Weird ideas about conceptual understanding

A paper I keep returning to is by Carol Bryan and colleagues. They surveyed the attitudes of maths teachers in the US, Australia and Mainland China and Hong Kong. The most notable finding relates views on the relationship between memorisation and understanding:

Though they all agree that memorization plays an important role in mathematical understanding, teachers from the four regions do not fully agree on what that role is or to what degree memorization is important. Furthermore, there was a re-occurring concern of whether or not memorization should come before or after understanding. For teachers from Mainland China and Hong Kong, memorization can come before or after understanding. However, for Australia and US teachers, memorization can only come after understanding. Nevertheless, memorization after understanding is held in higher regard than memorization before understanding (or ‘‘rote’’ memorization), though some of the teachers from Mainland China say that perhaps the latter could be an intermediate or transitional step towards understanding the mathematics.

I was reminded of this when I saw the comments following a tweet by Pritesh Raichura, Head of Science at Michaela Community School in London.

Michaela is an independent, government-funded school known as a ‘Free School’, similar to Charter Schools in the US. Last year, its first students completed GCSE examinations and the results were phenomenal. It is important for those who are unfamiliar with GCSEs to realise that these assessments involve essay questions and structured answers, not just factual recall. So Michaela students are capable of far more than simply recalling facts.

Michaela serves a diverse community and is open to all. Its students are not wealthy. I used to teach similar students in nearby Greenford and so I find Michaela’s achievements particularly impressive. You may think education academics would be interested in how they achieved their success.

The objections to Raichura’s tweet seemed to centre on the idea that understanding should come before memorisation and so drilling students is inappropriate. Even if you truly believe that this is what the research shows, would Michaela’s success not at least give you some reason to pause? Science is not the process of dismissing anomalous results because they do not fit your preconceived ideas. Science is about actively seeking anomalous results.

And we know there is often a disconnect between the small-scale studies of researchers and large-scale implementation in schools. Dismissing meaningful, large-scale success on the basis that it conflicts with smaller research studies – if such a conflict existed – would seem to have things entirely the wrong way around.

What does research actually say about memorisation and understanding? Well, as far as it goes, it seems to support a two-way, iterative relationship. So the East Asian teachers probably have this right.

Perhaps the US commentators are suffering from being locked into a ‘WEIRD’ perspective – western, educated, industrialised, rich and democratic. They may do well to seek more diverse viewpoints.

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8 thoughts on “Weird ideas about conceptual understanding

  1. luqmanmichel says:

    Yes, memorisation has a place either before or after understanding.
    I teach my students to memorise the Dolch words and they get to understand the words as they read sentences. Most of the Dolch words are used on a daily basis and they probably know its usage even before memorising.

    Memorisation has a place in learning to read.
    Stay safe.

  2. Joe Hamkari says:

    Hi,
    If the purpose of education is the why not the what then we have a problem.

    I agree that some learn the what (through repetition, practice or whatever) and then go on to the why, but this is not all or perhaps in fact many.

    While we mostly assess the “what” we then judge ourselves as successful but are not perhaps reaching our goals with a large number of learners. I also find GCSEs to be very much based in the “what” rather than the “why” (the level is such that we can rote teach any possible answer to a why questions and this, in fact, is done frequently). I can only speak from my experiences, but Schools that have the highest scores for all students in GCSEs have students who are given the chance and the opportunity to “learn” the answers, enough repetition will get most kids remembering enough (or they are selective in their intake). My cynical side would say the success of such schools is based not on “learning” but getting students to follow the instructions as to how to memorise the material. It is social acquiescence, not a learning paradigm shift. Many do start by setting up an ordered environment, with uniforms and the following of procedures.

    However, as a science teacher, I do know that content and knowledge is an important part of understanding. So am left feeling a little schizophrenic with my practice.

    I perhaps think we as educators have to identify exactly what content is needed to develop the understanding we are seeking rather than cramming kids with as much as we can with the hope that the “magic” of understanding will happen. We need to combine this content with proven methods to develop the understanding we all say we want. Of course, kids are different and perhaps the amount and level of content are different (another variable in an already overcomplex process!).

    I do not know the answer I just think we need to think carefully about what others propose as solutions. I also welcome any comments that will help me to advance my own understanding.

  3. Chester Draws says:

    I agree that some learn the what and then go on to the why, but this is not all or perhaps in fact many.

    Citation required.

    My experience is the exact opposite. When I teach Algebra, the biggest impediment to the students understanding higher level concepts is a poor grasp of the mechanics. I cannot teach “solving” to those that cannot add like terms. Those that can articulate why 3x + 4x = 7x but do it inadequately are worse off than those that do not
    “understand” why, but can nevertheless do it reliably.

    I am reminded that many students these days display what Barry Garrelick calls “rote understanding”. They can parrot back what the teachers wants them to say, and they are deem to have understood. Meanwhile a student who can do a skill 100% reliably, but lacks the ability to articulate it clearly, is deemed to lack understanding.

    When young, my daughter was asked to add two large numbers. Which she did quickly and efficiently. She was then asked to do it another way, to “show understanding”. She asked why, when her method always worked, would she do it any other way? As she didn’t have another way, because she didn’t need one, she was marked down as not understanding. Despite being clearly the best Maths student in her class, she was graded a curriculum level lower than others.

    Given a Year 9 student who knows his times tables completely by rote, and another who only knows how to get the answer a long way, I’ll take the rote one any day. Everything from that point forward will be done much more quickly and reliably, despite no understanding.

    • luqmanmichel says:

      “Given a Year 9 student who knows his times tables completely by rote, and another who only knows how to get the answer a long way, I’ll take the rote one any day. Everything from that point forward will be done much more quickly and reliably, despite no understanding.”

      Me too. I learned my time table by rote but as time progressed I understood the why.

  4. Joe Hamkari says:

    The experience of the what and not the why is from my personal experience of students over many classes and subjects. In fact, a number of maths students who were very good at solving known problems could not apply this knowledge to solve any issues that were not exactly like those they had been given. I acknowledge this is anecdotal but I do have a number of years experience. There are however multiple studies to show both the limitations and the uses of rote memorization.

    Perhaps what we are looking for is an agreed understanding of what the word understanding means and perhaps a view of what exactly we mean when we talk of the why
    I refer to an updated Blooms Taxonomy and see that understanding is just one step above remembering (which is the base level).
    See this image by Vanderbilt university centre for teaching.
    https://images.app.goo.gl/g5NUrUqypTh4tZAb8
    Found here
    https://cft.vanderbilt.edu/guides-sub-pages/blooms-taxonomy/

    It has application, analysis, evaluation and creation as being more cognitively taxing.
    So what level are we discussing when it comes to the why?

    There exists a dichotomy in learning methods (one of many really) one side of which is constructivist at its heart and another that is more teacher-directed. Constructivist say knowledge schema created by students create better learning. The other speaks of the cognitive load argument that says more learning occurs when we can limit the cognitive load which is done by teachers presenting materials in an organised form.
    My experience is that like all theoretical models life is generally a mix of both. We need to do both. Allow students to discover but also provide a basis for their discovery questions.
    Some tasks such as multiplication tables perhaps do need to be rote learned by many, but perhaps not by all.
    So my questions always return to;
    What will allow learners (and not when they get to university) develop higher cognitive skills?

    I am always looking for the best way to achieve this. We have been using the rote method in many schools for many years and I will ask can we not do better for more students?

    • Chester Draws says:

      I refer to an updated Blooms Taxonomy

      Bloom’s taxonomy is a Constructivist concept, so of course it supports Constructivism. “Understanding is just one step above remembering” because someone said so isn’t much of a proof. I think we need better than Bloom’s as a starting point, because it is already aligned to one side of the argument.

      My experience is that like all theoretical models life is generally a mix of both. We need to do both.

      You can’t do both. You either lead your teaching with a direct explanation or you don’t. You either support the learner thoroughly or you don’t. Cognitive load issues don’t go away just because you overload the student only sometimes. Constructivist techniques are still a very time inefficient way to teach, even if you only use them sometimes.

      You can alternate methods, but that’s different from mixing them. I use a variety of different teaching techniques without ever feeling the need for Constructivist concepts.

      I don’t believe in a mix of scientific and alternative medicine either, for pretty much the same reason — only one actually works. But the hucksters will often try to sell alternative medicine on the basis that we “need” a mix.

      (Experts need direct explanations less, but so long as we are discussing high school students, that isn’t really important.)

  5. I found it interesting that the article generalized about “Eastern” vs. “Western” ideas regarding abstraction and mathematics while comparing the Asian countries only to Anglophone western ones. I suspect that had they consulted teachers in continental Europe (particularly France), they would have found much stronger support for teaching mathematics as an abstract pursuit.

  6. Alice Flarend says:

    “it seems to support a two-way, iterative relationship. ”

    YES!! Doing memorization before any conceptual development or vice versa is not as effective as having the 2 build on each other. The two can build on each other.

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