What happened?

Whenever I travel I try to take as many photos as I can that are worthwhile but not find-able all over the web. No sense taking pictures that have already been taken. I look for unique pictures. Even more so, I like them to be though-provoking.

This one I took while walking on a dock in St. Thomas, USVI, a bit over a year ago. I can tell you that the hat was floating, although some people to whom I have shown the picture thought I had caught it in midair. That led me to consider how we describe what we see, and how we extrapolate in the process.

Anyone who describes the hat as in midair must presume some person, or some thing, as the thrower, unless perhaps it was windblown.  Even knowing it is floating, the question could still be asked: How did it get there?

So, if I was to ask you to write the history of this hat, as best you can, how would you respond?

FOILed again?

The statement in red here is one I came across while taking another look at Common Core standards. I decided to Google the statement. I did discover that Google will not let you use more than 32 words between quotes, so I Googled just the 1st sentence in quotes (so Google looks for an exact match), and got 14,100 hits. Listed below are just a few of them.

There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y).

2014 Mississippi College- and Career Readiness Standards for Mathematics

https://www.illustrativemathematics.org/standards

http://www.inghamisd.org/downloads/iisd_curriculum_math/feb_2013_cd_update_20130306_154552_1.docx

State of Nevada Dept. of Education

It's a bit misleading. The second sentence says "may have a better chance". Anyone who really and truly understands what binomial multiplication is and why it is, WILL have a better chance. A much much better chance.

It's also interesting that the statements refer to "a mnemonic device" and gives it a bad flavor. I suspect the device the authors had in mind was the old FOIL method for binomial multiplication, and the impetus behind the statements was the knowledge that there are teachers who taught only FOIL, and nothing more. (Could it be in use here only because F-O-I-L actually spells a word in English? Could this be why US math is falling behind?!? Imagine!?!) Do you know the meaning of FACE? Does it help or hinder in music studies?

I find it odd that a statement which embodies such a major thrust in educational philosophy is phrased in the context of binomial multiplication.  Multiplying binomials is something most people never do in their lives. However, if binomial (and polynomial) multiplication where actually dealt with carefully and fully they would be recognized as the foundation behind most algorithms for multiplication. Multiplying a pair of two digit numbers is actually a binomial multiplication. Multiplying 564 and 37 is actually multiplying a trinomial and a binomial. (I find it interesting that my spell-checker just marked "trinomial" as a misspelled word. For the record, it does not acknowledge "pf" as a possible error.)

The 4th grade standards include this: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

The arrays and area models referred to here are exactly the same devices that are, or could be, used to describe the distributive process applied when multiplying polynomials. It would be nice if our educational system would recognize this and capitalize this, and perhaps gather 4th grade and high school teachers together to carefully comb out more connections. I an willing to bet that a substantial number of 4th grade teachers are not tuned in to the distributive rule as it relates to polynomial multiplication. A few visuals are here.

Whatever happens with Common Core, let's at least recognize that it doesn't have all the right answers, but it could be spurring people into asking the right questions. I hope.

GeoGebra no later than middle school...

This is just a continuing saga of options that teachers and students now have that I hope they are using.  with the technology we have now students can be introduced to new things in dynamic ways making them accessible as never before.

I just made this in GeoGebra as an example of check boxes in case I needed one at the AMTNYS conference presentation next week.

A Heart for Mathematics

Basic geometry as work. Circles with radii 1, 3, and 6, centered at the origin and 3 units below. Two points rotating counterclockwise at the same rotational speed, one clockwise at twice that speed. Those points are connected to form a triangle, and two angle bisectors shown. The trace shows the path of the point of intersection of those medians, the incenter.

There is a lot of mathematics in this diagram. I believe that creations such as these can be used with young students to "hook" them into mathematics long before they know the math needed to write equations for such graphs. Although the equation would be heavy into trig, the creation of the graph requires no trig at all. Basic geometric concepts such as circle, line, bisection, rotation are just about all that is needed. 

A tremendous amount of mathematics can be learned by setting kids loose with GeoGebra and a basic knowledge of the GeoGebra toolbar. In many respects, kids would probably learn GeoGebra faster than adults.

Animations and GeoGebra will be the topic of my session at the AMTNYS (Association of Mathematics Teachers of New York State) annual conference in Syracuse next Monday.

 

Floating Midpoints

This image consists of a visual recording made while using GeoGebra which was then converted to an animated gif and then resized to fit the blogger window.

If only...

I wish I had this ability when I was teaching. For more details on animations and GeoGebra, come to my presentation at AMTNYS in Syracuse on Nov. 10. Click here for details. the sketch is online here.

Avoid those errurs, I mean, errors.

During 30+years in the classroom, I developed a great appreciation for the impact of the careless error. Sometimes called "avoidable" errors, they crop up frequently in the mathematics classroom, and often are the root cause of a problem's seeming impossibility to the student solver.

Such an error is much more powerful in a student's math class than, perhaps, in his English class. Readers slide past spelling errors, often not even noticing them, while perhaps having a bit of a stronger jolt by a punctuation error. But even a tiny arithmetic error can turn a simple mathematics problem into an insurmountable task.

We may call them avoidable, because they can and should be avoided, and would be fixed if not avoided if we would only do such things as check our work, proofread, in short: be careful.

I have put together a small collection of such errors as they have appeared out in public. This collection is merely meant as a reminder to us all that double-checking our work is not a sinful task.

The first one here was snapped during a TV news broadcast, while the second was from a newspaper. Neither is a major error, but they both were avoidable.

Many many times the individual who made the error never notices it, which is why, I presume, the media has people on staff with the responsibility of checking each other's work. The reason I include them here is that they are evidence of a systemic failure in that regard.

I have put these and a few others on a web page, which can be found here. I pose the task to you to identify the errors. After all, somebody has to find them before anyone will fix them. That is the nature of avoiding an avoidable error.