# Mathomat V5 Design Rationale

**Design rationale for the new Mathomat V5 template**

**By John Lawton, publisher at OLM. March 12, 2021**

In Mathomat V5 we are trying to "start where the teachers and their students are at" in terms of their concept image for angle measurement. We do this by providing a template with a large, simple, 180 degree protractor at its centre. The Mathomat V5 then presents students with 360 degree protractors to work with for those measurement tasks that need them.

OLM changed the protractor in Mathomat from a half circle to a full circle design 22 years ago, in 1999. Since the protractor in Mathomat is at the centre of its design as a geometry tool, this was an important change.

The new Mathomat V5 template with its 180 degree protractor is a reversion back to the half circle design that we used to have.

The reason for this change is that 180 degree protractors remain very popular as a teaching tool, leading me to think that the argument that a 360 degree protractor is a better learning tool for angle is not as straightforward as we been thinking it is.

**The argument for 360 degree protractors, and why I have become sceptical about it.**

There is plenty of evidence to support the use of 360 degree protractors in geometry teaching. For a start, any senior school maths teacher will tell you that the full circle protractor is an essential tool for understanding trigonometry. Using a full circle protractor in Mathomat means we can give trigonometry students a tool for visualising the unit circle as they learn. In earlier school years the full circle protractor offers a simple and direct way for students to measure and draw angles that are greater than 180 degrees (reflex angles).

Further support for the use of 360 degree protractors comes from education research studies. When these studies have commented on protractor design they invariably recommend full circle versions in preference to half circle types. This includes the following comments by education researchers;

- In 2000, in their foundational work on how students learn about angle, Mike Mitchelmore and Paul White (Mitchelmore & White, p.234) made the argument that a 360 degree protractor is superior to the half circle kind when tackling the difficult topic of angle measurement with students. This was argued to be because a full circle protractor design helps students more effectively in connecting up the dynamic action of turning with the notion of angles as having two (static) lines.
- Mitchelmore and White were drawing, in their argument above, from a still widely referenced foundation study in 1982 on angle by Gillian Close. As a result of her research Close (p.198) recommended that primary teachers replace half circle protractors with the full circle kind in their teaching of angle measurement.
- Helen Crompton, in her 2013 PhD research on student’s learning of angle and its measurement, was generally critical of protractors as a tool. However, Crompton (p.31) also made the point that full circle protractors, especially ones with rotating arms, were more effective as angle measurement tools than half circle protractors.

So, with the Mathomat template squarely based on a full circle protractor design (so to speak) since 1999, and with all of this intuitive and research based commentary backing that change, what’s the problem?

Well, I have always been somewhat uneasy about this situation. These concerns have grown over time, leading finally to the development of the new V5 Mathomat design.

In 2011, I decided to test the protractor design situation by asking teachers in a survey what type of protractor they would like to see in a new teaching aid we were developing (the R300T pencil case version of Mathomat). At the 2011 Mathematics Association of Victoria (MAV) annual conference we had survey responses from 33 teachers, of whom 30 were at secondary schools. Of these responses, 18 (55%) said they would prefer to have a 180 degree protractor in our new tool. Only 11, or 33% said they would prefer to have a full circle protractor with the remainder having no preference.

I further studied this situation anecdotally, through discussions with mathematics teachers. This has revealed some interesting comments about protractor design:

- One teacher I spoke to about the time of the 2011 survey said that she preferred to teach angle with a 180 degree protractor because her students thought it “looked like a protractor”. This indicates to me that this teacher believed that her students had a mental picture (
*concept image*) of a half circle protractor when being asked to use protractors as a tool. - A maths tutor I spoke to recently told me that she preferred a 360 degree protractor when teaching trigonometry at secondary school level, but that she would
*only*use a 180 degree protractor to introduce angle as a concept at grade 5. In other words this tutor believed that full circle protractors were actually inappropriate as a tool for introducing angle at primary school.

In my commercial experience, outside of Mathomat publishing, half circle protractors are overwhelmingly popular in relation to full circle versions. For instance, as Australian distributors for a number of international brands of educational stationery over the years, I have been involved in the sale of literally millions of protractors that have been used by geometry students. Amongst these products the half circle protractor is a juggernaut, accounting for more than 98% of those protractors sold. As manufacturers of drawing instruments used by government departments and professionals for technical use, the W&G instrument company made many different types of protractor, and the great majority were 180 degree versions.

In addition to being a commercial publisher, I am also a PhD research student, with a focus on the concept of angle in the teaching of geometry. Speaking as a researcher, there is a great deal more that I would like to understand about this situation. Especially, that the commercial survey of mathematics teachers that we did (aside from its small sample size) did not explain *why *a majority of secondary teachers preferred half circle protractors.

Despite the fact that we do not know the reasons for teacher’s choices, as publishers of Mathomat we need to make decisions about its design. I do not believe that we should continue to ignore the strongly stated preference by teachers for half circle protractors in our geometry template range, and as a result have developed the Mathomat V5 template. This product is based on the following conjectured case for preferring to teach with a half circle protractor over a full circle type:

** The case for teaching with a half circle protractor**

From the comments above, two arguments emerge that support the idea of teaching with a half circle protractor in preference to a full circle type.

**Argument 1: Half circle protractors are more practical**

There is a delightful comment by Amy Hackenberg-Hastings in her 2005 history of the classroom use of protractors, where she says: “Aspects of this story may assist those teachers who are trying to communicate mathematical ideas to students who may not have a background in abstract reasoning but who do have protractors wedged into their backpacks” (p.217). This captures the classroom context for protractor use – they are limited use physical tools that exist in a strictly limited space (a student’s backpack). School students obviously have some say over what goes into their backpacks, but so too do their parent/guardians and the subject coordinators at their school who specify what will be required for use in classrooms. It is possible to imagine the adults involved in this decision making process concluding that a half circle 100mm plastic protractor is the only acceptable form of physical angle measuring instrument that a middle or primary school student can realistically be expected to carry in their backpack. Research studies may find that full circle protractors are preferable in terms of learning outcomes, but if the purchasing process has decided that “it's a half circle protractor or nothing at all” then in my experience that is what mathematics teachers will decide to work with.

**Argument 2: Half circle protractors look like protractors**

There is an old saying that “quantity has a quality all of its own”. It may be that the sheer weight of numbers in favour of the half circle protractor design over full circle versions has created an environment in schools in which the word “protractor” invokes an image in student and teacher minds of a half circle protractor. I believe this is what the teacher I quoted above meant when she said that half circle protractors have the teaching advantage that “they look like a protractor”. I believe also that the maths tutor I quoted earlier may have only been prepared to use a half circle protractor when introducing angles because of the strength of this shared concept image for it with the many students that she tutors.

Concept image is something that needs to be taken very seriously in geometry education. This was first recognised by David Tall and Shlomo Vinner in 1981 when they argued that students actually learn geometry by developing a concept image for it. All of the abstract theoretical ideas (definitions) that we usually associate with geometry and mathematics are important, but they are not central to the learning process. This idea has continued to gain acceptance over the decades. In 1993 Efraim Fischbein recognised that when students were developing their concept imagery in geometry lessons they entered an entirely different realm of space, which he called a *figural concept *and which Marie Jean Perrin Glorian and colleagues (2018) in France refer to as *Geometric Space. *There has been momentum develop by educators in Canada (Alain Kuzniak in 2006) and in France (Joris Mithalal and Nicolas Balacheff in 2018 and Marie Jean Perin-Glorian and colleagues in 2018) to accept this idea of graphic space as representing an entirely new paradigm in geometry education. A *paradigm *(or fortress of ideas) is a big deal, and changing to a new one amounts to a revolution in how geometry is learned.

**Conclusion; half circle protractors and shared concept image**

The ongoing commitment to the use of half circle protractors by teachers that I have seen might be recognition of a dominant role played by the concept image of a half circle protractor in certain classroom situations. If this is the case then, as publishers, we need to recognise the power of that imagery and give teachers a tool that they can use to work with it. That is the design concept behind the Mathomat V5 template.

In Mathomat V5 we are trying to "start where the teachers and their students are at" in terms of their concept image for angle measurement. We then offer teachers a tool which allows them to further develop their students concept image through use of 360 degree protractors that are available within the same tool.

Our student learning materials that we plan to publish for the Mathomat V5 template will address that transition, including the very important process identified by Mitchelmore and White, referred to earlier, of connecting dynamic and static concepts of angle through use of a 360 degree protractor.

**References**

Ackerberg-Hastings, A. (2005). Protractors in the classroom: An historical perspective. In A. Shell-Gellasch & D. Jardine (Eds.), *From calculus to computers: Using the last 200 years of mathematics history in the classroom*. USA: The Mathematical Association of America (incorporated).

Bingobali, E., & Monaghan, J. (2008). Concept image revisited. *Educational studies in mathematics, 68*, 19-35.

Close, G. (1982). *Children’s understanding of angle at the primary/secondary transfer stage *Retrieved from London:

Crompton, H. (2013). *Coming to understand angle and angle measure: A design-based research curriculum study using context-aware ubiquitous learning.* (Doctor of philosophy). University of North Carolina at Chapel Hill, Chapel Hill.

Fischbein, E. (1993). The theory of figural concepts. *Educational studies in mathematics, 24*, 139-162.

Kuzniak, A. (2006). Paradigms and geometric workspaces. *Canadian journal of science mathematics and technology education., 6*, 167-187.

Mitchelmore, M., & White, P. (2000). Development of angle concepts by progressive abstraction and generalisation. *Educational studies in mathematics*(41), 209-238.

Mithalal, J., & Balachef, N. (2018). The instrumental deconstruction as a link between drawing and geometrical figure. *Educational studies in mathematics*(November), 1-16.

Perrin-Glorian, M.-J., & Godin, M. (2018). Plane geometry: For a coherent approach from the beginning of school to the end of college. from HAL

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational studies in mathematics*(12), 151-169.