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I teach to predominantly non-majors in college algebra, precalculus, and calculus.

How can one possibly incentivize or rationalize assigning practice problems outside of class when this software is readily available? Math/engineering folk need to understand the algebraic manipulation behind differentiation and integration, but not the social science or bio-tech majors.

A student asks me "Why do I need to know how to do this if Wolfram Alpha can just do it for me?". I honestly don't know how to answer anymore.

I take great issue with answers like "You won't have wolfram alpha on the test" or "Maybe the internet will break or the power will go out". I think we need to embrace it to allow us to go deeper, like the Ti8X's did.

WAlpha has also rendered any meaningful outside-of-class assessment useless, as I will just get regurgitated output, or "trained" results from the tutors in the math center. Any knowledge demonstrated on these assignments is fleeting, such is seen on the exams and in class.

How have you solved this problem?

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    $\begingroup$ hold on a second, did graphing calculators improve math education? I'm not convinced. $\endgroup$ – James S. Cook May 18 '16 at 15:53
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    $\begingroup$ Probably the internet will not go away soon, so figuring out how to cope would be smart. Oddly, the previous inaccessibility of information (and processing power) was "convenient" for inducing students to learn how to do things unassisted. Even in grad-level (math) courses, the "homework" I assign, if it is reasonable, should be eminently find-able on-line, and it would seem perverse to me to declare that out-of-bounds. Instead, wherever a good solution can be found is fine... for info and practice... but grades are entirely determined by in-class, unassisted exams. $\endgroup$ – paul garrett May 18 '16 at 16:31
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    $\begingroup$ I think there's a question before this of what the students would actually most benefit from in these classes. It may be that some calculus is actually necessary. But if you have enough control over the syllabus you could change the focus from calculations to more general descriptive/conceptual understanding built up but getting a computer system to do the number-crunching and plotting. $\endgroup$ – Jessica B May 18 '16 at 17:34
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    $\begingroup$ Many of the questions we give in those sort of classes are, in a sense, "pre-digested". By which I mean that we did the hard part already: rephrasing the question as an equation or other mathematical object. Perhaps that activity is where future classes should focus. (at least, more heavily.) $\endgroup$ – Adam May 18 '16 at 20:26
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    $\begingroup$ Math/engineering folk need to understand the algebraic manipulation behind differentiation and integration, but not the social science or bio-tech majors. Wait, what? Every math student needs to understand the math they're doing. $\endgroup$ – Ben Crowell May 21 '16 at 18:52
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First of all, I believe this question is quite similar to the question "How to give homework for integration technique?". I avoid the temptation of repeating my answer for that question. Instead, I try to give an answer from a different angle based on a recent experience I had in a numeracy class with adult students. One of the questions I asked was inspired with the way my son calculated $142 \times 39$ (that was one of his school procedural homework). Basically, he started from $140 \times 40$ and found his way through to the answer of the initial question. Following that, the question for the class was

Knowing that $140 \times 40=5600$ suggest a way to calculate $142 \times 39$.

Of course, they all had smart phones and on their phones they had a calculator. But, no smart phone was of any help to solve the problem at hand, they needed smart minds.

Mathematics is about solving problems not repeating exercises. For the latter you cannot beat Wolfram and the likes; for the former, it would be beneficial to know how to use them (if needed), but you need something more. Thus, it is one of our responsibilities (as lecturer/teacher) to give our students problems which need that something more.

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    $\begingroup$ Definitely the answer I was looking for. The intention and rhetoric are aligned. Its unfortunate that no textbook provides examples in this way, but maybe providing prompts like this alongside assigned problems could work. $\endgroup$ – goruda May 22 '16 at 17:27
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    $\begingroup$ I love this answer. $\endgroup$ – Jon Bannon May 24 '16 at 22:58
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Imagine you had to look up every word you wanted to use, because you had a poor vocabulary. This would get old, fast.

The trouble is, many people don't have a genuine need to internalize computational technique as a way to see.

As a researcher, I find one of the main values of manual calculation is as error correction for reasoning, and a source of intuition. When I am in love with a certain idea, carrying out calculations related to this idea is a way to slow down and become objective about the idea and to test it from many angles. I am constantly guessing what should happen and checking this against the results of the calculation, which I attempt to do accurately and honestly to bolster or debunk my belief.

At some stage, when trying things by hand to develop intuition and generate ideas that will bypass computation, it may become appropriate to use technology to continue this process. Usually, though, moving to technology without having developing feelings by calculating many pages by hand feels very icky.

I'd like, as an educator, to try to motivate students to calculate for similar reasons to why researchers do. Unfortunately, I often don't do a terribly good job of tying "getting the right answer" to "debugging reasoning". There ends up being very little student metacognition or conjecture to be tested...only whether the number matches the answer key.

Incidentally, this is similar to why we ought to care about grammar and even spelling...for metacognitive reasons. Grammar, despite its conventional aspects, provides logical checksums that can rudder thought. Subtle differences in meaning can be detected by one's critical consideration of turns of phrase. Mindlessly relying on spell check or denying the need for grammar has similar detrimental effects to the blind use of Wolfram Alpha and calculators. Of course, these things are helpful, but can easily erode one's mental acuity if one is not very careful.

Perhaps the way to go is (1) Focus on reasoning about mathematics, and thinking about what it means; (2) Giving problems that require thought and ideas, and that would benefit from accurate exploratory calculation.

Standard Calculus often fails this miserably, I'm afraid.

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    $\begingroup$ Wonderful answer. $\endgroup$ – Daniel R. Collins May 20 '16 at 15:01
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    $\begingroup$ Thanks, Daniel! $\endgroup$ – Jon Bannon Nov 27 '16 at 18:02
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It doesn't exactly solve the problem, but one way to make positive use of Wolfram would be to use some Wolfram Demonstration Projects.

There are some that attempt to gamify calculus calculation practice.

Some help with intuition. If the students can see how the ideas could be of use to society as a whole (say through higher-dimensional calculus), even if not directly to themselves, it might make them resent the course a bit less.

Others could possibly help to link the ideas to contexts they might be more interested in.

Also, I read somewhere (I've completely forgotten where) that 'why do we have to do this?' often is a symptom of 'we're struggling to make any sense of this'.

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You can also use Wolfram Alpha to look up definitions of words. So by reductive reasoning, there is no need for infants to learn vocabulary because when they are old enough to use a smartphone, people can always look up the meaning of words in Wolfram Alpha.

Being less reductive and less cynical, there is no need to learn a foreign language because you can use google.

Hey life is good! There is no need to learn anything anymore.

Students in college algebra can appreciate the fallacy in these arguments, and appreciate how vocabulary and translation parallel algebraic manipulation.

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    $\begingroup$ I'm not convinced by the assertion in your final sentence. Do you believe that pointing out parallels with learning vocab or foreign languages will have a sizeable impact on students who'd rather differentiate/integrate via WA? $\endgroup$ – Benjamin Dickman May 19 '16 at 21:05
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    $\begingroup$ Yes, I do believe this works. I started teaching undergraduate math courses before even NSFNet came into being. Now I often use this parallel and challenge students to be creative and imagine more things that they not longer need to learn. They really do come up with some funny things! $\endgroup$ – user52817 May 20 '16 at 3:06
  • $\begingroup$ I am thinking more in terms of things that have been supplanted by AI, web searches, etc.--learning to drive a car is soon to be an example. $\endgroup$ – user52817 May 20 '16 at 3:18
  • $\begingroup$ You might like to look at Profession by Isaac Asimov on this theme. However, I'm not convinced the general line of argument will do much for students. $\endgroup$ – Jessica B May 20 '16 at 12:35
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    $\begingroup$ I think there is a flaw in this analogy. Even though the internet gives us quick access to word definitions, it is not fast enough to match the functionality of actually knowing/understanding the word. I can't use dictionaries while having a conversation. Mathematically, however, Wolfram Alpha can complete certain computations FASTER and MORE CONSISTENTLY than even capable mathematicians. So, it is of greater use mathematically than conversationally. $\endgroup$ – Tyler Auer May 25 '16 at 11:38
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Show your students some simple questions that Wolfram Alpha cannot answer.

Wolfram Alpha currently cannot satisfactorily answer some problems stated in words. For example, the query "find the maximum area of a rectangle given its perimeter" is interpreted as "rectangle | area" and has the result "A = a b."

The query "find the maximum area of a rectangle given a perimeter" is interpreted as "rectangle | perimeter x | maximal area" and has the result "1/2 h (x-2 h) (assuming height h)." While this is a more informative answer than that for the previous case, it still does not explicitly state that the maximum area is when the rectangle is a square.

Now, perhaps a future version of Wolfram Alpha will be able to answer the question above. But I doubt that it will be able to answer a question such as "Given a fence of fixed length, find the largest rectangular plot of land that it can surround." To answer a question like this, Wolfram Alpha needs some common sense, for example, it needs to know that a fence is like a perimeter, and that a large plot of land is one that has a large area.

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The answer to this question is identical to the answer to the question "why should I learn trigonometry if I'm not going to be an engineer". We do not learn math for the sake of being able to solving equations. We learn math to train our minds and develop mathematical reasoning. Wolfram alpha will solve an equation but it will not train you to comprehend complex situations in real life.

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Different needs for different students

To answer your question in another way, consider what is most important for students to be able to do or understand. Do you have students that will ultimately be users or creators of mathematical knowledge (using or in the mathematical sense, of course)? Non-majors will need, at worst, access and skills to technology that can solve repetitive/rudimentary problems for them (e.g. calculating alpha levels for hypothesis testing). In addition, they will need abilities to collaborate and co-create solutions to problems with their peers.

Guiding questions

Are you creating opportunities for them to work together? Are you creating opportunities for them to use technology to solve menial tasks and focus on the big ideas/problems? Are there groups of students in similar areas of study that could explore big issues / big questions in their field? In this way, you can emphasize what role Wolfram Alpha plays in their learning: a tool that can solve "simple" questions, so that the complex, "big" questions can be focused on.

An empathetic analogy

If this forum existed years ago, and an instructor had asked the same question, had the same concerns, but replacing all instances of Wolfram Alpha with "calculators", what would your answer have been for him? Calculators were a game-changer for how mathematics was done. It changed how we teach mathematics: we could finally focus on certain types of complex, difficult questions more often. The same is true for Wolfram Alpha. So, with the understanding that technology-assisted learning is here to stay, what skills and abilities are most important for students to learn that TA learning cannot give them? The answer this question is the answer to yours.

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