Year: 2014

An unorthodox way of solving quadratic equations

This post concerns an unorthodox but logically correct technique for solving a quadratic equation via factoring. I showed this to some senior math majors as well as graduate students in mathematics; none of them had ever seen this before. Suppose that we want to solve

6x^2 - 13x - 5 = 0

without using the quadratic formula. Trying to solve this by factoring looks like a pain in the neck, as there are several possibilities:

(x + \underline{\quad})(6x - \underline{\quad}) = 0,

(x - \underline{\quad})(6x + \underline{\quad}) = 0,

(2x + \underline{\quad})(3x - \underline{\quad}) = 0,

or

(2x - \underline{\quad})(3x + \underline{\quad}) = 0.

So instead, let’s replace the original equation with a new equation. I’ll get rid of the leading coefficient and multiply the constant term by the leading coefficient:

t^2 - 13t - (5)(6) = 0, or

t^2 - 13t - 30 =0.

This is a lot easier to factor:

(t - 15)(t+ 2) = 0

t = 15 \quad \hbox{or} \quad t = -2

So, to solve for x, divide by the original leading coefficient, which was 6:

x = 15/6 = 5/2 \quad \hbox{or} \quad x = -2/6 = -1/3.

As you can check, those are indeed the roots of the original equation.

This technique always works if the quadratic polynomial has rational roots. But why does it work? I’ll give the answer after the thought bubble.

green_speech_bubbleThe original quadratic equation was

6x^2 - 13x - 5 = 0

Let’s make the substitution x = t/6:

6 \displaystyle \left( \frac{t}{6} \right)^2 - 13 \left( \frac{t}{6} \right) - 5 = 0

\displaystyle \frac{t^2}{6} - \frac{13t}{6} - 5 = 0

Multiply both sides by 6, and we get the transformed equation:

t^2 - 13t - 30 = 0

Although I personally love this technique, I have mixed feelings about the pedagogical usefulness of this trick… mostly because, to students, it probably feels like exactly that: a trick to follow without any conceptual understanding. Perhaps this trick is best reserved for talented students who could use an enrichment activity in Algebra II.

Opting Out of High-Stakes Assessments

In response to the growing movement of parents who have opted out of high-stakes testing, Michelle Rhee wrote a defense of the (commercial) enterprise in the Washington Post. This op-ed piece was brilliantly deconstructed, point by point, at http://curmudgucation.blogspot.com/2014/04/wapo-wastes-space-on-that-woman.html. I encourage you to read the whole thing. A few excerpts:

[Michelle Rhee]: No, tests are not fun — but they’re necessary. Stepping on the bathroom scale can be nerve-racking, but it tells us if that exercise routine is working. Going to the dentist for a checkup every six months might be unpleasant, but it lets us know if there are cavities to address. In education, tests provide an objective measurement of how students are progressing — information that’s critical to improving public schools.

Except that the current crop of Standardized Tests are not like stepping on a scale or going to the dentist. They are like trying to find out a child’s weight by waterboarding him. They are like having your teeth checked by a blind blacksmith. Because, in education, tests NEVER provide an objective measure of anything, because tests are made by people. Yes, tests are useful– but only good tests. And do you know what good tests are useful for? They are useful for providing information critical to helping further the education of students.

I am not a Systems True Devotee. STDs believe that we just have to create a well-oiled precision machine and it will spit out Smarterer Student Products like toasters off an assembly line. I would stop to further develop the point, but we’re only one paragraph in. These woods are dark and deep, but we have miles to go.

From this diving board, That Woman proceeds to register her stunned amazement that in various places, there’s a movement that is convincing parents to pull kids out of these tests! Really!!! These marvelous tests that will tell us how schools are doing!! What in the name of God are they thinking!?!?!!?

 

[Michelle Rhee:] This makes no sense. All parents want to know how their children are progressing and how good the teachers are in the classroom. Good educators also want an assessment of how well they are serving students, because they want kids to have the skills and knowledge to succeed.

Allow to help you comprehend this, O She. You are correct that parents and educators do want to know these things. Your mistake is in believing that they can only know this by looking at standardized test results.

Yes, the Great and Powerful Woman Who No Longer Has a Curtain To Hide Behind imagines a world where parents sit at home after eight months of school, wringing their hands and saying, “Oh, jehosephat, I wish we knew how Janey was doing in school. But we have no idea.” Meanwhile, at school, teachers sit and the lounge and say, “Yeah, I’ve been with this kid for eight months but I just don’t know how he’s doing. Thank God we’re going to be giving a high stakes high pressure badly written unproven standardized test soon so that I’ll know how it’s going.”

In That Woman’s universe, parents and teachers (sorry– public school parents and teachers) are dumber than dirt. In fact, the list of People Standing in the Way of Educational Excellence gets longer and longer. Parents, teachers, democratically elected school boards– reformy fans have an enemies list that keeps lengthening.

 

 

[Michelle Rhee:] We don’t need to opt out of standardized tests; we need better and more rigorous standardized tests in public schools. 

Yes!! When you’re doing something stupid and bad and non-productive, do it More Harder!!

 


[Michelle Rhee:] We also shouldn’t accept the false argument that testing restricts educators too much, stifles innovation in the classroom or takes the joy out of teaching. That line of thought assumes that the test is the be-all and end-all — and if that’s the perspective, the joy is already long gone. 

Here’s a multiple choice test for you, dear, exhausted reader. Select which statement best reflects the meaning of the above excerpt:

1) Do not assume that the test is the be-all and end-all. It will just be-all the way we decide to end-all teaching careers, school existence, and student futures.

2) You cannot claim that this year’s testing is sucking up all the joy of teaching, because we actually drained that lake long ago and killed the fish flopping in the mud with fire and big pointy sticks.

Jonathan Katz on Some Problems of Common Core Mathematics

Courtesy of Diane Ravitch:

Jonathan Katz taught mathematics in grades 6-12 for 24 years and has coached math teachers for the past nine years.

He prepared this essay for the New York Performance Standards Consortium, a group of high schools that evaluates students by exhibitions, portfolios, and other examples of student work. The Consortium takes a full array of students and has demonstrated superior results as compared to schools judged solely by test scores.

What is of special concern is his description of the mismatch between the Common Core’s expectations for ninth-grade Algebra and students’ readiness for those expectations.

Here is a key excerpt…

[The Common Core standards seem] to honor the idea of problem solving and the many ways a student might engage with a problem. It seems to value the process of problem solving, the ins and outs one goes through as one tries to solve a problem and that different students will engage in different processes.

To implement such a standard, a teacher would need to present students with problems that allow for and encourage different approaches and different ways to think about a solution—what we call “open-ended problems.” Yet, when you look at the sample questions from the Fall 2013 NY State document you would be hard pressed to find an example of a real open-ended problem. Here is one example in which a situation is presented and three questions are then posed.

Max purchased a box of green tea mints. The nutrition label on the box stated that a serving of three mints contains a total of 10 Calories.

a) On the axes below, graph the function, C, where C (x) represents the number of Calories in x mints.

b) Write an equation that represents C (x).

c) A full box of mints contains 180 Calories. Use the equation to determine the total number of mints in the box.

A situation is presented to the students but then they are told how to solve it and via a method that in reality few people would even employ (who would create a graph then a function to find out the number of full mints in the box?). If you are told what to do, how can we call this solving a problem? (This would have been a very easy problem for most students if they were able to solve it any way they chose which is what we do in real life.) In fact, all eight problems in the same of Regents questions follow the same pattern. Students are told they have to create the equation (or inequality or system of inequalities or graph) to answer the question. Thus there is no real problem solving going on—merely the following of a particular procedure or the answering of a bunch of questions. Why don’t we use problems where there is a real need for an algebraic approach? Why would we ask students to look at a simple situation then force them to use an algebraic approach, which complicates the situation? We should be helping students to see that the power of algebra is that is gives us the means of solving problems that we would have great difficulty solving arithmetically.

If we were truly trying to find out if our students are developing the ability to problem solve, we would never create questions of this nature. They would be more open-ended so students had the chance to show how they think and approach a problematic situation. But that can’t happen on a test where everyone is instructed to do the same thing so we can “measure” each student’s understanding of a particular standard. This is not real mathematics and a contradiction of the Common Core Standards of Mathematical Practice!

Why does this matter? The consequences are huge, and not just for students. Consider the message we are sending to teachers. Since students will be assessed on following given procedures rather than how they strategize and reason through a problem, then teachers’ lessons will become all about following procedures to prepare their students for an exam they must pass in order to graduate. This will simply perpetuate the same failing math teaching practices we had in the past, will compound the dislike that students already have for math class, and will not in any way help our students to develop mathematical thinking.

Source: http://dianeravitch.net/2014/04/02/jonathan-katz-on-some-problems-of-common-core-mathematics/

How the Texas Testing Bubble Popped

The Dallas Morning News recently ran a three-part long-form article on the passing of HB 5, which significantly rolled back the number of high-stakes exams that are administered in Texas. From the concluding paragraphs:

So in a relatively short time, a Legislature that had been the most all-in in the nation about high-stakes testing as the key tool for accountability became almost as all-out as federal law would allow.

As inevitable as it may look in retrospect, however, the shift was anything but at the time. Politics, policy and more than 30 years of history pushed hard against the change in course. As House Speaker Straus put it recently:

“We got as close as we could to something not happening, but it happened.”

HB 5 did not have my unequivocal support, as it removed the requirement that all high school students take Algebra 2 before graduating from high school. But, on balance, I think HB 5 definitely helps more than it harms.

Part 1: http://res.dallasnews.com/interactives/2014_March/standardized_tests/part1/

Part 2: http://res.dallasnews.com/interactives/2014_March/standardized_tests/part2/

Part 3: http://res.dallasnews.com/interactives/2014_March/standardized_tests/part3/

Statistics Done Wrong

I happily provide the following link to Statistics Done Wrong, a free e-book illustrating pitfalls when using statistical inference. From its description:

If you’re a practicing scientist, you probably use statistics to analyze your data. From basic t tests and standard error calculations to Cox proportional hazards models and geospatial kriging systems, we rely on statistics to give answers to scientific problems.

This is unfortunate, because most of us don’t know how to do statistics.

Statistics Done Wrong is a guide to the most popular statistical errors and slip-ups committed by scientists every day, in the lab and in peer-reviewed journals. Many of the errors are prevalent in vast swathes of the published literature, casting doubt on the findings of thousands of papers. Statistics Done Wrong assumes no prior knowledge of statistics, so you can read it before your first statistics course or after thirty years of scientific practice.

http://www.refsmmat.com/statistics/index.html

Statistical errors and their tendency to mislead

As a follow-up to yesterday’s post, here’s a recent article in the scientific journal Nature about the slippery nature of P-values, including a history about how reliance on P-values has evolved in the past 100 years or so: http://www.nature.com/news/scientific-method-statistical-errors-1.14700

While I’m personally familiar with many of the pitfalls mentioned this article, I have to admit that a couple of the issues raised are brand new to me. So I’ll refrain from editorializing until I’ve had some time to reflect more deeply on this article.