I generally work with students who would be considered above average in school. But every so often a student comes into my life for whom each new math concept is an exhausting struggle. Math is an endless menu of incomprehensible and unrelated steps to be memorized and catalogued. That there could ever be any purpose to, let alone any joy in, this cryptic jumble of numbers, formulas, and procedures is unimaginable.
These are the students that inspire creativity, awaken passion, and elicit reflection. They are the reason I teach and they are the students who make me want to be a better teacher.
Em came to be my student last year at the start of grade 7. A portfolio of sixth grade work revealed a math program that was largely focused on computation. Em had countless examples of multiplication and division of whole numbers, fractions, and decimals. While it was clear that Em attempted to dutifully follow the algorithm of the day, there were signs that something was seriously amiss.
Em needed to prepare for a private school entrance exam. The test primarily consisted of problem solving, pattern recognition, and general mathematical concepts. Em could only confidently answer 2 of the 50 questions on the diagnostic test. As we worked our way through the problem set, numerous content holes, flawed reasoning, and misconceptions were exposed. Em had managed to mimic the computational steps necessary to pass classroom tests and quizzes but had escaped any real mathematical learning. Em did not understand place value, could not order simple unit fractions, saw no relationship among equivalent fractions, did not understand the purpose of a decimal point, and lacked number sense.
Em and I worked together regularly for 6 months without any significant progress. I thought I had tried everything - visuals, manipulatives, real world examples, common language, even acting out problems. Just as I was about to give up, the connection I so desperately sought finally made an appearance.
We were exploring decimals when Em called the decimal point a period. While privately lamenting this student's misunderstanding, I wondered if perhaps there might be something to it. I asked Em to explain further. Em went on to tell me that the decimal point marks the end of the whole numbers and the start of the "smaller pieces", the pieces that weren't quite whole yet. In Em's mind, that was very similar to the way a period ends one thought but can also signal the start of a new one. From there, Em told an elaborate tale of a fantastical world of whole numbers and pieces, how they are kept apart by the will of the decimal point, how the wholes and pieces organize themselves into groups by size, and how these groups are either 10 times bigger or 10 times smaller than groups on either side. Em also described how the decimal can make numbers grow or shrink by moving its location and that it is always present even when there are no "pieces".
Em understood decimals better than any 7th grade student I had ever met. No standardize test in the world would ever ask Em to tell the story of the wholes and the pieces. Yet, that was the only way Em could confidently share her knowledge.
Since that time, classes with Em have been rather magical journeys into far-off lands where numbers and symbols come to life and tell their stories. Em is in 8th grade today and is struggling with the rigidity of her pre-algebra course. Her class has been studying the distributive property and combining like terms. Em confided that she just didn't get it. I mentioned something about helping the expression escape from its parentheses prison. Before long, Em had crafted a story about the number guard that stood watch outside the prison, the banning of subtraction, and the look-alike law.
And every homework problem was solved perfectly.