Focus Issue: Calculators in K-12

The use of calculators in the schools has been heavily promoted by the K-12 education community and is specified in many state standards. They are widely used in K-8. The reception at the 9-12 level seems to have been slower but graphing calculators are now widely used.

At the college level the attitude has gone from interest and flirtation with similar ideas a few decades ago to dismay. Among teachers responsible for training students for technical careers, outright hostility is common.

The Issue: Is it true that current calculator use (in K-12) is in some ways harmful? If so can we be more specific about this harm, identify how the harm is caused, and find ways to use calculators that are at least harmless?

Arguments For Calculators

• Avoid rote memorization and the drudgery of hand arithmetic;
• Improve understanding by allowing use without lengthy and distracting routine work;
• Provide early access to high-quality arithmetic and thus to more interesting examples and real-life applications;
• Improve number sense; and
• Improve alignment between early training and modern real-life activity.

The K-12 education community has conducted a great many studies that seem to support calculator use. Many consider the basic issue settled and believe problems will sort themselves out as schools move toward complete implementation.

Objections

Recall that the emphasis here is on college-level preparation of higher-achieving students for technical careers. Problems that are not urgent for general-level students can be severe in this context.

• Calculator-trained students typically have weaker symbolic skills and less facility with mathematical structure and abstractions.
• Most “applications” are so contrived that they have no credibility as “real-life” and some are literally used as jokes (A train leaves Chicago ...", Joe has a triangular garden …''). They are a major turn-off for students and calculators have not helped.
• Modern real-life mathematical activity uses computers, not calculators. The kinetic-tactile calculator interface does not prepare students for visual computer interfaces.
• Methodologies in K-12 “studies” are so flawed that reliable conclusions are impossible and reported conclusions are largely shaped by ideological preconceptions. Priorities are set internally and are unresponsive to college-level concerns.

There has been a lot of discussion of underlying causes of the first item (lower facility with symbols and abstractions) in particular. The ideas put forth have not been “demonstrated” but may be specific and cogent enough to be useful.

Show conjectured causes

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• Students working by hand internalize the mathematical structure encoded in the algorithms and this provides a starting point for later abstract work. The keystroke approach to arithmetic hides structure and does not provide such a foundation.
• Tests and course material increasingly emphasize numbers over symbolic work. This allows students to exploit calculator skills and gives credit for acquiring these skills but leads to deficits in symbolic skills.
• Texts and other materials have diffuse content and are packed with visual distractions and irrelevant material. Precise or concise definitions are avoided in spite of the fact that mathematicians evolved them to serve as landmarks to help organize learning and effective use.

Questions

We seek input on context, problems and solutions.

Context

This Project is primarily concerned with high-achieving students and preparation for technical careers. Deficiencies that would be realistic limitations for general-level students may be serious problems in this context.

• Are the concerns expressed above reasonable, in context, or do they seem misplaced or overblown?

Specific problems

This Project is focused on problem sets. Generalities such as “calculator-trained students have weaker symbolic skills” are not useful in formulating problems that would support symbolic skills.

• We solicit specific examples of important tasks that need improvement.

Examples: “can't find exact roots of quadratics”; “can integrate from 1 to 2 but not from to ”, “have great difficulty thinking of a complicated expression as a function, e.g. as a function of ”.

Ideas about improvement

The issue is not “Calculators: yes or no”. Calculators and machine-enhanced environments in general are a fact of life and will not go away. The issue is “Some current practices seem to be harmful. Are there other ways calculators could be used that would be beneficial, or at least harmless?”

• We solicit specific examples of constructive calculator use.