Teaching negative exponents

Continuing on with the exponents, I have another video about negative exponents. Check also the "prelude" to this, my video on zero exponent.

negative expon! ents: Learn Them with a Pattern!

In books we usually find the definition that x⁻ⁿ = 1/xⁿ or perhaps x⁻ⁿ = (1/x)ⁿ (you take the reciprocal of x and use the opposite of the exponent).

But again, I want to highlight that we DON'T have to just "announce" to students this definition or how negative exponents are done. We can justify it. In the video, you can see this done first with the help of a pattern, where we go from the positive exponents to zero exponent to negative ones.

I also show another way of showing your students why negative exponents have been defined the way they are. If you look at the shortcut for dividing powers with the same base (I used 6³/6⁵ in the vide! o), we can simplify as usual and arrive at 1/6². Ho! wever, t he shortcut says we can just subtract the exponents, so 6³/6⁵ also equals 6⁻². The same reasoning applies for any non-zero x and any positive whole number exponent.

This way, we teach our students not only the "how" but also the "why" of mathematics, and help them become better thinkers. Mathematics is not just about memorizing things and then learning to "spit" them out at tests - it also has a LOT to do with proof and logical thinking. And surely our kids need that today, more than ever.

negative exponents

The LA TIMES Cracks Out of Turn When It Doesn't Know The Shot

David Mamet*

I'm not sure what I was doing or where I was the day the LOS ANGELES TIMES went from being a newspaper to being a national leader in evaluating teacher quality. Perhaps it was supposed to be kept secret, like what was discussed and said at the meeting Dick Cheney had with Big Oil executives a decade ago. If so, several  TIMES reporters have blown it with a recent article, WHO'S TEACHING LA'S KIDS?

In it, three reporters, Jason Felch, Jason Song and Doug Smith, present ratings of "teacher effectiveness." In particular, they single out one particular fifth grade teacher, John Smith, and claim he is the least effective teacher for his grade level in his school. Mr. Smith's photo is at the beginning of the article that purports to know, based on kids' scores on administrations of a single standardized test, which teachers are helping their students, teaching effectively, and making a positive difference, and, of course, which, like Mr. Smith, are allegedly failing to move their students ahead. 


I wrote the following to these reporters today and will be fascinated to see if any of them respond. I know that were I John Smith, I'd be speaking to my attorney and considering lawsuits against several parties, not the least of whom are Jason Felch, Jason Song, and Doug Smith. As I am not, the best I can do is try to point out how wrongheaded, how irresponsible, and how ultimately counterproductive is both their article and the methods they employ to smear the professional integrity of many fine teachers who for any number of reasons may not "measure up." The professional integrity I call into question, however, is that of these reporters, their editors, and others who profit from the publication of this sort of cheap-shot, ignorant journalism. 

If you start with the absurd assumption that multiple-guess
standardized test scores tell us anything (let alone EVERYTHING) we
need to know about teacher effectiveness or student learning of
subject matter or all the other things that teachers and schools are
about (not all of which are good, but that's another debate entirely),
then it follows that the LA TIMES is as qualified as anyone else with
no expertise whatsoever in psychometrics to determine which teachers are "most
effective" and which are "least effective." Further, with the same
starting assumption, there's nothing unconscionable about reporters
and editors  from that noble publication choosing to print a photo of
a so-called "ineffective" 5th grade teacher and include the following
in the article:

Yet year after year, one fifth-grade class learns far more than the

other down the hall. The difference has almost nothing to do with the

size of the class, the students or their parents.

It's their teachers.

With Miguel Aguilar, students consistently have made striking gains on

state standardized tests, many of them vaulting from the bottom third

of students in Los Angeles schools to well above average, according to

a Times analysis. John Smith's pupils next door have started out

slightly ahead of Aguilar's but by the end of the year have been far

behind."

But if the assumption is false, then what the TIMES and its reporters have done is to pillory one 5th grade teacher on the wheel of meaningless test scores. They have, in fact, violated two  fundamental principles of psychometrics: never use a test designed to measure one thing (e.g., student achievement) to measure something it was not designed to measure (e.g., teacher effectiveness), and never use a single test score or measurement type to draw definitive conclusions (particularly not in the social sciences). Further, they have made the fundamental error of assuming that correlation (Teacher A's kids scores are higher than Teacher B's scores) equates with causation (Scores rose primarily BECAUSE of the superior teaching skills and methods of Teacher A).

In fact, the above-cited article is so fraught with error and leaps of logic (and bad faith) as to be utterly, irredeemably worthless, not unlike the test scores upon which its false (and probably libelous) conclusions are based. But then, the article's authors began with a patently incorrect assumption, and  they very likely had its conclusions well in mind to begin with.

So I am moved to ask: may we expect in the near future an article by the same reporters on which LA TIMES journalists are "most effective" and "least effective" based on how sales of the paper are impacted by their articles and reportage?

May we expect that the reporters will be taking, say, the tests given to high school kids in LAUSD (I assume all these journalists graduated from college) or perhaps the SAT or ACT (or, Darwin forbid! the GRE) and publishing the results in the paper? How about the politicians who pushed and voted for using these tests as fair measures of a host of things they were never designed to assess? (And I include in that list not only state and local officials, but every US senator, congressperson, US Department of Education secretary, and every US president from William Jefferson Clinton to George W. Bush to Barack Hussein Obama who has supported these tests as the measure of all things.)

Let's shed more sunshine on the test-based competence of our reporters and politicians. Publish and publicize their scores. Threaten, meaningfully, to hoist these folks by their own petard and we'll see some critical examination of the assumption that the tests are valid and reliable, as well as adequate measures alone of effectiveness or lack thereof. My suggestion is no worse than what politicians and reporters are doing now with kids, administrators, schools, districts, and, of course, everyone's favorite scapegoat, public school teachers.

Until such time, shame on Messrs. Felch, Song, Smith; shame on the LA TIMES. Shame on anyone and everyone who buys into the ridiculous test-mad nonsense that has this country by the throat.

Michael Paul Goldenberg

p.s.: Lest someone suggest otherwise, my last official GRE scores, taken in October 1991, are Verbal 800; Mathematics 780; and Logical Reasoning 720. I feel safe in putting those scores up against those on any comparable test of the three reporters, any member of the LA TIMES staff, any current legislator in California or the United States Congress. I've spent over 30 years preparing students for various standardized tests and debunking many of the myths surrounding them. I'll happily meet anyone on the standardized test battle ground, No. 2 pencils aready at dawn or high noon.

*For those wondering what David Mamet's photo is doing at the beginning of this blog entry, it has to do with the title of my post. Mamet is very fond of the language of con artists. Apparently, our intrepid LA TIMES reporters are not unfamiliar with both the short and long cons. Or perhaps it's just their editors, the publisher, and others with vested interests in destroying US public education.

measurement math problems

Grammar-Land Worksheets Part 1

I've mentioned one of the free resources we are using for grammar in a previous post, but I made some worksheets today to go along with the book and I wanted to share them with you all.  Grammar-Land by M.L. Nesbitt is a wonderful book, and a perfect choice for Charlotte Mason inspired families.  You can view it online or download the pdf file, or buy the book.  We read it on my Sony Reader (the best homeschooling investment I have made!)

At the end of each chapter are some exercises woven into the story.  I've taken those exercises and turned them into worksheets.  Without something printed out and ready to go we tend to forget to do the exercises and I really think they are helpful in cementing concepts.  So far I have 6 worksheets completed.  I believe there are 17 chapters in the book, and I hope to have them all completed by the end of the week.

Worksheet 1 for Grammar-Land or, Grammar in Fun for the Children of Schoolroom-Shire by M.L. Nesbitt (by Je...

Worksheet 2 for Grammar-Land or, Grammar in Fun for the Children of Schoolroom-Shire by M.L. Nesbitt (by Je...  

Worksheet 3 for Grammar-Land or, Grammar in Fun for the Children of Schoolroom-Shire by M.L. Nesbitt (by Je...

Worksheet 4 for Grammar-Land or, Grammar in Fun for the Children of Schoolroom-Shire by M.L. Nesbitt (by Je...

Worksheet 5 for Grammar-Land or, Grammar in Fun for the Children of Schoolroom-Shire by M.L. Nesbitt (by Je...

Worksheet 6 for Grammar-Land or, Grammar in Fun for the Children of Schoolroom-Shire by M.L. Nesbitt (by Je...  

math 4 today worksheets

brains in a vat

"I want now to ask a question which will seem very silly and obvious .., but which will take us to real philosophical depths rather quickly. [..]
The argument I am going to present is an unusual one, and it took me several years to convince myself that it is really right. [..]
It first occurred to me when I was thinking about a theorem in modern logic, the Skolem-Löwenheim Theorem ..."

Hilary Putnam about brains in a vat and related thoughts about the mind-body problem.

kind of sets

If you are looking for help or directions at NYSML

Here's how to find the people who can help you.

Our small army of volunteers will be wearing distinctive badges like the ones on the table above, made by our awesome sponsor Tough Traveler, in their Schenectady factory.



Some of them, like our Lead Zone Proctor Alexandra Schmidt, a math teacher from Hebrew Academy of the Capital District, also like to wear math t-shirts, so they'll be wearing the same light blue Albany Area Math Circle T-shirts that our students wear to help them stand out doubly from the crowd.



Our UAlbany student guides will carry pennants that look like this:



If you are looking for definitive answers to those all important questions like "Where is our team's classroom?" or "Where is the gym?" or "Where is the restroom?" or "Where am I supposed to be right now?" or "The light went out in our classroom? How do we get it turned back on" or "How long until lunch?" or "Why is there no proctor in our classroom yet?" or "I am an alternate, but I don't remember what team I'm supposed to be working with?" these folks can help you.

If your question relates to the UAlbany campus (like location of buildings and restrooms), your campus guides with the purple and gold Athena pennants are the best go-to p! eople. If you question relates to the NYSML contest (like "Where am I supposed to be right now?" the folks with the Tough Traveler badges are the best bet.) For the most challenging questions, look for volunteers like Alexandra who are wearing light blue math circle shirts AND Tough Traveler badges.



These awesome volunteers will either know the answer to your questions OR they will know how to contact someone who can answer them!

Thanks to all our great volunteers.

help with math

Pre-Algebra and My Dear Aunt Sally

Will you Please Excuse My Dear Aunt Sally? It seems she had indigestion and burped. Can you find where she is? This is how I had fun publishing my Unit on Pre-Algebra! This unit includes in-depth lessons on:

1. Order of Operations
2. Order of Operations with Exponents
Order of Operations with Integers
3. Writing AlgebraicExpressions
4. Writing AlgebraicEquations
5. Practice Exercises
6. Challenge Exercises

So can you find my Dear Aunt Sally? And will you excuse her?

factor tree calculator

I Love Me Some ExamView

I spent the last two days on school business building a set of common formative assessments in ExamView that are aligned to my department's skills list for our algebra and pre-algebra classes. Many of the questions are dynamic which means that for any given question type, multiple versions of the question can be created by the click of a button. As I was looking through the question banks, I realized that there were some skills that didn't have questions that fit what we will be trying to assess. Solution? Build it yourself.

The process is a bit tedious at first, but I was able to use previously made questions to figure out how to build a simple question regarding slope.

Step 1: Define your algorithms.

Step 2 : Create a graph using pre-defined variables.

Step 3: Define the distractors.

Yeah, gotta say I love this feature. Finding questions with good distractors is tough at best. But if you decide on an algorithm based on common misconceptions, you're good to go. In this problem, I d! ecided the common misconceptions for students calculating slop! e based on a graph were:

• Using Run/Rise instead of Rise/Run
• Confusing negative with positive slope
• Confusing the y-intercept with the slope

Of course, there may be some discussion on what the best misconceptions would be, but for the sake of practicing, those are what I went with.

Step 4: Decide how many of these type of question you want in your assessment.

I'm glad that I don't have to build all of these from scratch, but I like the fact that filling in the holes isn't too tough.

finding the vertex of a parabola