Session 5: Number Sense

This was an important session in that it confirmed so much of what I consider best practice in mathematics education.  The observations in the earlier section that low achievers over-rely on counting resonates with my experience.  the over-arching idea here is flexibility in approach to calculation and that one size does not fit all.  The multiple ways of approaching a calculation are important for students to see and give essential validation to individual students' methods.   However, not all strategies are equal and I disagree with Jo's dismissal of the separation of conceptual and procedural.  Whilst the separation may be artificial from a teaching point of view, from an assessment perspective, identifying errors that are conceptual or procedural is important in gauging the students' level of understanding.

I was interested in the William Thurston quote regarding compression and that without conceptual learning such compression was impossible!  The examples of number talks shown in the video clips often gave examples of the teacher framing the calculation talks within deeper concepts, such as the rules of arithmetic and seeking to explain a problem algebraically.

For ISZL, much of this session should serve as a positive reinforcement of what we already do.  I have recently seen examples of number talks in G1 and G4 that were very productive and pivotal in developing the flexibility we should aim for in approaches to calculation.  Our students learn a repertoire of mental strategies and written methods but we should not be complacent.  How consistent is our approach across grades for instance?  My recent foray into G7 was interesting in that so many of the students showed evidence of the kind of compression that Thurston talks about; they had assimilated strategies, number sense, knowledge of operations etc and were applying them to a range of problems.  What was significant to me was the variety of calculation methods successfully used.

A key feature of the examples of number talks in the session was the teacher modelling the spoken strategy using standard notation, and checking with the student that the recording was an accurate representation of their method.

The visual models employed were sometimes effective but at times I didn't think they accurately represented the strategy discussed, however, the message here is important: visual models are an essential component of helping students develop number sense...and we need to use a wide variety.

Number Talks - Session 5

Number talks have great potential in our primary maths lessons. There are many benefits in doing number talks including: students work in partners or groups of 3 to share their thinking and to problem solve, which allows students to feel more comfortable in taking risks and sharing ideas; more students have the opportunity to communicate their understanding; after students come together and share with the class, they can see that there are many ways to solve a problem; they get many opportunities to ‘play’ with numbers.

Students learn more about number sense. They can begin to see maths as a flexible subject and to start using and seeing numbers flexibly. They begin to realize and understand that there is not only one way to find an answer. Making mistakes is okay and encouraged – we can learn from our mistakes. The answer is not as important as thinking about and sharing the methods that they used. I still have students that think ‘faster is smarter’. Doing these number talks teaches students that there are more ways to show how to be a good mathematician.

Number talks work well because they teach number fluency and automaticity at the same time as teaching a conceptual understanding of number. It was also emphasized in session 5 how important it is for students to see visual representations as often as possible when learning mathematical concepts.

2 goals for the rest of this school year: provide opportunities for number talks and to try dot cards (or ten frames) in addition to calculation problems. I often have students share their thinking for solving math problems, however I have not used number talks as an explicity teaching method or as a consistent routine, nor have I spent such a long amount of sharing time on one problem. It is necessary sometimes to spend a significant amount of time on this so teachers and students can go deeper into problem solving and really get the sense of flexible numbers. At first I thought the dot cards looked more appropriate for younger ages, but as Jo Boaler said, it can be used all the way up to high school to help students develop their number sense. A quick search on google showed that there are many ideas for using dot cards and ten frames to deepen number sense.