### Site swap notation

Again this term I followed the path laid down the past three terms - to put up a correctly labelled 3 site swap and then ask if the students understood the diagram. I then put up a 51 pattern. I left space on the right end and asked students what comes next, an L or an R? Which color? Why?

The students were able to answer these 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 digressed into asking for the expansion of (x + 1)(x +1) and after an initial x² + 3 answer eventually was told that x² + 2x + 1 is the solution. 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.

I asserted that the students no more understood algebra than they did the strange patterns on the board. They could answer questions correctly, 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.

I noted this was just abstract symbols on a board being manipulated. Then I ran a 342 swap in what was still an abstract system on the board. I pointed out that both symbolic manipulations on the board, the site swap pattern and the the algebraic 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. And the news on retention is even worse, students retain nothing beyond plotting xy scattergraph points post-college algebra.

I then tied this back to physical science by noting that in this class we had used mathematics to find the density of soap, the speed of a ball, the acceleration of gravity... I went through the labs of the term. This also foreshadows the final examination format at present.

In the afternoon I built on the morning structure of putting up the correctly labelled site swap, but not saying anything. Just putting up the diagram. First 3, then 51, then a 3 with a single 342 swap. This was vertically aligned to the 3 so the swap could be seen more clearly. I said nothing as I put up the diagrams.

Then, and I did in the morning, I asked what letter the next blank would get. The students correctly inferred the LR pattern and selected the correct answer. I also asked what color was next for both 3 and 51. I added multiple blanks and asked specific students to answer including what number goes above each arc as I added the arc. Everyone could answer correctly. I asked if they understood the patterns. The students answered that they did. For hadn't they answered each question about the pattern correctly?

Then I put up (x +1)(x +1) and the students correctly informed me that the answer was x² + 2x +1. I asked if they understood algebraic multiplication. They also responded affirmatively. I then asked why the answer was x² + 2x +1, how do they know this to be the case? This was clearly a confusing question as they had already given me the correct answer I sought. I then showed them the diagram of a square piece of land with an edge length of x, what happens when the land increases by one foot on each edge. I drew out the diagram and then drew the separated pieces of the result.

I used this 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.

I noted too that on a pre-test students had difficulty with slope and intercept, and while some still do, many now understand these ideas more deeply.

I posited that mathematics without physical frameworks on which to hang the concepts was a meaningless activity. Then I demonstrated what the strange notation on the board meant.

The class wrapped up with the students trying to juggle 3 and 51 patterns.

*Site swap notation 3 and 51*

The students were able to answer these 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 digressed into asking for the expansion of (x + 1)(x +1) and after an initial x² + 3 answer eventually was told that x² + 2x + 1 is the solution. 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.

I asserted that the students no more understood algebra than they did the strange patterns on the board. They could answer questions correctly, 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.

*342 site swap*

I noted this was just abstract symbols on a board being manipulated. Then I ran a 342 swap in what was still an abstract system on the board. I pointed out that both symbolic manipulations on the board, the site swap pattern and the the algebraic 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.

*Salvin was the first to get a functional 3 up and running*

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. And the news on retention is even worse, students retain nothing beyond plotting xy scattergraph points post-college algebra.

I then tied this back to physical science by noting that in this class we had used mathematics to find the density of soap, the speed of a ball, the acceleration of gravity... I went through the labs of the term. This also foreshadows the final examination format at present.

*Rosalyn working on a site swap 3*

*Stacey*

*A sequence of Rosalyn and Stacey practicing site swap 3*

Rosalyn and Stacey

Skyler

In the afternoon I built on the morning structure of putting up the correctly labelled site swap, but not saying anything. Just putting up the diagram. First 3, then 51, then a 3 with a single 342 swap. This was vertically aligned to the 3 so the swap could be seen more clearly. I said nothing as I put up the diagrams.

LcRose

Then, and I did in the morning, I asked what letter the next blank would get. The students correctly inferred the LR pattern and selected the correct answer. I also asked what color was next for both 3 and 51. I added multiple blanks and asked specific students to answer including what number goes above each arc as I added the arc. Everyone could answer correctly. I asked if they understood the patterns. The students answered that they did. For hadn't they answered each question about the pattern correctly?

Rangpino

Then I put up (x +1)(x +1) and the students correctly informed me that the answer was x² + 2x +1. I asked if they understood algebraic multiplication. They also responded affirmatively. I then asked why the answer was x² + 2x +1, how do they know this to be the case? This was clearly a confusing question as they had already given me the correct answer I sought. I then showed them the diagram of a square piece of land with an edge length of x, what happens when the land increases by one foot on each edge. I drew out the diagram and then drew the separated pieces of the result.

Emelisse "Jan"

I used this 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.

Janice Stacia "Dass"

I noted too that on a pre-test students had difficulty with slope and intercept, and while some still do, many now understand these ideas more deeply.

I posited that mathematics without physical frameworks on which to hang the concepts was a meaningless activity. Then I demonstrated what the strange notation on the board meant.

The class wrapped up with the students trying to juggle 3 and 51 patterns.

Sequences of Dorothy Daniel demonstrating control of the 3 pattern both standing and sitting

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