Teaching

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.

Here is another installment from my review of the June 2012 New York State Math Regents exams.

On the left is a problem from the Algebra 2 / Trigonometry exam; on the right, a problem from the Geometry exam.

Notice that no scale is indicated on any of the graphs here.  That is, there is no indication of what “one unit” is equivalent to on any graph.

I admit that I’m pretty sloppy when it comes to labeling graphs, however I know some teachers make accurate labeling and scaling a point of emphasis when it comes to creating graphs.  Properly understanding the scale of a graph can be of crucial importance, especially when trying to with questions pertaining to specific numeric values, as in these above.

Tests should stand as models of mathematical content and practice for students; they should not reinforce bad mathematical habits, like ignoring scale.

I know very little about early childhood education, but have recently started to think more about it.  I greatly enjoy interacting intellectually with my nieces and nephews and find it fascinating to explore ideas like fractals, infinity, and ordering with them.  But I don’t really know anything about the theory of how children learn, what they should learn, or when they should learn, mathematically, or otherwise.

To begin exploring the idea, I thought about possible fundamental questions and eventually settled on this:  What are some important content-independent skills that children need to learn?

I posted the question on Google+, and Don Pata, MrBombastic, Jim Wilder, and Christopher Danielson all offered some great ideas.  Here’s the list we compiled through discussion, in no particular order.

Problem-Solving Perseverance — the ability to sustain focus and work through a problem to the end

Intellectual Discipline — the willingness to focus and invest energy on learning and development

Number Sense — an intuitive understanding of quantity:  magnitudes, relationships, and scales

Reflection — the ability to objectively self-assess, refine, and adapt

Communication — the ability to express information and emotion in a variety of ways, and appropriately interpret and process the expressions of others

Courage —  the willingness to fail

Curiosity — the habit of inquisitiveness and the ability to ask good questions

A good list to start with!  Thanks for all the help, and if there are other suggestions, please feel free to leave them in the comments.

You can see the original thread on Google+ here.

Here is another installment from my review of the June 2012 New York State Math Regents exams.

Below are a few examples of what I consider “bad” questions.  “Bad” here might mean poorly worded, poorly conceived, or irrelevant.  In addition, there is an example of a question with a problematic rubric.

First, a type of problem that occurs regularly, one that is a pet peeve of mine.  From the Algebra 2 / Trig exam:

The concept of “middle term” is artificial and depends entirely on how one chooses to evaluate the given expression.  This question does not test an authentic mathematical skill; it tests how well a student executes one particular method of evaluating this particular expression.

Next, an example of a poorly-phrased question, one that confuses mathematical terminology.  From the Integrated Algebra exam:

To “solve” a system of equations, one must find the ordered pairs that satisfy the given equations.  Apparently this question wants only the y-values of those solutions, but the phrasing confuses what it means to “solve a system” and to “solve an equation”.

Students can probably figure out what the question-writer wants to hear in this case, but the lack of precision will only exacerbate confusion about the word “solve”.

Here’s a problem on the Algebra 2 / Trig exam that is simply irrelevant.

This question tests one thing, and one thing only:  knowledge of an arcane and largely irrelevant notation, namely, degree-minute-second representation of angles.  Would anyone outside the nautical or astronomical worlds consider this even remotely valuable?

Lastly, this question from the Integrated Algebra exam is formulated in a reasonable way, but the official scoring guide poses some unnecessary problems.

This question asks the student to graph an equation and then, using the graph, determine and state the roots of the equation.  The correct answer is “2 and -4”, and with appropriate work, is worth three points.

However, if the student gives the answer “(2,0) and (-4,0)”, the student can only earn two out of the three points.  So if the student gives the coordinates of the points where the graph crosses the x-axis, rather than names the “roots” of the equation, there is a one-third deduction.

While I believe that the distinction between roots and points is important, losing one-third credit seems seems unnecessarily punitive here.  If we want to test student’s knowledge of vocabulary, there are better ways to do it than by sneaking it in at the end of an involved algebra problem.

Moreover, since the question requires that the student use the graph, the student is already being forced to interpret the problem in a geometric context.  Penalizing them for thinking of the roots geometrically, then, doesn’t quite seem fair.

Here is another installment from my review of the June 2012 New York State Math Regents exams.

Mathematically erroneous questions consistently appear on these exams.  Here are two recent examples, both from the Algebra 2 / Trigonometry exam.

According to the scoring key, the correct answer is (4).  This would be the correct answer if the angle were given as -50 degrees.  Notice, however, that no degree symbol is present.  This means the angle is actually -50 radians.  In degrees, -50 radians is equivalent to roughly -2864.8 degrees, which itself is equivalent to roughly 15 degrees.  Thus, the actual correct answer is (3).

The above problem could be considered a typo (although no correction was ever issued), but the most erroneous Regents questions demonstrate a real lack of mathematical understanding on the part of the exam creators.   Consider the following question on complex numbers.

None of these answers are correct.

The exam writers believe that (3) is the correct answer.  Given a complex number a + bi,  the conjugate is indeed a – bi, provided that a and b are real numbers.  But x is a variable, and there is no reason to assume that x has to be a real number.  If x = i, for example, (3) is not the complex conjugate of  -5x + 4i.  In this case, the conjugate of the original expression is i, while (3) evaluates to – 9i.

As emphasis on standardized exam performance continues to grow, a few points here or there can make a big difference in the lives of students, teachers, and schools.  The consistent appearance of erroneous mathematics on these exams calls into question their validity as a measurement of “student achievement”.