At the ATM London Branch conference on Saturday, Kate Gladstone-Smith from Langdon Park School in East London, presented her research into the nature of communication she had observed in maths classrooms and how this differed according to the set, the students were in. (Anyone not from the UK will need to know that in English schools teachers decide in advance how well students will do with a subject and place them in ‘top’ and ‘bottom’ sets (i.e. class/teacher groups) accordingly).
Kate used a sociological analysis known as Social Activity Method (SAM) devised by Paul Dowling of the Institute of Education, London. He suggests that a practice (in this case mathematics education) has discourse in one of four domains of action. If the content (e.g. solving an equation, constructing a proof) would be recognised as mathematical and the symbols and technical vocabulary recognised as mathematical (e.g. evaluate 3x+1=10), then this is ‘esoteric domain’ discourse. This is contrasted with ‘public domain’ discourse where the content and the symbols/vocabulary would not be recognised as mathematical (but nonetheless a discourse in maths education). Importantly, the task of the mathematics educator is to induct learners into the esoteric domain of mathematics.
Kate found that students only rarely maintain any discourse in the esoteric domain. The teacher would mostly restrict their discourse to metaphor (in solving an equation: “get rid of the x’s”) or make appeals to common sense knowledge (“What is a square? It’s like one of those ceiling tiles”). Perhaps unsurprisingly, lower achieving students had very little discourse in the esoteric domain, while higher achieving students had at least some. However, this was in response to the restricted discourse of their teachers, not necessarily to what they could achieve.
So, I billed this blog as being about HP Prime Wireless. Well, later in the day, I had my first opportunity to use the system in a classroom setting. The class was a group of teachers and maths educators and I gave them an activity to explore conics starting with the form x^2 + y^2 = 9 in the advanced graphing App. I could observe the class’s conics by monitoring their screens in the Connectivity Kit monitor. When I saw an interesting example (a larger circle) I would double click on the screen and show it to the class. I asked; “How did you make the circle bigger”, to which their were two response to the two times this happened; “I changed the constant” and “I zoomed out”. This immediately sets up a rich discussion about the relationship between graph and function and the scaling of the graph. I then said; “Has anyone found a non-closed curve?”, which led to a new burst of activity. When I saw one I could ask; “How did you do that?”. Here, the teacher discourse is generally just teacherly prompting. However, the student discourse is predominantly in the esoteric domain of mathematics. The HP Prime only gives access to esoteric domain mathematics (the graphs and functions in symbolic form) positioning students to make esoteric domain responses.
The second activity was a new way of doing a classic. I sent out a poll asking for shoe size and handspan data. My class entered the data on their handhelds and pressed send. Within seconds a whole class worth of data was available for analysis. In the poll results screen in the connectivity kit the points are plotted and a line drawn, showing an overview of a possible relationship. However, selecting the HP Prime emulator and sending the data to it, generates a new APP on the emulator with full two variable statistics facilities. So, we can see a relationship. We can see the correlation coefficient to see it is a weak relationship. Then we validate the relationship by seeing if my hands and feet fitted the model. I was a poor fit, so we could discuss why my hands/feet relationship was different from the group (they were all women, which suggested a new hypothesis to test). Issues of experimental design were discussed. Within a few minutes of setting up an experiment we were having a well framed and well informed discussion, entirely within the esoteric domain of statistics.
This was unexpected. Kate’s research suggests it is very difficult for teachers to sustain discourse in the esoteric domain they aim to induct their students into. Harder still for students to work in that domain. Yet by putting students into a setting where they work with technology that only communicates in this domain and by keeping the discourse framed by the technology based activity, the vast majority of discourse is generated in the esoteric domain. See my previous post for a description of the software and how to set up the polls and the monitor. Suffice it to say it is not difficult to set up. Inevitably there are a few teething troubles (notably my 13″ LapTop screen is just not big enough to see the screen of enough connected Primes). Also, it is amusing to see how classroom management techniques are still needed. Calling the class to order and announcing the arrival of a message with instructions is still necessary (even with teachers!). But, teachers using a HP Prime wireless kit could use it in almost any lesson, so they will quickly become fluent in the changed classroom environment.
Please get in contact to share your thoughts or if you would like to see the system in operation. (email@example.com). Click here to link to the HP Prime pages at hp.com to see pricing etc.