Aiming for Mastery?

The other day, a good friend of mine shared this picture with me asking me what my thoughts were:


On first glance, my thoughts were mixed.  On one hand, I really like that students are receiving messages about mistakes being part of the process and that there is more than one opportunity for students to show that they understand something.

However, there is something about this chart that seems missing to me…  If you look at the title, it says “How to Learn Math” and the first step is “learn a new skill”.  Hmmm…… am I missing something here?  If the bulletin board is all about how we learn math, the first step can’t be “learn a new skill”.  For me, I’m curious about HOW the students learn their math?  While this is probably the most important piece (at least to me), it seems to be completely brushed aside here.

I want you to take a look at the chart below.  Which teaching approach do you think is implied with this bulletin board?  What do you notice in the “Goals” row?  What do you notice in the “Roles” row?  What do you notice in the “Process” row?

Teaching Approaches - New

If “Mastery” is the goal, and most of the time is spent on practicing and being quizzed on “skills” then I assume that the process of learning isn’t even considered here.  Students are likely passively listening and watching someone else show them how… and the process of learning will likely be drills and practice activities where students aim at getting everything right.

I’d like to offer another view…

Learning is an active process.  To learn math means to be actively involved in this process:

  • It requires us to think and reason…
  • To pose problems and make conjectures…
  • To use manipulatives and visuals to represent our thinking…
  • To communicate in a variety of ways to others our thinking and our questions…
  • To solve new problems using what we already know
  • To listen to others’ solutions and consider how their solutions are similar or different than our own…
  • To reflect on our learning and make connections between concepts…

It’s this process of learning that is often neglected, and often brushed aside.


While I do understand that there are many skills in math, seeing mathematics as a bunch of isolated skills that need to be mastered is probably neither a positive way to experience mathematics, nor will it help us get through all of the material our standards expects.  What is needed are deeper connections, not isolating each individual piece… more time spent on reasoning, not memorizing each skill… richer tasks and learning opportunities, not more quizzes…

“One way to think of a person’s understanding of mathematics is that it exists along a continuum. At one end is a rich set of connections. At the other end of the continuum, ideas are isolated or viewed as disconnected bits of information. A sound understanding of mathematics is one that sees the connections within mathematics and between mathematics and the world”
TIPS4RM: Developing Mathematical Literacy, 2005

So, while I don’t think putting up a bulletin board (or not) is really going to do much, I really do hope we are spending more of our time thinking about HOW our students learn and WHAT our goals are for our students (see the chart above again).

I still wonder what it means to be good at math?  I wrote about my questions here, but I am still looking for ways to show others how important mathematical reasoning is for students to develop.  Skills without reasoning won’t get you very far.  Maybe more about this another time…

11 thoughts on “Aiming for Mastery?

  1. Charts on the wall that describe the processes of learning to learners may have some value However, I think there is an important difference between presenting such a chart/poster and “living-out” the processes on a regular basis using students’ work; hence classroom culture. Living-out could happen with a visualiser, to exemplify students’ work, for everyone to discuss what these words and phrases mean within the context of problems students have been working on. After all, processes such as seeking to solve a problem, reasoning, conjecturing, generalising and proving are the process skills through which students subject knowledge needs to be developed. Using and applying knowledge and modelling needs to become essential ingredients of students’ mathematical diets.


    1. Great points Mike. So how or when would you use a chart like this? I like that you are getting us to see that the decisions we make as teachers are much more nuanced than simply copying a poster somebody else did.


      1. I would be more interested in discussing with students what they think mathematics is and what they think mathematicians ‘do’. I would then ask them the same question several weeks later after having had a diet of inquiry/problem-solving type tasks to work on. I would then ask them to produce posters themselves to answer these questions and see what they come up with. It would then be incumbent upon me to keep giving them an inquiry/problem-solving diet – because that is how I believe we can help them to become evermore confident and competent mathematicians.


      2. Yes, of course learning processes are fundamentally important. However, I seek to avoid ‘transmitting’ such processes to students in the same way some teachers might transmit ‘their’ subject knowledge to students. This is because I believe everything has to start from the learners; this fits my pedagogic underpinning described by Gattegno as “Subordinating teaching to learning.” This, in turn has become one of the Guiding Principles of The Association of Teachers of Mathematics, which reads: “The power to learn rests with the learner. Teaching has a subordinate role. The teacher has a duty to seek out ways to engage the power of the learner.” To this I wholeheartedly subscribe. This does not mean, however, I will never ‘tell’ students something, specifically when helping develop students mathematical vocabulary, or when looking at conventions. Potting co-ordinate number pairs is one such example and whilst I would encourage students to ‘knowingly’ mis-plot co-ordinates, in order to see that a transformation of a reflection in y = x is created, I am going to confirm the convention is horizontal followed by vertical.


      3. Great theory to guide your actions.
        It’s so important to have beliefs, be able to reflect on them as you teach and be able to justify why you have made the decisions you’ve made.


      4. My experience tells me there are two important actions to bring to my teaching. These are, firstly, to develop my awareness and ability to notice ‘things’ in classrooms and to choose whether or not to act upon them; “Only awareness is educable” (Gattegno). This phrase was also used as a title for a short piece by John Mason which was published in Mathematics Teaching 127. Secondly, to be metacogniitve; to decide in-the-moment whether to intervene and when to stand back. At best interventions are helpful to learners, at worst an intervention might be seem to be an interference. Teaching is, I believe, all about balances and plate-spinning, metaphorically speaking.


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