For the past few weeks, with the Julia Robinson Mathematics Festival, I have been doing math work with different groups of 5 to 17-year-olds who have been separated from their parents at some border, transported to the South Bronx. From there they are bused to a downtown office where they are interviewed by an advocate as part of the next stage in their journey. For 90 minutes I work with kids I will never see again, each of whom, by definition, is a trauma survivor. Yet for those 90 minutes the ugly reality pursued by a morally bankrupt regime is put on hold and the universality of the relations between numbers and shapes is there for us to work with, together to see patterns, solve problems and discover insights we didn’t know we had the power to find.
I will really never know anything else about these too young children, but they have helped me see again why it is a civil right and a human imperative that every person deserves the opportunity to learn mathematics with understanding.
Thus, for example, the set of real numbers is infinite, and each real number can be associated with a unique point on the number line. It is an illustration of the truth from the world: when one more is added the total is one greater, whatever the quantity is called, however it is expressed. There is no value great enough that there can’t be one more (or small enough that there can’t be one less) And further: There are an infinite number of values between any two other values, only bound by our ability to measure.
The number line fixes the relationship between all possible values, and allows us to name new values by relating any two values in different ways – the sum of two values, the difference of two values, the product of two values or the quotient of two values (and later exponentiation). These relations just are.
In the same way, a triangle retains its shape and area no matter how it is slid, flipped or rotated. The relationships never vary, the forms conserve their shape. We can depend on them as perhaps we can depend on little else these days.
So for 90 minutes the children put aside their worries and fears. We face each other across a table and play Spit, challenging each other to run up and down the number line with the cards we are dealt, one more or one less without dispute, laughing when we are too late to a pile.
A smile lights up a face when a cat finally appears out of the 7 pieces of tangrams– five triangles, a square and a parallelogram.
A particular figure often needs a precise arrangement of pieces, yet this cat’s tail can be made with either the parallelogram or the two small triangles. Which should it be? Figuring that out, on your own time, with the opportunity to try and regroup, is a certain kind of freedom born from dependability.
What are all the ways you can organize 5 same-sized squares, each piece sharing at least one full side with another? There are 12 such ways. Can you make them all? Then can you use the resulting pentominoes to build a rectangle?
I haven’t yet succeeded. But with a colorful puzzle template it is still a challenge to flip, rotate and turn the pieces into a rectangle. We all have the power to turn 12 different shapes into one gorgeous rectangle and are pleased together when we do so.
What makes Mathematics joyous is not turning lessons into planning for a party or a game that disguises memorization of facts. What makes mathematics joyous is the continuing discovery of the eternal nature of the underlying relationships that are there for all of us to discover, the relationships that are the same for me and for you, no matter on what side of the border you happen to be.