## Introduction

The ratio or proportional reasoning is one of the key concepts presented to students during mathematics classes. However, it is not easy to transform mathematical definitions and concepts into a task that will be perceived equally by all students. To understand what problems might arise during this class, an assessment was prepared and administered to an elementary student.

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## Assessment

The assessment was based on a common task about piano keys:

The point of this task is to teach the student to understand the ratio concept and use proportional reasoning to find the right solution. The student is also expected to use tables to relate quantities and compare ratios and understand the ratio a:b (where b does not equal 0). The student’s work can be presented with symbols or verbally; it is also possible for the student to engage both of these methods in understanding the presented concept.

However, the student’s response has shown that he did not understand the concept of ratio but rather mixed it with a difference. The answer to the first question was 33, and the answer to the second question was 238. Thus, the student decided to use an additive approach, while he should have focused on the multiplicative approach. Moreover, the student decided not to use any tables to find the ratio between the black and white keys, while it was the main part of the task. The part to the whole ratio was entirely omitted by the student.

## Instructional Supports

A precise definition of the ratio is crucial for a student’s understanding of the concept. Although mathematicians who also work as schoolteachers might find these concepts not that hard to explain, they usually experience challenges while presenting the topic to students (Watson, Jones, & Pratt, 2013). The teacher can use blackboards or interactive whiteboards to explain that ratio is a “multiplicative comparison between two numbers”, the relation between one quantity to another (Small, 2015, p. 63). It is advisable to illustrate the definition, e.g. using a picture of six white circles and four black ones, indicating, that there are three light circles for two dark ones (Small, 2015).

As can be seen from the assessment, the student decided not to use diagrams. It may be useful to ask him to use tables, graphs, or manipulatives to understand the ratio of black keys to white keys. The task also should have drawn the student’s attention to the fact that it is not necessary to know the actual size or the quantities, but it is more important to be able and know how to compare those (Watson et al., 2013). If the student has a hard time understanding that difference and ratio are not the same, it is advisable to explain that difference can be expressed in the units presented in the task (e.g. keys), while ratio can also be demonstrated using their quotient or pairs of numbers (Watson et al., 2013). A common mistake that the students fail to notice during their work is the decision to use addition when scaling, and, therefore, failing to maintain the ratio.

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Additional supports can include the use of multiple strategies to explain the difference between rate and ratio and to show why the additive approach does not work in such tasks. If some students express doubts about ratios and their need to solve real-life problems, it is possible to use different examples, including measurement of car performance (liters/mile (kilometer) (Booker, Bond, Sparrow, & Swan, 2015). Double line and tape diagrams can be used to explain the concept. A prompt, excellent response should not be anticipated from the student as these mathematical concepts and methods demand abstract thinking and are a less common approach to the problem that students have learned earlier. Any discussion of the problem among the students should be encouraged because a verbal representation of a complicated task might help students understand what ratio and rate are, and how they are used.

Drawing and making graphs and tables should also be encouraged by the teacher, as visualization can be an extremely helpful tool in mathematics classes. The teacher might stress the importance of proportional reasoning, explaining that a $15 off the price and a 15% discount can provide different profit to the customer depending on the item’s price (Booker et al., 2015). Not all students are able to associate fractions with ratios and rates, while this approach is often crucial for a successful accomplishment of the task (Small, 2015). Nevertheless, students then tend to see fractions and ratios as equal units or as subsets of one another, which can also lead to serious mistakes during classes (Small, 2015). Therefore, a diverse representation of the concepts is obligatory to ensure that students perceive these concepts correctly or are able to understand them partially.

## References

Booker, G., Bond, D., Sparrow, L., & Swan, P. (2015). *Teaching primary mathematics.* Sydney: Pearson Higher Education.

Small, M. (2015). *Building proportional reasoning across grades and math strands, K-8*. New York, NY: Teachers College Press.

Watson, A., Jones, K., & Pratt, D. (2013). *Key ideas in teaching mathematics: Research-based guidance for ages 9-19*. Oxford: OUP Oxford.