The role of students' drawings in understanding the situation when solving an area word problem

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The role of students’ drawings in understanding the situation when solving an area word problem

Manuel Ponce de León Palacios, Jose Antonio Juárez López

To cite this version:

Manuel Ponce de León Palacios, Jose Antonio Juárez López. The role of students’ drawings in un- derstanding the situation when solving an area word problem. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands.

�hal-02435316�

The role of students' drawings in understanding the situation when solving an area word problem

Manuel Ponce de León Palacios¹ and Jose Antonio Juárez López²

1 Popular Autonomous University of Puebla State, México; [email protected]

2 Meritorious Autonomous University of Puebla, México; [email protected] Keywords: Graphic representations, drawings, visualisation, word problems, geometry.

Introduction and theoretical background

The use of drawings when working with geometry word problems will emerge naturally in students;

however, the strategic use of representations to solve the problem does not. The contribution of visualisation to mathematics education, and especially in geometry, is undeniable; however, it is worth noting that to make effective use of this tool in the solution of word problems, the student will need specific prior knowledge and certain cognitive skills (Schnotz, 2002). Mathematics relies heavily on visualisation because it deals with abstract objects (Arcavi, 2003), the possibility to "see"

the mathematical objects and express numerical information through graphical representations helps students understand concepts and solve problems (Edens & Potter, 2007).

Although the drawings with pictorial characteristics do not seem to be related to performance in mathematical modelling and the correct solution, they can serve as a preliminary step to more schematic representations (Rellensmann, Schukajlow, & Leopold, 2016). This sequential process from pictorial to schematic drawings contributes mainly to the understanding of the situation and the task, which has shown to be particularly useful for students who have more difficulty making the transition between the real world and the mathematical world.

Method

Research question: What is the role of the drawing that students do when they solve a problem of geometry in which they are asked to find the area of a figure that is not presented explicitly but results from the relationship of the elements?

Participants: 20 students of the 9th grade from a public school in Mexico (12 girls and 8 boys) with an average age of 14 years.

Procedure

For the research, a worksheet was developed including one area problem: A dog is tied to a chain that allows a maximum range of 2 metres, attached to a ring that moves in a bar in the shape of a right angle whose sides measure 2 metres and 4 metres. What is the area of the region that the dog can cover?

The worksheet was applied in a single session with all the participants without any previous intervention. This study only considered the analysis of the drawings made by the students on the worksheet to solve the problem.

In the analysis of the representations, two processes were carried out: a classification process and a qualitative analysis of the drawings. Three levels of classification of the drawings were made considering (1) level of abstraction, (2) relationship with the statement of the problem and (3) the explicit inclusion of information in the drawing that is mathematically relevant to solve the problem. The qualitative analysis consisted of a finer revision of the characteristics of the representations considering the students’ proposed figure; the information included; the proportionality of the presented elements; and the transitions between pictorial, schematic, operations and result domains.

Results

None of the participants was able to draw the expected figure or reach the correct result. Most of the students showed some difficulties, inadequately representing the proportions of the objects in the situation and paying excessive attention to mathematically irrelevant details, such as flowers, clouds and the dog's house.

The analysis of the drawings showed how the students carry out the transition between the real domain posed by the problem and the mathematical domain. Some students make a gradual transition, while others move directly from the representation of the situation to mathematical operations. Given the above, we can classify the transitions into three groups: (1) pictorial to pictorial with data, (2) pictorial to schematic, and (3) pictorial to numerical.

Conclusion

Although the use of drawings in solving geometry problems emerges naturally in students, effective use of graphic representations must be worked on through activities in the classroom. If the drawing activity is kept away from mathematics classes, it is very likely that the students may assign a purely decorative function to the drawings. The representations generated by the students can be a handy tool for observing their reasoning in such a way that allows us to understand better what they know and how they know it.

References

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, (52), 215–241. https://doi.org/ED419696

Edens, K., & Potter, E. (2007). The relationship of drawing and mathematical problem solving:

“Draw for math” tasks. Studies in Art Education, 48(3), 282–298.

https://doi.org/10.1080/00393541.2007.11650106

Rellensmann, J., Schukajlow, S., & Leopold, C. (2016). Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students’ mathematical modelling performance. Educational Studies in Mathematics, 95(1), 53–78. https://doi.org/10.1007/s10649- 016-9736-1

Schnotz, W. (2002). Commentary: Towards an integrated view of learning from text and visual

displays. Educational Psychology Review, 14(1), 101–120.

https://doi.org/10.1023/A:1013136727916