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Math You Can See
April 13, 2017 | Volume 12 | Issue 15
Table of Contents
With Math, Seeing Is Understanding
Helping children visualize math is critical to their success in the subject. I recently observed a 5th grade class starting a lesson on area and perimeter. I turned to a girl who was in my class four years earlier and reminded her that she knew the topic. "Yes I do!" she said excitedly. "The perimeter is where you sit along the outside of the rug in morning meeting, and area is the inside of the rug, where the squares are. That's from 1st grade," she said confidently, circling her fingers in the air to represent her thinking.
Visual cues, like this one I use with my six- and seven-year-old students, stick and show that envisioning math helps children learn in lasting ways. We teachers can do more to give students internal ways to see the structure of mathematics—to understand types of units and what it means to move between them, and to pull apart and combine numbers. But math instruction is changing.
At my school, in the early grades, we encourage children to use their fingers, something that feels so natural to them, to better understand numbers and the numbering system. We might talk about how a "high five" involves using a whole hand, which is really a unit made up of five fingers; while a thumbs-up involves just one segment of that five-part unit. We then go on to using things like beads on a string and, later, place-value disks, which are like poker chips, to help children see and work with numbers, units, and place value.
We then move from concrete items to encouraging students to use and draw math models to solve problems, sometimes referred to as a pictorial approach. The tape diagram, for example, is one visual model that relies on rectangles to represent parts of a ratio. It helps our students picture the information in a problem, seeing how the parts and the total relate, and then build a simple pathway to a solution.
Research shows that working back and forth among these visual, relational models and the more abstract lexicon of math symbols helps students make sense of concepts (Pashler et al., 2007). In the younger grades, "high fives" are quickly replaced in class with "high sevens" as seen in the photo.
Figure 1. High Sevens
Students see 7 as a unit of 5 plus 2 more (5 + 2 = 7), as well as 3 less than the larger unit of 10 (7 + 3 = 10, or 10 - 3 = 7). My role as their teacher is to help integrate the abstract representations of these number sentences with their concrete representations, which can be the key to improved learning (Paschler et al., 2007).
Currently, my four-year-old niece has been obsessed with giving me "high fifteens," consisting of slapping "high tens" and then an extra "high five." When I ask her about it, she says it's made of a 10 and a 5. Even her language points to her building of units: a ten and a five. As a dedicated aunt, I wonder if her teachers will help her connect the concrete understanding she already has with the abstract notations she will see in her math books.
This same process of connecting symbols with meaning has happened for my students when using the tape diagram. Inevitably, every year I work with some students who read a word problem like this one and say, "I don't know what to do."
In a class of 24 students, 5/6 are boys. How many boys are in the class?
Typically, they mean they don't know which operation to use first. When a student draws his tape diagram to visualize the relationship between the information in the problem, he will then say something like, "Oh! So I need to divide 24 into these 6 groups and then put 5 of those groups back together." Once they understand this concept, the student will proceed to use equations, math symbols, to solve the problem. Many students need the visual representation as the bridge of understanding between the word problem and the symbolic notation they used to solve the problem.
Figure 2. Solving with a Tape Diagram
Of course, we also teach children how to solve problems using standard algorithms. But even then, it always helps to be able to envision the problem and not just solve it through memorization. Students can go back and forth between visual representation and the algorithm once they know both. The latter becomes a quick shortcut for a process that now has clear meaning for them.
As teachers, it's natural to teach math the way we learned it or emphasize the method we use ourselves. But we have to think about how children learn best. If we do that, I'm confident we'll continue to move toward this more visual approach to teaching math, and I'm certain it will lead to student success.
Pashler, H., Bain, P., Bottge, B. A., Graesser, A., Koedinger, K. McDaniel, M., & Metcalfe, J. (2007). Organizing instruction and study to improve student learning. IES practice guide. NCER 2007–2004.
Marianne Strayton was a classroom school teacher for 19 years. She now works with students in grades K–5 who need further support in math at Woodglen Elementary School, part of Clarkstown Central School District, in New City, New York. She also served as a writer for Eureka Math/EngageNY.
ASCD Express, Vol. 12, No. 15. Copyright 2017 by ASCD. All rights reserved. Visit www.ascd.org/ascdexpress.
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