### Site swap notation

Again this term I followed the basic path laid down the past four terms. This term is I put up a correctly labelled 3, 51, and a 3 with two 42 site swaps.

I left blanks on the right side and had the students tell me what went in each blank. The students were able to answer these questions based on the pattern already set up on the board. Note that I included the orange 3R lines to indicate that the pattern of threes also existed R to R. I had not yet tipped my hand, so to speak, that L and R were left and right.

I asked each student if they understood, in each class some nodded or agreed that they understood. Some of the student chose to remain silent, but I could not coax from them that they lacked understanding.

I then digressed into asking for the expansion of (x + 1)(x +1). In both classes a variety of solutions were suggested by the students. In the morning section the class could not agree on which of the solutions was correct, with most students leaning towards x² +3x+2. The afternoon class eventually let themselves be swayed by a couple students who insisted the answer was x² +2x +1.

I asked the class what this means, this x² + 2x +1, but the class was non-responsive. I then showed that if a square piece of land has a side length of x and the side lengths are increased by one, then the new property is x² + 2x +1 in area. I did this graphically. This term I learned that only a couple students in each section could recall having seen this areal explanation for x² + 2x +1.

I asserted that the students no more understood algebra than they did the strange site swap patterns on the board. Some students could answer questions correctly about the site swap diagram, I could assess them and measure their learning, I could show they met a student learning outcome. And yet they would have no real understanding of site swap diagrams. Or algebra for that matter.

I noted this was just abstract symbols on a board being manipulated. I pointed out that both symbolic manipulations on the board, the site swap pattern and the algebraic scratching were meaningless. Of no particular use, and of especially no use to those not in science, technology, mathematics, or engineering. That the next time the students would see a quadratic equation was when their high school aged children had trouble on their algebra homework.

Only then did I demonstrate the meaning of the site swap notation. As soon as I began counting off the red ball landings in my right hand, I could see the faces of the students light up, suddenly the abstract had become the tangible, the incomprehensible had become comprehensible. Without a physical system, mathematics is incomprehensible. Only a few rare individuals thrive in the world of pure mathematics and learn purely abstractly. Yet education insists on teaching mathematics as if everyone were a mathematical genius. And then mathematics teachers wonder why so many struggle in mathematics. Or why so many students "hate" mathematics by the time they reach college.

I used this introduction to explain that this what the course had been about. In laboratory one we found that the volume and mass varied according to a linear equation the slope of which was the density. In laboratory two the slope was the velocity of the ball. In laboratory three fitting the data to d = ½gt² generated the acceleration of gravity. In laboratory four the slope confirmed the conservation of momentum. In laboratory five a linear equation provided the coefficient of friction. In laboratory six the water cooled at a rate predicted by Newton. In laboratory seven the slope provided a conversion between meters and arcminutes. In laboratory nine the slope was the speed of sound. In laboratory eleven the slope was the index of refraction for water. In laboratory 12 the slope was the resistance.

We had used mathematics, but only in the context of a specific physical system. One equation at a time. Not thirty problems, odd numbers only, bereft of context. And, no, we had not "solved" any equations. This is an outgrowth of renaissance Italian competitions where two mathematicians would make up lists of math problems, trade problem lists, and in the village square see who could solve the other mathematician's problems correctly. Whomever solved the greater number correctly, won. The crowd undoubtedly placed bets. These problem lists were intentionally designed to play to "tricks" and special techniques that the one mathematician hoped the other mathematician would not know.

Math textbooks to this day are full of the lists of these sorts of problems, as if learning all these tricks and special techniques has any particular value. Ultimately, the result is large swathes of the population who do not like mathematics and, at some level, do not trust mathematics. The goal of a broadly literate population, including a mathematically literate population, guaranteed not to be achieved.

*Site swap diagrams*

I left blanks on the right side and had the students tell me what went in each blank. The students were able to answer these questions based on the pattern already set up on the board. Note that I included the orange 3R lines to indicate that the pattern of threes also existed R to R. I had not yet tipped my hand, so to speak, that L and R were left and right.

I asked each student if they understood, in each class some nodded or agreed that they understood. Some of the student chose to remain silent, but I could not coax from them that they lacked understanding.

*(x+1)(x+1) generated four different solutions*

I then digressed into asking for the expansion of (x + 1)(x +1). In both classes a variety of solutions were suggested by the students. In the morning section the class could not agree on which of the solutions was correct, with most students leaning towards x² +3x+2. The afternoon class eventually let themselves be swayed by a couple students who insisted the answer was x² +2x +1.

*Ashlyn juggles a 3 pattern*

I asked the class what this means, this x² + 2x +1, but the class was non-responsive. I then showed that if a square piece of land has a side length of x and the side lengths are increased by one, then the new property is x² + 2x +1 in area. I did this graphically. This term I learned that only a couple students in each section could recall having seen this areal explanation for x² + 2x +1.

*Swister was first to get a site swap 3 up and juggling*

I asserted that the students no more understood algebra than they did the strange site swap patterns on the board. Some students could answer questions correctly about the site swap diagram, I could assess them and measure their learning, I could show they met a student learning outcome. And yet they would have no real understanding of site swap diagrams. Or algebra for that matter.

*George*

I noted this was just abstract symbols on a board being manipulated. I pointed out that both symbolic manipulations on the board, the site swap pattern and the algebraic scratching were meaningless. Of no particular use, and of especially no use to those not in science, technology, mathematics, or engineering. That the next time the students would see a quadratic equation was when their high school aged children had trouble on their algebra homework.

*Vivian*

Only then did I demonstrate the meaning of the site swap notation. As soon as I began counting off the red ball landings in my right hand, I could see the faces of the students light up, suddenly the abstract had become the tangible, the incomprehensible had become comprehensible. Without a physical system, mathematics is incomprehensible. Only a few rare individuals thrive in the world of pure mathematics and learn purely abstractly. Yet education insists on teaching mathematics as if everyone were a mathematical genius. And then mathematics teachers wonder why so many struggle in mathematics. Or why so many students "hate" mathematics by the time they reach college.

*Lillian and Nemely juggling, Harvey observing, Dexter working on bounce juggling*

I used this introduction to explain that this what the course had been about. In laboratory one we found that the volume and mass varied according to a linear equation the slope of which was the density. In laboratory two the slope was the velocity of the ball. In laboratory three fitting the data to d = ½gt² generated the acceleration of gravity. In laboratory four the slope confirmed the conservation of momentum. In laboratory five a linear equation provided the coefficient of friction. In laboratory six the water cooled at a rate predicted by Newton. In laboratory seven the slope provided a conversion between meters and arcminutes. In laboratory nine the slope was the speed of sound. In laboratory eleven the slope was the index of refraction for water. In laboratory 12 the slope was the resistance.

*Darlene juggling, George, Patricia, and Vivian in the background*

We had used mathematics, but only in the context of a specific physical system. One equation at a time. Not thirty problems, odd numbers only, bereft of context. And, no, we had not "solved" any equations. This is an outgrowth of renaissance Italian competitions where two mathematicians would make up lists of math problems, trade problem lists, and in the village square see who could solve the other mathematician's problems correctly. Whomever solved the greater number correctly, won. The crowd undoubtedly placed bets. These problem lists were intentionally designed to play to "tricks" and special techniques that the one mathematician hoped the other mathematician would not know.

*Darlene*

Math textbooks to this day are full of the lists of these sorts of problems, as if learning all these tricks and special techniques has any particular value. Ultimately, the result is large swathes of the population who do not like mathematics and, at some level, do not trust mathematics. The goal of a broadly literate population, including a mathematically literate population, guaranteed not to be achieved.

*Gilbert showed control and mastery, Praislyn watching*

If my students have enjoyed physical science, if they have glimpsed the uses of mathematics in context, if they now see science as a process of examining data and drawing conclusions based on mathematical analysis of that data, then I have indeed succeeded. None of my students are science majors, thus my best hope is to kindle a fire of interest and understanding of mathematics and science.

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