Working Like a Mathematician
Angela Luchetti & Monica Hay
Mt. Carmel School, Yass, New South Wales
Angela and Monica were two of 40  50 teachers from primary and secondary schools who participated in a six day professional development program titled Working Like a Mathematician in Pattern & Algebra. The course was presented under the PD from MC banner for the Catholic Education Office, Canberra Goulburn Diocese.
This article reports the school's work in the period between Days 1 & 2 and Days 3 & 4 of the course. The report offers considerable insight into the value of practical, active professional development, especially when presented in stages with classroom trialing sandwiched between sessions. But the course only stimulates. It is these creative teachers and their colleagues who have produced the successes reported here. 

Teaching at a Central school (K10) allowed us to have two representatives participate in the Working like a Mathematician course. This has enabled all our students to participate in Mathematics lessons where the students can be engaged in investigations and challenges.
It has been an exciting journey since the initial two days of the course and many investigations have been trialled, and are now used regularly in our classrooms. We conducted a staff professional development afternoon to showcase the new resources, provide a summary of our two days and make staff familiar with the Maths300 web site.
In the Primary department Year 5/6 decided to launch straight into the PredictACount activity. The students were excited by the prospect of being able to check their answers with a calculator and it was difficult to stop them once they had started. We were able to extend into decimals and fractions and all students experienced success.
This activity was then followed by the constructing a bridge support section with newspaper rolls task (Match Triangles). The students enjoyed putting the bridge together on the floor as a group and were quickly able to work out a solution if the bridge had 100 rolls. This activity challenged the more mathematical students in the class as they needed to pair up and explain their theory to another class member who was still working on a solution.


...engaged in investigations and challenges.
...now used regularly in our classrooms.
Students were excited by the prospect...
...and all students experienced success. 
The Year 5/6 classroom has taken on Working like a Mathematician day every second Friday. (See Mathematician's Friday.) The students have borrowed Lab coats from the Science department and included their own special dress up features. It has been amusing to see their perceptions of what a Mathematician looks like!
It was during one of these Friday sessions we decided to tackle the Domino Challenge. After some initial discussion about dominoes it was established the students had had varying levels of exposure to them. We spent some time laying the dominoes out and compiling a class list of their features. (See also Domino Trails.) One of the students suggested that a whole number can be represented on a domino. The observation enabled me to launch into the next step of demonstrating how an addition algorithm can be made using three dominoes. Working in pairs the students were asked to make a simple addition algorithm using three dominoes. All the students were able to successfully complete this component of the investigation.
The next step was to ask the students if they could find a combination that involves carrying. Some students found this difficult, however, once some pairs started to put together a combination, others were able to observe these solutions and construct a combination themselves (even if it was the same!).
The final part of the investigation involved the students arranging all twentyeight dominoes into nine sums which all show a correct addition. I suggested that for their first attempt at the investigation it might be easier to not attempt any carrying. Twenty minutes into the investigation and still no completed solutions  time for some hints! The total number of dots on the 28 dominoes is 168, so the total number of dots in the answer row must be 84. One strategy is to set up the answer row first and work backwards. It is also important to note that the total number of dots in a noncarry problem is always even.
In our given time frame six students were able to successfully arrange all twentyeight dominoes into nine sums. Many of the students found they were nearly there only to find the last dominoes remaining didn't fit together to make a correct addition, so back to the drawing board!
This investigation is ideal for a variety of age groups as it encourages the application of knowledge about factors, primes and multiplication by 0 and 1. The answer row, and the recognition that each domino can only be used once in a particular multiplication, is the key to the solution.
