to remember a math curriculum that emphasized rote learning of math facts?

You may recall that in the previous post a comparison was made between the topics covered in the current student support materials and those that were in “Investigating School Mathematics” the textbook series used for most of the 1970s. You may also recall that it was found that the topics were rather similar. In a similar way, the language used in the teacher guides was also examined and, once again, striking similarities were found between that language and that in use today. Let’s look at some examples, all from grade 4.

We will begin with the general introduction to the book. Skim through the quote below, taken from the front matter of the teacher’s guide:

It is intended that each day’s lesson in which the child is presented with a new concept be divided into four parts: Preparation, Investigation, Discussion, and Using the Exercises. The preparation usually should be kept fairly short, and care should be taken to see that this work does not preempt either the Investigation or the Discussion. Generally, the Preparation should do nothing more than provide the children with that readiness which they need before they begin the Investigation. The Investigation presents the rudiment of the concept treated in the lesson and should be the “main event” in terms of pupil activity and involvement in the unfolding of the concept.

In general, it is expected that the Investigation be done by the children either independently or in small groups. Think of the Investigations student-centred activity. It is fully anticipated that the students will grope, question, search, and explore. Investigations are designed to provide for individual differences; that is, the child is frequently asked to perform a certain task as many ways as he can, or to find how many ways he can do a certain thing. By presenting the child with this type of challenge, at least some degree of success is assured. That is, your slowest student will find that he can do something more than one way, while your more able children will find many ways to do a given task. Thus, as you guide the children through an investigation, it is important for you to recognize that they will achieve in widely differing ways, and that you should give recognition for all levels of achievement. Perhaps the most important thing to remember in working with the children during the Investigation is to encourage them to do the thinking arid exploring.”

Critics of modern-day math education, who vividly “recall” spending significant time memorizing math facts and performing endless drills may find it hard to come to terms with the fact that the above quote came from the text series in use in the province in the 1970s. In particular the “Investigations” may come as a surprise. Yes, even four decades ago, there was an acknowledgement among the teaching veterans that students need adequate time to explore new situations and to try and get them to fit with their preexisting cognitive structures.

It does not end there. The whole idea of students talking about and discussing their math work is not new, as evidenced by this quote from the same book:

“Following the Investigation, the children are given an opportunity in the Discussion section to talk about what they did and to summarize the mathematical ideas in the lesson in preparation for working independently in the Using the Investigate section. Generally, the beginning discussion exercises are designed to stimulate the children to talk about what they did in the Investigation. You should encourage them to discuss the various methods that they used to investigate and explore the concepts. Also, you should follow your teacher’s guide carefully to ensure that whatever mathematics ideas are to be developed in the section are actually summarized an understood by the children.”

And, finally, the current ideas we term “Differentiated Instruction” are also not new. Consider this quote from the front matter of the teacher’s guide:

Minimum, average, and maximum assignments are provided for each lesson other than review lessons. These assignments are given to assist you in providing for the individual needs of the children. It is not intended that you give the minimum assignment to the slower children, the average assignment to the average children, and the maximum assignment to the more able children. Rather, these designations are given to assist you in making individual assignments according to needs, abilities, and time available for each individual child. For example, if time is short and you need to move rapidly through a particular lesson, you may choose to use the minimum assignment for all children. The minimum assignment will, in general, provide the children with sufficient practice and mastery of skills to move ahead to the next lessons. On the other hand, you may sometimes choose to use the maximum assignment with slower children over a period of two or three days. Also, it is highly likely that you will not want to assign the maximum assignment to the more able children, since quite often they need less practice than some average and below average children. For example, when your more able children demonstrate the ability to perform a particular skill with great efficiency, they should not be made to drill excessively in that skill. In some cases, an asterisk is placed beside an assignment to indicate that the lesson could be omitted without loss of continuity. “

But, enough with the general talk. What follows are images of separate pages from the grade teacher’s guides. Each image contains a page from the student text as well as teacher’s notes. Notice that in each case, while there is ample opportunity for drill and practice, strategies are also presented that emphasize concept development and connections.


Notice in the above case that the students are actively encouraged to relate multiplication with an area model.

Here’s another example.


Notice that, once again, the focus is on strategies. In particular the investigation notes encourage teachers to help students to use already-known facts to help find the ones they need to learn.

Here’s an example from grade 3. Notice, once again, there’s a balance between concept development / exploration and drill.


Notice, from the investigation notes, that a degree of insight and creativity is encouraged.

Now this is not to try and pretend that things like drill and practice do not / did not have a valid place in the curriculum. It is, rather, to point out that things today are not as radically different from the past as some would have us all believe. There is ample evidence of, for example, opportunities to practice with multiplication facts. See the image below–taken from grade 4–for example. The page from the student book clearly is intended as basic drill and practice.


But look a little closer. In particular look at the teacher notes at the right of the student page. Notice that, rather than just assigning the exercise the teacher was presented as a valid option for making a game out if it.

See also the example below from grade 3. Straight up “drill and kill,” right?


But drilling is not everything. Look at the teacher note to the right of the student page above. It’s indicated by the circled “1.” It’s acknowledged that the students don’t need to recall all the facts with speed, and furthermore, that it’s also important that students know strategies that can be used to find facts they do not know rather than having to look them up. So, too, for other grades. From the grade 4 book covering that same ideas comes the page represented below. There are several things to notice.

First, note the paragraph indicated with the circles “1.” From reading it there’s no doubt regarding the importance that’s been set on the students accurately stating the multiplication facts. Notice that quickness is now an emphasis–it wasn’t in grade 3.


But there’s more. Now look at the paragraph that follows, indicated by the circled “2,” and see that the students are expected to develop strategies for determining facts they do not know rather than just learning them by rote. Clearly there’s a balance afoot. Now, finally read the discussion indicated by the circled “3.” It becomes pretty obvious that the strategies emphasized by today’s curriculum were just as important four decades ago when “Investigating School Mathematics” was implemented.

So there it is. There are many, many more examples that could be presented but these will suffice. The long and short of these past three posts? 1. The curriculum today is similar in depth and scope to that of the past 4 decades. 2. the present-day math classes do not rely on the type of “discovery learning” that they may have been led to believe exists and, 3. the methods that were recommended decades ago are not so very different from the ones in vogue today.

Of course that’s nowhere near the end of the story. This post has only discussed what’s recommended, not what’s actually implemented in each classroom. Today, and in times past each school and each individual teacher placed their own interpretation on the recommended curriculum. In the 1970’s, no doubt, some classes used the methods advocated by the textbooks and others the drill-and-kill approach. The same is true today, and perhaps instead of wasting time longing for times that probably never existed we should be more focused on what actually works.

Three Myths About Math Education: Part 2-Discovery Learning

In the previous post a comparison was made between the math curriculum currently taught in grades 3-6 in my province (NL, CA) and the one used in the 1970s. It was noted that, while there’s an assumption that the scope of the content has been significantly expanded, the evidence does not support it. In fact, the present curriculum is not significantly different in terms of content from its predecessors. It was also pointed out though, that, perhaps the real issue is not with what is taught but, rather, with how it is done.

Besides the (erroneous, as it turns out) assumption that the current curriculum includes too much material—it’s often been called “a mile wide and an inch deep”—there’s also the assertion that current teaching practice is based on what some call “discovery learning.”

This is wrong on at least two levels:

  1. There is a misunderstanding about what is meant by the term.
  2. The teaching methodologies recommended today are not significantly different from the ones that were in vogue when most of today’s adults went to school.

This post, and the next, will justify the above statements. First, the misunderstanding. When many people hear, in the media, “the current curriculum advocates a discovery learning approach” they conclude that the teaching and learning of mathematics is left to the students’ own ingenuity; that students are expected to learn everything essentially unassisted. This, in turn, leads to a vision of frustrated students vainly trying to glean, in a noisy classroom, that which mathematicians took many centuries to build. Nobody can argue that this is anything but silly. After all, for just about everyone, learning mathematics is difficult and frustrating at the best of times. A situation in which students are left to discover all of it themselves it is nothing short of impossible.

That said, the above is not what is expected anyway. So, what is?

Let’s look at some of the language used in the curriculum guides, starting with a sentence right from page 1 of grade 4: “Students need to explore problem-solving situations in order to develop personal strategies and become mathematically literate. They must come to understand that it is acceptable to solve problems in a variety of ways and that a variety of solutions may be acceptable.”

At first blush–focusing on the term “explore”–this can be taken to mean, “leave the students to it. Anything goes.” That’s not the case, however.  Contrary to popular opinion this statement does not mean that students must explore novel situations unassisted. It means, rather, that in the real world problems are complex and rarely boil down to a situation in which a pat algorithmic single solution can be implemented. The world is not a series of predefined story problems, but, rather a complex set of interacting priorities, situations and limitations and students need to be given something more than just a few multiplication facts to memorize. But there’s more–students also need time. Simply presenting a “problem” situation is not enough. When confronted with a novel situation sufficient time and support needs to provided so that the student clearly understands what it is they are facing. Too often, “problem” situations are briefly presented and then the student is led immediately to THE solution. While this might seem to make a degree of sense–after all the best solution to anything is generally far from obvious–the fact remains that jumping too quickly from problem to solution typically results in learning that is superficial at best.

When it is said that students need to explore the situation it is not meant that they should be left to stumble erratically to a solution, but, instead be allowed sufficient time so that they clearly understand the situation and then be led, through various means, to an acceptable solution that results in the greatest possible growth mathematically. It is expected that a sufficient degree of guidance needs to be given by the teacher and that this will vary somewhat from student to student owing to: differences in maturity level, knowledge background, previous mastery level and so on. It’s not simple.

There should be no doubt that the current math curriculum favors understanding over rote memorization. Consider, for example, this quote from the grade 3 guide, where multiplication is first introduced: “Strategies for multiplication facts are a focus in Grade 4 and should not be the emphasis in Grade 3. The focus here is on understanding the meaning of multiplication and division and one to the other.” Fine, understanding is key.

I know what you’re thinking. “Hey, but surely you’re not saying that a basic knowledge of math facts is unimportant. That’s stupid on about as many levels as I can think of. Besides not being to do everyday calculations, how about being able to do further math? After all, how do you expect someone to factor x2 + 14x + 48 if they don’t know 6 x 8 = 48?” As a side – note, one quick way to arrive at the solution to the above is to find two numbers whose sum is 14, the middle coefficient and whose product is 48, the last term. they are 6 and 8 so the solution is (x + 6)(x + 8)

Admittedly, you might be led to think that multiplication facts are unimportant by merely skimming through the guide. Consider this quote, for example, which comes from page 17 of the grade 4 guide: “By decreasing emphasis on rote calculation, drill and practice, and the size of numbers used in paper and pencil calculations, more time is available for concept development.” So what’t it saying? After all, to say the least, this one is open to interpretation. On the one hand it could easily be taken to mean that rote memorization of things such as multiplication facts, is not something that should be happening at all (a contention that I most certainly DO NOT agree with, in case you are wondering). On the other hand, you could choose, as I have, to take this to mean that teachers should ensure that rote memorization is not seen as the one and only thing that is emphasized.

It’s likely that this is closer to what is intended in the guide as evidenced by a quote on the same page, “There should be a balance among mental mathematics and estimation, paper and pencil exercises, and the use of technology, including calculators and computers. Concepts should be introduced using manipulatives and be developed concretely, pictorially and symbolically.” If this is true the real issue, then, is in finding that balance, not of eliminating the less fun activities, such as memorization. In other words it’s important that students learn math facts but doing this in the absence of other things is pointless.

It’s really too bad that the term “discovery learning” has made its way into the popular vocabulary, especially in light of the fact that the word “discovery” does measure greatly in how math, and for that matter just about everything, is taught. Perhaps, with that in mind, it’s wise then that we distinguish between two different uses of the term:

  • (the bad use) Discovery is what happens when students arrive independently and essentially unassisted, at some major conclusion. This is something that is very difficult to cause to happen in a classroom. Sure, some students may be able to make it on their own but they are very mich the minority.
  • (a better use) Discovery is what happens when cognition happens, through whatever means—presentation, direct teaching, performing various teacher guided learning activities and, yes, guided discovery. This is something that is difficult to manage but doable by a teacher who is suitably prepared. What it involves is a teacher who knows when to take the reigns and when to hand them over.

There’s one more thing. In no part of any guide is a “discovery learning” approach recommended. That part is just a misinterpretation of the real intent–to balance “the basics” with understanding and application.

There’s still the second issue: that is, the notion that the situation was radically different a generation ago.

They weren’t. Next post.


Three Myths About Math Education: Part 1-Inch Deep and a Mile Wide

No school subject experiences more public controversy than does mathematics. Whether it’s the start of the school year, the fallout from a letter to the editor of a local paper or, perhaps, the release of an international measurement such as PISA, numerous occasions present themselves throughout the year that stirs up the intense negative feelings the general public seems to harbour toward mathematics.

These criticisms tend fall into one of two broad categories:

  1. Issues with what is taught.
  2. Issues with how math is taught.

Let’s start with Item 1, the what.

There seems to be an unchallenged assumption that math education is trying to do too much. In fact it’s become popular to characterize the math curriculum used in my province (Newfoundland Labrador, Canada) as “an inch deep and a mile wide,” a sentiment typically voiced by those apparently longing for the so called “good old days,” times when all students allegedly sat quietly in rows and assiduously practiced nothing but their math facts; sum after sum after sum, inching closer and closer to perfection one supposes.

Kind of reminds one of a pre-Christmas Carol Bob Cratchit, does it not? That and everything that goes along with it, you know, thrifty middle class that doesn’t ask too many questions, stays out of the way of the privileged ones, doesn’t rock the boat too much, especially with that pesky liberal minded thinking.

But, that’s a digression…

Let’s come back to reality and take a closer look at that apparently too-unfocused curriculum.

A few weeks back I took the time to dig out, from the education library stacks, the student textbooks and teacher resources currently in use in my province for grades 3 to 6. Why those grades? Simply put, there wasn’t the time to go through all of the grades. A quick glance through grades k-2 showed no great surprises anyway. As I was already intimately familiar with what’s there from grades 7-12 I decided that no personal gain would come from a review those so I settled for the middle ground of grades 3-6.

I then scanned through the materials noting the major topics and organized them under general headings such as Measurement, Geometry and such. The list, once completed, did not seem all that diverse.

I then took it to the next level and decided to do the same for the curriculum that had been experienced by those typically levelling the criticisms: the adults. This posed a bit of a challenge as the math curriculum has been changed several times in recent history. The current one dates from around 2009, the previous from around 10 years previous to that, then the late 1980’s, the early 1980’s and, finally the early 1970’s. Since I was quite familiar with all but the last one, having been a teacher / program development specialist during the years they were in use, and knowing that they were not substantially different, I decided to go for broke and take on the earliest one I could find in the archive.

After all, the so-called good old days that the loudest opponents seem to be calling for did seem to be set in those times.

The series was called “Investigating school Mathematics,” published by Nelson and was in use in this province (NL, CA) during most of the 1970’s. I dug the teacher and student materials out of the archives and repeated the exercise.

The lists were then matched and tabulated side by side using the best match between terms. The results are below.

Here’s Grade 3:

Grade 3
1973-74 2009-10
Patterning (increasing & decreasing patterns)
Counting & Measurement (length, rulers, half-units, area & volume with squares & cubes, liquid measure) Measurement (time, calendar, ruler, perimeter, mass)
Number Theory (odd / even, multiples, factors, primes, place value) Numbers to 1000 (representing, ordering & estimating)
Addition & Subtraction (relation between the two, large numbers, regrouping, money) Addition & Subtraction (relation between the two, 2-digit addition / subtraction with regrouping, estimating)
Fractions (parts, naming, comparing)
Geometry (edges, faces, vertices, segments, angles, triangles)

Geometry (parallel & intersecting, angles, quadrilaterals, parallelograms, polygons, symmetry)

Geometry (point pictures, symmetry, translations, functions, bar graphs, negative numbers)

Geometry (polygons, prisms, pyramids, cylinders, cones, spheres, sorting)


Multiplication & Division (number line, sets, repeated addition, commutative, multiplication-addition, multiplication algorithm, estimation, division by rectangular arrays, repeated subtraction, number line, quotients as factors, relation to multiplication., finding quotients, remainders, checking) Multiplication & Division (repeated addition, grouping, division as sharing, repeated subtraction, arrays
Data Analysis (collecting, bar graphs)

There’s fairly similar, aren’t they. For the one in the 1970s you may have been expecting something that was heavy on the basic computations and memorizing of math facts, perhaps. While, for the earlier one, there are more words next to the Multiplication & Division section you will notice that the focus is on various strategies by which the process is understood and used instead of the “drill & kill” you were probably expecting or, for that matter, think you remember. It’s essentially the same for everything else. The one notable thing is the inclusion of a Data Analysis (focusing on reading bar graphs) in the newer curriculum.

Here’s the table for grade 4.

Grade 4
1973-74 2009-10
Patterns (in tables, in charts, solving problems and equations with patterns)
Numbers & Measurement (estimating, perimeter, distance on maps, area from counting squares, volume from cubes, liquid measure) Measurement (time, dates,  area & volume, cm2 & m2)


Numbers & numerals (place value, inequalities, large numbers)

Number Theory (even/odd, factors & products, primes, function machine)

Numeration (Place Value, number words, large numbers, number lines, inequalities, ordering)
Addition & Subtraction (function machine, regrouping, magic squares, money, estimating ) Addition & Subtraction (L-R, and R-L, regrouping, renaming, estimating)
Multiplication & Division (relation between, mult.-add. principle, Multiplication facts, function machine, multiples of 10, Multiplying 2 and 3 digit factors, problem solving Dividing through subtraction, estimating, 3-digit quotients, 2-digit divisors) Multiplication & Division (relation between, Multiplication facts review to 7×7, New to 9×9. skip counting, doubling & halving, multiples of 10, patterns in a multiplication Table, 3-digit multiplication, dividing through subtraction, estimating quotients)
Fractions (number pairs, parts of a whole, representing length, equivalent fractions, adding, mixed numbers) Fractions & Decimals (fractions of whole and group, comparing & ordering fractions, decimal tenths & hundredths, estimating, making change)
2D Geometry (points, segments, rays, angles, Parallel, pyramids, symmetry)

3D Geometry (cylinders, cones, circles, tangents, congruent and symmetric figures)

Motion Geometry (ordered pairs, stretching & shrinking, translations, negative numbers)

2D Geometry (symmetry, tangrams)

3D Geometry (prisms)

Data Relationships (pictographs, bar graphs, Venn and Carroll diagrams, solving problems)

Once again, the two versions are remarkably similar. Once again look to the multiplication and division section. In both cases the students are expected to know the multiplication facts up to 7×7 at the start of the year and up to 9×9 at the end of the year but in neither case does it seem that s single minded focus on drill is evident. Rather, it seems that the focus is on building in an overall understanding.

Grade 5 follows.

Grade 5  
1973-74 2009-10
Sets, Logic and Patterns (exploring sets and patterns) Patterns (modelling patterns, increasing & decreasing patterns, solving problems and equations)
Numbers (place value, different bases, old numerals)


Numeration (representing numbers, expanded form, rounding, large numbers, decimal place value, rounding decimals, comparing decimals, estimating with decimals, adding & subtracting decimals with regrouping)
Decimals (and fractions, adding and subtracting, with metric units)  
Equations & Operations (function machine, function rules, solving equations, multiplication-addition principle)  
Estimation (note appears throughout; not done separately)
Computing (area, perimeter, average) (note:  area & perimeter done in measurement unit)
Dividing (estimation, 2-digit, larger)  
Number Theory (factor trees, primes, Union, Intersection, GCF, LCM  
  Multiplication & Division (strategies, relating facts, by tens hundreds etc., halving & doubling, estimating, multiplying 2-digit numbers, multiplying using base 10 blocks & arrays, estimating quotients, using subtraction to divide, dividing by sharing, remainders as decimals, interpreting remainders)
Fractions (length, naming, comparing fractions, ratio, LCM, mixed numbers and addition, subtracting, measurement)

Multiplying and Dividing Fractions (whole numbers and fractions, dividing by faction, function machine, equivalent fractions, lowest terms)

Fractions (and area, equivalent fractions, on a number line, comparing fractions, decimals)
2D Geometry & Measurement (points, lines, segments, congruence, parallel, measuring angles, symmetry, congruent Triangles)

3D Geometry (space figures, volume, surface area, POV)

Graphing and Geometry (Symmetry, Rotation, Enlargement, Tessellations)

2D & 3D Geometry (lines & faces, parallel, intersecting & perpendicular, sorting quadrilaterals)

Motion Geometry (translations, reflections, rotations)

A New Math System (other kinds of arithmetic e.g. clock arithmetic)  
  Probability (lines, experiments, comparing probabilities, solving problems)
  Data relationships (gathering data, double bar graphs, solving problems)

Measurement (length, perimeter, area, volume in cm^3 and m^3, litres & ml)

Again, the two are more alike than they are different.

And, finally, here’s grade 6

Grade 6  
1973-74 2009-10
Sets, Logic and patterns (classifying figures, exploring similarities and differences, dot matrix printers) Patterns (identifying patterns in numbers, in tables, creating tables using expressions, equivalent equations)
Numbers (exponents, bases, scientific-notation) Numeration (millions, billions, large numbers, millionths, billionths)
Number Theory (primes, factor trees, sets, GCF, LCM) Number Theory (factors, multiples, primes, integers, order of operations)
Whole Numbers (function machine, distributive principle, equations  
Ratio and Percent (ratio, equivalent rations, scale drawings) Ratio & Percent (ratio, equivalent ratios, percent, scale diagrams)
Fractions (equivalents, lowest terms, Adding & Subtracting Fractions, Multiplying & Dividing fractions) Fractions (modelling fractions, improper fractions & mixed numbers, comparing fractions)
Decimals (multiplying & dividing, rounding) Decimals (money, multiplying and dividing by 1-digit numbers)
Geometry (perimeter, circumference, area, Pythagorean, volume

Geometry (congruence, construction, symmetry, angles, tessellations, cross sections

Geometry & Graphing (ordered pairs, reflections, rotations, translations)

Geometry (angles, angles in triangles & quadrilaterals, area, perimeter volume of rectangular prism, classifying triangles, sorting polygons, congruent polygons

Motion Geometry (reflections, translations, combining transformations)

Probability (experiments, outcomes) Probability (experimental vs. theoretical probability
Positive & Negative Numbers (meaning, adding & subtracting)  
  Data relationships (questionnaires, databases, experiments, line graphs)

Regardless of how closely you read the above tables it’s obvious that the curriculum of today is not so very different from the one experienced those who’d now be in their 40s and 50s. In particular, there’s no evidence that that today’s covers significantly more ground.

So where does the notion that “today’s curriculum tries to do too much and we should move back to the way it was,“ come from? Frankly I am at a loss and can only speculate that it is an erroneous thought, the product of selective memory, and one that that has never been challenged. As you can see, though, there’s scant evidence that supports that it’s much different from the one experienced by those fronting the accusation.

But, perhaps you’re thinking, “No, it’s not the what I was worried about. It is, rather, the how. Today’s math education is focused too much on “discovery learning” and it’s silly to expect that students, by themselves, can be expected to rediscover what took genius mathematicians many lifetimes to figure out.”

Let’s deal with that in the next post.

Reconsidering Programming in Schools as a Mandatory Course

Earlier this week I heard a piece on the CBC St. John’s morning show regarding the assertion that computer programming is something that should be taught in our public school system. Later on I read much the same article off the CBC NL website. The spokesperson for Code NL asserted that existing public school courses within our province were “a joke” and noted that in giving people the training in computer skills we could move away from our reliance on natural resources.

While, at least on the surface, this sounds reasonable, the reality is much more complex.

First some of the assertions are inaccurate. The spokesperson stated that existing courses were “a joke” and, besides sounding rather condescending to both teachers and students alike, without data to back up this assertion it must be considered to be only a personal opinion. The belief that the courses are only offered “in the Metro” area is also false. They are, for example, available in Gander, Clarenville, Corner Brook, Stephenville, Goose Bay, as well as in a host of smaller communities both locally and through CDLI. There is academic life “beyond the overpass.”

But there are more important things that should be stated in reply to the story.

Chief among these is the fact that schools do not exist for the sole purpose of preparing people for the world of work. While that is certainly ONE of the aims of public education it is important to also realize that the full picture is much broader. Schools exist because we wish to have individuals with the attitudes, skills and knowledge necessary to lead happy, worthwhile lives—at home, at work and within the community at large. Yes, of course we need people who contribute to the economy—after all, bills, both public and private, have to be paid and for that we must all do our part: earn money and pay our taxes. That said, it’s important to remember that as a society what we really need are people who lead good, personally meaningful lives, and who also live out their duties to the community.

Added to this is the reality that we live in a diverse, vibrant society. Young people come to school with varying interests, abilities and values. Sure, we are all citizens of a single community, a single province and, at least at first glance, it makes some degree of sense that an intricate knowledge of those little electronic gadgets that so dominate our lives seems to make some sense. But just think about our already busy schools and consider the value of additional mandatory publically funded courses in:

– Plumbing, because running water and sewer are vital parts of our public infrastructure;

– Carpentry, because, shelter is important, especially in our nasty cold environment;

– Cooking, because we all have to eat on a regular basis;

– Embalming, because we’re all going to need it.

Of course not! That’s silly in the extreme. Schools cannot be expected to do everything and, besides, one of the benefits to living in a large diverse society is that we have the critical mass needed to ensure that levels of expertise exist, to the necessary extent, across any given community.

We don’t all have to be able to do everything.

So, too, with programming: It’s a vital part of our economy and its effects within our personal lives are too broad to even summarize. Still, we don’t all need to be programmers to appreciate the technology or to use it effectively.

There’s something else: it’s naïve to assume that taking a course or two, in school, in programming, is something that will prepare a young person for a career in that field. Programmers do much more than just write code. Sure, that’s a vital part of the enterprise and, besides, it’s fun to write code bits and have computers do clever things. That said, the fact is that only a few of the students who would be forced to take that mandatory course (or courses) would see the value in it and, thus, put in the required effort. The result would likely be a halfhearted thing leading to jaded teachers and students; in sum a waste of time and money.

The reality is that computer science is not something that can be sparked and ignited like your backyard barbecue. It is, rather, a complex skill that takes many years of personal investment of both time and effort. Besides knowing the basics of a code’s “language” the programming professional also understands logic and structure. Most importantly the programmer sees it all within a complex, disciplined problem-solving framework, something that only happens in an environment specifically created to doing just that—namely a computer science academic unit or a well-run enterprise dedicated to that pursuit…

…and specifically NOT a public school that is already over-burdened with unrealistic expectations from its governing agencies and from the public at large.

Still, the sentiment is a valid one, albeit a bit misdirected. Instead of trying to create yet another course, along with its attendant monetary costs (and they will be steep; computer hardware and software, along with the required training is costly; a bottomless black hole into which one pours money) perhaps those interested in promoting the cause of programming should do what others with similar interest have done, and continue to do: forego advocacy for outreach.

Instead of publically shaming governments and schools for not teaching the stuff, work alongside of the various partners: government, districts, the university and the NLTA.

Instead of asking them to do what you feel is important, offer free workshops for students and teachers. Visit schools and participate in professional development activities. Focus in integrating some of the skills and knowledge within the existing educational framework. Add vitality rather than simply grafting on something else to an already overburdened structure.

Electric Vehicles in St. Johns–Let’s be Rational

There’s been considerable debate regarding a recent RFP to purchase a pair of electric vehicles as part of the St. John’s city fleet. As expressed by councillor Dave Lane, who spearheaded the proect, the plan was to make the new vehicles available to the parking enforcement unit and to treat the whole thing as a pilot project and to see where things went.

Initially this did draw considerable, polarized, interest with those for it noting that much could be learned from the acquisition and use of the vehicles and, besides, the cost of operating them should be considerably cheaper. Those against typically viewed the whole thing as not worth the bother; playing with newfangled toys. The debate was brought to a head, though, by a letter written to the council by former mayor Andy Wells in which he slammed the plan as a waste of taxpayers’ money and concluding that electric vehicles are, in general, ” ‘driveway jewelry’ for the eco-affluent, who benefit from public subsidy to indulge their guilt about living in a fossil-fuel-dependent society,”

Then the fight started, the polar opposites moved yet further apart and both sanity as well as reason, it seems, exited the building.

Still, though, it’s impossible to ignore the fact that, year by year, electric vehicles (EV’s) are becoming more and more prevalent. Sales, while not exactly meeting the growth targets guessed at 4 years ago (they were expected to triple each year) have still been showing decent growth. What’s more, most of the major manufacturers are in on the industry. Presently, models are available from GM, Toyota, Nissan, Fiat, Daimler, Mitsubishi and of course Tesla to name just a few.

With a little time on my hands I investigated the costs associated with the requested purchase for the city. I asked just one question: does it make financial sense? In other words should the expected reduced cost of operation translate to a lower cost of ownership. Not to spoil the rest of the post but the short answer is “no” but it’s worth reading on, if you have the time and interest.

Let’s look at the cost for just one car and let’s leave the cost of the charging station out of it altogether since the car doesn’t really need a dedicated charging station as such; all it needs is access to a 110 V or a 220 V (preferred) standard outlet. Since the EV is to be used just around the city, all of the costs should be based on that type of driving.

Now—what car to choose? While no doubt some users would love to cruise around in something very nice such as the luxurious and trendy Tesla S,  we have to be more pragmatic here and choose something better suited to the job at hand. It’s for parking enforcement and won’t be carrying significant cargo. It’s also bought on the taxpayers’ dime so it therefore needs to be inexpensive. For the sake of argument let’s choose the Nissan Leaf.

Nissan Leaf (Wikipedia)

Let’s stick with the base model. According to Nissan’s website it can be had here for $33,788.00. Not exactly cheap but the expectation is that what we lose up-front we’ll gain back in the long term with lower operating costs.

Let’s figure them out. Let’s start with charging the battery. According to Nissan, the battery capacity is 24 kWh so you might assume that it therefore will take that much electricity to charge it. You need to take into account, though, the simple reality that no process is 100% efficient. You may notice that when batteries are charging and discharging they warm up. This means that some of the energy is being wasted as heat. You can expect, therefore to need more than 24 kWh.

Based on some data found here I am assuming 85% efficiency this means that to fully charge the battery you must therefore supply (24/0.85) or 28 kWh of energy. At NL Power’s current rate of $0.1178 / kW h this means a full charge will cost $3.30.

Next you need to determine how far a charge will get you. Nissan’s stated figure is of 135 km for a full charge.  The US EPA, however lists a much more conservative value of 117 km. Given the fact that batteries function les efficiently in cold temperatures, however, it makes sense to question the validity of even this figure. Fortunately, some low temperature data are available. The website has listed some empirically derived figures of how the range per charge varies with temperature so let’s use those.

We also need to use temperatures that are realistic for this setting. Average temperatures by season can be found here. Let’s assume that the vehicles will be driven 20,000 km/year and, further, let’s assume equal distances in each season. Since the temperatures will be different in each season let’s take that into account. The table below lists the anticipated costs for driving 5000 km in each season, as well as the yearly total.

Season Temperature (degrees C) Range (km) Cost for charge Cost for $5000 km
Winter 1 70 $3.30 $236
Spring 8 75 $3.30 $220
Summer 19 77 $3.30 $214
Fall 13 77 $3.30 $214
Yearly total $884

Table 1: Yearly charging cost for Leaf, assuming 20,000 km

Now let’s compare the EV to something reasonable. Since we started with a small Nissan EV let’s compare it to a small conventional Nissan, the Versa. Once again, let’s stick with the base model but equip it with an automatic transmission to make it more functionally equivalent to the Leaf. According to Nissan’s website that vehicle should come in at $17,165.00.

Nissan Versa Note (Wikipedia)

Now we need to find the cost of fuel for 20,000 km. Based on US EPA figures the Nissan Versa is rated at 7.6 L/100 km in city driving so you can expect to use 1520 litres to drive the 20,000 km in a year. The variability in gas pricing makes it impossible to provide a definitive single cost so upper and lower figures will be used instead.

According to Gas Buddy the yearly low was $0.95/l and the high was $1.44/l. This then gives two yearly fuel costs.

Based on $0.95/l
“low fuel”
Based on $1.44/l“high fuel”
Yearly fuel cost $1444 $2189

Table 2: Fuel costs for Versa, assuming 20,000 km

Now the maintenance. Data on this are available in US$ from and are presented in the table below (converted to Canadian dollars). Interestingly enough the figures are roughly the same and could realistically have been omitted from the calculation.

Vehicle Leaf Versa
Repairs and Maintenance $4844.20 4857.94

Table 3: Maintenance and repair

So, finally, let’s look at the total five year cost for the two vehicles. Insurance and licensing will be the same for both so we can omit them.

Item Leaf Versa (low fuel) Versa (high fuel)
Purchase Cost $33,788.00 $17,165.00 $17,165.00
Fuel $4420.00 $7220.00 $10945.00
Repair and Maintenance $4844.20 $4857.94 $4857.94
Total $43052.20 $29242.94 $32967.94

Table 4: Five year cost of ownership, based on purchase with no resale.

Clearly, presented this way, there’s no contest. Based on straight up purchase the leaf will cost anywhere from around $9000 to around $14000 extra to own over the five-year period.

Now, you may be crying foul, “wait a minute, you don’t ditch the car after 5 years. The Leaf will be worth more at the end of that period so this is not a fair comparison.” Fine. Let’s factor in depreciation. Once again the estimates came from autoblog.

Leaf Versa
Original Cost $33788.00 $17165.00
Depreciation $21317.32 $9115.47
Resale Value $12471.00 $8049.53

Table 5: Expected depreciation and resale values

Set’s just do the total cost table 4 above over again but use depreciation instead of purchase cost.

Item Leaf Versa (low fuel) Versa (high fuel)
Purchase Cost $21317.32 $9115.47 $9115.47
Fuel $4420.00 $7220.00 $10945.00
Repair and Maintenance $4844.20 $4857.94 $4857.94
Total $30581.32 $21193.41 $24918.41

Table 6: Five year cost of ownership, based on purchase with resale

The Leaf is still considerably more expensive, even when you consider a worst case scenario for gasoline.

From a strictly cost-based perspective, then, it does not make sense to procure and use the EV’s if the assumptions used are valid.

That’s not really the end of the story, though, is it? It’s not my intention to be negative here, just reasonable, and since the main argument put forward was based on cost it needed to be pointed out that it was likely invalid. That said, there are far more compelling reasons that may still make the plan a good idea. Consider these:

First, you need to consider the overall environmental impact of EV’s. They have the potential of being much cleaner and environmentally friendly. Assuming that the batteries are correctly recycled and re-purposed (and there’s cause for some optimism in that area, see here.) then the real environmental issue is related to the source of the electricity. Right now in NL, unfortunately that’s just a bad joke as the electricity is as non-green as it gets, coming, as it does, from a dirty thermal generating plant. Later, though, when the feed is switched over to the Hydro-based Muskrat Falls plant that will be an entirely different matter; much greener. Simply put, right now electric cars are just contributing to the pollution coming from the Holyrood Plant bit that will change in a few years—right about the time those vehicles are ready to come off the road as it turns out.

Second, you need to consider the value in foresight and planning—something often badly absent from the NL milieu. (As an aside a good friend often half-jokes that the NL Government’s idea of long-term planning is, “what’s for supper?” His words, not mine.) Based on the best available data it does seem likely that EV’s will become more and more prevalent as time goes on. To what extent? I would suggest it is impossible to ascertain that right now. It’s still worth considering. As such not only the city but also the province needs to devote a reasonable amount of time and effort in gathering pertinent empirical data regarding use costs, reliability, safely and infrastructure needs. In that light, the proposed plan, if altered and fleshed out appropriately as a rigorous pilot project, and not just a vague idea, can easily be seen to have significant merit.

So maybe the best advice to those involved should be this: plan it all out a bit better, look again at the timelines and goals, and maybe see if partnership assistance is available from the province. Don’t just mess around driving from meter to meter and, asking the traffic enforcement officials, “how’s it going with the new EV’s?” from time to time. No, devise a proper plan and implement it. Log everything: kilometers driven, time needed to charge, energy transferred in the charge, times required, maintenance and repairs–everything. Put it up there where we can all see it and benefit from it. In that way, properly implemented, the project does have the possibility of yielding information that can be used by consumers and governments alike.

Learning Resources: Where’s the REAL Commons?

Have you ever wondered why, over thirty years after personal computers became affordable, and over twenty years after the widespread adoption of the Internet, digital technologies have still not reached their full potential? There are some generally good reasons why this is so and it’s not primarily due to resistance to change. Let’s examine a typical course.

The Setting

Consider a reasonably popular and somewhat universal course of study: First-year University Physics. This course is taken by students primarily interested in pursuing careers for which a knowledge of that discipline is a necessity—all jokes aside, that is it is manly a course people take because they need to. It is a gateway to a career in oil & gas, mining, engineering, aviation and, yes, the very few who wish to become physicists also take it but they must be considered a minority. The students who sign up are typified by a wide range of interest and ability. Some of them have studied physics in high school and come to the course with a solid background—that is, well-developed laboratory/inquiry skills, mathematics skills and a decent grasp of the fundamentals. Still others have next to no experience in the area, a weak grasp of mathematics and, sadly an interest level that leans more in the direction of “Mom/Dad wanted me to do this,” rather than “This is cool.” The majority, as you would expect, find themselves clustered closer to the middle of these extremes, that is, they have some background and experience as well as enough motivation to make them show up for class and put at least some effort into performing the various required activities (being attentive during class, performing the lab work and making their way through the written assignments as best they can). Overall, to a few the course is pain to be tolerated, to another few it is a total joy; the essence of their existence, but to the majority it is a right of passage; a series of tasks to be done with care but not necessarily with the burning love and passion felt by the instructors and other members of the faculty. Simply put, the audience is reasonably competent and serious but by no means a young version of the faculty

The content of the course is a wide-scale survey of the discipline as a whole. To the extent that it can it tries to provide an overview of the various areas in which the discipline has stepped into. Over two semesters–two courses actually–it includes:

  • A non-matrix approach to statics (forces at rest).
  • A non-calculus approach to mechanics, including potential and kinetic energy, impulse and momentum as well as Newton’s Laws of motion (and maybe Universal Gravitation) with a particular focus on the 2nd.
  • Static electricity including the concepts of fields ( but without the use of field equations), charge and electric potential.
  • Current electricity including Ohm’s Law and Kirchoff’s rules but with a focus on DC circuits and a serious limitation in terms of complexity—the circuit analysis rarely involves the use of simultaneous equations.
  • An introduction to waves, including basic coverage of sound and light. Wave phenomena such as the Doppler effect, diffraction and interference are introduced with a minimum of mathematics.
  • Perhaps: An introduction to special relativity and quantum mechanics, fluid mechanics and geometric optics.

The course endeavours to serve as a bridge in many ways. It tries as best it can to be accessible to students who do not have a previous background while, at the same time, not boring those who do. It does try to impart a fair degree of disciplined thinking while at the same time, encouraging further study. All in all the managing of the course can be described as quite a balancing act.

But here’s the thing: like most (but not all) scientific disciplines it is reasonably universal. That is, the background required by students does not tend to vary much by geography. Unlike, say, history which is impossible to separate from the local culture, first-year physics can be assumed to be more or less the same just about anywhere.

The Issue

This brings us to the big issue: even though there is the potential for a large audience for it, there does not exist a high quality integrated set of digital teaching and learning resources for that course. There are, rather, collections of good efforts that must be assembled and then put to use at each institution, each doing as best they can despite limited human resources and budgets. All things considered this is a great loss.

The same is just as true in other subject areas including Pre-Calculus and Introductory Calculus, Chemistry, Biology and Earth Science, along with possibly Psychology.

Now, before this gets too far let me hasten to explain why this discussion is dwelling on just STEM. It is solely because those disciplines are reasonably global in nature, that is, there is more-or-less worldwide uniformity on what is taught and how it is taught. This is simply not the case for other first year courses such as English (or whatever you wish to call the study that centres on the most popular language in the region), any of the fine arts, liberal arts or social studies. In all of those disciplines the local context matters far too much for anyone to get very serious about talking about a global approach to learning resources. But let us leave that for another time and just return to the ones for which it is the case.

So what is the extent of available learning resources for STEM? Here’s a partial list.

  • Commercially available print-based resources including textbooks and self-study guides. These tend to cost in the range of around $200 each and are generally of good quality. They are logically organized, well-illustrated, complete and correct (contain modes of thinking in-keeping with the established canon). For the motivated student who reads well they serve as excellent and compete resources. For those less motivated they often lie unused, as evidenced by the many so-called “used” (Irony, yes) books out there with unblemished spines.
  • Instructors’ notes and personal websites. Once something you could only get if you could afford the photocopying fee, thanks to scanners, word-processors and most importantly electronic Learning Management Systems these are becoming increasingly accessible. The quality varies widely, owing to the lack of formal peer-review processes that typifies other areas of academic life, but at least in my experience leans toward “very good” more often than “lackluster.” Notes tend to be short-form representations, lacking in the commentary and elaboration available in books. They also tend to be more to the point and, unlike the texts, do tend to be carefully read by students.
  • Communitarian resources such as Wikipedia. Over the past decade these have significantly improved both in terms of scope and quality. For any given topic that one would find in a first-year STEM course the entries tend to be complete and useful. There is no guarantee, though, that the depth of treatment is the same as is expected in the course. Instructor guidance is definitely a must if Wikipedia is used as a source.
  • Other web-based resources. A significant number of piecemeal efforts exist. These do an excellent job on portions of a course but do not try to be a single point of contact. A good example of this is the University of Colorado’s Phet site, which has developed a huge array of Java-based science simulations. Taken one by one any of the Phet resources does an excellent job of exploring the topic it intends to but it has to be left to the instructor to decide which ones to use, how to use them and how to link them in with the rest of the course resources.

So a wide variety of useful resources does exist so what, then, is the big deal.

A Simple Vision

Let’s think for a moment what it could be like online when a student accesses the course.

The course home provides an overview of what’s in the course along with a summary of progress to date. This includes a list of tasks completed, along with appropriate achievement indicators (grades, etc.), upcoming events and deadlines as well as uncompleted tasks, along with suggested resources and activities. It’s worth noting that just about any online Learning Management System (LMS) such as Desire2Learn, Blackboard or Moodle can do this right now.

For any given course organizer (whether it be lesson, topic or learning outcome, for example) course resources are provided in a variety of formats including:


  • Print Materials, and preferably in a format that lends itself well to display using either paper or an electronic reader such as an eBook reader or tablet device.
  • Multimedia presentations—that is, an electronic version of an in-class presentation, complete with visuals and audio—that could be created with software such as Adobe Captivate.
  • Interactive simulations (where applicable) in which students could investigate topics of study. These should be similar to the ones already available from Phet but with the added value of having guidance on what you are looking for; the simulation has a built-in lesson plan. In some, but not all, cases (investigating DC circuits, for example) these could replace activities generally done in the lab.
  • Laboratory resources in the form of videos, analysis software and handouts that would be used in conjunction with lab activities. Students would still be expected to go to the lab but because these would replace the lab manuals and demos from the front of the room. Students would have more autonomy, meaning that at any given time various activities could be managed at once in the same location.


For any given course organizer the course would also host a variety of assessment/evaluation tools including these:


  • Traditional written assignments. These could be printed off, completed on pencil and paper, scanned, and then uploaded to the assignment drop-box for that item, where they would be graded, probably by a TA.
  • Online assignments, similar to the above but with the submissions and solutions done online. This is similar in form to the existing open source LON CAPA program currently used worldwide but with several important additions: (1) integration with the LMS instead of just stand-alone (2) provision for viewing of solutions, not just answers.
  • Interactive, Simulation based assessments. Instead of just working in pencil and paper the student would perform actual tasks online and be assessed on them. For example, the student could use an interface to work through an exercise traditionally done with paper and pencil or could use a drag and drop interface to assemble, test and analyze a circuit. These tasks could be done, for example by tweaking existing java based simulations or built from scratch using the simulation features in software such as Adobe Captivate.


Overall, you may notice that none of the items mentioned are too far-fetched. While this could have been listed: “The development of a completely immersive online lab based learning environment for physics,” it was not, owing to the extremely prohibitive cost (probably in excess of $50M).

The course assemblage mentioned has a much more modest cost, probably in the vicinity of $1M or so, with the majority of it going to the programming efforts of getting the interactive pieces up to a sufficient quality. While it is unlikely that any given institution could be expected to foot this sort of development bill, when you consider the fact, already mentioned, that this course is one that would have worldwide appeal it is rather amazing that it does not already exist.

The Barrier

Think about the numbers for a moment. Consider just doing the course in English and thus limiting it to primarily English – speaking countries (of course we really want this done in all popular languages but lets look at a limited, simple case here, just to make a point). This would potentially give a market in which millions of students would wish to access it. Currently those students are expected to purchase either new (at around $200/copy) or used (at around $100/copy) traditional textbooks for the course. What if, instead, this money which, at a conservative count would be around $50M per year (assuming that only half the students purchase the text and most of them buy used) were instead invested to the development of online resources? If the figures given were correct, the development costs would be recouped in such a short while as to be insignificant!

This, of course, makes no sense. Commercial publishers are not stupid enough to pass up such a lucrative cash cow so why has this not been done? I would suggest it is the sum of three interacting causes.

Educational institutions are unable or unwilling to fund the development of high-quality course content. It costs money—lots of it, and in these times when all institutions are facing increasing pressure to keep costs down any requests for additional funding are unlikely to be met with anything other than skepticism. To develop course content requires (1) time for the subject matter expert—likely an already over-burdened instructor (2) time for an instructional designer as well as (3) various multimedia/programming professionals who assemble the content into the various types mentioned in previous posts. Generally there is little or no money available to put the IDs and multimedia specialists on the projects and requests from the instructor for release time are met with the response, “we are already paying you your salary and we assume that the development if class-related materials is included in that already.” Simply put, the administration does not have the extra funds to pay the people and the instructors do not have the extra time to prepare the content that would be needed to take it to the next level.

Educational institutions do not cooperate to share the development burden. It has already been suggested that, while individual institutions are likely unable to fund the development of high-quality materials, collectively, the human and monetary resources exist when you consider that, at least for the courses mentioned, most institutions are, in effect, teaching much the same courses. If instead of each institution doing its own thing, they cooperated and jointly developed the materials it would apportion to very little.

The fact is, though, that this is one of those things easier said than done. To pursue a joint venture there must be (1) an overall plan (2) formal coordination and management of the project and (3) buy-in. The fiercely competitive atmosphere that exists between institutions coupled with the absence of a formal unifying body means that it’s hard to get this done, especially when you realize that this whole topic is nowhere near the top of most educational administrators’ lists of priorities. Still, it is a shame as the Internet has already demonstrated how well it is suited to cooperative development projects, as evidenced through successful development projects such as Mozilla as well as more communitarian development such as has been done with Wikipedia.

Commercial educational publishers are unable to implement an effective business model. Ask just about any administrator that holds the educational purse strings for education this question, “Why don’t you allocate money towards the funding of teaching and learning resources?” Chances are this will be the response: “Because that is the job of the educational publishers. We can’t afford to do it ourselves but they can because they can access a much larger market.”

Fine; after all, why waste taxpayers’ money when there’s a much better way?

Now go ask any executive with any of the major publishers the same question. Chances are, this will be the response: “Because we cannot recoup the development costs. Not only are institutions unwilling to pay the license fee, even though it is significantly less than they used to pay for textbooks, but, worse, our experience has been that people will always find a way to obtain and use our materials, regardless of copyright. We just can’t win, no matter what we do.”

Put the three together and you get the situation we face today. Despite the huge potential that Internet-based resources hold for improved teaching and learning in first year courses much—not all, mind you, but still the majority—of that potential remains untapped, with little sign of any widespread, sustained effort to do much of anything about it.

Suggested Solutions

This is not to suggest that the appropriate response is to just accept that things are the way they are for good reasons and the best that we should all do is to learn to accept the status quo. While it is unlikely that any revolutionary change is likely to happen in the short-term, significant benefits can still be realized from some straightforward actions. If sustained, some of the items below are likely to go a long way towards realizing the dream of much more optimal usage of digital technologies in service to teaching and learning. Here are three items which, taken together, hold every possibility of resulting in widespread improvements.

Existing and prospective faculty need to continue to work toward positive change. Large scale changes take time. Not only do materials need to be developed but, more importantly the two book-ending sets of activities need to be done right. (1) The preliminary work of understanding the current problems and planning appropriate responses need to be done well. Likewise (2) the follow-up activities of fine-tuning introduced measures and modifying them in light of unexpected contingencies is something that cannot be forgotten. In situations where the general consensus is that things are just fine as they are, the general response is not just one of stagnation but even worse, is one of gradual decline. Things break. Things change. If there is no response, what’s broken remains broken and sustained change brings the reality increasingly further and further from the classroom. If, on the other hand, the general consensus is “We need to make things better” then the meaningful improvements—that is the ones forged from REAL need—will slowly be realized in a spirit of collegiality.

Leaders (Deans, Directors and College Presidents and perhaps government officials) should move for more inter-agency cooperation. While there are few formal opportunities for collaboration, the academic world is rife with opportunities for informal exchanges: presentations, conferences and such. If, at those occasions the topic was, brought around to the whole idea of cooperatively developing teaching and learning resources, in time the interest would build and, along with it, the ways and means of getting it done. People who acknowledge a need tend to see the opportunities for finding the means by which to solve the problem—a positive off-shoot of the generally unhelpful confirmation bias. In short, talk about it and a way generally tends to be found for getting it done in a way that everyone can live with.

Publishers need to work more closely with institutions. Despite the fact that they work closely with some faculty members—after all most current texts are authored by faculty—publishers tend not to have good two-way relationships with the learning institutions their whole business model is built upon. Generally the only formal relationship is through the bookstore and the general attitude is one of client service, that is, the university states a requirement, generally in the form of a syllabus and then evaluates the available resources. This activity is generally muddied somewhat by the publisher’s efforts to sharpen their competitive edge through either the provision of some free goodies or through the haranguing of either the dean or the individual committee members. Often the relationship shakes out something like this: Faculty view the publishers as greedy & grasping and publishers view the institutions as needy & cold-hearted. All in all not a great atmosphere in which anyone can be expected to thrive.

It does not need to be that way. Institutions have great need for improved resources, especially as students gravitate more and more toward the Internet and away from print-based materials. Likewise the publishers are faced with an ever-diminishing pool of revenue as more and more of the old-style core business of just feeding the thirst for basic knowledge is met more and more through existing resources such as Instructors’ own websites and Wikipedia. What’s needed, then is a more sincere and productive dialog in which the publishers gain a better understanding of how to meet current needs while universities find better ways in which to ensure that the publishers’ financial expectations are met.

Overall, the situation is far from desperate. Despite the many shrill cries of doom and gloom our modern educational institutions are by no means in a sorry state; far from it. First year students do as well as they always have—in many cases even better. Enrolments, overall, tend to be strong, and the product—students who achieve and thrive—tends to be good, as evidenced by the continued relative success that all still seem to enjoy.

That said, the situation as always can be improved. The great potential that the Internet holds for education is far from being realized. An overall attitude that is conducive toward positive improvement coupled with a willingness to strive collectively, achieving the small gains that, measured together will result in great strides is just what we all need.