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At the school I am student teaching at starting in 2 weeks (Algebra 1), students are strongly encouraged to rely on their calculators for nearly everything, from any operation with fractions to matching graphs with equations. I am fairly sure that even the most vehement calculator proponents would agree that this is unhealthy, for instance, in the cases where using the calculator works "well enough" but doing pieces by paper or in your head will vastly increase the speed.

I recognize some questions that are similar have been asked; however, few focus on actual pedagogical techniques that are used at the high school level. Most are aimed at convincing students at the undergraduate level that they should use their calculators less, ways to talk about it during lecture, or the amount of calculator use students should be allowed. So, my question is threefold:

  1. What are ways that I can design assignments that discourage the over-reliance on the calculator and encourage the development of heuristics? Additionally, how would you change your answer, considering that the current class is structured on the availability of a calculator because "they are allowed on the state test"?
  2. What are class activities (such as group work, games) that can be done to help students develop heuristics and mental models that do not rely on calculator usage yet are still appropriate for early high school? (For instance I remember playing a sort of "beat the calculator" type game as a young elementary student that was designed for the full class.)
  3. What are other (direct?) instructional techniques that I can use that help students see the utility of avoiding their calculator if unnecessary? (possibly covered here)
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  • $\begingroup$ I always like to give really large numbers for permutations and combinations which cannot be calculated on a typical calculator which makes getting an actual exact whole number answer impossible. Leaving an answer in terms of a factorial, For some reason, made my students uncomfortable at first and led them to become more comfortable with having what I call a "conceptual answer" rather than an "exact answer" in turn leading them away from the gut reaction to grab a calculator $\endgroup$ – celeriko Apr 12 '15 at 16:23
  • $\begingroup$ Why is this important? $\endgroup$ – futurebird Apr 19 '15 at 21:09
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For some ideas, look at resources relating to NZ Achievement Standard AS91027. In this external assessment, calculators are banned. It is a "Level 1" standard which is at a similar level to US Year 10.

A description of the standards in student language is here: http://www.studyit.org.nz/subjects/maths/math1/2/

The standards documents, with previous assessments and marking schedules are here: http://www.nzqa.govt.nz/ncea/assessment/view-detailed.do?standardNumber=91027

A third party resource with teaching materials and example assessments is given here http://www.mathscentre.co.nz/Booklets/ There should be a download button to the bottom right of AS91027 Apply Algebraic Procedures In Solving Problems

As for motivation to not use calculators, we have "if you can't do it without calculators, you fail!". If all your state tests allow calculators, it is not so easy to motivate.

I don't know of any particularly innovative pedagogical techniques surrounding this standard. It is designed purely to ensure minimum proficiency of basic algebra necessary for further study. By contrast, the other 12 standards strongly emphasise contextualised mathematics, making them easier to motivate, but calculators are required.

As for designing assessments, the big thing is that the decimals are at most 2 decimal places (for money), and most things work out to nice numbers. Everything is designed to be solved by pure mental arithmetic, not even pencil and paper algorithms should be necessary.

I personally agree with you that mental heuristics should be emphasised, but it is not emphasised in the curriculum. Many heuristics are very contextual, making them hard to teach. Also most students don't understand the point of getting something wrong by over 50% but within "Order of Magnitude", and test markers don't give credit for it.

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A general anti-calculator strategy is to give the students the answer and then adjust your questioning accordingly.

Suppose you were teaching multiplication and was trying to get them to do $23\times27$. You could just outright give them the answer $621$ and ask them to convince you the answer couldn't be anything else. How do you know it can't be $400$? or more difficult $641$? The calculator would be redundant or at least used creatively in that situation.

I appreciate that this is a simple example but I hope it gives you ideas.

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    $\begingroup$ 23 * 27 is a great example of using (a+b)(a-b) = a² - b² as a shortcut: (25-2)*(25+2) = 625 - 4. This can be much faster than using the standard FOIL algorithm for multiplication. $\endgroup$ – Jasper Mar 13 '15 at 6:48
  • $\begingroup$ @Jasper I agree. Giving the answer away allows experimentation and clever ways like the one you describe. $\endgroup$ – Karl Mar 13 '15 at 10:15

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