It is the next step up from your number skills when get your number skills in place then you can start generalizing into algebra. (Interview 1, August 2012)
Grace also, like John, sees the importance for students to understand number, and the need for them to have gained sufficient fluency in using number before they can begin to generalise using algebra. When asked if she thought whether good number skills would guarantee success in algebra, Grace did not believe that this would necessarily be the case. To support her assertion, Grace cited her experience of teaching students that had worked well with number, but who had later forgotten their number skills, defined here as operations with number, when presented with the abstraction of algebra.
I have seen it in good classes where they have really really sound number skills but suddenly we venture into the abstract area they forget their number skills. They probably do it less than kids who are not good with their arithmetic skills but you still get the same things coming through. (Interview 1, August 2012)
In working with students who do not have good number skills, Grace believes that regular revision is important. In her position as Head of Mathematics at the school Grace has advised her teachers to include more practice of arithmetic skills in Years Seven, Eight and Nine classes.
I think too it is something I am trying to address at the moment with arithmetic skills as well is that you do a little block of it and then you don’t do it for a while. I have [been] trying to get staff to do a lot of immersion with arithmetic skills in Seven, Eight and Nine. And I think we will probably then lead that into going back and revising our algebra skills as well. (Interview 1, August 2012)
Grace’s theme of practice for teaching number was seen to be Teaching/learning balance.
Table 33 Teaching/learning balance: Grace's theme of practice in teaching number.
TEACHING/LEARNING BALANCE
Code Strategy Orientation
34 Allows students to make errors Discovery
47 Students work with others Discovery
37 Closed questions Transmission
40 Teacher assessment Transmission
27 Connections made between ideas Connectionist
In the first observed lesson Grace began the lesson with a dice game that gave students the opportunity to investigate three digit numbers using an eight or ten-‐sided dice. They had to find out how many three-‐digit numbers they could create that would meet the requirements of the task. The game was both personalized and inclusive as all the students in the class could play it. Closed questions assisted Grace in assessing whether or not students were on the right track with the game. As the game progressed Grace asked questions of the students, which prompted them to think more deeply about what they were doing in terms of finding the biggest difference or the smallest difference. Students were able to correct their own errors as they played the game. The game allowed students to think about the
construction of three-‐digit numbers and how place changed digits’ values as they were moved.
In the third lesson observed by the Researcher Grace introduced formulas that had a practical use. The numbers involved in the calculations were decimal and squared numbers and this called for calculator use at this time.
You should be doing body mass index with this formula. This is an actual formula that they do use to calculate body mass index. And then we have got our speed formula and this is the actual formula you use to calculate speed. If you haven't met these already in Science and Phys Ed you are going to in the next couple of years. (Transcript of Lesson 3, October 30, 2012)
Grace was committed to making improvements in number skills and number sense of the students in her school and this was supported by her contribution to the group discussion on the subject of calculator use at the focus group meeting, held towards to the end of the
preventing a student from dealing with early algebra, then a calculator could be used. Grace did not share this view and stated that to neglect the practice of mental arithmetic skills would not serve students well in the future years of schooling, particularly if they intended to study mathematics at Stages Two or Three in Years 11 and 12. In Western Australia the examination papers for courses at the Stage Two and Stage Three have a calculator free section of the examination paper.
I think we still persevere because we know the end point is calculator free exams. We do, do remediation across Seven to Ten and yes some of their assessments are modified yet where possible we are trying to improve their mental maths skills. (Focus group meeting, November 2012)
During the final interview and in the reflective questions submission at the conclusion of teaching the unit on beginning algebra, Grace reflected on the need to review her number work with students before she began to teach such a unit of work in the future.
One thing I would probably do differently before the start of the unit that I was working with is build in number. I didn’t do it and the kids didn't suffer because they are fairly bright and retentive anyway. I probably would have built in some revision of order of operations square numbers and that sort of stuff. And negative numbers. (Interview 2, November 2012)
I was able to remind them of things on the fly. But if it was a different level class then I would probably make it more formal … have some mental maths beforehand. Have one lesson we will do all those things beforehand or have some sort of game that to play that reminded them of those things. (Reflection questions response, November 2012)