From circles to ellipses?

Many people seem to think of circles as simple things and ellipses as "oh, that's math and I was no good at it...".

Here is a bit of a perspective on that created in GeoGebra.

If you know nothing of ellipses beyond "oval", start by just using the control buttons at the top. If you are comfortable with ellipses, but a bit of a skeptic, experiment with the sliders as well. If you really want to know what is happening, download the complete file here.

Regardless of where you stand, GeoGebra does give a great dynamic approach to mathematics.

Let's make mathematics great again...

This is a magical question from the New York Regents Exam in Geometry (Common Core) of June 2016. Based on a NYSED decision, both (1) and (3) are accepted as correct, which means that item III is both always true and not always true. Magic!!!

For the visually-motivated. here is a quick GeoGebra file based on this question.. Unfortunately, due to space limitations, I had to put limits on the dilation scale factor and the translation components (computers do not like to accept infinity as a number!)

This question was one of 3 in this exam in which multiple answers were deemed acceptable (after initial scoring, grading, and graduations were all over.)

I have two concerns over this fiasco:

1. A student seriously confused by question 14 (not be the mathematics) might have, in following the directions to indicate the BEST choice, left the answer blank. That would have been correct, as there cannot be multiple BEST answers. They would get no credit, yet a student who was clueless and put down a random selection would get credit.
2. Many teachers use old regents exams as study/review/practice for exam time. What will NYSED do to ensure that future use of these exams is predicated on information regarding these mess-ups? It is now three questions (14, 22, and 24) on this exam that have been acknowledged as invalid.

Be aware that on question 14 two answer selections are accepted as correct, on question 22 all 4 answer selections are being accepted as correct, and on question 24 all 4 answers plus a blank are being accepted as correct. See these links: question 24, questions 14 and 22

The Cubic Regression Rules!

The actual exam for the June 2017 New York Algebra 1 (Common Core) has not been released, and teachers are strictly forbidden to communicate via the web on the exam until after June 30, so as I begin my yearly review (as a NY citizen who happens to be a retired teacher), all I can comment on is the conversion chart. Here I have plotted it and included (dotted line if you look closely) the cubic regression curve based on the data contained in the conversion chart itself.

I have always looked at level 1 as the failing range, and level 2 as the "safety net" where Special Ed passes but non-Special Ed does not. That might have changed by now. Needless to say, a regular ed- student has to get to level 3 to be considered "passing"

I noticed that, in comparison to the conversion chart for last June, every raw score from 4 to 21 received a scaled score 1 or 2 points higher this year.  No raw score from 24 and up got any boost this year, and a few lost a point on the conversion side.

Here are June 2016 and June 2017 plotted together:

Needless to say, the cubic regression is pretty much the same. I do wonder why NYSED prefers it so much.

Here I have added (in green) the conversion chart for Algebra II from 2016:

Ever since NYSED started using the raw-score to converted-score approach to scoring these exams, there has been battle between those who liked the old "straight-line" approach and NYSED itself. In case you are wondering, the straight-line approach is based on the concept that if you answer 50% of the test correct, your score is 50. Answer 75% correct, get a 75, etc. If I added the graph of the straight-line approach, it would be a straight line from bottom left (0,0) to upper right (86, 100). 

What the heck- here it is:

It is easy to see that almost everywhere the scaled score exceeds the 'straight-line" score, with a minimal gain at the top end, and increasingly large gains as one heads from the bottom towards that magical mystical number known as "65".

I am partially from the "old school". When I took the Regents Exam in algebra 1 (I believe it was properly called "9th Year Mathematics" back then) the exam consisted of 30 short-answer questions (2 points each) and 7 Part 2 ten-point questions, of which you had to answer 4.  There was a bit of a top end hammer since a student capable of answering all 7 part 2 questions could get credit for no more than 4 of them. (If a student answered more than 4, only the first 4 answered would count)

Thinking back to those tests, the typical 9th Year Mathematics exam contained a dozen multiple choice questions, each with 4 choices. The random guesser would, on average, get 3 correct out of 12, contributing 6 points towards passing. That student would have to earn 59 points out of the remaining 76. In last year's Algebra I exam there were 24 multiple choice questions, with 4 choices each, so the random guesser would average out with 6 correct, for 12 raw score points. To get to the minimum passing score that student would need 15 out of the 36 remaining possible points. Which is more difficult, 59 out of 76 or 15 out of 36? Hard to tell, recognizing that a student who has to guess on all of them probably doesn't know much of the course.

The other end does intrigue me: the student who gets all the multiple choice correct, be it by guessing or by knowledge or some combination of the two.

In 9th Year Mathematics the multiple choice got you 24 points on your way to a minimum passing grade of 65. In Algebra I last year, the multiple choice gets you 48 points on your way to a minimum passing score of ....27. For that matter, a student who can successfully answer 11 out of the 30 multiple choices gets 22 points. Now that student guesses  on the other 13 multiple choice questions will, on average, get 3 or 4 correct. Even at 3, those student now has 6 more points, and has now earned a passing raw score of 28. the student has "passed" the exam without going beyond the multiple choice questions. If that student had guessed by filling in choice 2 or choice 3 for all guesses, he/she had a solid advantage. (There were 5 answers of "1", 7 answers of "2", 7 answers of "3", and 5 answers of "4". I will have to see if it is a trend to put correct choices in the middle slots.)

What does appear to be a trend to me is a trend to make exams in such a manner that passing scores are in easier reach while top scores are harder to obtain. Some may say it appears to narrow the "achievement gap", but appearances can be deceiving.

Bezier for Young People

Bezier Curves are generally not confronted by k-12 students at all. Their equations can be complex, and in the absence of dynamic geometry, the topic in general can be mind boggling. Historically, algebra has been the driving force for graphing. With dynamic geometry, that can be reversed.

If you can mentally stomach (how's that phrase?) the idea of a point on a line steadily sliding from one location on that line to another location on that line, you've mastered the necessary skills. Visually, a point sliding from one place to another would look like this:

Being able to picture this is all you need to know. Here it is

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.

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.

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"?

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.

Home, home on the SPREAD, where the deer and the antelope play....

I truly wish that New York had higher quality exams, as they are intended to be used to measure not only student performance but teacher performance as well.

The analysis continues with question 20 from the January 2017 regents exam in Algebra I (Common Core)

At issue here is choice (2), that refers to the spread of the data. Here, in Algebra I, the word spread should not be used.  Spread can be represented many different ways, from range to interquartile range to standard deviation. To the best of my knowledge, standard deviation is not part of the Algebra I knowledge base. Even so, range and interquartile range do not go hand-in-hand: it is possible for a set with a smaller range to have a larger interquartile range.

This choice should most probably have used the word "range" instead of spread.  For the record, the ranges are equal in these two sets, but the interquartile ranges are not (7 to 10, or 3 years for soccer players and 9 to 11, or 2 years for basketball players. The standard deviation for soccer is 2.05798 and for basketball is 1.81137. So using range, choice (2) is false, while using the other two measures, choice (2) is true.

Could it be the case that the word "spread' was used when the word "range" should have been used?

New York has to make better tests!

The above is the first section of a question from the January 2017 New York State regents exam in Algebra II (Common Core).

I believe the question needs the word "tsunami" rather than "tidal".  The presence of the word "tidal" in this context illustrates the need for improved "proofreading" in the creation of these exams.

Continuing on, here is question 24 from the New York State Geometry (Common Core) regents exam:

I suspect that the word "cone" here should read "inverted cone". When used by itself, the word "cone" refers to this:

Rarely have I seen a water cup used "point up". Let me correct myself: I have never seen a cup used that way.  Should a student solve the question as written, they could be perfectly correct and get an answer not listed. That situation should be avoided at all costs on a state exam.

Let's look at question 8 from the January 2017 NY regents exam in Algebra I (Common Core):

I found this question misleading, since the USPS charges 49 cents for up to 1 ounce and 21 cents for each extra ounce or fraction of an ounce. The best mathematical model would be 

\({\rm{Cost}} = 49 + 21(w - 1)\)

where w is the weight of the letter in ounces and the costs are measured in cents.

I suspect the question writer was trying to come up with a "real world' application of recursive functions. My advice would be to look again. Question 20 on this test would have made a much better model, as postage must take into account portions of an ounce but mp3 sales would not.

Now comes question 14 from the same Algebra I exam;

The mathematics in this question is basically asking "Which of the following is equal to 6(16)^t ?" The rest of the verbiage is due to the attempt to make the problem "real world".

Can't we just ask math questions to test math knowledge? 

Clarification needed!

Here is question 24 from the New York State Algebra II (Common Core) Regents exam from January 2017. Please look at it closely!

Now that you have read it carefully, take note that the domain of this question seems to run from -2 past 5. Also take note that if x is less than 1 or greater than 5, one of the sides must have a negative length. Since lengths cannot be negative, this graph can NOT be a model for the volume of a box, hence the question cannot be answered. 

A few thoughts to ponder....

Here is a question from the New York January 2017 Regents Exam in Algebra II (Common Core).

Please read it carefully, then answer some questions below.

1) How many rabbits were there four weeks ago?

2) What do t and P(t) have to do with it?

3) Suppose a student thought as displayed in this chart. Would they get it right?

now	5 rabbits
in 28 days	10 rabbits
in 56 days	20 rabbits
in 84 days	40 rabbits
in 112 days	80 rabbits
In 98 days	Between 40 and 80

On a different note, here is question 21. Read it carefully, then answer a couple questions below.

1) Can you tell me who gets away with no credit card payments for 73 months? Could I stretch it out another 300 months?

2) If this is supposed to be a "real world" question, can you tell me what world that is?