In 2014, I designed an interactive web app called Drawing Area that won the WGBH Innovation Math Challenge. The app is based on a curriculum unit I created in 2001 and revised over the years. The premise behind Drawing Area is simple: students find the area of a figure by re-drawing the figure on grid paper using rectangles and right triangles.
I created Drawing Area because, as a classroom teacher, I observed students struggling to complete two tasks.
When finding the area of a figure, the general strategy is to decompose the figure into rectangles and triangles, find the area of each rectangle and triangle separately, and then combine the individual areas to get the area of the entire figure. But some students don’t see how to divide figures into rectangles and triangles, and some students don’t see how they can figure out the dimensions of those rectangles and triangles once they have them.
What I observed wasn’t a lack of effort, motivation, or mathematical understanding — it was a lack of basic spatial sense.
Many students and most math teachers can see how to divide a figure into rectangles and triangles just by looking at it. We have no idea how we know how to do this, and there doesn’t seem to be any thinking or conscious effort involved. We see, we know, we do. The students who are unable to divide figures into rectangles and triangles simply don’t see what the rest of us see. With enough repetition and practice, they learn to recognize certain patterns they can use to cope and muddle through, but those patterns are often unreliable and don’t scale.
Why is this important?
Whether or not students are able to find the area of complex figures by decomposing them into rectangles and triangles may not seem very important — but how students experience a lack of spatial sense in school reflects a much larger problem that has profound implications for how we approach education reform.
Imagine you are a student sitting in math class. When the teacher draws a figure on the whiteboard, your classmates instantly know how to divide the figure into rectangles and triangles. The rectangles and triangles in the figure appear so obvious, they are genuinely shocked you don’t see them, too. One by one, they patiently show you how to draw in rectangles and triangles, but it all looks like sorcery to you — lines and shapes appearing out of thin air.
You feel alone and isolated. Now imagine that same experience repeated day after day for years. This is how many students experience school.
As an educator trained in constructivist learning theory, I design learning experiences that enable students to construct their own theories and make sense of the world around them. But what happens if a student is unable to make sense of an experience because he sees the world differently than I and most of his classmates do? This happens far more often than you may realize.
Karen Kilbane argues that a fight-or-flight response can be triggered when we are unable to make sense of an experience and predict what will happen next. Our brain interprets confusion and contradiction as a threat, and it reacts by going into survival mode. Students who exist in a constant state of fight-or-flight in school shut down and block things out. This makes it nearly impossible for them to learn, and undermines their self-image as learners. Students who are under siege and think they must be stupid are simply not going to perform very well.
Information-processing and microworlds
People who lack the spatial sense to decompose figures and deduce missing dimensions literally see the world differently. We see with our brains as well as our eyes. When we look at a chair and see a chair, it’s because our brain is processing and interpreting the sensory information our eyes collect. Therefore, to develop spatial sense, we must re-train how the brain processes information — without repeatedly telling the brain it’s wrong and triggering its fight-or-flight response. How do we do that?
One approach is through the use of microworlds. A microworld is a small, self-contained learning environment governed by a set of internally consistent rules. Students learn how a microworld works by exploring and playing in it. Lessons learned in a microworld are often transferable to the real world. Although Drawing Area is not as open-ended as a full microworld, it has several key features characteristic of microworlds.
In Drawing Area, students are guided through three steps to find the area of a figure. Since I want to focus on step 1 where students develop spatial sense, I’m going to describe all three steps, and then go back and discuss step 1 in more detail.
Students re-draw a figure on grid paper using rectangles and right triangles in step 1, and then find the area of each individual rectangle and right triangle in step 2. Because the rectangles and right triangles are drawn on grid paper, students can easily figure out the dimensions of each rectangle and right triangle using the grid, find the area of each rectangle by counting squares, and see that a right triangle covers half the area of a rectangle.
Immediate feedback throughout step 2 enables students to quickly pinpoint and correct any errors, developing a sense of mastery. Once the area of each rectangle and right triangle has been found, students add those areas together in step 3 to find the area of the entire figure. In the full curriculum unit that Drawing Area is based on, students also learn how to draw and find the area of figures by subtracting rectangles and right triangles.
Developing spatial sense
How do students develop spatial sense in step 1? Students are asked to re-draw a figure with labeled dimensions on grid paper. They draw the figure by placing and re-sizing any number of rectangles and right triangles anywhere on the grid. Because there is no one right answer, students have room to explore and discover. At the same time, rectangles and right triangles automatically snap to the grid, which gives students some guidance and helps them recognize the types of moves that are likely to lead to a solution. Students who need a little extra guidance are shown an outline of the figure on the grid they can use to draw over.
When observing students using Drawing Area for the first time, students who lack spatial sense are very unsure of themselves. They place a rectangle on the grid, and pause to compare their drawing with the original. They re-size the rectangle, and then pause to compare it again. It frequently takes a few dozen individual moves — with a number of false starts — for them to draw the figure on grid paper, where a student with spatial sense will draw the same figure in under ten moves.
The students who lack spatial sense are using trial and error to draw the figure. Trial and error is an effective strategy when students can make a move, and then determine if that move has moved them closer or farther away from a solution. Once students are able to decompose a figure using trial and error, the development of spatial sense is rapid. Instead of making a move and then evaluating the result after the fact, students start to visualize and evaluate moves in their head, choosing the optimal one.
They develop enough spatial sense to predict the result of a move before doing it — and soon they are seeing rectangles and triangles in figures without grid paper just as easily as their classmates do.
How do microworlds work?
Making sense of the real world can be challenging because there are so many complex systems interacting at the same time. As a teacher, whenever I try a new instructional strategy, it can be hard to tell if the strategy is effective or not. Individual students respond differently; I have an extremely limited view of what students are thinking and feeling; and by the time I collect any meaningful data, too many intervening factors will have already muddled the picture. Trying to pick out a clear signal from all that noise is next to impossible.
Microworlds can help us tune our antennae by immersing us in a model of the real world that is more transparent and has fewer systems.
Because the microworld is more transparent, we have a more complete picture of everything that happens; and because the microworld is simpler, there are fewer interactions for us to tease apart. This makes it possible for us to learn how the microworld works simply by exploring and seeing what happens when we do something. Then, if the microworld mirrors the real world, we can transfer what we’ve learned back into the real world.
While Drawing Area is not as open-ended as a microworld, it’s still a transparent model of the real world students can explore and discover. As long as Drawing Area is simple and transparent enough for students to decompose figures through trial and error, they can begin tuning their spatial-sense antennae for the real world — and change how their brains see geometric figures.
The role of teachers
A microworld can help students process information more effectively by highlighting the information students should zero in on, but it can also help with the brain’s fight-or-flight response.
To re-train our brains, we need lots of feedback. In Drawing Area, the primary source of feedback is internal. We make a move, and then evaluate whether that move is helpful or not based on what we see and think. A secondary source of feedback is only triggered when we think we’re done or ask for help. The source of that feedback is Drawing Area itself, and we use it to tune our internal feedback loop.
By removing the teacher as a primary or secondary source of feedback, the re-training process feels less threatening and judgmental.
So, what role do teachers play when learning occurs in a microworld? Teachers should ask questions that encourage students to reflect on, articulate, evaluate, and generalize their own thinking. They can help students accelerate their own learning by nudging students in a new direction, posing intriguing what-if scenarios, and encouraging students to solidify and build on their thinking.
Once we’ve decomposed a figure into rectangles and right triangles, we need to find the dimensions of those rectangles and right triangles before we can calculate their areas. Students with spatial sense can often deduce those dimensions intuitively, but many students cannot.
If a student wants to re-draw figure A on grid paper using rectangles and right triangles, she might start by drawing a 6 x 9 rectangle. She knows the dimensions of this first rectangle because they are given in the diagram.
Next, she knows she wants to draw a second rectangle, but its width isn’t given. She aligns the bottom of the second rectangle with the bottom of the first rectangle and gives it a height of 5.
Then, she uses the grid to measure the width of the entire figure, which is 7. The second rectangle is too narrow. So, using trial and error, she keeps adjusting its width until the width of the entire figure is 14.
Most students who are able to find the dimensions of rectangles and right triangles through trial and error will naturally start to deduce those dimensions without trial and error. Their brains automatically detect patterns, construct theories, and test those theories. But some students with low confidence don’t make that transition; they are so concerned with getting the “right” answer, they don’t give their brains permission to notice errant patterns and theorize.
As a teacher, I would ask those students to guess the width of the second rectangle before working it out through trial and error. Just take a wild, random guess. Then, once the entire figure is drawn on grid paper, I would ask them to use the grid paper to measure the second rectangles’s actual width. Is there any way we could have deduced the width of the second rectangle now that we know what it is?
I have found that simply making a prediction, and then evaluating that prediction later on, is enough to stimulate the brain’s curiosity.
Implications for instructional design
This article is a tale of two stories. The first story is relatively straightforward. As classroom teachers and instructional designers, we need to figure out what our students are thinking and seeing. Don’t assume they are lazy or unmotivated. Students who have checked out may be utterly lost and in flight; disruptive behavior may be a reflexive response to confusion. I have seen disruptive and unmotivated students be remarkably calm and on-task when the environment makes sense to them.
I developed Drawing Area because I observed students as they attempted to find the area of complex figures, and I could see that they didn’t see figures in the same way I did. Then, ignoring conventional wisdom — which says it’s normal for students to struggle with area — I applied a little first principle thinking. What should students look for when decomposing figures and deducing the dimensions of rectangles and right triangles? How do I design a learning environment that highlights the information students need, and provides feedback without triggering a fight-or-flight response? After that, it was a matter of my own trial and error.
Implications for transforming education
The second story is far more important and complicated. I’m not the first teacher to notice that some students lack spatial sense, which prevents them from decomposing figures. Microworlds became popular over thirty years ago.
Why didn’t someone design a tool for students to develop spatial sense long before I did?
There is nothing particularly ingenious or creative about Drawing Area. The solution was fairly obvious once I had defined the problem. Yet, generations of students have been abandoned to struggle with their lack of spatial sense — and many other ways of seeing — on their own.
I believe one of the root causes of our failure to transform education is the dominant theory of action in education today: If students have grit and are empowered to pursue their own interests and design their own learning, then learning is natural and automatic. The only reason why learning might not be natural and automatic is if students aren’t “developmentally ready”.
Drawing Area puts the lie to this theory. I didn’t attempt to motivate students or give them a reason for finding area. I immersed them in a learning environment that enabled them to remove an obstacle to their sense-making, and they responded enthusiastically. Their brains did exactly what brains are designed to do, figure stuff out.
If I had given students a reason to persevere, some of them would learn how to decompose figures and deduce dimensions using brute force. But they would do it by memorizing patterns — which means they would be less fluent than their peers with spatial sense, and they would encounter numerous situations where their patterns fail or do not scale. While “successful”, this experience would reinforce the core belief that we are not very good at some things.
Immersing students in a learning environment that enables them to discover more effective ways of seeing through trial and error creates an entirely different experience. In the end, these students have the same spatial sense their peers do, and they walk away knowing they aren’t less capable, they simply see the world differently. This creates a sense of confidence and resilience. Students learn to explore more options and pay attention to how their own brains are working.
We learn we are capable of anything once we have the right perspective.
Feedback, new ideas, and guinea pigs
If you get a chance, please try out Drawing Area. I’d love any feedback. While I’d like to earn a living through my curriculum, I’m happiest knowing that people are using and benefiting from my work. Don’t be afraid to reach out if you’ve got something you’d like me to build or help you build. I might just say yes. And I’m always on the look out for guinea pigs to test my latest ideas. :)