3 Developing mathematical language
Becoming mathematically resilient requires students to notice what they are struggling with and to express their questions, thoughts and ideas. To be able to do so, students need to have the opportunity to learn to read, interpret and apply mathematical vocabulary. As with learning any language, this involves identifying words and expressions, using them in different contexts and phrases, and giving meaning to the words and expressions. To learn a language effectively you need to regularly hear it, see it, read it, write it and get practice at speaking it.
Activity 1 explores how to deal with mathematical vocabulary. It requires students to devise their own mathematical dictionary, identify mathematical words and statements that need clarification, write their own explanations and sound these out with another student. In doing this they will also learn how to make sense of mathematical language for themselves, which can also help them get ‘unstuck’ when learning mathematics.
The mathematical context of all the activities in this unit is triangles, and in particular, the similarity and congruence of triangles. However, the approaches taken in the activities can be applied for all topics the students study.
In Part 3 of Activity 1, students are also asked to reflect on their learning in Parts 1 and 2. This is repeated in most of the activities in this unit. The purpose of this is for students to become more aware of what works for them when learning, and become more active in their learning as a result.
Before attempting to use the activities in this unit with your students, it would be a good idea to complete all (or at least part) of the activities yourself. It would be even better if you could try them out with a colleague, as that will help you when you reflect on the experience. Trying the activities yourself will mean that you get insights into learners’ experiences that can in turn influence your teaching and your experiences as a teacher. When you are ready, use the activities with your students. After the lesson, think about the way that the activity went and the learning that happened. This will help you to develop a more learner-focused teaching environment.
Activity 1: Making your own mathematical dictionary
This activity requires students to exchange their ideas with other students – they could be working in pairs or small groups to prompt more ideas and responses. Think about timing for the activity so that it does not overrun and all parts are addressed; for example, say, ‘In the next ten minutes I want you to find as many …’
You can find some examples of entries in a student’s dictionary in. Resource 2.
Part 1: Making the dictionary
Tell your students the following:
Look at the chapters in your textbook about triangles.
- Write down a list of geometric words that are being used.
- On your list, are there any words that are also used in everyday life? What is the same and what is different between the geometric meaning and the everyday meaning?
- Write down your own explanation of the geometric words. It might help to add a sketch or drawing as well.
Part 2: Giving meaning to mathematical statements
Tell your students the following:
Again, in groups or pairs, look at the chapters in your textbook about triangles.
- Write down any mathematical statements you come across that students might find difficult to understand. For example: ‘two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle (SAS congruence rule)’.
- Identify the words that do not make sense for you in the context of the statement. From the example above, this could be the word ‘included’, which you might understand in everyday language but might find hard to explain in the mathematical statement.
- Discuss and write down what the meaning of these words could be. Try and be as precise as you can.
For this part of the activity it might be helpful to have some examples ready rather than getting the students to find them. You can take them from the text that they are looking through. This will save time as they do not need to search for examples.
Part 3: Reflecting on your learning
Tell your students the following:
This part of the activity wants you to think about your learning so you can become better at, and feel more comfortable, learning mathematics.
- What did you find easy or difficult about Part 1 of this activity?
- What did you like about this activity?
- What mathematics did you learn from this activity?
- What did you learn about how you (could) learn mathematics?
You may also want to have a look at the key resources ‘Using local resources [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ’ and ‘Using groupwork’.
Case Study 1: Mr Agarwal reflects on using Activity 1
This is the account of a teacher who tried Activity 1 with his secondary students. He teaches in an English-medium school.
The students were quite happy starting on this activity; they opened their books and looked for words they felt were new to them or typical to geometry.
As I walked around the classroom, I noticed they found out from each other or from reading the explanations in the textbook the meanings of some of the words, and leaving blanks for the ones they did not know. There was actually much noise from students asking each other what the words might mean. Perhaps next time I would say they are only allowed to ask the students sitting next to them, not in front or behind. On the other hand, the noise was mathematical talk and they did find out more by being able to ask more people.
After a while, I reminded them they also had to do the other questions in the activity, which were to attempt to write it down in their own words, make a sketch and then think and write down whether they had seen that word in any other context than mathematics, and what it would mean there. The students found the last question about meaning hard: their use of the English language is not extensive and I encouraged them to look for the meaning in the dictionary so that they could see how closely related, or not, the everyday word was to the mathematical word. For some of the students I needed to translate the words into Hindi.
It was an extremely productive exercise for both the students and me: the students had an opportunity to really work on the language aspect of mathematics, in writing, reading and verbalising it. For me, it made me realise to what extent the language used in mathematics is alien to the students and a barrier to their learning of mathematical ideas. I really want to spend more time and attention on the language learning of mathematics in the future – if the words have no meaning for them, then how can they learn about them? For example, I have started compiling a mathematical English–Hindi dictionary that we have available in the classroom.
Reflecting on your teaching practice
When you do such an activity with your class, reflect afterwards on what went well and what went less well. Consider the questions that led to the students being interested and being able to get on, and those you needed to clarify. Such reflection always helps with finding a ‘script’ that helps you engage the students to find mathematics interesting and enjoyable. If they do not understand and cannot do something, they are less likely to become involved. Use this reflective exercise every time you undertake the activities, noting, as Mrs Agarwal did, some quite small things that made a difference.
 Pause for thought Good questions to trigger such reflection are:
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2 Teaching for mathematical resilience