# 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:

5

“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.”

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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.”

• (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.

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.

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.

 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 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.