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I am interested in educators real-world experience or being pointed to any research in this area.

I have a student whose arithmetic skills are weak for his/her age. The student counts on his/her fingers when attempting to multiply, struggles to add two integers with a total greater than 20, cannot compare the relative size of fractions with different denominators, etc. These are skills which I understand "most" students would hope to master several years earlier than this student. I believe this student has fallen into a comfortable routine of (i) finger-counting or (ii) guessing or (iii) giving up on a question. Not to be critical of finger-counting, it is just that even the most basic of exam questions are inaccessible to this student.

I have used a number of tools such as Cuisenaire rods (which I think are extremely powerful and flexible teaching aids) to develop his/her number sense and skills in basic arithmetic. However, I decided to enter the student for an exam where basic electronic calculators are permitted, i.e. there are no non-calculator papers.

My sense/ instinct is that if a "number/arithmetic sense" is not developed then handing a student a calculator is pointless because the student has not feel for expressions and questions like "the ratio of the lengths is 2:5" or "this cost has increased by 20%" or "the sale discount is 15% what is the selling price?" or "how much should we distribute £1m grant money if we want to distribute it equally between 7 organisations?" etc. etc. My sense is that "teaching how to answer the question" is a very "fragile" form of learning since a question can be rephrased slightly and the student can feel he/she hasn't been taught how to answer that question.

So do people have evidence that routinely using electronic calculators helps or harms development of basic/fundamental mathematical problem-solving skills?

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    $\begingroup$ There have been many articles and books on this subject over the years: scholar.google.com/… $\endgroup$ – Daniel R. Collins Mar 25 '17 at 16:33
  • $\begingroup$ Before electronic calculators, we had slide rules. Did people back then lack mathematical understanding because they depended on their slide rules too much? $\endgroup$ – Ben Crowell Mar 26 '17 at 0:49
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    $\begingroup$ Not an answer to the question, but I can say that "arithmetic skills" can be poor while a good understanding of size and general understanding of question you gave can be well developed. I guess that for these people the electronic calculator can be a blessing. $\endgroup$ – Dirk Mar 26 '17 at 17:08
  • $\begingroup$ @Ben Crowell: Slide rules were almost never used in school math classes, except perhaps physics. Students used the logarithm, power, and root tables at the back of the book. My 12 June 2006 post What has technology subtracted? may also be of interest. $\endgroup$ – Dave L Renfro Mar 27 '17 at 19:46
  • $\begingroup$ I can see the point that pre-electronic calculator much early-years maths was the learning of counter-intuitive algorithms to multiply multi-digit numbers and add/subtract fractions. I acknowledge this activity left many people with a fear and loathing of maths. So electronic calculators making these skills (relatively?) redundant levelled the playing field and did not "eject" the less naturally numerate. $\endgroup$ – Clive Long Mar 27 '17 at 20:13
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I am going to venture an answer here:

Try the calculator. I have had students whose minds were opened by less likely tools, and you never know. You shouldn't be restricted by thinking logically and linearly, viz. foundation prepares for scraping the sky. Neurology is more complicated than that.

I have seen my students' eyes opened by other "tools." Also, the calculator may act as a much-needed crutch and allow this student to lean a bit. This comfort may just replace the comfort of finger counting, but it may help them reach a breakthrough moment.

As was mentioned above, there are papers and books on the subject, but my gut says that the calculator should be included in your strategy. It will probably not be clear initially, or ever, if it "helped" or "harmed," as you put it. But, maybe, learning to use the calculator (not as trivial as one might expect, especially for a struggling student) could help push this student to that next level.

Remember, using a calculator requires one to mentally translate mathematics from one form to another, via mental effort.

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    $\begingroup$ Quote "Remember, using a calculator requires one to mentally translate mathematics from one form to another, via mental effort.". Good point. I am so used to doing this that I have forgotten the effort to learn how to do this. $\endgroup$ – Clive Long Mar 27 '17 at 20:06
  • $\begingroup$ You're not alone. Metacognitive thinking is very elusive. I find that a calculator with larger viewing screen, which keeps your work on screen (e.g. TI-83) can actually help students in a number of ways: (1) understanding and following their own thinking (2) improving their understanding of the order of operations (3) giving context to dicsuss concepts such as "terms," expressions, ratios, etc. $\endgroup$ – Kevin Glynn Mar 28 '17 at 1:59

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