Monkeys or Mathematicians (Math is More Than Memorization)

Pedagogy trumps curriculum every time.

Dylan Wiliam

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)