Talking Maths in the Esoteric Domain: HP Prime Wireless

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. ( Click here to link to the HP Prime pages at to see pricing etc.

EAL in Maths? Problem Solved!

Where we were working in South East London, a number of students would arrive in England for the first time in the middle of secondary school. They would have very little English language and would try to get into local secondary schools. The schools would turn them away because they assumed that these students would end up with poor grades and compromise their exam statistics. So, a unit was set up to support these students make the transition to school. I got together with Gwyn Jones to produce a course designed to teach the mathematics content of GCSE with the minimum of language, but developing the key technical vocabulary of maths and of school while they learnt. The materials were supported by online interactives to see the maths dynamically and practice the ideas in an open format. There was a very low language pre-test, so that the student could show what they already knew, a tracker sheet to choose the maths they now needed to work on, a large collection of activity sheets to develop the maths and a post test with the same language demands of a normal maths test to show the schools how good they were.

In the very first group of students to use the first version of materials there was a student who had just arrived from East Africa. He had been rejected by every school in the borough. He took the pre-test and got 100%. He worked on the advanced materials and did the same on the post test. He took his work as a portfolio back to the schools and immediately found a place. Within 18 months he had an A* in GCSE maths.

We are proud to announce that we have now redesigned and updated this course and made it available to schools. Called Access to Mathematics it comes as one of our course boxes (like our well known gifted and talented courses; Wondermaths and Illuminate). There is a comprehensive teacher guide with notes on running the course. Ten copies of the comprehensive student book (120 pages) and access to the online interactives, test, answers, etc. in the Access to Mathematics web site. Priced at £195 this gives access to mathematics for all of your students for whom English is an Additional Language from those who have just arrived with no English to those who appear to have conversational English, but cannot access or succeed at maths in lessons.

Everything is described diagrammatically, putting the maths into a visual structure. Two colours are used to emphasise the structure and the maths is practised through this structure, gradually peeling it away to leave the formal symbolic maths. The course worked well supervised by non-specialist teachers as it is designed largely for self-teaching. However, with access to a specialist teacher, the materials could be used for a whole range of learners where reading and language demands of any sort are an issue.

Once you have the box, further copies of the students books are available in packs of 10 priced at £45. So, you can use them as a standard class text if you want. The overall content is covers about 90% of a higher level GCSE.

We are very proud of this publication. We have so often seen excellent mathematicians languishing in low achieving sets simply because they are still learning English and find accessing conventional books difficult. Now, they can quietly and quickly show everyone how much they know and can do, while learning the essential school language that they need.