"Eyes on Progressive Education" Blog
A Green Acres School journal that chronicles progressive education in action, the research that supports and informs our practice, and ways in which we live out our mission each and every day.
Truly Understanding How Numbers Interact
Peter Klam, Middle School Head
One of the things about my job that I most enjoy is gaining a deeper understanding of what teachers of other disciplines do. I spent 19 years teaching Language Arts, and I can have cogent discussions about semi-colons or design a rubric for shoelace tying if necessary. But I never knew much about how to structure a science class around an experiment or how to develop a simulation about the Cold War.
Imagine my delight when I realized that when multiplying by fractions, the result is a smaller number using the same logic that results in multiplying by one giving the same number and multiplying by higher numbers yielding ever higher numbers. So 7x1 is the same as thinking about “seven, one time” and 7x3 is the same as “seven, three times.” With fractions, 7 x ½ is the same as thinking about “seven, a half time—that is, a half of one time.” When I read it, it sounds confusing. But when I picture it in my head it makes total sense, in a way that it never did when I was 12 or 13.
When I was in middle school, I was content to just learn the algorithms of the processes without thinking about the deeper meaning behind them. For me, 7 x ½ really just meant “seven divided by 2” because that always yielded the right number. Why did it yield the right number? I didn’t know; and it didn’t matter because my teachers didn’t care whether I knew or not—they were content with getting the right answers from me consistently.
But that isn’t really math. Math isn’t about learning algorithms and applying them, it’s about understanding how numbers interact together. It is a logical system capable of modeling our world because for the most part, numbers interact via algorithms in predictable ways that help us to understand how different elements of the world around us (assigned numerical values based on set criteria such as measurements or counting) interact with one another.
That’s what I love about the Green Acres approach to math. It isn’t necessarily about accelerating kids forward to the point where they are on the fringe of their intellectual capability. Often, this is done by teaching the algorithms as quickly and efficiently as possible and expecting the student to come to a more coherent understanding on her own. Rather, it is about challenging them deeply by leading them to fully grasp the modeling capabilities of numbers and how they interact via the four fundamental algorithms—addition, subtraction, multiplication, and division. It is a way of pushing their intellectual capability that will pay dividends as their deep understanding both of numbers themselves and of how they interact allows them to quickly grasp and deeply understand newer and ever more sophisticated concepts as they make their way through higher math.
Deep understanding of math like this is often called “number sense.” As the name suggests, students who have number sense understand the sense behind the numbers—both the values that they represent and also the many ways that they can combine differently to produce different results.
From a teaching standpoint, the challenge of number sense is that it is difficult to measure. Students who complete a problem could be demonstrating mastery of process and algorithm rather than number sense, while students who miss a problem might just have made a simple error or might be demonstrating a deeper misunderstanding. This is where showing work and writing to explain are so crucial to our math classes. In showing their work and writing to explain, our students are articulating their number sense—their understanding of how the numbers come together to become the answer given. Whether that answer is right or wrong, the work shown and written explanation provide a much more clear and definitive window on the child’s developing number sense. And that allows our teachers to teach to the skill so that when our students are moving on to high school, they are poised to flourish with a deep and sophisticated understanding of numbers and operations.
We consistently hear from high school math teachers and administrators in independent schools and magnets that our students are very well prepared for success in math because their foundations of understanding and knowledge are so comprehensive and solid. It may seem counter-intuitive in this increasingly fast-paced world, but spending more time digging deep and less time covering ground prepares our students for success in math whatever the program, and whatever their trajectory within it.
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