math-sample-lesson

A Little Lesson

Written below is a small lesson on standardized math problems in the hopes to provide a slightly better understanding of how I work with students on changing their expectations of standardized tests while developing their critical thinking skills.

The SAT and ACT math questions require a shift away from the kind of processing taught in school: “read all the directions before beginning, think about how to solve the problem, then once a solution has been envisioned then, and only then, can you safely write out the solution. Holistic solution thinking can become a very powerful limitation on our ability to think critically. Since early childhood, we are taught to never act before you know exactly what you are doing on a math problem--don't you dare!SAT-math

But the test’s engineers develop problems to specifically control that processing style. The goal is to have students look upward, stare at the page, or frantically scribble number nonsense, all while barely taking a breath for three to five minutes on a single problem. Ever happen to you before? It’s certainly happened to me. The mind either races or stagnates as students become highly frustrated and/or overwhelmed from taking in too much information at once.

If for the typical test taker, reading and solving problems holistically renders the progression of math questions totally counterproductive as the “difficulty” increases, then is there a different way to process that can bolster success on the difficult problems, ease the effort on the medium ones, and help ensure the easier ones have less silly errors? I think so. Mechanically, it is easy to do, but psychologically difficult to put into practice. This is how it’s done: begin to perform and thoroughly finish any calculation, organization of data, or visualization that you can as soon as you read enough information to do so (in other words, start doing calculations before reading the entire problem, let alone before reading the question). The first thing that comes into the mind when this processing style is the question, "How do I know that what I’m doing is right?" There seems as though there could be so much unnecessary effort. But the test engineers ensure that this is not so. Problems are carefully crafted to have extremely contained operations.


Let's compare these two approaches with an example.

Simply click on each step in the list below to see how we can process math differently:

  • “In a certain store, the regular price of a refrigerator is $600. How much money is saved by buying this refrigerator at 20 percent off the regular price rather than buying it on sale at 10 percent off the regular price with an additional discount of 10 percent off the sale price?”

    Feel it? That stretch to get your mind around all the information, let alone generate a solution? Alright, click on the next step and now let’s read just until we can do any math operation.

    First, the typical process(read all the way through without pause):
  • “In a certain store, the regular price of a refrigerator is $600. How much money is saved by buying this refrigerator at 20 percent off the regular price...”

    Now, a new process(read just enough to process one thing):
  • “...rather than buying it on sale at 10 percent off the regular price...”

    Finished...read a little more...
  • “...with an additional discount of 10 percent off the sale price?”

    Finished...read a little more...
  • How much money is saved  by buying this refrigerator at 20 percent off...rather than buying it on sale at 10 percent off...with an additional discount of 10 percent off the sale price?”

    Finished...read a little more...


So there are, give or take, four major operations in the problem. This problem is medium difficulty. A hard problem may have up to seven major operations. That means that the difference between a seventieth percentile and a ninety-ninth percentile is . . . three steps. How powerful is this play on our processing style? Add to the context a little rushing (the anticipation that there is or will not be enough time), and our typical style of processing math becomes an extraordinary hurdle for a test taker's psyche to overcome. Ironically, the “stuckness” typically absorbs far more time than the act of slowly going through each step.

There are many, many other behavioral and processing concerns, as well as a number of specific themes within problems, that should be learned and recognizable to the test taker. But, however heretical, there is certainly a cogent argument that a simple shift in a student’s mentality from reading entire problems holistically to reading them incrementally can vastly improve his or her scores.

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