Each month in 2020, Cambridge Mathematics published a blog that reported on the CM Define It app – a survey research tool investigating perceptions of existing mathematical definitions amongst those in the maths education community. Some of you might have signed up to CM Define It and offered your views on the weekly definitions presented, for which we thank you. The next step will be for us to look at the data collected through the app to see how it can inform the glossary embedded in the Cambridge Mathematics Framework. In this blog, I reflect on last year and on the reasons for the development of the glossary app.
So why exactly did we even consider the idea of investigating mathematical terminology and definitions? When speaking to the writing team, they raised several important issues and ideas. Although the team had been reading literature in the areas of mathematics education, mathematics, psychology and the learning sciences for over five years, they found that there was no single, holistic and accessible source of them that would support all individuals who needed to understand and use mathematical definitions. Their research also highlighted that different countries and cultures have different terms for mathematical concepts. One good example of this is how different cultures define a trapezium. As my colleague Rachael wrote in one of her blogs, “In the US … a trapezium is a four-sided polygon with exactly one pair of parallel sides … whereas in Canada a trapezoid has an inclusive definition in that it’s a four-sided polygon with at least one pair of parallel sides” (Horsman, 2017).
Figure 1: Where in the world is a trapezoid a trapezium? blog
There are many other reasons why we believed it was important to investigate the perceptions of mathematical definitions. One of these was how important language is in mathematics. Riccomini et al. (2015) state that language and communication are vital in learning, understanding and applying mathematics. They also write that in order to communicate using mathematical language, several elements must be present; these include sound mathematical vocabulary, numerical fluency and comprehension skills. Without the necessary mathematical language and vocabulary skills to access mathematics, students can be faced with barriers preventing their understanding of the subject. Rubenstein and Thompson (2002) (as cited in Riccomini et al., 2015, p. 238) identify 11 categories of potential difficulties that exist for students learning mathematical vocabulary. These include:
- meanings of mathematical terms being context-dependent (e.g. the term “foot” meaning 12 inches or the bottom bit of the bed)
- meanings having more precise implications (e.g. “product” being the solution to a problem in multiplication or something that a company produces)
- terms holding meanings that are very specific to mathematics (e.g. polygon, parallelogram, imaginary number)
- terms having multiple meanings (e.g. side of a triangle vs. side of a cube)
- technical meanings specific to various disciplines (e.g. “cone” suggesting the shape or food)
- everyday homonyms (e.g. pi vs pie)
- words that are related but different (e.g. circumference and perimeter)
- challenges with translations (e.g. mesa vs table)
- irregularities in the spelling of words (e.g. obeluses vs obeli)
- different ways of expressing concepts (e.g. 15 minutes past or quarter past)
- using informal terms rather than mathematical terms (e.g. diamond vs rhombus)
In a world where mathematical language competency is important, learners are faced with multiple difficulties in accessing and using mathematical language. These challenges are even greater when learners are presented with many different definitions of the same word – a situation demonstrated by the CM Define It app or simply by conducting your own search. How do learners decide which definition is stronger or more appropriate for them?
Teachers may help students with understanding mathematical definitions, but they also need to select definitions that are appropriate for their students. Ball et al. (2008) suggest that “choosing and developing useable definitions” is a challenge that is very specific to teachers (as cited in Mosvold & Fauskanger, 2013, p. 49). They further propose that the activity of selecting and developing appropriate definitions goes beyond simply reciting mathematical definitions and touches on the ability to understand many different, complex definitions and know which ones are most useful and accurate. Mosvold and Fauskanger (2013) found that most of the 15 interviewed teachers believed that knowing definitions is an important part of teachers’ knowledge. However, teachers themselves could face difficulties when searching for and choosing mathematical definitions; for instance, knowing which sources of definitions will be best suited to their students or recognising which definitions are misleading.
Let’s use a practical example. As you are reading this, think about what you need to work out this instruction:
Construct a rhombus with side lengths of 5 cm.
Firstly, you need to be able to read and comprehend English. Secondly, you need to know what “construct” actually means. My colleague Rachael says that “construct” doesn’t just mean to draw – it is asking you to use geometrical equipment (such as a ruler and a compass in this instance) to “accurately construct the shape,” not to draw it freehand or just with a ruler. You also need to know what a rhombus is (a quadrilateral with equal length sides) and what 5 cm means. So, what happens if you have mostly worked with other metrics? In a nutshell, there is a lot more to this instruction than meets the eye.
The barriers to accessing and using appropriate mathematical definitions can impact both learners and teachers. This was another factor that encouraged the Cambridge Maths team to explore perceptions of mathematical definitions. We wanted to find out if certain users (such as teachers, resource developers, researchers) preferred specific sources of definitions and if they preferred some definitions more than others. This spurred us to develop the app in a way that allowed the users to choose the groups of learners they work with, from novice, intermediate and advanced. As different users often work with different learners (primary teachers may work with more novice learners, whereas lecturers might work with more advanced learners), we thought it was important to capture this information in the survey app.
Figure 2: Categories of learners in the CM Define It app
Hopefully, this reflective piece answers the question of why the CM Define It app came to life. I hope that it also demonstrates how important language and mathematical vocabulary are in learning, teaching and understanding mathematics.
So, when you are presented with a key word and its definition, think about how you decide if the definition is appropriate for your needs. Is it accessible to you and/or to those who will be taught it? Perhaps consider the difficulties that it may present and think about how it could be made simpler for the learners who will use it. Is it accessible to those for whom English is an additional language? Does it consider how other cultures view or talk about the key word?
Let’s all consider how important language is to the field of mathematics.
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