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.

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