January 21, 2014

Monkeys or Mathematicians (Math is More Than Memorization)

Pedagogy trumps curriculum every time.
The recent parent-driven push for a “return to basics” shift in math curriculum in Alberta is not unexpected. Our post-industrial society remains regrettably focused on relaying and assessing content over process. The deeply embedded desire to quantify student thinking for the sake of a neat, uni-dimensional continuum that claims to represent student potential results in the inevitable association of learning with factual and procedural recall. Quite simply, we've designed schools to train and measure our children. We group them by age, divide their days into standardized units and test them at regular intervals in order to compare them to their peers. Memorization is easy to measure in math so we convince ourselves we’re holding kids accountable by measuring their recall. This also allows us to rank and sort students effectively without actually engaging them in conversation, something PISA has effectively mastered. However, making a judgement about the quality of an entire math curriculum based on data snapshots from a moment in time is not only irresponsible it's ridiculous. Advocating that because memorization scores have dropped, an entire curriculum should re-focus on memory work is incredibly shortsighted. We've already been there. It wasn't awesome.

Drilling students on basic math facts and the memorization of assigned algorithms for the past several decades has overwhelmingly killed interest in mathematics, hampered intellectual development and misused teaching opportunities (more on the complexity of assigning "standard" algorithms here). Math facts worksheets have been painful for all but those gifted in regurgitation (and not the good kind of worthwhile pain that is connected to something important). As Paul Lockhart writes, “I don’t see how it’s doing society any good to have so many members walking around with vague memories of algebraic formulas and geometric diagrams and clear memories of hating them.” The test results may have been good but it is safe to say that a fair majority of “old system” graduates continue to dread the thought of trying to explain math to their children.

The inquiry-based shift away from an emphasis on memorization was in acknowledgement of the fact that the cultivation of deep understanding is now recognized to be much less straightforward than simple transmission and regurgitation. "Inquiry offers a way of thinking about things that does not begin with isolated bits and pieces, but with webs of relationships," (Friesen & Jardine, 2009) meaningfully connected through experience and conversation. Although powerlessness has become a surprisingly seductive habit, society is slowly recognizing that it is no longer acceptable for high school graduates merely to be able to do what they’re told. Today’s students will have opportunities to collect, synthesize, analyze, connect and design and will need to be able to make use of these opportunities in order to be considered effective 21st century citizens.

Proponents of “back to basics” math education love to make comparisons between the importance of memory work and the necessity of “practice” in sports. It’s a good analogy. But advocating for a "back to the basics" math curriculum is very much akin to advocating for "less gameplay, more passing practice" approach to sport. When kids don't get to play the game, passing loses it's appeal pretty quickly.

We read to kids and we encourage them to write things before they can spell. It's what gives the importance of spelling proper context, makes it meaningful. The same should be (and is) possible in mathematics. Inquiry-based practice retains the rigour of strong work in mathematics without limiting kids to one perspective or strict procedural recall. Children work on problems using a variety of strategies. They are held accountable for justifying their thinking to their peers in class discussions. My experience with an inquiry-based approach to mathematics has never been that “basics” are ignored. It just structures learning so that things like arithmetic come up in authentic mathematical contexts. The emphasis is on conceptual understanding, not just procedures and practice of them. The result is that kids can not only think but can articulate their thinking. This video represents a range of grade 4 students explaining their solutions to a problem undertaken in class. Providing these students with the opportunity to share their thinking with peers and allowing their peers the opportunity to question their thinking led to some incredibly valuable conversations about efficiency, best practices, and worthwhile work in mathematics.

Inquiry effectively frames a question or topic of investigation, carefully guiding students to a solution but not directing their every move. I’ve continued to use the analogy of bringing students to a mountain and teaching them to climb but providing them with the freedom to discover their own way up rather than dragging them all up the main route. The approach involves checking in with students on a regular basis, discussing their experiences, missteps and difficulties, and supporting them as they work their way up (see Galileo Teaching Effectiveness Framework). Introducing students to a topic and subsequently abandoning them to their own devices is not inquiry, nor is it effective teaching. Nowhere in the current Alberta Math Curriculum does it state that students should “teach themselves.” High school students whose only strategies for adding 4+7 are counting up, or using their fingers are the result of ineffective teaching practice, full stop.

I recently posted the following quote to twitter:

In response I received the comment: “That way, arts graduates can appear to be competent math teachers.”

No. The opposite. The new approach asks teachers to challenge the curiosity of students with problems proportionate to their knowledge, guiding them toward strategies, conjectures, and conclusions that are mathematically sound and that they can justify with confidence.
Instead of: “Find the perimeter of a 4cm × 9cm rectangle,” we are asking: “Find a rectangle which has unit sides and a perimeter of 100. How many answers are there and how do you know you've got them all?” 
Instead of: “Find the area and perimeter of a 3cm × 8cm rectangle,” we are asking: “If the area of a rectangle (in cm²) is equal to the perimeter (in cm), what could its dimensions be?” or “If the area of a rectangle is 24 cm² and the perimeter is 22 cm, what are its dimensions? How did you work this out?”  (via nrich.maths.org)
Instead of worksheets like this:
Students develop puzzles like this: 
And instead of videos like this, they make videos like this  
As an educator, it is no longer enough simply to hold the answer key. In order to effectively guide students through the process of thinking mathematically, teachers need to have wrestled with the same problems themselves and be familiar with the range of mathematical possibilities and conceptual connections that each problem might elicit. In grade 4, in lieu of repeated memory work we determined what was worth memorizing (ex. students agreed that it was particularly helpful to have memorized what we called "friendly numbers" - pairs that we add to give 10) and why.

I want to acknowledge that the task of moving beyond sequential, transmission-based math instruction can be overwhelming and intimidating, particularly without support or access to powerful exemplars. As Sharon Friesen writes in Back to the Basics, “it’s not just that teachers [and parents] don’t like math; they don’t know what’s happening because they can’t remember what the real work really is, All they remember of math is the equations and the rules and the facts they’ve memorized - the surface activities with all the relations forgotten...” Math teachers province-wide, most of whom are graduates of the “drill-and-kill” system, have recently taken on the difficult and uncomfortable task of attempting to introduce mathematics as the complex web of relations it is, often with very little support. It is fair to say that inquiry-based pedagogy is still new and inconsistently executed. The answer to any resulting challenges however, is not a throwback to an outdated curriculum or oversimplification of the complexity of mathematics.

WISE Math, an organization actively advocating for "back to the basics" mathematics instruction, have written on their main page that "in order to become a competent piano player, a child must practice regularly and memorize piano scales, " stating that the same is true for mathematics. I do not disagree. Children who practice piano scales however, do not do so at their music lessons. In piano class, they play, they get feedback, and then they go home and practice scales because it makes a difference. They recognize that it affects the music. The new curriculum does not prohibit children from practicing math facts at home. It just acknowledges that there's more worthwhile work to be undertaken in the classroom. WISE interestingly has an article linked to their website called "'Drill and kill' no way to teach math in 2011" which concludes with a quote I thought particularly apt:
"Do we need a high level of proficiency in math teaching? Sure. Is our goal one of raising student achievement and equipping kids with the math tools they need to function effectively in society? Absolutely. But we do this through effective, reflective practice, not by blindly adhering to outdated approaches that have characterized instruction for the past century."
Also, this TED talk by 13 year old mathematician Jacob Barnett is well worth a watch:

Further reading 
(a by no means comprehensive snapshot of academic research and writing I have found of particular value relating to mathematics education for the development of deep understanding)

Baroody, A.J. & Ginsburg, H.P. (1990) Children's mathematical learning: A cognitive view. Journal for Research in Mathematics Education. Monograph, 4, 51-64
"Cognitive research indicates that it is essential to distinguish between meaningful learning and rote learning. It is not enough to absorb and accumulate information. Children must be given the opportunity to assimilate mathematical knowledge - to construct accurate and complete mathematical understandings."
Caliandro, C. K. (2000). Children's inventions for multi digit multiplication and division. Teaching Children Mathematics, 6 (6), 420-424.
"The procedures [the children] developed were meaningful to them and flowed out of their deepened mathematical understanding. Their procedures will not be forgotten. Memorized procedures, in contrast, are frequently forgotten and have to be reviewed again and again."
Carroll, W. (1997). Invented strategies can develop meaningful mathematical procedures. Teaching Children Mathematics, 3 (7), 370-74.
"By encouraging children to invent and use their own procedures, teachers allow them to use a method that makes them focus not simply on practicing computation but also on developing strategies for which computational approach to use[...] The reward of seeing students make sense of mathematical situations and the resulting appreciation of children's thinking and capabilities more than make up for the difficulties."
Friesen, S. (2008) Raising the floor and lifting the ceiling: Math for all. Education Canada. 48 (5), 50-54.
"I think it is important to note that students were not left alone to 'discover' the math for themselves. Rather, a series of lessons were designed to scaffold the student learning, ensuring that students uncovered and connected the underlying key concepts, worked through procedures related to measuring and calculating angles and arcs, length and perimeter, area and volume, congruence and similarity, and scale factors. They were asked to reason, to conjecture, and to justify conclusions."
Friesen, S. (2006). Math: Teaching it Better. Education Canada, 46 (1), 6-10.
"While the task of creating classrooms in which students understand abstract and difficult mathematical ideas, see relevance in the mathematics they are learning, and achieve mathematical competence seems daunting, as a mathematics community we are further down the road in knowing what to do to achieve these goals. We have made demonstrable progress by working together - mathematicians, mathematics educators, and teachers who understand that mathematics reform is a complex matter. There are no easy answers." 
Jardine, D.W., Friesen, S. & Clifford, P. (2003). “Behind every jewel are three thousand sweating horses”: Meditations on the ontology of mathematics and mathematics education. In E. Hasebe-Ledt & W. Hurren (Eds.), Curriculum intertext: Place/Language/Pedagogy (pp.39-49). New York, NY: Peter Lang
"All of us at the table knew, beyond a shadow of a doubt, that this solution was correct. But, equally, none of us knew at all why it was correct. One boy insisted, with an insistence that we all recognized in ourselves, "That's just how you do it, ok?" [...] Many of us in this classroom had, over the year, talked about that odd feeling of having learned, having memorized a procedure and knowing how to do it beyond question or hesitation, and yet suffering the terrible silence and feeling of cold and deathly immobility if anyone should have the audacity to ask a question about it."
Kamii, C. & Livingston, S.J. (1994). Young children continue to reinvent arithmetic - third grade: Implications of Piaget's theory. New York, NY: Teachers College Press
"Children's first methods are admittedly inefficient. However, if they are free to do their own thinking, they invent increasingly efficient procedures just as our ancestors did. By trying to bypass the constructive process, we prevent them from making sense of arithmetic"
Madell, R. (1989). Children's natural processes. Arithmetic Teacher 32 (7), 20-22.  
"It is hard to follow the reasoning of others. No wonder so many children ignore the best of explanations of why a particular algorithm works and just follow the rules [...] The early focus on memorization in the teaching of arithmetic thoroughly distorts in children's minds the fact that mathematics is primarily reasoning. This damage is often difficult, if not impossible, to undo."
Steffe, L. P. (1994). Children's multiplying schemes In. G. Harel & J. Confrey (Eds.). The Development of Multiplicative Reasoning in the Learning of Mathematics. (pp 3-39.) Albany, NY: State University of New York
"[Children] can, indeed, be told to do something, but they cannot be told to understand [...] It is a drastic mistake to ignore child-generated algorithms in favour of the "standard" paper and pencil algorithms currently being taught in elementary schools. Other than the work already cited, there is solid evidence that imposing the standard algorithms on children yields discontinuities between children's methods and their algorithms (Easley, 1975; Brownell, 1935; McKnight and Davis, 1980). even when they are to some extent based on operative arithmetical concepts, the standard algorithms become essentially instrumental for the children (Skemp, 1978) and pose a serious threat to the retention of insight (Fredenthal, 1979; Erlwanger, 1973).
Stein, M. K. (2007). Selecting the right curriculum. NCTM Research Clips and Briefs. Retrieved from: http://www.nctm.org/news/content.aspx?id=8468 
But, efficacy for what? It is important to note that students tended to perform best on tests that aligned with the approaches by which they had been taught, repeating the well-worn finding that students learn what they are taught.  Combined with the findings from the analyses of curriculum materials cited earlier, the research examined here suggests that students taught using conventional curricula can be expected to master computational and symbolic manipulation better, whereas students taught using standards-based curricula can be expected to perform better on problems that demand problem solving, thinking, and reasoning.
(Also, Cathy Fosnot's entire series Contexts for Learning Mathematics, Christopher Danielsen's Talking Math With Your Kids, the University of Cambridge's nrich.maths.org and the Galileo Educational Network's Math Resource List)


  1. Deirdre, you have put into such eloquent words the many many conversations we have had, as we bump up against misconceptions in education, especially in mathematics. Exploring math as a field of study vs. isolated facts and algorithms makes it not only more engaging and relevant to students, but also to teachers as we bump up against the misunderstandings we carry with us from our own math education.

  2. This is beautifully written, Deirdre. I think this really can be applied to any subject in school, not just math. I love Friesen & Jardine's version of "back to basics" - curiosity, wonder, and spending some time asking what the field or discipline needs from us.

  3. Hi Deirdre,

    First, I would like to point out that my colleagues and I certainly want math to be taught with understanding! You would be hard-pressed to find a mathematician/math professor who does not want kids to understand math! However, we also want children to memorize times tables (which does not imply that they cannot understand multiplication) and to master standard algorithms. Furthermore, we want those algorithms to be taught with understanding.

    I am commenting on your blog because I want to address the issue of what you have listed as "research" articles. Having been at this for awhile, I have learned that what passes for research in education is often simply opinion. I suppose this makes me rather angry which is why I'm taking the time to post here.

    Of the articles you've cited, several do not involve any type of study at all (eg. Math: Teaching it better, Behind every jewel…).

    Several are based on case studies (eg. Raising the floor (one grade 7 classroom), Baroody (interviews with individual children)).

    None involve randomized controlled studies and sample sizes are all very small, with children treated as the experimental units as opposed to classrooms. As one example, the Caliandro article is based on a study in which there is no control group.

    Faulty conclusions are drawn in these articles regularly. For instance, the Caliandro article appears to be a study of procedures that students use if not given the standard algorithms. However, the authors make several conclusions that do not follow from the study they conducted. For example “Their procedures will not be forgotten. Memorized procedures, in contrast, are frequently forgotten and have to be reviewed again and again." This is an opinion and by no means follows from the study they’ve conducted.

    Probably the most prevalent problem in the algorithm articles is the false dichotomy that authors set up between standard algorithms and understanding. They seem to imply that standard algorithms are something to be memorized and not understood. Is it possible that the authors don't understand why the algorithms work themselves?

    For example, in the Caliandro article we see "In the standard multiplication algorithm, single digits are multiplied without regard to values the digits represent or why it is possible to break multidigit numbers apart and multiply single digits”

    Are the authors unaware that the standard multiplication algorithm hinges on the the distributive law???? If we want students to know why algorithms work, why not advocate for explaining why they work (this is not difficult to do) in conjunction with mastery, instead of campaigning against the standard algorithms?

    There are many other problems! As for Kamii, I am quite familiar with her work and have critiqued it before, as have several other mathematicians. Here is a critique written by Bill Quirk, which lists some of the problems: http://www.wgquirk.com/kamii.html

    Hung-Hsi Wu also mentions issues with Kamii's work in his piece Basic Skills vs. Conceptual Understanding: A Bogus Dichotomy in Mathematics Education: http://www.wgquirk.com/kamii.html

    I strongly feel that all graduate students in education should take a rigorous survey sampling course taught by a qualified statistician. Maybe this would eliminate some of these problems. On the other hand, a bit of critical thinking from an average citizen would quickly reveal the issues with most articles.

    Best regards,
    Anna Stokke

    1. Hi Anna!

      I received an email notification of your comment on my post Sunday morning but it did not appear on my blog and I couldn’t figure out why. I just realized that it had been filtered into the spam folder. I am not sure how this might have happened so my apologies for your comment’s belated appearance on the blog.

      I am grateful for your clarification that WISE wants kids to develop mastery of basic facts with understanding. I think in many ways we are not at odds and want to take the opportunity to clarify that I am not campaigning against algorithms at all. My concern is quite simply that when rote memorization shifts to the forefront in math education, it often becomes a substitute rather than supplement for the cultivation of deep understanding. I am simply advocating for pedagogical design that effectively allows students to develop an understanding of “why” as well as a mastery of “what” and “how” and that accounts for their individual differences by beginning with a question instead of a series of prescriptive instructions. While inquiry-based pedagogy is often misconstrued as “open exploration” or “discovery learning”, it is neither of these. Inquiry-based learning, effectively undertaken, is carefully and intentionally designed. It remains open as well as bounded and provides space for both repetitive practice and problem solving (see Galileo Educational Network).

      With respect to your comment about the articles I shared for further reading, please note that they were described as “a snapshot of academic research and writing.” Certainly I am aware they did not all constitute research specific articles. Of those that did, it is worth noting that your criticism of their validity is from the perspective of a mathematician. Your preference for RCTs is no surprise given your background. However as you are aware, educational research does not deal exclusively with numbers but with learning processes and children. While empirical evidence undoubtedly has its place, no amount of data can provide the information about classroom learning that field notes, interviews, conversations, photographs and recordings can. Issues of validity in qualitative research are not linked to sample sizes and control groups but to whether the researcher has made practices visible and therefore auditable and whether or not their work adheres to professional norms of peer review. This was the case for all of the research-specific studies linked above. While the nature of qualitative research is that it remains open to review, it is also worth noting that the criticisms you shared were webpages that do not appear to have been subject to peer review and could thus be perceived as less valid than the articles they critique.

      I agree that all graduate students in education would benefit from taking a sampling course from a statistician. I would add that perhaps all graduate students in mathematics with an interest in teaching might consider a course in educational research and qualitative methodologies.

      As Sharon writes at the conclusion of her “Teaching Math Better” article mentioned above “we have made demonstrable progress by working together - mathematicians, mathematics educators, and teachers.” I am grateful to have had the opportunity to engage in this conversation. Thanks for your comment!!


  4. Lots of interesting commentaries and opinions here. What we really need to see, though, is an actual rigorous study of kids in a classroom, where discovery math is used and memorization and teaching of standard algorithms isn't used, and the results from that class are positive, where we actually . Can you please link to that actual study? Not a textbook summarizing it and writing a commentary on math, but to the actual research report. That's what I'm hoping to find in this debate. No luck so far.

    1. I think this is the type of study you're looking for?:

      Thornton, C. A. (1990). Solution strategies: Subtraction number facts page. Educational
      Studies in Mathematics, 21(3), 241-263.

      If you us one of the databases at the University of Calgary Library you'll find more such research.

  5. Hi Dierdre. I hardly got past your first line -- about the "parent-driven push". Your tone suggests that you think parents are ignorant of what's educationally best for their children and, not having Education degrees, are simply unqualified to address the issue as "stakeholders".

    As you surely know I have gotten deeply involved in this issue and I must say that I find it shocking to see the frequency with which those who've bought into trendy pedagogical theories speak down at parents in this way. You don't even bother to disguise your contempt for these supposed troglodytes, do you? Or perhaps you are simply unaware of it, it is such a seamless part of your worldview.

    Just so you know, I don't share this contempt. My view is that parents are the most important stakeholders and we do well to listen to their voice.

    I just clicked on the link you provided to Dr. Nhung Tran-Davies (the lowly "parent" in question -- a medical doctor). Her petition has attracted over 3500 signatures in a few short weeks. Many, even most, of the signers have added their own stories in their comments. For a petition dealing with the teaching of an academic subject in public schools, I would say that this is a remarkable phenomenon. Perhaps unheard-of in a jurisdiction the size of Alberta.

    In contrast, at the time I am writing this comment I see 3 (THREE) comments on your blog, though I see it is getting plenty of exposure through your friends in the Twitter-verse. Hmmm ... a 1000-to-one ratio.

    Now, obviously, truth is not determined by taking a straw vote. But let's just say that you're not doing a very good job convincing people. All your fine words weigh very meagerly against the effects people see on their own children, in front of their own eyes. You cite a lot of folks with fine-sounding theories. But theories are worthless if they can't stand up to empirical tests. Their children are the data points in the current empirical test.

    Proof ... meet pudding.

    One more thing you should consider. Perhaps you haven't bothered to do this yet: Go to the petition site, and just start reading all the public comments left by those who've signed. Notice how many EDUCATORS have signed the petition. Not a few. Not even "many" -- a LARGE fraction of those signing the petition are your fellow teachers. HUNDREDS of teachers. Perhaps over 1000. Many of these are parents, and see the same data points as the other parents. But even those who don't mention their own kids provide educational reasons for their support of Nhung's petition.

    I would think, even if you must look down your nose at the lowly parent-class, at least you would listen to your fellow educators. But perhaps you also have contempt for the many teachers who aren't so eager as you to "get with the program". Perhaps you think all those fellow teachers are troglodytes too.

    Again, I don't share this contempt. I think teachers are heros in this story and neither should their voices be ignored.

    Dr. Robert Craigen
    Assoc. Prof. of Mathematics, University of Manitoba
    Co-founder of WISE Math ( wisemath.org )

    1. Hi Dr. Craigen,

      Thanks for your comment. There's a lot there to consider and I want to make sure my response is thoughtful but I did want to clarify right away that I have absolutely no contempt or lack of regard for parents or parent voice in education. My introductory sentence was simply an acknowledgement of the petition and the dominant voice behind it. I do not agree that making children memorize multiplication tables in math classes is productive to their learning. Parent and other teachers' perspectives, although based on different experiences, are as important as mine to this debate. I remain open to further conversation on the topic and welcome feedback on the post. I think this debate is important and am sincerely hopeful that it can remain civil.

      With regards,


    2. Thanks for your reply, Deirdre; I look forward to your full response.

      I address the one point you make here about memorization.

      Memory is the seat of learning. No memory, no learning. Those opposing memory work would do well to -- at minimum, acknowledge its importance (I presume you do not disagree, but I note that there is no hint of it in what you have written.) And, having acknowledged this, to explain why memorization of times tables is not a particularly good use of memory.

      I can tell you why, to the contrary, it is a good use of memory.

      Early-years learners are passing through a rapid acquisition phase during which important facts are committed to memory, establishing the foundation upon which later synthesis of cognitive understanding takes place.

      The ease with which these particular basic facts are automatized at this age contrasts the difficulty faced by those who must do so later in life -- students who fail to memorize these elementary facts at this stage may struggle mightily with them later, and many will simply fail in the effort and never master them.

      The goal of education is UNDERSTANDING. And understanding cannot grow in a vacuum -- it requires context. Context necessarily entails factual content about things, processes and relationships, committed to long-term memory.

      Our long-term memories are very well suited to storage and retrieval of such trivia. (Long-term) memory is not the ANTITHESIS of understanding -- it is an integral part of understanding. Understanding is functionally impossible without a well-stocked memory. I don't think any of this is controversial in the least.

      WNCP Math and the like forces children to adopt inefficient and ad-hoc mental approaches to even single-digit arithmetic, and in multiple ways.

      How different is memorizing multiplication tables from WNCP's rote work? "Rote" means "repetition". The demand for children to revisit the same "strategies" by rote in WNCP is simply astonishing.

      Grade 5 students are expected to "apply mental strategies" and use skip-counting, doubling, halving and looking for patterns, to "determine answers" (i.e. NOT facts they have yet mastered) "for basic multiplication facts to 81" (WNCP K-9 Framework, p104).

      "to 81" = "the 9x9 times table".

      Five years is FAR too long to force children to obsess over the mechanics of single digit arithmetic without closure! The 2008 NMAP report warns against "any approach that continually revisits topics year after year without closure", an indictment of WNCP Math, which specifies NO point at which these are to be mastered.

      Now it doesn't take a PhD in brain function to know that working memory -- particularly for abstract facts -- is extremely limited. Famously, humans can reliably hold about seven digits in their heads but few can do the same with fourteen digits. It is a very small working space.

      A year ago a paper published in cognitive psychology quantified the rather obvious consequences. A team of researchers led by Dr. D. Ansari of UWO separated children according to whether they performed single-digit arithmetic with automatic recall or strategies. They collected longitudinal data on the same children years later from their SAT Math scores. Those who had, years earlier, committed the fundamentals to memory, significantly outperformed those who had been trained to use ad-hoc approaches.

      Some contend that those ad-hoc approaches constitute "understanding". In fact the WNCP Framework itself, in outcomes specifying that children "demonstrate understanding of" such-and-such a concept, the corresponding achivement indicators specify that the children are to show multiple strategies.

      Yes, some call it "understanding". But I call it "clutter".

    3. Thank you! My response wouldn't fit, I've posted it as a new blog.

  6. Deirdre,

    Your thoughtful words are inspirational to me as an educator. Children desire and deserve work that is deep, thought-provoking and worthwhile. I only wish that growing up I could have had even one math teacher as wise as you about the subject - Perhaps then my interest in math wouldn't have been killed by the lethal "old" math that forced me to memorize complex algorithms with no knowledge of why it was necessary.

    Thank you for sharing your important words.

  7. Hi Deirdre,

    Your passion and ability to see the beauty in mathematics, as well as in children's understanding of it, wherever that may be, is inspiring as an educator and makes me hopeful as a parent. As I watch my daughter engage in the world of quantity, number, and the manipulation of those things, I hope she has teachers in her life that will know her well enough to guide her through these concepts and charm her curiosity of them as well. As an educator, I see children walk through the door with a complete variety of background knowledge, and understanding of numbers. These same children think in very different ways, from each other and from me as well. Myself, as someone who was 'successful' in the current school system, I have come to realize that the way I have learned things is not always representative of the majority of our population. The care and accuracy of with which you speak addresses this variety and allows for different types of learners and thinkers to find their own success. Thank you for the inspiration.


  8. Hi Deidre,
    As I sit hear re-reading your post, which I have read several times and recommended to many of my Teacher Candidates at UBC, I am reminded of Jo Boaler's book "What's MATH Got To Do With It: How Parents and Teachers Can Help Children Learn To Love Their Least Favourite Subject. In discussing the "Math Wars" she writes "And this is where the crux of the disagreement lies, with one group of people believing that students need to spend a lot of time practicing, and the other group believing that it is better to understand an idea than it is to practice by rote" (Boaler, 2009, p. 37). On behalf of a proponent of the latter group, I want to thank you for championing our cause with great passion.

    Warm regards, Jen

  9. AWESOME post!! love it!

    Mr. Craigen, you said "Just so you know, I don't share this contempt. My view is that parents are the most important stakeholders and we do well to listen to their voice", is this true in your courses in the university? Do you listen to the parents of the grad students in your class?

    While I agree that parents are important, I wonder why you put them above the the students in your class? How much do you allow parents to influence your doctorate courses?

    Next you address the survey as being signed by 1000s of Albertans. I have asked you many time for this proof? You always dodge the question. I am sure that this question is being signed by many outside the province, but I don't have any proof so I am wondering if you can step up and bring actual evidence?

    Also you claim that 1000 of her colleagues have signed this petition? I remember back to my undergrad class on statistics and how we need to know more than just simple numbers to create a correlation. Are these so called 1000 teachers from a certain demographic, community, etc?

    Dear Anna,

    I find it ironic that it is your opinion that most Educational papers are more on opinion than fact. You want students to understand math, however you force it to them through memorization. I think the point (and I may be wrong) is that we can't assume because a student has memorized a multiplication chart that they understand mathematics.

  10. Deirdre,

    What I appreciate most about your post and following dialogue is that you never lose sight of the fact that we teach children before subjects. Perhaps what’s most interesting about teaching is what children bring to the disciplines that we take up. A great problem of concern here is the negative feelings and self perceptions that so many students seem to experience under a curriculum that values a single form of knowledge. What the current curriculum successfully does is engage children in the actual “doing” of mathematics; that is exploring relationships, wrestling with concepts and thinking critically about the how and why behind mathematical concepts, as opposed to arriving at one answer, one way. The excitement that erupts in my classroom when children discover relationships, uncover something previously hidden or experience an “ah ha” moment is incredible. Deep understanding is borne in these moments.

    Thanks for such a thoughtful piece.