A literacy teacher shared with me about a recent reading conference with one of her second graders: the little girl sat diligently by her side, accurately pronounced aloud all the sentences across three pages of a Magic Treehouse book and then burst into tears, proclaiming loudly, “I said all the words, but I don’t know what they meant!” Sobs. The frustration of sagging comprehension. The child knew that there was some meaning in there that she was missing, and she craved after it.
Now what in math? Just yesterday, I visited a fifth grade classroom where a student calculated that 1/3 + 7/12 was worth 7/36, while his partner across the table said the sum was 11/12. When I asked about the discrepancy, the first student offered that they might be both correct, as these could be equivalent fractions. He did not cry. He was not concerned. He had no expectation that the math ought to make any sense whatsoever.
When students memorize only procedural skill in mathematics, they miss the backstory, they miss the opportunity to question their own calculations, they miss the meaning: What do these numbers represent? How are we combining them? What would be a reasonable solution?
Making Sense of Math
In order to support students believing not only that math makes sense but that they themselves can be sense-makers as mathematicians, we need to slow down, ask deeper questions, offer and analyze models, , and practice transfer.
Slowing down is difficult. Most states’ scopes and sequences expect students to surf through piles of concepts at break-neck speed. If instead we gave ourselves permission to dwell with a few important ideas each year in school, to investigate those in depth, to dissect and question, more students would understand more deeply. Let’s take time to talk, to engage learners in conversations about whether 7/36 might be equivalent to 11/12: What do these fractions mean? How could we represent them visually? How close are they each to zero, to one? How could we prove that they are equivalent – or not? And can an addition problem, adding two rational numbers, have more than one correct answer? These are important mathematical questions. And our job as math teachers is not simply to communicate procedures but to convey the beautiful logic behind mathematical ideas. To do so takes time.
Mathematical comprehension is facilitated by multiple models, rich representations that offer learners three and two dimensional experiences with the concept at hand. Yesterday, in planning with a fourth-grade teacher, we talked about how she might demonstrate the meaning of fractions for her students. She knew from experience that a numeral over another number, separated by a line, was essentially meaningless to most of the nine and ten year olds in her care. We talked about how she might create opportunities for students to see, feel, touch, count, so that they could more deeply grasp the concept of a fraction. Give them two water bottles, one blue and one silver: ½ of the water bottles are silver. Show them ten shoes, six of them with laces: 6/10 of the shoes have laces. Hand them seven pencils, two of which are purple: 2/7 of the pencils are purple. These are a few examples of the sort of real world concrete representations that go a long way towards building students’ grasp of the important – and too often misunderstood – concept of fractions. Mathematical ideas, if not tied to concrete models, can leave students swimming in a sea of meaningless numerals.
Without understanding concepts, students might still master procedures, even produce proficient scores on assessments. Yet my young friend yesterday demonstrated the folly of knowing procedures without knowing the why: he handily and accurately multiplied two fractions instead of adding them, as the problem prompted. But because the meaning was lost on him, the discrepancy between 7/36 and 11/12 was of no concern. He even had a possible explanation: equivalent fractions.
When a learner truly understands a mathematical idea, she is able to apply it to novel contexts, with different numbers and situations, as well as to explain why her solution is correct. For example, once we agree that 7/36 and 11/12 are not equivalent fractions, might we be able to design a test for fractions’ equivalency? Or find fractions that are equivalent to 7/36 and to explain why? Or to describe how we got 1/3 + 7/12 = 7/36 in the first place and what we learned about whether that is correct? Procedural skills assessed on well-behaved end of unit tests that focus merely on the one taught concept can certainly produce 100% mastery scores, yet this does not mean that learners understand. In order to find that out, we need to ask better questions, seek multiple representations, and offer learners time to think.
Understanding is as important in math as it is in reading. When we slow down, talk in depth, model for meaning and practice transferring ideas into alternate contexts, we encourage all learners to believe that math is meaningful and that they themselves are meaning-makers. We can teach students to want mathematical meaning so bad they could cry.