Earlier this year, we had a post here on teachers as a problem or a resource (see http://www.devmathrevival.net/?p=1021). Technology — calculators in particular — presents another problem/resource discussion. Is the use of calculators a good thing, or an evil contribution to an ignorant population of math students?
For example, an article in USA Today mentions calculators as part of a discussion on math illiteracy related to pushing too much math too soon (see http://www.usatoday.com/news/opinion/forum/story/2012-07-09/math-education-remedial-algebra/56118128/1). I don’t usually cite a USA Today item, as the publication presents so many examples of bad statistics and mathematics. One line in this article did resonate: Nobody in a high school math class could tell the teacher what the answer is for 8×4 was — without using a calculator.
To some extent, we are still in the “back to basics” movement (basic skills). People who complain about calculators usually mention basic skills or facts as a goal of mathematics education. We also have colleagues who see nothing wrong with intense use of calculators in math classes; and, we have entire colleges who ban calculators from math classes. The question, then, is why use calculators? Why not use calculators?
We need to answer this question within our framework for education in general, and math education in particular.
Education is about a process that creates a qualitative and quantitative change in the capacities of the student.
If a student leaves a class, or a college, with the same capacities with some added skills, we have not educated the student — we have provided some training. Training is all about skills; education is about capacities. This is the reason why college graduates do better in jobs and quality of life measures.
Mathematics education is about a process that creates qualitative and quantitative change in the mathematical capacities of the student.
Knowing the answer to a problem like 8×4 is not an issue of capacity. However, needing to use a calculator to find the answer to simple problems often means a lack of mathematical capacity. Capacities are based on understandings and connections; a specific missing fact is not a matter of capacity. Having a grasp (call it an intuitive grasp) of number relationships begins the network of quantitative structures that make up mathematical capacities.
At some point in reading this, it is likely that you thought of the word ‘memorization’. When calculators are not allowed in classes such as developmental mathematics, we often justify it by saying that students need to memorize basic facts. My guess is that students in such classes store number facts in special locations in their brain with an index like “stuff I have to remember verbatim in order to pass”; I would like to see good research on this learning issue. I want the number facts stored in a more complex way related to indices such as “factors”, “multiples”, “sums”, “differences”, “divisors”, and “properties of numbers”.
In my own classes, I require a calculator for all students. This happens to be a department policy, though I would do the same thing if it were my choice only. The issue is not ‘memorization’ — the issue is ‘understandings’ (as part of capacities). Allowing the calculator implies that I need to observe students and provide feedback about the goals of a math course (understanding). This is admittedly tricky, and I know that I do not provide enough feedback to enough students.
A professional use of calculators is to focus on the contributions to learning. The presence of the calculator provides learning opportunities that I value — such as understanding the difference between (-5)² and -5². As you probably know, the confusion between these forms is common and problematic; I have students (this week, in fact) who have learned to state the correct words (memorized) but enter it incorrectly on the calculator.
Another example: One of the most common relationships in the world (natural and societal) is repeated multiplying. These exponential relationships require sophisticated methods to solve symbolically. However, a numeric and graphical exploration is within reach — IF we use a good calculator. Exponential relationships, in fact, are behind many of the general education goals in colleges (science, economics, and politics as examples). Without a calculator, we are saying that a student needs to complete the advanced symbolic work of a strong pre-calculus course in order to be generally educated. This is exactly the approach of many universities, including a large institution located a few miles from my college. Pre-calculus is not general education; it is STEM education, and using that course for general education is part of the larger problem in college mathematics.
One final thought on learning opportunities with calculators — with calculators, we can present reasonable approximations for ‘real world problems’. The world is messy; few calculations out there deal with integers only, and many involve very large numbers … or very small numbers. It might not actually help students transfer what they are learning, but it feels better in class.
Can calculators be a problem in a math class? Obviously yes — depending on many factors. NOT using calculators is also a problem; knowing how to use technology is an employment skill, and also can support learning mathematics. Not using calculators puts mathematics in a make-believe world that has no connection to a student’s life; after all, almost all students have cell phones that they use as a calculator … some have a smart phone with a ‘math app’. We might argue that a spreadsheet is a better mathematical tool than a calculator; as a learning tool, a spreadsheet has a learning curve and some limitations that make it more difficult.
Calculators, then, are both a good thing (resource) and a bad thing (problem). The important decision is not ‘calculator’; rather, the important decision is ‘learning as building capacity’.
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