**Cultivating Diagram Drawing Skills for Math Word
Problem Solving Among 1st-Grade Elementary School
Students: Making the Link Between Concrete and
Abstract Representations**

*Yuri Uesaka*

**Abstract.**

Using diagrams is a very efficacious strategy for students in problem solving. However, students tend to experience difficulty in using it as it demands abstraction. Examining how to cultivate a solid foundation for diagram use from an early stage in real educational contexts is important. Thus, in this study, a new 45-minute instruction for 1st-grade elementary school students was collaboratively designed between teachers and a researcher. The way to solve math word problems with diagrams was taught, and making connections between semi-concrete representations (arithmetic learning blocks) and abstract representations (diagrams with circles to represent people/items) was facilitated through explanations and collaborative exercises (peer interaction). The instruction was provided in two classes, which were treated basically the same except in one group concrete objects (blocks) were actively used before drawing diagrams during collaborative problem solving. The studentsâ€™ performance was compared to that of a class without such instruction. The results demonstrated that both spontaneous diagram use and performance in the difficult type problems taught during the instruction were higher in the classes that received the instruction. The class performance in the extension problem was highest in the class that received instruction with the use of concrete objects in collaborative learning situations. The results suggest the importance of providing skills instruction to young learners, and of providing them with the experience of solving problems collaboratively with concrete and abstract representations, which lets students make the link between abstract and concrete representations deeply and internalize the skills more.

Thanks for the presentation. Very interesting.

I wonder what you think about the situation where the diagram consists of realistically rendered objects, as opposed to abstractions on the one hand, and actual physical objects on the other. I am thinking, of course, of Hyperproof diagrams, an example of which can be seen here https://www.gradegrinder.net/Products/lrds-index.html.

The obvious conjecture is that the more “realistic” the diagram, the more students would benefit. I wonder though, whether there is a smooth increase from abstract to concrete, or whether physical objects are genuinely different, and provide a discontinuous effect. Do you have thoughts on this?

Thanks again for the presentation.

Thank you very much for your thoughtful comment.

In my definition of diagrams, the Hyperproof diagrams and pictures in which people are depicted on paper (which are â€œrealistic diagramsâ€) are kinds of â€œconcreteâ€ diagrams because extra information that is not related to the inference required (e.g., face, hair and hand so on) is included. In contrast, such information was excluded from the black or circle diagrams in my study. When students draw diagrams by themselves, the dimension of abstraction is important as abstract diagrams are easier for students to draw and easier to mentally manipulate.Â (However, understanding and using abstract diagrams require some preparation of students – particularly students in the early stages of learning. For example, it is necessary for them to understand what is represented and how to use such representations.)

I do not mean that more concrete diagrams are not useful or effective. In some cases, realistic diagrams work in the same way as more abstract diagrams (e.g., picture of a length of road works just as effectively as line diagrams in solving some math word problems). Semi-abstract diagrams can also bridge the transformation (and necessary perceptual apprehension) between concrete and abstract. When we consider studentsâ€™ spontaneous use of diagrams, the cost to construct effective diagrams is important (Uesaka & Manalo, 2011).

Uesaka, Y., & Manalo, E. (2011) Task-related Factors that Influence the Spontaneous Use of Diagrams in Math Word Problems. Applied Cognitive Psychology.

And yes, I agree with your idea that the dimension of the abstractness is on a continuum!

This is a very interesting and useful study – the findings have very important implications for early childhood education practices:)