Stop Erasing, Start Visualizing: A Parent's Story to Geometry Success
A parent's insightful experience helping their daughter conquer geometry frustration by applying the same visualization techniques she used in chess
This short story illustrates the difference between visualizing a task before starting and diving in without a clear mental picture.
My daughter, Emma, slumped over her geometry textbook, with a furrow in her brow. "Dad," she sighed, "can you help me with this? I can't even draw the diagram." She pushed the book towards me, pointing at a problem involving intersecting lines and adjacent angles. I glanced at the problem, a familiar knot of intersecting lines and labelled angles. "Sure, honey. But tell me what you understand so far. Start drawing based on your understanding." I handed her a pencil and a fresh sheet of paper.
Emma hesitantly began sketching. As she drew, discrepancies between the problem description and her evolving diagram became glaringly obvious. She had drawn two lines intersecting, but the angles she labelled didn't correspond to the problem's description. "Okay, Emma," I interjected gently, "let's talk about adjacent angles again. Remember, they share a common vertex and side." We spent a few minutes reviewing the concept, using her existing drawing as a starting point. A lightbulb moment flickered in her eyes, and she nodded in understanding.
With renewed confidence, she erased a few lines and started redrawing. However, significant differences remained. I watched her closely, realizing she wasn't visualizing the drawing in her mind first, like I habitually do. She was reading the first sentence of the problem, picking out a piece of information, and then adding it to the drawing without a comprehensive understanding of how all the elements fit together. It was like trying to assemble a jigsaw puzzle without looking at the picture on the box. After much drawing, erasing, and redrawing, she managed to create a diagram, albeit a messy one, that she could translate into an equation and solve. Even then, the drawing didn't accurately represent the problem's parameters. It was a functional, but flawed, representation.
This struggle inspired me to show her the power of visualization. We erased the drawing completely. "Emma," I asked, "how do you plan your moves in chess? You're getting pretty good, so you must be doing something right." I knew she spent time thinking ahead, visualizing potential outcomes, and considering the implications of each move before actually making it.
She thought for a moment. "Well, I try to see what will happen if I move a piece here or there. I think about what my opponent might do next." "Exactly," I said. "You create a mental image of the board before you move, you don't actually physically move the pieces on the chess board. Can you imagine what mess you would create on the board if instead of making the moves in your mind you would physically do them? You see them and you keep the potential moves in your mind as possibilities. Try applying the same approach to this geometry problem. Build the drawing in your mind. Make adjustments mentally before you put pencil to paper."
Here is a way of enhancing and using the power of visualisation specific to geometry:
Just close your eyes and allow your mind to become clear just like a blank sheet of paper and while you have your eyes closed simply start seeing that blank sheet of paper and as you see it, there is no need for you to do anything with it, but all you need to do is to do is to keep the concepts of what you want clear and while your keep that clarity of the two adjacent angles also keep the clarity of the ratio between them, all while you allow yourself to see clearly the data that you already know just notice how everything is drawing itself to match the problem perfectly. And as you wait for a little bit more, more and more details appear until you know that it's time to start putting it on paper.
In under a minute, she opened her eyes. "I see it!" she exclaimed. "I have it in my head." "Excellent!" I responded. "Now, start drawing with the clearest elements. Then gradually add the less defined parts, referring back to your mental image." With a newfound clarity and purpose, she swiftly sketched the diagram. This time, the resulting drawing almost perfectly matched the problem's parameters. It was clean, accurate, and reflected a deep understanding of the problem.
The crucial question is: how do you recognize the existence of a better, more constructive approach like visualisation first? Paying attention to your feelings is key. With Emma, I simply asked her to compare how she felt when drawing without visualizing versus visualizing first. The difference was palpable. "When I just started drawing," she admitted, "I felt confused and frustrated. But when I pictured it in my mind first, it felt...easy. And good!" The visualizing approach brought a sense of calm and control, while the other left her feeling lost and overwhelmed.
So, you gain the triple benefit of feeling great, working efficiently, and accurately representing reality. Visualisation is a powerful possibility. Just as my daughter initially struggled to think ahead in chess, practice made it easier, and the inherent pleasure of successfully strategizing kept her engaged.
Note: Although visualisation is not the only way of learning, it's one of the most prevalent and effective ways. Here are a few more ways of learning that might be useful to take in consideration building on the chess analogy:
Breaking down complex problems: Just as in chess, geometry problems can be broken into smaller, manageable parts. Instead of tackling the whole diagram at once, one could focus on individual lines, angles, or relationships, then combine them. This is like analyzing individual chess pieces and their potential moves before considering the whole board.
Working backward from the goal: In chess, you often think about the desired end-state (checkmate) and work backward to figure out the steps to get there. Similarly, in geometry, knowing the desired outcome (e.g., proving two angles are congruent) can guide the construction of the diagram and the selection of relevant theorems.
Experimentation and Trial-and-Error: Chess involves trying different moves and seeing how the opponent reacts. While a neat final diagram is ideal, one shouldn't be afraid to sketch multiple versions, experiment with different approaches, and learn from mistakes. The "messy" diagrams are part of the learning process and you can play with the problem until you are satisfied of your understanding.
Verbalization and Explanation: Explaining the thought process aloud (to yourself or someone else) can help you to solidify your understanding. This is like analyzing a chess game with a friend, discussing different move options. Verbalizing forces your to clarify your thinking and identify gaps in your knowledge.
Connecting to Prior Knowledge: Actively connecting new concepts to previously learned material helps build a stronger foundation. Just as a chess player draws upon experience from past games, one can leverage her understanding of other geometric principles to solve new problems.
Seeking different perspectives: Sometimes, getting stuck on a geometry problem requires a fresh perspective. Asking a friend, teacher, or online forum for hints or alternative approaches can be helpful. This is similar to analyzing a chess game with a more experienced player who can offer insights.
I can't help but going even further that this story and tell you that there is a deep satisfaction in planning, thinking ahead, and setting goals, whether it's on a chessboard, with a geometry problem or in visualising what you want. The satisfaction comes from you thinking in a way that makes you feel great, trusting that when you feel good the clarity, the opportunities and inspiration will come and flow through you.
Thank you for reading!