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I notice that in current mathematics education they always have sections teaching about finding the missing member(s) of the sequences e.g. in this way:

$1,2,4,8,16$ , the next term is what?

Someone would argue that the next term is $32$ since they claim that the pattern of this sequences is about the non-negative integer powers of $2$ .

Someone would argue that the next term is $31$ since they claim that the pattern of this sequences is about the circle division by chords.

But in fact this type of questions are completely wrong since e.g. by the principle of interpolation, since we can find infinitely many numbers of the patterns of the existing terms, the values of other terms will be very depending on the patterns assumed and therefore have infinitely many possibilities, i.e. you can fill any values of other terms.

But the current mathematics education always assume this type of questions should have exact answers, that means they break the conscientiousness of mathematics.

Why the current mathematics education still hold this misleading education of the sequences?

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  • $\begingroup$ I think that teaching about finding the missing member(s) of the sequences is necessary, because it's highly probable that your students will stumble upon such question in the future. That is, you should teach them that these questions (in the general form, it's much different, for example, in the context of polynomials) are utter nonsense. $\endgroup$ – dtldarek Mar 16 '14 at 20:03
  • $\begingroup$ Yes, these questions are important and valuable, for the reasons cogently explained by Jim Belk. $\endgroup$ – Neil Strickland Mar 16 '14 at 20:51
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Theoretically, there's no way to determine the next term in the sequence $$ 1,\quad 2,\quad 4,\quad 8,\quad 16,\quad\ldots $$ It literally could be anything.

At the same time, it is a vitally important skill to be able to look at this sequence and say "it looks like the powers of 2". This answer is correct in the sense that any mathematician looking at this sequence would have that response, and a student who doesn't have that response when looking at this sequence has a serious gap in their knowledge.

Why is this an important skill? Because it's very common in mathematics that you don't know the answer to a question, but you can make a lot of progress by guessing the answer and then trying to prove it. If there's a sequence that you're trying to understand, and the first few terms are $1,2,4,8,16$, then the obvious thing to do is to try to prove that the $n$th term is equal to $2^n$.

This is a special case of the critically important skill of pattern recognition, which is necessary for solving almost any difficult problem. Indeed, in mathematics research, forming the appropriate conjectures is in many cases more difficult than proving these conjectures. This is a "soft" skill, in the sense that there's no precise definition of "the right conjecture", but it's still vitally important.

Indeed, recognizing patterns in sequences is so important that there is an entire website, the Online Encyclopedia of Integer Sequences, that is designed to help with this. The idea of the website is that you go there, type in the first few terms of a sequence you have found, and the encyclopedia tells you whether anyone else has ever studied that particular sequence. This website is commonly used by mathematics researchers, and can be a vital tool for discovering patterns and making connections between different areas of mathematics.

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    $\begingroup$ I agree that pattern recognition is critically important skill. But I don't think that this type of question helps in training it. In fact in any real situation you would have a context which points to the feasible set of possible solutions. Also, when unsure, you could (in most cases) calculate the next term. However, if only the bare numbers are given, then this becomes "guess the teacher's password", which useless by itself and might have even detrimental effects on learning other things (e.g. student might be more eager to guess rather than to think). $\endgroup$ – dtldarek Mar 16 '14 at 21:23
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    $\begingroup$ Define "simplest sequence"... Given by a linear recurrence of minimal order? Defined by a polynomial? Some guess at the generating has to be added for this to make sense. $\endgroup$ – vonbrand Mar 16 '14 at 22:15
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    $\begingroup$ While I completely agree with your answer, I can't help but think about a classic by Reuben Hersh: math.unm.edu/~rhersh/independent.pdf $\endgroup$ – András Bátkai Mar 16 '14 at 23:01
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    $\begingroup$ Just to add a detail: beyond the database there is also software for finding/guessing closed form expressions of sequences, see eg arxiv.org/abs/math.CO/0702086 (IIRC, Rubey, one of the authors, is an MOer). $\endgroup$ – quid Mar 26 '14 at 20:30
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    $\begingroup$ @AndrásBátkai: I'm not sure what the portrayal of the incident is actually meant to imply. I hope the last paragraph was sarcastic and he realized his mistakes, because otherwise I would seriously detest learning at his academy and would be glad that they didn't get in. $\endgroup$ – user21820 Jun 6 '14 at 10:23
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After reading the other answers and comments, it seems that there is a solution that combines the best of both worlds. Give students problems where patterns arise, then ask them to guess the pattern and prove it.

The level of question/proof depends on the grade. This post gives an example of a pattern that algebra students should be able to prove.

Example for undergraduates: There are $n$ parking spots on the side of the road. You can fill them with Lincoln town cars (which take up 2 spaces) or Yugos (which take up one space). How many ways are there of completely filling up all $n$ spaces?

Calculus example: Find $\int_1^\infty x^n e^x dx$.

This type of example encourages pattern finding as a means to an end.

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As the others have stated, the key issue is having students defend their pattern extensions with their reasoning or their assumptions. Now tie this to the question based on mistake analysis, and have students see if there are any errors in the alternate solutions. This could lead to rich mathematical discussions and an increased awareness of alternative solutions for discrete mathematics preparation.

Another issue, as I see it, is to tie patterns with term or figure numbers in tables; to physical objects; to graphs; to formulae; to words explaining their formulae. I would start with physical patterns like grid fragments with a shaded border of increasing size. The sequence generated would perhaps count the number of shaded squares in successive figures. The method they use to 'count' the squares would generate their formulae. There are multiple answers here but are all transformable into each other since they count the same number but in different ways. This takes the mathematics as scratches off the page and teaches them that abstract looking equations are representative of real things and have meaning.

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As @JimBelik aptly says, such problems are about pattern recognition. An important qualification of this is that the question might/should ask about the next number produced by the simplest pattern(s). That is, the "formally correct" notion that "of course there is no unique next number" is in fact not useful in practice. After the observation that "in principle" there is no unique next number, one should still pursue the issue of recognizing relatively-simple patterns.

Such questions could/should be asked in a form that requires more than just the bare number, of course. "Give a reason". "Give a simple formula".

(I disagree with the premise of the original question that there is something innately "unconscientious" about this. In the example given, the simplest pattern is powers-of-two, as noted by Jim Belik. The second-simplest is arguably polynomial (successive differencing) interpolation, which gives 31. What's the problem?)

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    $\begingroup$ The problem with this approach is that there is practically now way to prove you got the simplest pattern without an explicit, usable definition of simple. You could try for example with Kolmogorov complexity, but it would be insanely hard to prove that it is the simplest. On the other hand, if such definition is provided, then it would constitute the context I was talking about in one of the comments. In fact, appending zero might be actually the simplest solution, as those sequences are usually short, and "a few numbers and then 0" contains less entropy than "the power function with base 2". $\endgroup$ – dtldarek Mar 17 '14 at 10:00
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    $\begingroup$ @dtldarek, yes, if such questions are treated as a formal game with object being to "win", there are many ways to "game" the system. But I think that treating mathematics as a meaningless formal game is mostly non-productive, as it tries to separate it from the "sense" that we try to cultivate generally. "Sense" instead of "rules" is valuable, even if not as easily formalizable, etc. $\endgroup$ – paul garrett Mar 17 '14 at 12:28

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