Is Kindergarten too old?

Here is a very simple dynamic graph, showing a point sliding down a radius in the same amount of time that it takes the circle to do one rotation around its center. Simple to see and explain.

It is the explanation part that makes this valuable. No matter what grade level, it could be used. From simple use as an introduction to the words circle, radius, rotation, and speed (imagine if children started learning these words in kindergarten) to a PreCalculus class being asked to generate parametric equations so that they could continue the graph on their graphic calculators.

The file itself can be found here.

If you wish to read more about this type of structure, check out this at Wikipedia.

Be creative in your use!

Here is a sketch that could basically be used with any grade, from a visual with elementary students, to a "can we make it ourselves" with middle school students, to a model for exploration for upper levels.

With an elementary class, I would leave out all the text, and create a step-by-step show, from first circle to tangent line to second circle to midpoint to trace, but not using sophisticated language. With middle school students I would use the basic geometric language and do a step-by-step as well. Upper students who are familiar with Geogebra could be shown the graphic and asked to recreate it. Those unfamiliar could be guided through it. Precalculus students could be challenged to determine an equation that could be graphed on a graphic calculator.

Adjustments to the file are easily made.

The main point is that this technology should not just be used as crutch with old curricula, but should also be used as an avenue for new approaches to mathematics education.

The complete file can be found here.

The Magic Formula From Heaven

Here is a statement that absolutely scares me. It is from Lesson 19: Interpreting Correlation from  the NYS Common Core Mathematics Curriculum.

It is not necessary for students to compute the correlation coefficient by hand, but if they want to know how this is done, you can share the formula for the correlation coefficient given below.

It demonstrates, in a nutshell, a massive hypocrisy in the implementation of Common Core math here in NY. What starts out, in the lower grades, as a powerful attempt to make students consciously aware of what they are doing, has morphed into the old and disastrous "give them a formula or a calculator and they will be happy." When doing basic arithmetic the emphasis on understanding by expecting multiple methods sometimes seems like overdoing it. Here, understanding is just tossed aside.

Take a look at this Wikipedia page (https://en.wikipedia.org/wiki/Pearson_correlation_coefficient), read it carefully, and then tell me if anything related to this concept actually belongs in Algebra I. If that does not convince you, check out the Wolfram page. For a bit more, check out this one.

What really bugs me is that using a calculator to compute the Pearson correlation coefficient bypasses any attempt to teach understanding. No student will come out of this with any knowledge other than "this is what they told me in school".  That is exactly the situation that "teaching for understanding" was supposed to avoid.

Why not let students actually get the opportunity to explore topics such as least-squares regression using software such as GeoGebra? Here is a small visual example of what GeoGebra can do. The labeled points can be dragged and the sliders control slope and y-intercept of the line. The best fit is the line that gets the sum of the squares as small as possible.

We should keep in mind that there is no need to have a line of best fit if we can see all the data plotted in front of us. A best fit line can help us if it can be understood as a model of prediction. If that is the goal, then it would make sense to have some strong connection between the modelling line and the correlation coefficient. Check out what New York State students experience and see if that connection is solidly made.

Does mathematics need to be more wordy?

The January 2017 New York regents exam in Algebra I (Common Core) contains a question with a model response that I do not get.

Here are the directions:

Here is the question and the model response:

The student has shown in 2 steps how to convert one equation into slope-intercept form, and you can see that the equation ends up identical to the first equation in the question. In answer to "Is he correct?" the student answers "No." In explanation, he states that the two equations are for the same line.

The model response scoring states: Score 1: The student wrote an incomplete explanation.

Mathematically, this model response nailed it. For some reason it only gets half credit. Was it not verbose enough? Is there a minimum number of words required?

You can see the entire group of model responses here.