I was working alongside a Year 6 pupil in a recent tutoring session who, when tackling the problem 537 ÷ 10, used her whiteboard to jot down the calculation in ‘bus stop, first working out how many 10’s in 53, and carrying over the 3. She struggled over the question a while longer, counting in the 10 times table on her fingers. before writing her final answer as 53r7. When asked to share why she had chosen to solve the calculation using this method, she replied, “Because that’s how we have been taught to divide large numbers”. This is far from a unique scenario and our Action Tutoring maths sessions have many similar pupils – those who view mathematics as a collection of formal rules and strategies to memorise, rather than a system of relationships to investigate and better understand.
Despite this, the Department for Education (DfE) announced in February that thousands of eight and nine-year-olds in England will trial a controversial new times tables test, which will be made compulsory in 2020, despite opposition from teaching unions and parents. Ministers argue that the check will help children to know their times tables off by heart before starting secondary school, but critics say primary school pupils are already over-tested and this will only add to rising levels of math anxiety in primary school pupils. Mathematics facts are clearly important, but forced memorisation through repetition, practice and timed testing can be unnecessary, ineffective and stressful. This view is underpinned by the fact that the Conservative Education minister Nick Gibb (the main proponent of the new tests) refused to answer a multiplication question on television in light of the public ridicule faced by Labour schools minister Stephen Byers, when he incorrectly worked out that 7 x 8 came to 54. Such occasions only serve to highlight the limitations of rote memorisation without the corresponding development of ‘number sense’. In her book, About Teaching Mathematics, Marilyn Burns describes pupils with a strong number sense in the following way: “they can think and reason flexibly with numbers, use numbers to solve problems, spot unreasonable answers, understand how numbers can be taken apart and put together in different ways, see connections among operations, figure mentally, and make reasonable estimates.”
So how can we, as maths educators, make shifts in our tutoring practices to foster more ‘sense-making’ in mathematics and less memorisation and overly rigid adherence to formal strategies? One approach is the increasingly popular practice of ‘Number Talks’. These five- to fifteen-minute conversations aim to build number sense by offering a short, structured way for students to talk about calculation problems. During number talks, pupils are asked to communicate their thinking when presenting and justifying solutions to problems they solve mentally.
Sherry Parish, author of Number Talks: Helping Children Build Mental Math and Computation Strategies, recommends the following structure (this video clip shows a number talk in action):
- The teacher or tutor presents a calculation to the class; the pupils are given thinking time to mentally solve it.
- The students start with one fist to their chest; they make a “thumbs-up” on their chest to show that they have found an answer. They then use the remaining time to try to think of another way/strategy which they then indicate by putting up a thumb and a finger, and so on.
- The teacher or tutor asks children to volunteer their answers and all given answers are recorded on the board without indicating which is correct.
- The teacher asks a child to “defend their answer” or “explain their strategy”.
- All strategies are recorded on, using visuals where possible.
- The children agree on the “real” answer.
Many schools have reported that by providing a space in each lesson to collectively reason about maths not only builds mathematical understanding but has led to the development of more accurate, efficient and flexible mathematical strategies in pupils. ‘Number Talk’ also provides pupils with opportunities to clarify and communicate their own thinking, consider and try alternative strategies, investigate and apply connections and make decisions about the most effective strategies. Although they are usually incorporated into a whole class teaching context, this approach translates well into our tutoring sessions, particularly when working with 2 or 3 pupils. Embedding number talk into a session could enable tutors to better understand and assess pupils’ existing knowledge of a maths topic as well as being a valuable as a tool to review, practice and reinforce concepts and build mathematical fluency. If you are interested in exploring the idea of ‘Number Talks’ further or would like to pick up a few ideas for incorporating these into your maths tutoring, I recommend the following reads:
Number Talks Overview from Maths Perspectives Teacher Development Centre
Watch one of the video clips that are featured in Sherry Parrish’s book
Number Talk Session 1: A 17-minute video of the first number talk with a primary class. Good example of how to introduce the reason for number talks and classroom protocols.