Note — Again this week I’m revisiting some material about mathematical thinking first ‘processed’ during the Cognition course to see how it threads together with this course’s material on inquiry based learning. I’m sorry there are no videos or pictures to break up the text!]

In his book How Humans Learn to Think Mathematically, David Tall states that the key competence underpinning the learning of math is the ability to compress processes into “thinkable concepts” (p. iii) or “known facts” (p. 20) which can be manipulated “in their own right” (p.20). Compression is the brain’s way of getting around the limited capacity of its working memory which cannot hold more than five to seven pieces of information at one time (Kandel, 2008) and also makes possible the development of knowledge structures (Tall, p. 107) which can be further compressed or blended flexibly to solve highly complex real world and mathematical problems or to creatively invent new branches of mathematics (p. 121).

Tall says that the “human brain has become a mathematical mind” by using three biologically built-in abilities: we recognise similarities, differences and patterns; we learn by repeating sequences of actions until they become automatic; and we use language to categorise, to name, to create symbols, to define, to formulate rules, and to reflect. In math, language also enables us to understand the dual nature of compressed objects which are at the same time processes we can do and concepts we can think about (p. 337).

Tall observes that students’ reliance on such tactics as rote memorisation and counting strategies, although a successful strategy in the short run, will eventually limit their capacity to move farther “up the ladder of higher orders in math” (p. 192). For example, the student who uses repeated addition to perform long multiplication instead of compressing these calculations into known math facts can still manage to solve questions like 296×45. However, the mental demands of trying to use finger counting or strokes on a page to work out 683÷92 become overwhelming.

Throughout his book, Tall uses examples like the one above to illustrate why so many students become stuck first in long division, then in fraction and integer calculations, and also in high school algebra and problem-solving. Some students simply resist letting go of “the limited imagery and complicated embodied procedures” (p. 189) which they developed early to help them learn arithmetic. Rooted in what Tall calls “common sense” learning, these understandings are familiar and comfortable, but they do not serve a person engaged in more complex math tasks very well because they’re so inefficient. As well they fuel the desire to find a less stressful way to accomplish the same task rather than a determination to do the compression work needed to overcome, rather than sidestep, the old barrier to new learning.

Tall’s analysis of why people either succeed or struggle with math dovetails nicely with some of the ideas encountered in this course. Let me take them in turn:

A. Old understandings and practices tend to persist even if they have outlived their usefulness or fly in the face of the preponderance of evidence.

  • McNeill, Lizotte, & Krajcik, p.5: Students “may draw on other knowledge and beliefs to explain phenomenon rather that the data at hand.”
  • Donovan & Bransford, 2005, pp. 1-2: ‘Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp the new concepts and information, or they may learn them for purposes of a test but revert to their preconceptions outside the classroom.’
  • In math this will express itself as clinging to hopelessly cumbersome or incorrect processes rather than letting go and doing the work needed to get to the next level. I used to tell students to imagine they were out in the middle of the ocean clinging to a tiny rubber ring. It’s talking all their energy just to keep their noses out of the water, but within swimming distance is a beautiful, big raft. The only catch is that to get to the raft, they must let go of the ring and swim like heck. (I am beginning to think this is a lesson that needs to be learned by math teachers!!)

    B. To internalize new processes students must “compress” them –i.e. turn them into “thinkable concepts” Tall, p. 20) (math processes) or strategies (inquiry processes). According to McNeill, Lizotte, & Krajcik helping them do this requires deftly balancing such variables as individual and group readiness, the deliberate application and removal of scaffolds and “social supports” (p. 22) (i.e. teacher intervention), and the complexity of the material to be learned through inquiry. It’s also important that the teacher not make assumptions about students’ understandings of process or concept words. This is one of the things that I really like about Dan Meyer’s work. Because his techniques are evolving as he teaches classes of students who struggle in math, he has learned to back as far away as possible from all assumptions (even about such second nature aspects of math work as using letters instead of word or numerical labels) and allows the students opportunities to build up these understandings on their own.

    C. Mathematical thinking requires “the ability to repeat increasingly sophisticated actions until they can be performed automatically without a conscious thought” (Tall. p iv). But if there is only mindless repetition (application without learning), the kind that is fostered by what McNeill, Lizotte, & Krajcik called “continous scaffolding,” then the students will not as readily make the links and leaps required to get to the next level of independence.

    D. All three of these sources value the systematic development of knowledge. This definitely flies in the face of the current view shared by many constructivists that this value has been rendered old-fashioned and outmoded because of instant access to the internet and the rate at which information in such fields as science is growing and changing. Inquiry based learning may grow out of natural curiosity or from “baiting the hook” (from Meyer, as was mentioned in my last blog), but it isn’t haphazard. New understandings grow out of perceived limitations in previous learning. The 21st century change in our attitude towards knowledge should be that we must encourage students to consider whatever they know to be ‘the best so far’. Embracing the inquiry process can help them become open to adding to their knowledge or revising it or replacing it when old understandings don’t do the job any more.

  • Tall (p.3) :”Technical skill is essential, but a full range of mathematical thinking requires more than the use of routine skills in familiar situations: it requires the ability to deal creatively with unfamiliar problems. This in turn requires the learner to organize personal knowledge in ways that are flexible and richly connected.”
  • Donovan & Bransford (pp. 1-2): “To develop competence in an area of inquiry, students must (a) have a deep foundation of factual knowledge, (b) understand facts and ideas in the context of a conceptual framework, and (c) organize knowledge in ways that facilitate retrieval and application.”
  • McNeill, Lizotte, & Krajcik (pp. 4-5): A lack of sufficient understanding of a topic may make it impossible for students to understand what counts as appropriate and sufficient evidence or distinguish evidence from theory or personal views.
  • E. Metacognitive skills are an essential part of effective learning and investigation. Learning has a lot in common with long distance running. Both student and athlete need to know who knows where they are and where their destination lies. This ability to look back and look forward enables the individual to take responsibility for monitoring his own progress towards the goal.

  • Donovan & Bransford (pp. 1-2): A “metacognitive” approach to instruction can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them.”
  • McNeill, Lizotte, & Krajcik (citing Wood, Bruner, & Ross, 1976; Stone, 1998, pp.18-19) In order for scaffolding to be successful, a child must have some prior understanding of what is to be accomplished.
  • Tall, p.iii:”… the development of mathematical thinking depends on the increasing sophistication of perception and action as the child reflects on successive learning experiences. …the successive experiences that we encounter in life which fundamentally affect the ways in which we build current ideas on those we have met before.
  • This is what makes the act of using one’s mind in a deliberate way to unearth mysteries, make connections, and turn loose collections of skills and facts into knowledge that becomes a tool through which to understand the next experience. This is want makes for mindful learning.


    Donovan, S. & Bransford, J. (2005). Introduction. How students learn (pp. 1-2). Washington DC: The National Academies Press. Retrieved June 5, 2010, from EDIM 513 Unit 1 Topic A at

    Kandel, E. (Speaker). (2008, September 01). Mapping memory in the brain. Retrieved December 5, 2009, from

    McNeill, K. L. Lizotte, D.J., Krajcik, J., & Marx, R.W. (n.d). Supporting students’ construction of
    scientific explanations by fading scaffolds in instructional materials.
    The Journal of the Learning
    Sciences. Retrieved June 5, 2010, from

    Tall, D. (2008). How humans learn to think mathematically. Unpublished manuscript, University of Warwick. Retrieved December 4, 2009, from

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