My latest contribution to the New York Times Learning Network is a math lesson written in response to the editorial “Is Algebra Necessary?”, and contains a number of algebraic activities that can be explored using New York Times content and resources.
N Ways to Apply Algebra with the New York Times
Teachers and students can mathematically model mortgages, car costs, election predictions, and much more!
Algebra may not be necessary, but it sure is useful!
Here is a brief review of the August 2012 New York State Math Regents exams, as part of my on-going series examining the quality and validity of these tests.
Only the Integrated Algebra and Geometry exams were offered in August 2012, presumably due to financial constraints. Here are a few thoughts regarding some selected questions from those tests.
First, some issues that are probably minor to most, but important to me as a mathematician and a teacher. Here is number 26 on the Geometry exam.
What does it mean to “use” a point in drawing a triangle? Obviously, the author meant that the given points were to be the vertices of the triangle, but why not say that? Technically, I could use a point by drawing a line segment through it; as such, I could draw any kind of triangle using the given points.
Number 3 on the Integrated Algebra exam includes a particular pet peeve of mine.
There is no value of y in the given equation. In fact, that’s the entire point of a variable: it can take any value! What the author means is “What value of y makes the following equation true?” , so why isn’t that the question? In addition to asking a question that actually has a correct answer, this would also model for students the correct way to think about variables and equations.
Here’s another problem from the Integrated Algebra exam, number 36:
There’s nothing mathematically wrong with this particular question. However, the Integrated Algebra exam is the first of the high school mathematics exams in New York City. It is meant to be taken after an introductory algebra course, which means that most students will take this test in 9th or 10th grade.
Thus, it is strange to note that this question, on the 9th grade exam, is virtually the same as the highest-valued question on last year’s Algebra 2 / Trigonometry exam. The Algebra 2 / Trig exam is highest level math exam in the state. It is usually taken in 11th or 12th grade.
So where should this question be? Is it part of the 9th-grade curriculum? Or the 11th-grade curriculum? The authors of these exams seem a bit confused about this.
In addition, the August 2012 Geometry exam contained one of the most embarrassing math Regents questions I’ve ever seen, but I’ll address that in a separate post.
Here is another installment from my review of the June 2012 New York State Math Regents exams.
I tend to be rather critical in my evaluation of these exams, pointing out poorly constructed, poorly phrased, and mathematically erroneous questions. However, there have been some minor improvements of late.
First, it seems as though, in general, the wording of questions has improved slightly. To me, questions on the June 2012 exams were more direct, specific, and clear than in the recent past.
There were also some specific mathematical improvements. For example, although graphs were often unscaled, they seemed generally more precise, avoiding issues like this asymptote error.
There were considerably fewer instance of non-equivalent expressions being considered equivalent. The problem below avoids the domain-issues that plagued recent exams.
Perhaps it’s just luck, but we’ll give the exam writers the benefit of the doubt for now.
And the Algebra 2 / Trig exam definitely demonstrated a more sophisticated understanding of 1-1 and inverse functions, which is good to see in the wake of this absolute embarrassment from last year.
Perhaps someone has been reading my recaps?
Let’s hope we see continued improvement in the clarity and precision of these exams. If these exams are going to be play such an important role in today’s educational environment, it seems of utmost importance that they be accurate and well-constructed.
Here is another installment from my review of the June 2012 New York State Math Regents exams.
Below is a problem from the Integrated Algebra exam. Which of these graphs represents a function?
Have you identified the function? Well, you’re right, because all of these graphs could represent functions!
What the question presumably intends to ask is “Which of these graphs represents y as a function of x?” Under this interpretation, the correct answer is (1). But in (2), we see a graph that represents x as a function of y. So it, too, represents a function.
Indeed, even graphs (3) and (4) could represent parametric functions. For example, (4) could be written.
This plane curve is a function of t.
I doubt this makes much practical difference in the outcomes on this exam, but precision is important in mathematics; it should be modeled for students on official assessments. And those writing these important exams should be familiar enough with the content to write precise and accurate questions.
Here is another installment from my review of the June 2012 New York State Math Regents exams.
Below is a problem from the Integrated Algebra exam that highlights the artificiality of so-called “real world” problems.
In order to solve this problem in a high school algebra class, a crucial assumption must be made, namely, that every point on the target is equally likely to be hit. This means that the dart is just as likely to hit a spot near the bulls-eye as any spot near the edge.
Math teachers end up spending a lot of time training students to make these assumptions, probably without ever really talking explicitly about them. It’s not necessarily bad that we make such assumptions: refining and simplifying problems so they can be more easily analyzed is a crucial part of mathematical modeling and problem solving.
What’s unfortunate is that, in practice, students are kept outside this decision-making process: how and why we make such assumptions isn’t emphasized, which is a shame, because exploring such assumptions is a fundamental mathematical process.
Is it a reasonable assumption that every point is equally likely to be hit? Well, if the thrower is skilled, the dart is probably more likely to land near the bulls-eye. Would gravity make the lower-half more likely than the upper half? Discussing these, and other relevant factors as part of the modelling process can be engaging, fun, and highly mathematical.
But when standardized tests with “real world” problems are the focus of education, students usually end up getting trained to not ask these questions.
