The differences between test prep and academic tutoring, part I

Score improvement should be the primary goal of any test prep program.  Students will certainly learn concepts and ideas along the way, but that learning must always prioritize points over concepts.  This prioritization shapes fundamentally how I will approach a given concept with a student.  For example, a recent ACT featured a question similar to the one below:

First, look at the question from a math-content perspective.  The definition of a logarithm tells us that if x and b are positive real numbers and b does not equal 1,

log_b(x)=y is equivalent to b^y=x.

Using this definition, we can use our understanding of negative exponents to solve by computing the following:

So E is the correct answer.    

Next, let’s look at this question from a test-strategy perspective.  Our primary goal here is to earn the point, not necessarily to understand the mathematical concept being tested.  Under the MATH menu of the TI-84 Plus calculator is the logBASE option.  

We can use that function to enter the log as given, plug the answer choices in for x, and check each answer choice until we find the answer that yields -3, which will still be E.  In this case, the multiple-choice test format and the calculator have helped us bypass the actual math of the problem.

Which way is better if both approaches earn the point?  The best way to do a problem is the way that occurs to a student in the moment of the test and the way that will yield a correct answer quickly.   That approach will necessarily depend on the individual student and the individual question.  Some students spend many hours drilling logarithm and exponent rules, thus the content approach provides an instant, easy point for this question.  For most students, however, the strategy approach wins.  If a student without prior understanding of logs and/or negative exponents takes the time to learn those rules, the return on his/her time invested is low because these concepts appear infrequently on the ACT.  The same student can increase his/her calculator competency relatively quickly, so that student’s time will be better spent, and the return on investment of that time will be higher, by learning the functions of the calculator strategically.  

There are many ACT questions that can only be solved through the application of specific content knowledge and many that can be solved through a less content-centered and more strategy-minded approach.  The most prepared student can apply various strategies to various problems throughout the test to get the most total points.