The danger of “either/or” teaching

Ending the debate: Which teaching approach is most effective?

Tierney Kennedy

In 1997 a landmark study in the UK decided to define effective teaching using student improvement data… if you haven’t read it, the findings might be surprising

When I first started researching effective teaching I remember being horrified at the lack of student data that was used in educational research papers. Mostly it seemed that researchers just came up with a theory, wrote the background for why it might work, got a few example quotes from a handful of kids and then published the papers. And nobody ever checked to see if the theory actually worked to improve results before pumping it out en masse in our classrooms. Seriously scary stuff! But then I came across a study by Mike Askew, Dylan Wiliam and their team[1] from the UK involving over 2000 students and 90 teachers, and everything changed.

In a nutshell, they found that teachers’ beliefs had a profound influence on how effective they were.

They classified teachers into three groups:

  1. Discovery-oriented beliefs: learning of mathematics happens when students are ready for it and discover it for themselves. Think: play-based maths, everything in a context, everything hands-on.
  2. Transmission-oriented beliefs: learning of mathematics happens when teachers explain clearly and concisely and when students practice routines and procedures. Think: direct / explicit explanations.
  3. Connectionist-oriented beliefs: learning of mathematics involves both valuing pupils’ methods and teaching strategies with an emphasis on establishing connections within mathematics. Think: both exploring and explaining rather than either/or.

So what did they find?

While very few teachers fit exactly into one type,

it was clear that those teachers with a strong connectionist orientation were more likely to have classes that made greater gains over the two terms than those classes of teachers with strong discovery or transmission orientations”[2].

 What did the highly effective teachers do?

  • They used challenging problems for all students, not just the high-achievers. These were not word problems and this specifically did NOT involve teaching strategies to simplify or code the problems to make them easier. It did involve working out maths that the students had not yet been taught and requiring the kids to think hard about tricky maths.
  • They started with students’ own methods, and dealt with the underlying misunderstandings as they occurred, by helping students to think these through. They did NOT go back to explain the method again but helped the students to change their own minds.
  • They used students’ methods as the basis for building connections to the more efficient, formal strategies.
  • Then they practised the formal strategies to solidify the connections.

What does this mean for us?

It’s dangerous to think that either Discovery/Constructivist or Direct/Explicit approaches hold all the answers. If kids just explore randomly, they are unlikely to discover the underlying patterns and principles that make maths work. If we only ever tell kids what to think, when are they going to have to think for themselves and build new connections?

We need to stop being afraid to ask challenging problems:

  • Instead of saying, “it’s ok to be wrong”, say, “Great! Now we know it’s hard enough to grow our brains.”
  • Use challenging problems to introduce new topics.
  • Use challenging problems for all kids, not just high-achievers.

We need to think of problem-solving as an experiment to find out what works and what doesn’t

  • Instead of leading students so quickly to the right answers, allow time for students to test out their own ideas and figure out what doesn’t work.
  • Once we have found an idea that does work, link this with the formal explanations and strategies and make connections to what students already know.

We need to explain and practice once students have done the thinking:

  • Instead of explaining at the start, explain towards the end. Practise skills to build fluency. Use these skills next time in a new way when solving another problem.
  • We need to practise skills regularly to practise retrieval of strategies rather than simply executing strategies in the same way on the one day. For more information, look up the work by Rohrer and colleagues on Interleaving.

We need to focus on the connections instead of covering content:

  • Focus on similarities, differences and relationships between big ideas rather than on trying to do everything.

 We need to be “both and” teachers, not “either or” teachers. We need to help kids create connections.


[1] Askew, M., Brown, M., Rhodes, V., Johnson, D., & William, D. (1997). Effective teachers of numeracy. Final report. London: King’s College. Download the entire paper here:

[2] Quote from the Paper presented at the British Educational Research Association Annual Conference in 1997 from the same study:

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