“On two occasions I have been asked [by members of Parliament], ‘Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?’ I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.“ – Charles Babbage
This quote reminds me so much of many of my students who just can’t understand how they got an answer wrong because, “That’s what the calculator said the answer is, so it has to be right,” with no consideration whatsoever about whether they entered the information correctly in the first place. Especially when we are working with real-world data and applications, I ask them to estimate the solution first, even if it’s as simple as, “Should the solution be positive or negative?” or “Will be solution be greater than or less than 100?” Knowing where mistakes will occur helps guide my questions.
One of the most frequent errors occurs when students have substituted negative numbers that are raised to a power. They don’t remember that parentheses are needed to obtain a correct calculation. For example, recognizing the difference between and is tricky for them. Most often this occurs when substituting values into an expression. To combat this problem, I tell my students to always recopy any problem and put parentheses in FIRST before substituting. For example:
would look like this - 4( )( ) THEN substitute the given values for a, b, and c. This has helped eliminate at least some of the errors that occur when raising a negative number to a power.
Calculator use is so inconsistent in my district. The elementary schools don’t use them much, the middle schools use them some (the middle school state assessment has a “calculator section” and a “no calculator” section), Algebra I students are required to use a graphing calculator for the high school assessment. The students go on to Geometry, Algebra II, Trig, and Calculus and use a graphing calculator all the time. We offer college classes to our seniors, but to qualify they have to pass the college’s placement test, which does not allow the use of a calculator. We have trig students who can’t pass the basic math portion of the placement test and therefore don’t qualify to take College Algebra or Prob & Stats because they’ve come to rely on the calculator and forget how to do basic math.
I’m all for calculator use in the classroom, but I believe that basic skills needs to be reviewed throughout high school and that estimation must be a huge part of our instruction as well.