In order to work towards a solution for a problem, students must recognise key language items (e.g. terms or phrases) that can be translated into mathematical symbols. Following this learning, they then need to perform this translation.
Students will do this in two situations:
- in the context of the topic they are currently learning (i.e. where the type of symbols to be used is predictable and thus there is scaffolding provided)
- in novel situations where they must decide on which prior knowledge to use (i.e. without scaffolding) (HITS 6: Multiple Exposures).
Teacher-led discussion of translating words to symbols will teach students how to read a worded problem and identify which words can be translated (O'Halloran, 2000). It is important for the teacher to explicitly share their thinking and decision-making as they identify what information is crucial for the mathematical translation and how this translation is done.
The strategy below could be used for any problem where students need to translate words into symbols.
Understanding the strategy
The strategy involves four steps:
- Teacher presents a problem and either reads the problem or asks students to read the problem.
- Teacher guides students through discussion and questioning to focus student attention on key phrases that enable translation to symbols.
- Students attempt to translate the key phrases to symbols and share their reasoning in pairs.
- Teacher models translation of key phrases to symbols. Alternative approaches for translating to symbols (e.g. by choice of different letters, etc) can then be discussed as a class to promote students' fluency.
Example of translating words to symbols
The example below is from a Year 7 lesson on algebra (VCMNA251).
Mary and David were comparing their ages and found that Mary is twice as old as David and the sum of their ages is 57. Find the age of each person.
Teacher models thinking aloud
Teacher models thinking to begin solving the problem, saying aloud to the students:
- What do these terms mean? The term 'sum' means 'addition', so the phrase 'sum of their ages' means the result of adding the two ages.
- What do these key phrases tell us?
“Mary is twice as old as David” suggests that one person’s age can be written in terms of the other person’s age.
“The sum of their ages is 57” suggests that an equation can be written involving both ages and the total of 57.
- What can we use to solve the problem – algebra.
In pairs, students are given the opportunity to translate into symbols using algebra. Students could be scaffolded by asking them to give a statement which represents the age of one person using a pronumeral.
Student discussion and working
The teacher provides an opportunity for students to discuss their reasoning.
To support students in discussing their reasoning it can be useful to ask students how they represented the problem using algebra.
Students could be scaffolded by asking them to give a statement which represents the age of one person using a pronumeral.
Some students may write 'let m=Mary's age', but it is more precise to say 'let m = the number of years in Mary's age', which emphasises that the letter stands for a number.
It can be useful to use pronumerals which are not the first letter of the name or object involved in the problem, to remind students that the letter stands for a number rather than a person or object.
Students who have a 'letter as object misconception' may write an (incorrect) algebraic statement which simply uses a letter to abbreviate a name (MacGregor & Stacey, 1997); hence they might write 10 apples as 10a, which is not algebra, as the a does not represent a number.
Model the use of a pronumeral to represent the age of one person:
Let p = the number of years in David's age
Ask students to write down the age of the other person in terms of the same pronumeral:
Mary's age is twice David's age, so 2p
Note that the alternative, which is also correct is to denote Mary's age by a pronumeral:
Let m= the number of years in Mary's age
Then give David's age in terms of the same pronumeral:
David's age will be m/2
Discuss with the students that while the working out for both alternatives will look different, they are equally accurate and come to the same solution:
David is 19 years old and Mary is 38 years old