Monday, June 22, 2015

Meet the "New Math"...Same as the "Old Math"?

The title is partly flippant, but mostly a music geek's Who reference ("Won't Get Fooled Again"). We spend a lot of time debating "new" math reforms (that are in some cases over 50 years old), and we spend a lot of time thinking about the things that are "old" (time-tested, essential, or that truly matter). Where we stand on these issues says a lot about our own beliefs as educators, and things we learned and internalized in our own schooling.

I honestly thing we need to focus on the common ground that we all share in our beliefs about math teaching and learning. Polarities like "new"/"old" and "discovery"/"traditional" don't tell the truth about how things actually are.

What follows is a short module designed to get us examining our own assumptions about math teaching and learning. Its purpose is not to inflame. Its intended audience is anyone who cares about math teaching. If that is you, please feel free to comment.

Meta moment: the text below is also in this Google doc if you want to add to it:

On “Back To Basics” vs. Reform Mathematics in Schools

The debate over what constitutes “good” math instruction has flared up again recently in Alberta, with the publication of a recent CD Howe institute report, which gives recommendations.

Questions we all must answer for ourselves, as mathematics educators:  

  1. What do we mean by “basics”?
  2. What factors have influenced your own personal definition of the things that are “basic” to math teaching and learning?

Further to #2, reflect on your own personal experiences with school math. To a large extent, your answer to the question “what is math?” was informed at a young age by your school experiences.

3.  Did you see math as creative, vibrant, beautiful, interesting and alive as a student?

4.  Does it matter whatsoever if you experienced math as creative, vibrant, beautiful, interesting and alive as a student?

“Discovery” Learning: What does that mean?

What, if anything, do we mean by “discovery” learning?

See if the Wikipedia article helps us here.

Here is an article that seems to be the main one used as fuel against “pure” discovery learning. (Or, learning that is done with minimal teacher guidance).

In your opinion, what are some math concepts that can be “discovered” through exploration, and careful teacher guidance.  

I will give your the relationship between any circle’s circumference and its diameter. What can you give me? (Examine Ontario TIPS unit here.)

A lot of time has been spent unravelling what we mean by “direct instruction”, “explicit instruction”, and “inquiry learning”.

Is “guided discovery” the same as “direct instruction”?

As a teacher, what is your own personal definition of direct instruction?

Here is John Mighton, founder of JUMP Math, on what he sees as “guided discovery”:

Fluency, Automaticity, Memorization:  3 Sides of the Same Coin?

My assumptions:
-more practice with more number facts will lead to being fluent with them
-encountering number facts frequently through meaningful work, problems, and games will lead to them being committed to long term memory
-flexibility with facts, gained through things like number talks, will lead to more connections being made between numbers, and facts
-K-3 is a time to build a deep base with number sense
-teaching through problem solving, as specified by the Ontario Curriculum, is the best way to build procedural and conceptual understanding
-direct instruction does not always mean talking to the whole class at the same time
-mini lessons on key mathematical concepts, given at a key time in the instructional sequence, pack the most punch

Assumptions that are often made by “back to basics” advocates:

-number facts should be committed to long term memory ASAP, to “get them out of the way”. Grade 4-5 for memorizing times tables, for example, is seen as far too late
-standard algorithms should be taught as early as possible
-encountering problems without prior scaffolding is too much of a cognitive load on students
-forcing students to explain their thinking gets in the way of actually doing math
-direct instruction should take up most of our time.

Here is a Jo Boaler article on fluency that is popular right now:

Develop your own list of assumptions. What common ground do you see? What things do we all agree on?


  1. For the differences between old and new math see: -

    As far as "common ground" parents confronting school administrators are patronized and placated by school officials who will agree with parents and say something like, “Yes, students should learn math facts and procedures (and we do this!). Yes, teachers ought to actually teach, (and we do this!). And yes, students should do drills (and we do this!)” This is all followed with: “We use a balanced approach,” which is often followed with: “We’re saying the same things; we’re in agreement”

    The purpose of these bromides is twofold: 1) to make everyone feel good, and 2) to make parents go away. Pressed to define what “balance” means, the reform camp will say, “Show why things work first to gain understanding; then use the understanding to teach traditional mathematical operations!”

    Such statements reveal internal biases about priorities — priorities that intrinsically lack balance. Whether understanding or procedure comes first ought to be driven by subject matter and student need — not by educational ideology.

    And in answer to the statement that we’re all saying the same thing: No. We’re not saying the same thing at all.

    Why don’t those arguing for better math education (and who insist they are using a balanced approach) look at what those students are doing who are succeeding in pursuing majors in science, engineering or math? If they did, they would see students learning standard algorithms and practicing many drills and problems (deemed dull, tedious and “mind numbing”) and other techniques that they believe do not result in true, deep, and authentic understanding.

    But such an outcome based investigation is not occurring. Some parents whose children are not doing well in math believe what they hear from school administrators that, “Maybe your child just isn’t good at math.” Parents who recognize the inferior math programs in K-6 for what they are get their children the help they need. Unfortunately, parents who lack the means have fewer options.

  2. There needs to be some clarification here regarding "old" vs "new" math...that is rather misleading in terms of what is occurring in the media these days. What really needs to be examined, is why this debate is occurring. Dr. Stokke's CD Howe report was NOT, calling for a return to the basics. What her report was asking for, was to have effective instruction utilized in the classroom when it came to teaching arithmetic. She is asking for a move forward to the fundamentals.

    The crux of this argument centres on a demoralizing decrease in math facts for our students, and the fundamental problem here, is that they are not being taught their foundational facts effectively. This is not merely my opinion, that is what the evidence based research is saying, as highlighted in Dr. Stokke's report, and in other education reports coming out of New Zealand, Australia and the UK. The evidence suggests that Direct Instruction, daily practice - most effective when using pencil and paper (worksheets of all sorts of varieties), mastering of times tables and long division, and using columns for addition and subtraction, are mandatory for a strong foundation of arithmetic.

    Under the proliferation of discovery/inquiry/constructivist/student centred learning, these methods have fallen by the wayside, in favour of using multiple strategies to estimate and discover the answer. These convoluted strategies have proven to be frustrating for kids, as they have no foundational knowledge to draw on, and are left completely overwhelmed and demoralized by this exercise. My eldest was instructed to use lattice multiplication to learn her times tables...when she was 9 years old! Completely inefficient and a terrible waste of time. This is what the present day classroom looks like across North America. And the dramatic increase in Kumon and other tutoring centres has to be acknowledged here. Interesting to note how increased enrollment in these centres really took off when IBL/DBL proliferated in the classroom. And the finger is squarely pointed at the policy makers who are creating these policies and curricula. Those in the trenches, i.e. teachers, are seeing the fallout of these strategies, yet based on the policies are procedures being mandated at a provincial level, they are powerless to try and help. We have failed both our teachers, and our students for allowing this to happen.

    Tara Houle

  3. Well said, Tara and Barry, I haven't much to add, but I'll stretch it out over a few comments here as I think I've got a point or two worth making, and I am incapable of being concise.

    Indeed, there seems to be a condescending approach to parents' concerns as if the problem is merely one of perception, and that the shift that's happening isn't happening.

    I'll illustrate with the debate we had with education profs, math "consultants" and ministry officials in Alberta last year. They were saying, "Look! We ARE teaching the basics! See? it's in the curriculum!" And one ministry official pointed helpfully to the two outcomes on page 98 here:

    This serves well to illustrate the difference between what the two sides are saying.

    The ministry official's "point" was that here you see students "determining" products of single-digit numbers in Grade 5. Now, they reason -- isn't that the 9x9 times table? So these are the (multiplication) "math facts", which are a big chunk of "the basics" ... so QED, "we're teaching the basics".

    You have to be a bit indoctrinated to agree with this. Indeed, the 9x9 times table facts -- memorizing them, that is -- are part of what parents and the public at large mean by "the basics". But what is shown on this page is most definitely not "the basics".

    For one, it is a highly complex and heterogeneous collection of ad-hoc methods to obtain what ought to be committed to heart, for automatic recall. The basics are called that because they are simple, elegant foundations upon which to build other things. They are not meals of themselves, but raw staple ingredients. This outcome over-elaborates that simplicity, treating them as a big main course meal requiring complex thinking, numerous tasks, decision-making processes and ... well, counting on fingers and so on. It's a mixed bag.

    The officials came back to us saying, "Oh, but we DO expect students to remember the facts. It just isn't written there because it goes without saying!"

    Oh, it does, does it?

    When a parent turns to that page here's what they see (yes, take off your educator's glasses for a moment and think what is objectively there, not what fits into the current dogma): Workarounds.

    Not just one or two, but a whole page FULL of miscellaneous workarounds.

    What is a workaround for? Well ... it is is for compensating for NOT having the math facts memorized. If they are committed to heart ... and tested to ensure this is mastered ... who needs a workaround?

    Further, one can go to provincial assessments and see that multiple procedures are required. Students NOT doing so are marked down for not meeting the outcome. There is no evidence anywhere in the testing that memory of math facts is expected; there IS overwhelming evidence that multiple ad-hoc procedures are required.

    Now, when I raise this with educators often they say "Well, what's wrong with that? it's good to know multiple ways to do things, isn't it?"

  4. Uh ... okay there's two things to say here. First, this skirts the thrust of the discussion we had with them (and we're having with you, Matt): The point is that we're NOT saying the same thing.

    Second, there is a problem with multiple procedures, made worse by failing to teach auto-recall to the point of mastery on this material. Rather, there are two problems:

    A. There is a failure, when multiple procedures are used as the basis for all four operations, to develop appropriate schemata for these operations. Anyone who cares about the issue of students understanding math ought to pay attention to this, it's very important: A student must understand what multiplication is at conceptual and procedural levels. I don't borrow much from Piaget, but I think he was right on this point, that successful learning of complex knowledge comes through the student developing an appropriate mental schema for the concept in question. That is, a simple, overriding conception that "chunks" (in the sense used by psychologists) the complex idea into a simply grasped idea encompassing it. In the case of addition, for example, any child who's mastered the algorithm can look at an addition of two numbers, whether 1 digit, 2, 3 or 50 digits, whether or not there are decimals, whether there are 2 or 10 summands, and have a single overriding conception of what that sum will entail -- regardless of whether s/he is ever required to perform it: Their mind should visualize something like a cascade of single-digit sums, joined by a carry step. That is an appropriate schema. Powerful, elegant and informative. It is immediately clear how much work it involves, the student does not have to pause to think how to do it, and would have confidence that the sum could be performed without much fuss, though perhaps some labour.

    A student who knows only multiple procedures cannot even conceive the doing of sums beyond the "toy" size, or when s/he does, what comes to mind is this jumble of possibilities, like an untidy attic. Out of this clutter s/he must conceive that there is a pathway to it. That pathway may involve the assembly of mismatched ideas and steps, like some sort of abstract Rube Goldberg Machine
    Not a schema. And little *actual* understanding of the sense useful for proceeding to higher levels of abstraction.

    B. It is contrary to the unalterable aspects of brain function. Besides distracting from the central ideas of the operations with clutter, and directing attention to meaning at the wrong level for proper understanding, using such workarounds overload working memory. Does it really? Well, I've heard educationists try to dispute this point. However, brain scientists are quite in agreement about this, and a number of clever studies have been done to determine whether or not memorization of math facts really has the benefits advocates claim.

    Here is a recent such experiment done by a fellow in your backyard, Matt: Dr. Daniel Ansari of the University of Western Ontario. He was on a team that used MRI imaging to peek at students brains to see which areas fired up during single-digit math, more-or-less cleanly separating them into groups who used auto-recall versus those "working it out" -- i.e., "strategies" to get these elementary facts. Speed doesn't appear to have been an issue: I take it for granted that there was no observational difference in time taken by the two groups. At issue was only what sort of brain process was used. Then students wrote the PSAT and the two groups' Math SAT score were compared. The group using recall scored significantly better on high school math than those using "strategies". A brief video in which Dr. Ansari discusses their findings and educational consequences:

  5. Cognitive scientists generally explain such results in terms of a simplified model of memory. Your working memory is not well-designed to handle abstract information. For most people it's capable of juggling about 7 symbols (like a phone number) while retaining some basic function. Beyond that (like remembering two phone numbers), if we can do it at all, it only happens at significant deterioration or complete loss of other functions of working memory. Unfortunately, these include basically all the functions we care about as educators: Creativity, rational processes, decision-making, intuition, judgment and understanding. The top half of Bloom's, if you like.

    So ... like ... how can we even function in an abstract subject like math if working memory is so limited? Easy. Long-term memory, in contrast, is for all intents and purposes unlimited. Nobody has ever probed the upper bound to what we can pack into it. And unlike juggling information in working memory, once that information is well-established in permanent storage (memorized "by heart"), it is retrieved instantaneously when needed. And, just as importantly: without significant effort, and without impairing our higher functions that rely so critically on working memory.

    A simple demonstration of this: Try remembering the following 35 letters and spaces -- in working memory, don't try memorizing it.


    Got it? Now try the following 35 letters and spaces:

    Why is the second one easier to remember than the first, and even to keep in working memory? It's because you have so much stuff packed into long-term memory, and practiced to the point of "chunking" that this new information glues effortlessly into place and you know it because of how it relates to what is already known. That is essentially what language acquisition is ... in it's lowest terms, and how you function in language: a massive database of stored information, and instant, effortless retrieval that gives you immediate access to meaning, interpretation and procedure so that all processes are UNCONSCIOUS beneath the one that matters.

    That is the point of mastering the math facts: freeing us from lower processes in order to enable higher ones. As Alfred North Whitehead said,

    "It is a profoundly erroneous truism ... that we should cultivate the habit of thinking of what we are doing.  The precise opposite is the case.  Civilization advances by extending the number of important operations which we can perform without thinking about them."

    He was talking about mathematics, primarily.

  6. When we discussed the 2013 changes to the MB curriculum and insisted on auto-recall, the ministry wanted to put "obtains within 3 seconds answers to..." on the corresponding outcome. We said, no -- we know of no research supporting value of speed of calculation by itself, the critical point is de-cluttering working memory with auto-recall. The ministry eventually agreed on wording requiring this.

    We also managed to get the standard algorithms mentioned (very offhand, tepidly, and too late in the outcomes, but they are there now) in MB. In AB they eventually added the word "recall" to that outcome -- it now stands there, oddly, as an afterthought, in the middle of a page full of workarounds for NOT memorizing those math facts. But adding the word is a tiny step in the right direction.

    They called it a "clarification" (so did MB) -- a transparent ploy similar to what you appear to be doing here, Matt.

    The other 6 WNCP provinces still have neither in their curriculum. In any case all 8 WNCP provinces still require multiple ad-hoc procedures and workarounds at least til the end of Grade 5. 5 years is FAR too long for students not to have any coherent understanding or work habits for elementary single-digit arithmetic. But it IS long enough to inculcate life-long habits of mind when performing this work. Habits that can be helpful ... or harmful. Training (yes *training*) students to always think about the lowest level of meaning and use complex workarounds for simple facts is about the equivalent of training students learning to read to *always* keep their finger on the word while reading.

  7. A very good point about people saying the same things. Everyone is claiming to do a balanced approach. But what they are actually doing is more important than what they say the do.

    Take the example from TIPS. Jumping into empirical relationships between the circumference and diameter of a circle is an odd choice. There is no chance of turning this into mathematics with a proof until Calculus is well understood. The whole theme of developing math empirically and never addressing a proof is at odds with the key idea of math.

    If instead the subject was developed from squares to rectangles to various regular triangles to any triangle to any polygon it would offer a lot of room for both guidance while still leaving a lot of scope for having gaps for students to explore. Relationships between areas of squares and rectangles are relatively simple to provide a grade school proof for and there are lots of nice intuitive picture proofs of Pythagoras.
    The idea that interesting facts can have proofs that work without tedious repeated measurement is a huge part of what excites some people about math. Working from something as simple as the relationships between sides of squares and their perimeters all the way to circles' radii and circumferences offers a great deal of scope to learn math. But the Ontario curriculum goes out of the way to avoid such an approach.

    I really hope the author here regularly provides their students with an opportunity to fill out an anonymous survey on where repeated measurements of relationships sits on scale between creative, vibrant, beautiful, interesting and alive and incredibly tedious.

    I am a parent of an Ontario student. He loves math, has had some great and some less than great teachers but who reports that few of his fellow students enjoy the way math is being taught with open ended questions and rather tedious measurement exercises. If all teachers taught with guided discovery the way Lockhart of Measurement fame does it would be fantastic. Unfortunately this is not what happens.

  8. With the guided discovery what do they qualify as sufficient review at the beginning of the lesson? All lessons can't be increasingly challenging as they go. Extensions are possible yes.

    In the first link it seems disastrous. Memorizing division facts and not understanding division? No thanks. Great stuff to think about.