Thursday, April 20, 2017

Saturday, April 15, 2017

Why? Why? Why?

Here is question 3 from the January 2017 New York Regents Exam in Algebra I (Common Core),
This question shocked me when I first saw it.

To get a taste of what such a correlation coefficient is, I suggest you take a look at Wolfram. New York State's EngageNY has an "introduction" to correlation coefficients (find it here) that says "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" and then shows

It should be noted that this formula the bounds of summation and variable subscripts are omitted, so it is a meaningful formula only for those "in the know", and I suspect that very few Algebra I students fit that description. 

Is this a case of presenting a "magic button" on a calculator as a key to answering a complex question? Is that how math should be taught?

New York's modules include the following:

Take note of the phrase "use technology".  That essentially means the student should plug in the numbers, hit the necessary buttons, and find a result. It is more like following a recipe, and most people recognize that a recipe becomes unnecessary after it has been followed a number of times: not because it is not being followed and not because it is understood, but because it is remembered.

Here is a question from the August 2016 Algebra I Regents;
I would suspect that after enough repetitions of the recipe some students might have caught on and realized that the choices offered make this question a bit easier than it would have been had the data been a bit different and the choices involved some seemingly random 4-digit decimals.  Perhaps that might be why questions such as this have, over the years, been categorized as "cookbook" problems.

The next question here is from the Algebra I Regents exam from August 2015:
There is absolutely no way any student will successfully answer this question without following a recipe on their graphic calculator. I do not know why NYS left all the space on the page as all that is asked of the student is to write down one equation and then a two digit decimal together with one word.

Here is a similar question from the January 2015 Algebra I exam:
The big difference here is the 60% increase in time entering the data (16 values instead of 10) and the explanation of part (b). But again, it's enter two columns of data, hit a couple more keys on the calculator, and read off a result.

A similar process is involved in the next question, the big difference is that it asks the student to pick an item from a different line in the calculator's display:
Do all these questions belong in Algebra I?

Are these items here only because of the lobbying efforts of the companies that sell the graphic calculators?

I would much rather see students get the flavor of least squares analysis by using FREE software such as GeoGebra. For an example, check out the Least Squares Demonstration here.

While correlation coefficient seems to be an easy testable item (when proper calculator is present), it should be recognized that it is merely a measurement of how well a regression line actually "fits" the data.  The true mathematical questions begin with the regression line itself: what it is, why it is, why it is useful, when is it useful,  how we know it is the "best" fit, what do we do when this is not useful, and many others. For a sense of the current state of regression analysis, just visit Wikipedia.

Please take note: correlation coefficients and a slew of its tag-a-longs could fit in high school mathematics, but not in Algebra I.  Dynamic geometric approaches to concepts such as least squares could be well developed in an Algebra I course, using software such as Desmos or Geogebra or other equivalent packages. 

The biggest problems with high school mathematics occur when students are expected to know and do things mindlessly. Recipes need to be avoided. Magic buttons on calculators need to be avoided like poinson needs to be avoided.

The true value and meaning of the quadratic formula comes only to one who tires of completing the square. Completing the square obtains its true value and meaning when one tires of trying to find factors (especially when they do not exist!). The real meaning of factoring comes to one who is fed up with constantly having to guess and check. There is a hierarchy to the knowledge and skills of mathematics. Jumping too quickly to a higher level does a disservice to students.

Imagine what would happen if we limited single variable quadratics to finding intersections with the x-axis on a graphic calculator, and skipped over everything mentioned in the previous paragraph! It makes just as much sense as tossing regression analysis into Algebra I only because it can be done on a graphic calculator. Jumping too quickly to a higher level does a disservice to students, especially when that jump takes place largely do to the mere availability of a handheld calculator.

Tuesday, March 28, 2017

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.

Sunday, March 26, 2017

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.

Tuesday, March 7, 2017

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.

Wednesday, March 1, 2017

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.

Tuesday, February 28, 2017

Is understanding a language too difficult?

Here is a list of words:
    1. noun
    2. verb
    3. adjective
    4. adverb
    5. subject
    6. predicate
    7. direct object
    8. indirect object
    9. clause
    10. pronoun
I could continue the list, but these will do.

You might ask, so what about these words? In answer, I can state with conviction that the knowledge of these terms, their meanings, and their usages, was key in the development of my personal understanding of our language.

Even though I spent years teaching and learning mathematics, I still use the concepts of adjective and noun when discussing fractions. Many of you might remember the terms "numerator" and "denominator", which are fancy words for the ideas of "how many?" (adjective) and "of what?" (noun).

The simple fraction \({\textstyle{2 \over 3}}\), properly read "two thirds" gives us a quantity (two) and a denomination (thirds), thus answering completely the question "How many, of what?".

This morning I made a discovery: none of these words is used, at all, in the January 2017 New York state regents exam in English Language Arts (Common Core) 

I found that atrocious.  How can one even discuss the "art" of language without involving its grammatical structure? 

Basing an entire curriculum on grammar would be a waste of time, just as doing nothing but arithmetic under the name of mathematics would be useless drudgery. But to totally ignore grammar amounts to designing a skyscraper while ignoring the hidden structure that holds it up.

Here is question from the New York regents Comprehensive Exam in English from August 1978:
How would students react in this day and age?

I guess it won't matter too much, as long as the gun is around to take care of the grizzlies.