Charlie Harber is the Deputy Lead Adviser for the HfL Primary Mathematics Team.
Do pages and pages of repetitive questions deepen our children’s understanding?
Teaching to mastery requires a mind shift on many different levels and presents many challenges (differentiation, whole class teaching, culture and ethos to name but a few). It is underpinned by a range of theories… You can’t explore mastery for long without encountering ‘Variation’ theories. There are two distinct variation theories (conceptual and procedural) which develop symbiotically to grow deep conceptual understanding in learners. These theories are used widely in the high performing East and South East Asian countries. They are exploited skilfully by teachers to expose the underlying structures of mathematics and allow children to ‘self-discover’.
If you have completed 25 questions on a topic does that mean that you have deeper conceptual understanding than someone who has only completed three?
Of course not, it depends on the quality of the question and the depth of the explanation. Quantity does not lead to quality learning. Through a carefully designed sequence of questions/tasks, teachers can present learning opportunities (each question should be a learning opportunity).
The sequence has to be carefully planned otherwise the children might just follow the pattern without developing real understanding. This is sometimes where
misconceptions are formed. Through a combination of discussion, conceptual exploration and good questions, the children begin to recognise the key relationships. They need questions that help them apply and test this emerging knowledge – opportunities which force them to reflect and discern ‘what is the same and what is different ‘about the cases shown. Discussion exposes emerging misconceptions, shares the excitement of a conjecture and develops comprehension. Conceptual exploration nurtures the need to understand; supports explanation, reasoning, proof and scaffolds developing vocabulary. It is a careful balancing act – enough questions to allow sufficient variation but not too many so that it dissolves in to overly-repetitive practise for the sake of filling time and books. There is a time and place for further, more repetitive, practise but not at the stage where pupils are encountering new ideas and building deep understanding.
Consider these phases of learning…
Step 1: Revisiting and reconstructing Start low. A successful teaching sequence should always start well within the children’s fluency range. This allows the children to reactivate existing knowledge and successfully identify where this new learning links with their current understanding (assimilation or accommodation). Low number allows the children to explore the concept with a range of concrete materials (conceptual variation), without the pressure of ‘what is the answer?’ In many cases, they should already know what the numerical answer is; their attention should be on ‘why’ this is the case.
Step 2: Predicting and exploring
Does the conceptual understanding hold true for slightly more challenging numbers?
These numbers start to push their understanding. The focus at this stage is to allow for the translation of their conceptual experiences from step 1 and application to new numbers. But this should still be within the range in which manipulatives can continue to support the understanding.
We need to know why.
How about even more challenging numbers?
These questions have to be carefully considered; they need to expose the pattern clearly – but also allow for conceptual exploration.
Step 3: Confirming and explaining Once the pattern has begun to be established and proved, can the children apply it to calculations which aren’t easily modelled using concrete apparatus? This could be channelled by using some partially completed examples – preventing the children from beginning to develop misconceptions. Generalisation should be emerging and the focus in teaching now switches to tighter mathematical explanations.
Step 4: Applying Investigating the inverse or an inverted situation reinforces the learning and encourages flexibility in thinking. It is a final chance to expose misconceptions.
Each of these steps should involve a limited number of questions, but sufficient to focus upon the critical elements. How many questions is enough? There is no hard and fast rule. Instead, it is dictated by the response of the learners.
Why were the examples above chosen?
Step 1: Revisiting and reconstructing Through a variety of concrete manipulatives (linear, area and set represented), children explore equivalent fractions to firstly ½ – linking back to previous learning and then apply this knowledge to other fractions. The early introduction of a non-unitary fraction prevents the children from making assumptions and also strengthens understanding that the multiplicative relationship is applicable to the whole of the fraction, not just the denominator or the numerator.
Step 2: Predicting and exploring At this stage, the children have begun to establish what the relationships might be, but they haven’t yet formalised it into a ‘rule’. They still need to explore more which will provide pupils with the opportunity to apply their emerging understanding to less familiar fractions – starting with a unitary fraction. The fraction can still be constructed accurately with a variety of different models. The children should be predicting what the solution is prior to construction. It is vital that no judgement is passed by the teacher as the children make their predictions and that the class feel comfortable in presenting a range of predictions. The role of sharing and discussing possible answers is a powerful motivator for extended exploration and true investigation is there to ensure that the children are still considering the whole fraction not just individual sections.
Step 3: Confirming and explaining In this example, missing sections have been used (blankety blank) to encourage the children’s deeper thinking about the multiplicative relationship. It is possible that, up until now, the children have always expressed this as a doubling pattern or that they think you have to always use the same multiplicative relationship in a sequence of equivalent fractions. The example above shows that a ‘times 2 and times 3 pattern’ can appear within the same sequence. An example chosen to burst the possible misconception: “we can just double the denominator and numerator.”
Step 4: Applying Do the children really understand that the multiplicative relationship works using the inverse relationship and that you can simplify fractions (you can divide as well as multiply)? Now, they should be in position to firm up their class rule ensuring that it uses correct mathematical vocabulary and explain it conceptually.
Once you have completed the questions – there is still the potential for more learning. As Jane Jones (Ofsted’s National Lead for Mathematics) said, “the answer is just the beginning”. You could present the questions as digits only. But in a column rather than linear format; words only; a mixture of digits and words; worded problem e.g. how many different ways can we solve this calculation…? But that is a discussion for another time.