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"?

Sunday, February 5, 2017

Better Value: Teach Well, or Teach Again?

I passed by an algebra class and they were working on exponential functions, they never see it in geometry and then we expect them to remember it in algebra 2. It never happens and we just have to reteach the whole topic from the beginning!!!
The above is quoted from a message submitted by a high school math teacher to a mathematics newsgroup last week. The comment addressed the three fundamentals of education: what do we teach? why do we teach it? and when do we teach it? It also reminded me of the distinction between learning and educating. It seems at times like the "education" fan club forgets how often learning takes place outside of the world of education, and the "self taught" crowd loses sight of how helpful schools can be in guiding an individual's learning.

Mathematics is not unique insofar as how it is learned or how it is taught. Mathematics is unique in how it is perceived. For some crazy reason society places the burden of initial education in mathematics in the hands of elementary teachers. That is not to speak down on elementary teachers. On the contrary.

Consider, for a moment, foreign language education. In most places in out society foreign language instruction is held off until the post-elementary years. Why is that? Most likely it is because of the impression that the typical elementary teacher is not skilled (trained?) adequately to teach a foreign language. We hold off French classes, don't teach Spanish, delay Chinese instruction, and so on because individuals skilled (trained?) to teach them are assigned only to older students in post-elementary years. In some parts of the country there is a push to get these teachers into the elementary schools, where, in isolated instances it was done, and then undone because of budgetary constraints.

Take a moment to recognize that musical instrument instruction has acquired such a respect that most students are taught individually or in small groups by a special instructor. Ever come across an elementary teacher required to teach students how to play the clarinet?

Yet, society operates on the supposition that elementary teachers will have the skills (training?) to instruct in mathematics. The same subject which people disdain with the "I hate math" and "I am no good at math" is being introduced by elementary teachers with no more training in mathematics than they have in any other subject. 

Suppose we had a "race to the moon" approach in mathematics education. Imagine placing enough math specialists in elementary schools and adjusting priorities so that students would all leave elementary schools with the solid grounding necessary for handling secondary math. Imagine elementary schools leaving elementary school with a number sense so strong and a spatial perception so good that middle and high school algebra and geometry classes become common-sense subjects to them. 

Some students slide through mathematics in school with ease and it can be easy to fall into the trap of saying that all should be able to do it if only they would try harder. Quite often academic success is aided by support the student gets outside of school, and a good learner with some motivation and a good mentor (or sometimes all alone) can succeed even in the absence of a school or teacher.  But those are the exceptions, and we should not use them to support a less than adequate system.

Here is part of an article by John A. Dossey from The Arithmetic Teacher from 1984:

Please take note that the issue of improving the teaching of mathematics in elementary schools has been around for a long time. The vast majority of those who were teaching when the article referred to here was published have retired by now. yet the issue still exists.

We have spent at least 35 years recognizing this need and nothing has really changed. Maybe it is time to create a grass-roots movement making the case that any and all students in our society deserve the absolute best we can give them and not accept anything less.

If we were giving the students the absolute best, the scene described in the opening quote here would probably not take place: solid fundamentals create a mind where mathematics makes sense, and if something makes sense, it will be remembered. Plus, if all students were more solidly prepared, the geometry teachers would have the time to incorporate exponential functions into their courses.