Complex math problems always raise all kinds of questions for me as a parent, teacher, and mentor. One of the most important is: what does it mean to generalize, and why is this an important skill for a learner? You might now be thinking, “Generalize? Why do you even wonder about that?” Good question!
The last question in a recent problem asks:
“What patterns are you noticing?
1. How could you answer the question if you knew only that there was an even number of boxes, but not the exact number?
2. How could you answer the question if you knew only that there was an odd number of boxes, but not the exact number?”
THIS is generalization. When you generalize, you create a “rule”, or you make a statement, based on your experience that a process or a problem works a certain way. In the Apples problem, the solver was asked how they could answer the question no matter what number of boxes was given; when they can successfully do so, based on the patterns they are seeing, they are making a generalization.
I think you might see why, for me, as a teacher, this seems like an important skill. In the classroom, for sure, we want children to learn to find patterns and determine the rule that fits them. It leads to deeper understanding, to efficiency, and gives children the ability to “automate” some of their thinking, so that they can move on to learning bigger and better things. For example, if I know that when I subtract a larger number from a smaller one, I get a negative number as an answer, then it helps me quickly make sense of answers and be sure they’re right, but also allows me to solve more complex problems, where subtracting is just one step of many.
Generalization, though, applies to all kinds of places beyond school, which is why it’s a big deal. As a parent and mentor, I want my kids to learn to generalize in the same manner they do in school, based on repeated experiences with the world around them. It’s actually something we do with people around us all the time, probably without even thinking about it. Any time you ask something like, “The next time you do ____, will you do it the same way?”, you’re helping others to see patterns in their actions and the world around them, which lead to generalization. Just the other day, my daughter asked me why, when we build LEGO cars, I usually use longer pieces on the bottom. She and I then talked about how, in my years of experiences with LEGOs, I’ve learned that my car is less likely to break if it’s built with longer pieces as the base. This is a generalization I’ve made through experience, and one that she will make too over time (she totally didn’t believe my explanation!).
One note of caution: generalization is a really important thing to learn to do well, but, like anything else worth learning, it takes practice. One of the things that can happen as we try to generalize, is that we OVER-generalize. When we over-generalize, we make a rule without enough evidence or experience to know it will always be true. We all have done this at some point - in learning, in math, in life. As we help our children learn to generalize, just keep in mind that making generalizations that reach too far is just a part of learning how to do it well. When we over-generalize, the key is to talk about it (with questions like “Well, if that’s always going to be true, what if…?”), so that we can continue to learn, practice, and grow.
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