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.

Monday, February 27, 2017

January 2017 New York regents exam question 24 is not nice

Here is a question from the January 2017 New York regents exam in Algebra II (Common Core)
Please take note that, according to the statement, the height of the box is "x-1".

That fact alone is enough to force any modeling function to only use x values larger than 1.

Similarly, the width being "5-x" forces the modeling function to use values for x that are less than 5.

This graph includes values for x that must be excluded, so it is not a good modeling function.

In addition, words such as "fastest" when used without a reference create an implication of time as a parameter. Combine the lack of a time factor in this question together with the issue of a box with a changing volume (the first sentence in the question refers to "boxes" where as the rest of the question refers to a single box), and confusion in the mind of the test taker is to be expected.

This question could have been written quite differently and perhaps have been restored.

When the day comes when these tests can be truly computer-based, a question such as this might be accompanied by a visual such as that shown here:

In the meantime perhaps New York could stop its insistence on testing questions being "real world". 

Sunday, February 26, 2017

2017 NYS Algebra II (Common Core) question 5

Here is a question from the January 2017 New York regents exam in Algebra II (Common Core)
Based on the information given here, this question cannot be answered. If we place the graph of the the linear equation given, we would see this:
We might guess that the point (1,1) is a solution, but without anything regarding function g(x) other than a partial graphical representation, we have no way to check our answer.

Secondly, by the inclusion of choice (4), a guess could be made that the graph given in the question was intended to be parabolic and that the student was supposed to take an educated guess at the coordinates for a few of its points so as to derive a quadratic function. There is nothing in the question to rule out a situation such as this:
The function in red as shown here agrees with everything that is given in the problem.  In this case, none of the four choices can be accepted as a solution.

In this question any answer chosen by a student would be incorrect. To expect any student to reason out an answer would be encouraging them to base arguments on unknown information and would be encouraging them to make convenient but unsupported assumptions in the absence of facts.

Wednesday, February 8, 2017

If you do not try to avoid careless errors, will you avoid any errors?

The clip above is from the Albany Times Union from today (Feb. 8, 2017), which happens to be the grand opening day for Rivers Casino in Schenectady, NY. I note a little urban attitude in the clip, with its subtle inference that only farmers would use a measure in acres. That attitude alone does not surprise me, and it could be more my personal attitude showing through. But that is not why I include this clip here.

Read it really carefully, and you will note that its key thrust is that 50,000 square feet is approximately 1.5 acres. To quote our President, "Wrong!!!"

An acre calculates out to 43,560 square feet. I say "calculates out" because it is initially defined as an area of one furlong (660 feet, or one-eighth of a mile) by one chain (66 feet, or one-tenth of a furlong). Do the arithmetic, 66 times 660 equals 43560.

Calculating further, dividing 50,000 square feet by 43,560 square feet gets us approximately   1.147842. So, if the Times Union intended to use 1.15 acres, with 1.5 as a typo, then it is a sign that the TU has to strengthen their proofreading. On a different hand, if 1.5 was used because the writer and proofreader just did not know, then the TU has a bigger problem. Possibly, the writer might have "known" that 1.5 is correct in the same way that Donald Trump "knows" that over 3 million votes cast in November were illegal votes.

No matter what the cause(s) of the error was(were), some people will undoubtedly look at it and claim "no big deal". That is the part that is scary, because what is "no big deal" to you might be a big deal to someone else. This error was an error of 30.68%. Suppose there was an error that big on your tax bill or your car payment or your grocery store checkout or your casino hotel bill? Would that error all of a sudden become a "big deal"?