Newest questions tagged proofs - Mathematics Educators Stack Exchange most recent 30 from matheducators.stackexchange.com 2019-06-27T05:10:06Z https://matheducators.stackexchange.com/feeds/tag?tagnames=proofs&sort=newest http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://matheducators.stackexchange.com/q/16681 6 Writing up a proof that assumes what is to be proven? Ross Millikan https://matheducators.stackexchange.com/users/12340 2019-06-02T03:54:48Z 2019-06-07T21:23:55Z <p>I was working on <a href="https://math.stackexchange.com/questions/3248162/i-need-help-with-proving-vector-space-given-the-operations-confused-by-complex/3248165?noredirect=1#comment6679998_3248165">this question</a> on math, where (among other things), the OP was asked to prove that <span class="math-container">$$x \oplus y=\sqrt{x^3+y^3}$$</span> is associative. </p> <p>After some prompting, the offered proof was <span class="math-container">\begin{align} x \oplus (y \oplus z) &amp;= (x \oplus y) \oplus z\\ \\ &amp; \implies x \oplus\sqrt{y^3 + z^3}\\ \\ &amp; \implies \sqrt{x^3 + (\sqrt{y^3+z^3})^3}\\ \\ &amp; \implies \sqrt[3 ]{x^3 + y^3 + z^3} \\ \\ &amp;\implies \sqrt{x^3 + y^3} \oplus z \\ \\ &amp;\implies(x \oplus y) \oplus z\end{align}</span><br> How big of a sin is this? The algebra is right, but the implications start with what is to be proved. If we just deleted the conclusion at the start and changed the implications to equalities, I would be happy. As it is we start with what is to be proved and have implications between expressions, not between sentences. I have seen this kind of write-up a number of times and would like to know how to advise the poster.</p> https://matheducators.stackexchange.com/q/15434 8 How to motivate students to do proofs? matqkks https://matheducators.stackexchange.com/users/1567 2019-04-02T12:47:04Z 2019-04-04T22:14:37Z <p>I am finding it difficult to motivate students on why they should how to prove mathematical results. They learn them just to pass examinations but show no real interest or enthusiasm for this. How can I inspire them to love essential kind of mathematics? They love doing mathematical techniques. Any resources or any answers would really help me.</p> <p>Additional information: 1st year undergraduates who dislike proofs. The university mostly thinks the maths courses are there for supplying other subjects. The students don't like proof based courses.</p> https://matheducators.stackexchange.com/q/15322 2 Learning proofs in introductory analysis courses mathnoob123 https://matheducators.stackexchange.com/users/12036 2019-03-06T18:56:34Z 2019-03-10T14:35:33Z <p>I have browsed the website a lot and I encountered many similar questions but not a question that asks the same question as I intend to. </p> <p>In introductory undergraduate classes in Analysis, usually, Rudin is assigned which has a pretty straightforward way of introducing topics. The author provides the definitions, then concludable theorems along with their proofs and at the end exercises. </p> <p>My main concern is with the proof for the theorems. When I am going through the proof for the theorems, what should my aim be? </p> <p>Should I be making sure that I understand each step, that's to say, that I will be able to understand and explain the proof <strong>provided</strong> that I am given the proof?</p> <p>OR</p> <p>Should I try to read the proof again and again so as to be able to rebuild the proof from memory and intuition alone (and of course understand the proof in the process)?</p> <p><strong>Possible Constraint I would like the reader to keep in mind:</strong></p> <p>I am aware the optimal strategy is to try to approach the proof myself first, along with coming up with counterexamples for the conditions in the theorems, however, given the fast pace of these courses and the fact that an undergraduate is not just taking the analysis course, really minimizes the time a person can devote to such sort of analysis.</p> https://matheducators.stackexchange.com/q/15205 6 A question from a young student to mathematicians Jhdoe https://matheducators.stackexchange.com/users/11884 2019-02-12T09:45:12Z 2019-02-12T19:16:23Z <p>I'm a young math student. And I live with the effort of always wanting to understand everything I study, in mathematics. This means that for every thing I face I must always understand every single demonstration, studying the basics every time if I don't remember them. And this makes it impossible for me to prepare the exams, because I can't go on, I fix myself on wanting to derive by myself a theorem and I lose days in it. And so I ask mathematicians if it is always necessary to be able to prove everything, or we must accept what the theorems say and give it for good. If possible I also ask you some advices to help me study, knowing my problem.</p> https://matheducators.stackexchange.com/q/15142 6 Is it a problem if a senior student majoring in mathematics could not prove the quadratic formula? Zuriel https://matheducators.stackexchange.com/users/6166 2019-01-30T15:54:50Z 2019-02-08T14:39:24Z <blockquote> <p><a href="https://i.stack.imgur.com/inq3u.png" rel="noreferrer"><img src="https://i.stack.imgur.com/inq3u.png" alt="enter image description here"></a></p> </blockquote> <p>According to a recent experiment conducted by user <a href="https://matheducators.stackexchange.com/users/117/steven-gubkin">Steven Gubkin</a>, nearly one half of his students in a senior level Real Analysis course do not have any idea how to prove the quadratic formula. Is this a problem in our education of students majoring in mathematics? Or are we alright with students obtaining bachlor's degree in mathematics without knowing the proof of the quadratic formula? </p> https://matheducators.stackexchange.com/q/15040 15 Should theorems be proved to students who are not majoring in mathematics? Zuriel https://matheducators.stackexchange.com/users/6166 2019-01-03T23:22:46Z 2019-01-22T14:00:09Z <p>My impression to students majoring in mathematics is, whenever we teach them a theorem, a proof should be given in the class, or at least as a reading assignment. However, how about students not majoring in mathematics? One extreme is, proving everything, treating them as students majoring in mathematics. The other extreme is, not teaching any proof at all, only introducing the conclusions and their applications. Of course one can always teach some proofs while omitting others. Now the question is, what proofs should we teach and what proofs should we omit? What are the factors to consider?</p> https://matheducators.stackexchange.com/q/14956 7 is it appropriate or beneficial to mention weird results in math? Lenny https://matheducators.stackexchange.com/users/10335 2018-12-22T07:43:06Z 2018-12-22T22:55:44Z <p>Is it appropriate to mention weird/exciting results in math (or use as cautionary tales why one cannot apply mathematics naively) in say high school level?</p> <p>Examples of these results include the sphere eversion which turned into a meme (yes, I'm aware it's a true result). Banach-Tarski paradox, Vitali Sets, "1+2+3+4 = <span class="math-container">$-\frac{1}{12}$</span>" and Hilbert's Hotel.</p> <p>Additionally, I feel as though <span class="math-container">$e^{i\pi} = -1$</span> demonstrates that math despite seemingly random ties up at this beautiful equation.</p> https://matheducators.stackexchange.com/q/14765 7 Is there any example of a "forwards/backwards" induction? Steven Gubkin https://matheducators.stackexchange.com/users/117 2018-11-09T03:02:11Z 2018-11-09T17:43:13Z <p>I like to make the "dominoes" analogy when I teach my students induction.</p> <p>I recently came across the following video:</p> <p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="nofollow noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p> <p>In this video, a sequence of concrete block wall caps are set up like dominoes on the top of a wall. The first wall cap is knocked down, setting off the domino effect. The blocks are spaced so that they are resting on each other when they fall, but just barely. So rather than resting flat each block is supported slightly by its successor. When the last block falls, however, it falls flat (having no subsequent block to rest on). This causes the block behind it to slip off, and lay flat, which causes the brick behind it to slip off and lie flat, until all the blocks are lying flat perfectly end to end.</p> <p>Is there any instance of a similar phenomena occurring in mathematics? I am thinking of a situation in which you want to prove both <span class="math-container">$P(n)$</span> and <span class="math-container">$Q(n)$</span> for <span class="math-container">$n = 1, 2, 3, \dots, 100$</span> (say). If you are able to prove: </p> <ol> <li><span class="math-container">$P(1)$</span></li> <li><span class="math-container">$\forall k \in \{1,2,3, \dots, 99\} P(k) \implies P(k+1)$</span></li> <li><span class="math-container">$P(100) \implies Q(100)$</span></li> <li><span class="math-container">$\forall k \in \{ 100, 99, 98, \dots, 3,2\}, Q(k) \implies Q(k-1)$</span></li> </ol> <p>Then it will follow that both <span class="math-container">$P(n)$</span> and <span class="math-container">$Q(n)$</span> are true for <span class="math-container">$n = 1, 2, 3, \dots, 100$</span>.</p> <p>If an example is found, it could be a great example for teaching because it would force students to think through the logic of why induction works rather than blindly following a certain form of "an induction proof".</p> https://matheducators.stackexchange.com/q/14751 14 Why do inequalities flip signs? [closed] Lenny https://matheducators.stackexchange.com/users/10335 2018-11-07T20:10:59Z 2018-11-09T18:02:20Z <p>Is there a mathematical reason (like a proof) of why this happens? You can do it with examples and it is 'intuitive.' But the proof of why this happens is never shown in pedagogy, we just warn students to remember to flip the inequality when</p> <blockquote> <ul> <li>multiply or divide by a negative number both sides</li> </ul> <p><span class="math-container">$$-2&gt;-3 \implies 2 &lt; 3$$</span></p> <ul> <li>take reciprocals of same sign fractions both sides</li> </ul> <p><span class="math-container">$$\frac{3}{4} &gt; \frac{1}{2} \implies \frac{4}{3} &lt; 2$$</span></p> </blockquote> https://matheducators.stackexchange.com/q/14549 3 How is it correct for a lecturer to prove and "explain" a proof while explicitly knowing students are not familiar with logic itself? Turkhan Badalov https://matheducators.stackexchange.com/users/9168 2018-09-12T10:31:26Z 2018-09-13T20:39:26Z <p>I often see a situation when professors use words "logic", "mathematical proof" and even prove logically while actually knowing that students are not even familiar with logic itself, i.e. no formal understanding of equivalence, implications, inference rules, etc.</p> <p>How are students supposed to understand such "proofs" because they can actually accept as a proof any "intuitively" explained reason why a theorem is true and never even suspect they were deceived unless they know the exact definition of an argument and true (sound) argument? Is knowing logic always taken as granted like a prerequisite? Even so, shouldn't the lecturer at least designate some time for explaining basic logic?</p> https://matheducators.stackexchange.com/q/14514 2 Undergraduate Math Seminar topic KMC https://matheducators.stackexchange.com/users/10348 2018-09-02T22:40:55Z 2018-09-05T21:31:30Z <p>** Edit</p> <p>Thanks everyone for some great suggestions. I should have been more clear though. I am actually looking for a college level proof that pertains to algebra or leads to algebra in some form. My professor has already shot down any proof that would be easy enough to teach to a high school algebra I or II class. </p> <p>I am currently in my final year studying mathematics education. I am doing a 1 credit math seminar project. I will be writing a paper and presenting on this topic. I have chose integrating mathematical proofs into high school math. However, I need a detailed proof for this topic. Anyone have any suggestions? I am looking for a proof that I can expand on and talk about for nearly 30 minutes.</p> https://matheducators.stackexchange.com/q/14409 7 Should students be given partial scores when they gave an incomplete proof by contradiction? tonychow0929 https://matheducators.stackexchange.com/users/10174 2018-07-31T08:51:24Z 2018-09-19T18:40:20Z <p>In a quiz, there was a question asking students to show something doesn’t exist. A lot of them gave proofs by contradiction.</p> <p>Initially, I designed the marking scheme so that an incomplete proof by contradiction would lead to 0 marks, because I thought that exhausting all cases is very important in a proof by contradiction. Otherwise, there may not be any contradiction at all (another case is valid). Some students may impose an (perhaps, obviously contradictory) assumption and claim that their proofs are complete.</p> <p>However, I found that a lot of students missed some cases, or imposed extra assumptions which were neither given nor proved. Some of them were certainly with loss of generality. I gave them 0 marks for that question. I’m not sure whether marking in this way is too harsh.</p> <p>Should students be given partial marks if they gave an incomplete proof by contradiction?</p> https://matheducators.stackexchange.com/q/14402 3 Why are proofs written in flowery language incomprehensible? Ooker https://matheducators.stackexchange.com/users/8875 2018-07-28T03:35:48Z 2018-07-28T21:32:03Z <p>Let's take an example in Wu-Ki Tung, <em>Group theory in physics</em>:</p> <blockquote> <p><strong>Theorem 3.4:</strong> Irreducible representations of any abelian group must be of dimension one.</p> <p><strong>Proof:</strong> Let $U(G)$ be an irreducible representation of the abelian group $G$. Denote by $p$ a ﬁxed element of $G$. Due to the abelian nature of the group, we have $U(p)U(g) = U(g)U(p)$ for all $g ∈ G$. According to Schur’s Lemma, $U(p) = λ_p E$. This applies to all $p ∈ G$. Hence, the representation $U(G)$ is equivalent to the one dimensional representation $p → λ_p ∈ C$ for all $p ∈ G$. </p> </blockquote> <p>I would rewrite it as:</p> <blockquote> <p>Because in an abelian group every irrep commutes with another irrep, then according to the Schur’s Lemma, they are just multiples of the identity operator, <strong>forcing them to be one dimension only</strong>.</p> </blockquote> <p>In my opinion, the rewrite is much better. There is no need to prove the theorem anymore, but they are blended into a flowery but self-contained text with no single symbol, and readily to connect to the next idea. 7 sentences in 6 lines reduce to 1 sentence in 3 lines, but the true intuition is still preserved, and I don’t see how this is vague or makes the formalism lost at all. In other words, we have the best of both worlds.</p> <p>But in the discussion <a href="https://www.reddit.com/r/math/comments/92fb8w/making_intuitive_formalism_and_concrete_flowery/?st=jk4cphfr&amp;sh=3c9f4d63" rel="nofollow noreferrer">Making intuitive formalism and concrete flowery text</a>, it is said to be awful or incomprehensible, even to those who understand the topic. Do you know why is that?</p> https://matheducators.stackexchange.com/q/14310 7 Teaching logic through "high school algebra"? Steven Gubkin https://matheducators.stackexchange.com/users/117 2018-07-04T13:31:18Z 2018-07-04T17:43:17Z <p>I am going to be teaching a discrete math class in the fall. One of the major goals of the course is a solid understanding of the basics of logic: the precise meanings of "and", "or", "not", "implies", "if and only if", "there exists", and "for each" and how to logically manipulate statements involving these.</p> <p>Often Discrete Math tries to illustrate all of these concepts in the context of set theory, graph theory, combinatorics, elementary number theory, etc. This has the disadvantage that the student is learning new content at the same time they are trying to master the basic logical ideas. </p> <p>Logic is ubiquitous in mathematics, so I figure that I should be able to illustrate all of these concepts in a <strong>familiar</strong> mathematical setting: namely "high school algebra".</p> <p>An example: does the first line imply the second line, does the second line imply the first line, or both ?</p> <p>\begin{align*} x^2 &amp;= 2x\\ x &amp;= 2 \end{align*}</p> <p>I am sure that I could come up with "algebra problems" which illustrate all of the concepts I am trying to convey. For instance, partial fraction decomposition is a fairly logically complex idea (you want to find the only values of some variables ($A$, $B$, $C$ , etc) which make an equation true <strong>for all $x$</strong>).</p> <p>My questions:</p> <ol> <li><p>Does anyone have any textbooks which would fit well with what I am trying to do? This could either be a book on logic or discrete math which already does what I am proposing, or a book on high school algebra which pays careful attention to logic.</p></li> <li><p>Has this idea been tried before? In particular, is there any relevant research on this approach?</p></li> </ol> https://matheducators.stackexchange.com/q/14147 2 Ideas for high-school proof class? R.B. https://matheducators.stackexchange.com/users/9971 2018-06-07T04:51:10Z 2018-06-07T12:08:53Z <p>I have a math degree and have been hired to teach a proof class at a summer program. Our goal is to help the students learn the material they need for school (they take an algebra class separately) while also helping them improve their problem-solving skills more generally and find joy in doing math. They are mostly 16-17 years old and test into Algebra I or II in the program.</p> <p>I've observed and helped with this course in previous years, but the curriculum is not well-defined and there aren't many past resources for me to use. So, my question is: what are your ideas for discussing proof, logical reasoning, etc. with a group of students who may struggle with math and don't necessarily like it very much? I will be focusing on the development of mathematical self-confidence and new ways of thinking, more so than specific content knowledge.</p> https://matheducators.stackexchange.com/q/13988 6 Questions similar to Wason Selection Task Steven Gubkin https://matheducators.stackexchange.com/users/117 2018-04-20T23:42:26Z 2018-04-20T23:42:26Z <p>The Wason Selection Task (described by Pete Clark <a href="https://matheducators.stackexchange.com/a/412/117">here</a>) is a great problem for getting students to grapple with all of the intricacies of logical implication.</p> <p>I will be teaching a discrete mathematics course for the first time in about a month, and I would love to have a sizable repository of similar questions.</p> <p>Does anyone have a link to a big list of similar questions? If not, I am happy to hear your ideas about similar problems which might stimulate a similar amount of thinking about the other logical operators.</p> https://matheducators.stackexchange.com/q/13935 2 What is the correct symbol to use for ending a counterexample? Jordan https://matheducators.stackexchange.com/users/9808 2018-04-10T15:22:41Z 2018-04-10T16:09:26Z <p>I am familiar with the tombstone symbol, "$\blacksquare$", that is used to signify the end of a proof. However, it is my understanding that an example isn't technically a proof. For instance, one can't just find an example of a proposition being true and then claim the proposition to be true. </p> <p>So, I was wondering if there is a symbol that should be used to signify the end of a counterexample?</p> https://matheducators.stackexchange.com/q/13901 -2 Proving basic Theorems and properties in high school [closed] user50896 https://matheducators.stackexchange.com/users/9777 2018-04-02T23:20:53Z 2018-04-04T03:59:19Z <p>Why high school teachers do not emphasize knowing the proofs of properties and theorems in math. In my 40 years of teaching prospective high school teachers, I rarely found students who can derive formulas they learned in school. For example properties of logarithms, changing from one base to another, Law of Sines and Law of Cosines proving major Geometry theorems. These "formulas" and theorems can be presented as problems to solve so with help, students can discover them on their own. Then they would "own" the theorems.</p> https://matheducators.stackexchange.com/q/13686 1 Does studying elementary number theory improve one's proof skills and ability to understand algebra and analysis? [closed] user9495 https://matheducators.stackexchange.com/users/0 2018-02-28T15:35:17Z 2018-03-01T05:50:35Z <p>I'm taking a number theory course and don't know whether it's worth it. I currently can't understand algebra and real analysis and decided to take # theory to see whether this would help me prove and understand set theory and cardinality and sequences and the prerequisites for analysis and galois theory. Should I expect to get better at proving after this course? </p> https://matheducators.stackexchange.com/q/13594 18 Inability to work with an arbitrary mathematical object Brendan W. Sullivan https://matheducators.stackexchange.com/users/80 2018-02-13T23:31:41Z 2018-02-26T22:57:46Z <p>This question is motivated by student responses to homework and quiz problems I have recently posed in an undergraduate real analysis course. I will share some examples and observations first, to present the main ideas, before posing a question.</p> <ol> <li><p>A homework problem asked students to consider an arbitrary set $A\subseteq\mathbb{R}$ that is nonempty and bounded below. From that set, define $B=\{b\in\mathbb{R}\mid b\text{ is a lower bound for }A\}$. The students were asked to prove that $\sup(B)$ must be equal to $\inf(A)$. I noticed two kinds of mistakes. They were not widespread, but they were common enough that I made comments about them during class time.</p> <ul> <li><em>Proof by example,</em> such as: "If $A=(2,\infty)$ then $B$ must be $(-\infty,2]$ and notice that $\inf(A)=2$ and $\sup(B)=2$." This is a gross misunderstanding of <strong>arbitrary</strong> versus <strong>specific</strong>.</li> <li><em>Referring to "$b$" without instantiation,</em> such as: "$b$ is a lower bound for $A$ so all $a$s are upper bounds for $B$." This contains a kernel of truth (indeed, <em>every</em> element of $B$ is a lower bound for $A$) but it demonstrates a misunderstanding of a <strong>universal assertion</strong> versus a <strong>specific instance</strong>.</li> </ul></li> <li><p>A quiz problem asked the following: "Suppose $A\subseteq\mathbb{R}$ is nonempty and bounded above. Let $s=\sup(A)$, and let $\varepsilon &gt; 0$ be arbitrary. What can we say about the interval $(s-\varepsilon,s]$? How many elements of $A$ lie in that interval? None? 1? At least 1? Infinitely many? What can we say <em>with certainty</em>? Consider illustrating with examples."</p> <ul> <li>I expected all students to at least point out that there cannot be <em>zero</em> elements of $A$ in the interval $(s-\varepsilon,s]$; if so, that would violate the fact that $s$ is the <em>least</em> upper bound for $A$.</li> <li>I expected the better students to further point out that $s$ may be the <em>only</em> element of $A$ in that interval. For example, if $A=\{2\}$ is a singleton, then this is true regardless of $\varepsilon$.</li> <li>I also expected the better students to further point out that there may be <em>infinitely many</em> elements of $A$ in that interval. For example, if $A=[0,2]$, then this is true regardless of $\varepsilon$. </li> <li>Instead, <em>more than half of the class</em> submitted an answer that amounted to nothing more than saying "there are infinitely many real numbers in any non-trivial interval." Zero consideration or even literal mention of that arbitrary set $A$. They treated this as if the original question was: "Does an interval have infinitely many elements?"</li> </ul></li> </ol> <p>This worries me for several reasons. One of the main reasons is that I think I have an "expert blind spot" regarding the ability to consider and work with an arbitrary mathematical object because that practice is so natural to me. I'm truly baffled why a student would consider an example as a reasonable solution to problem #1, yet I'm sure there was a time in my educational career where I would have easily made the same error without yet understanding the big issue. </p> <p>But problem #2 worries me the most. Is there something about the way I posed the question that obscured its intent so much that students could have genuinely thought I was cryptically assessing them on their knowledge of the infinitude of the reals? I think that they lack strong skills in mentally working with an arbitrary set, that they need to consider specific examples to make mental progress, and that they are not yet in the habit of responding to a question like this by <em>creating their own examples</em> to consider while reflecting on the problem.</p> <p><strong>Question: How can I, as an instructor, help students to work with problems like the examples above, knowing that the students do not have good, practical habits for working with arbitrary objects? Beyond simply telling them to do so in the moment, how can I encourage them to create and test examples for themselves, to not "over/under assume" about what they're given, or otherwise just to correctly interpret given information about an arbitrary object?</strong> </p> <p>Upon reflection, I seem to have addressed this in the past by just trying to "model good behavior" when presenting in-class examples and when working with students one-on-one. However, I wonder whether there are particular activities I could use, or problem types to assign and assess, that would better promote the kind of thinking and behavior that I want my students to develop.</p> <p>A good answer will contain suggestions for activities or assessment tools. If there is any research about this, I would love to know about it, as well. If it helps narrow the scope, I am especially interested in undergraduate math majors learning to write proofs in their advanced courses (e.g. real analysis, linear algebra, etc.).</p> <p><em>Meta comment:</em> I could not find a good tag that properly encapsulates the main issue of this question. Would "abstraction" be a reasonable tag to create? This question is mostly about a student's ability to abstract from specific cases to general concepts, and I'm sure there are and will be other questions related to that ability.</p> <p><strong>Followup comment:</strong> I want to give a shoutout to #2 above as a really good problem to give students when learning about suprema before learning about sequences. Today, in my class, we proved the Monotone Convergence Theorem by defining the proposed limit to be the supremum of the set of the terms of the sequence. Given an arbitrary $\varepsilon&gt;0$, I asked the students: "What can we say about the interval $(s-\varepsilon,s]$?" Some students laughed upon realizing this is precisely the quiz question from a few weeks ago and said, "There must be <em>at least one</em> element in there!"</p> https://matheducators.stackexchange.com/q/13494 12 How to write proofs on the board in the classroom jdods https://matheducators.stackexchange.com/users/9365 2018-01-24T13:02:51Z 2018-01-25T09:17:32Z <p>I'm teaching an introductory analysis course, and I am seeking some feedback on how proofs should be written on the board in class in order to maximize learning. I realize that there is an opinion-based component to this question, but would appreciate any citations of research on the subject (not necessary) in addition to arguments on how student learning might be impacted.</p> <p><strong>My question is regarding the actual style that the proof is written on the board including but not limited to the following points :</strong> </p> <ul> <li>Should theorems and proofs always be written on the board in complete English sentences or is it ok to use abbreviations and symbolic shorthand?</li> <li>Is it ok to use slightly different notation than in the textbook?</li> <li>Is it ok to use slightly (or even completely) different arguments than what is used in the textbook?</li> </ul> <p>I'm including a couple theorems from Wade's introduction to analysis as examples. I'll type up the theorem statement and proof exactly as in the text and then how I might write it on the board in class.</p> <p>Here is a <em>nearly</em> word for word as the theorem and proof appear in the book:</p> <hr> <p><strong>2.12 Theorem.</strong> Suppose that $\{x_n\}$ and $\{y_n\}$ are real sequences and are convergent, then $$\lim_{n\rightarrow\infty}(x_n+y_n)=\lim_{n\rightarrow\infty}x_n+\lim_{n\rightarrow\infty}y_n.$$ <strong><em>Proof.</em></strong> Suppose $x_n\rightarrow x$ and $y_n\rightarrow y$ as $n\rightarrow\infty.$ Let $\epsilon&gt;0$ and choose $N\in\mathbb N$ such that $n\geq N$ implies $|x_n-x|&lt;\epsilon/2$ and $|y_n-y|&lt;\epsilon/2.$ Thus $n\geq N$ implies $$|(x_n+y_n)-(x+y)|\leq |x_n-x| + |y_n-y|&lt;\frac\epsilon2+\frac\epsilon2=\epsilon. \quad \blacksquare$$</p> <hr> <p>Now here is how I might actually write it on the board in class:</p> <hr> <p><strong>Thm 2.12</strong> $\quad x_n\rightarrow x, y_n\rightarrow y \quad \Longrightarrow \quad x_n+y_n\rightarrow x+y.$ </p> <p><strong><em>Proof.</em></strong> Given $\epsilon&gt;0$, choose $N$ s.t. $|x_n-x|&lt;\epsilon/2$, $|y_n-y|&lt;\epsilon/2$ $\ \forall n&gt;N.$ \begin{aligned} |(x_n+y_n)-(x+y)|&amp;\leq |x_n-x| + |y_n-y|\\ &amp;&lt;\frac\epsilon2+\frac\epsilon2=\epsilon. \qquad\qquad \blacksquare \end{aligned}</p> <hr> <p>The main differences is that the text is more "wordy", and I will write everything out more "symbolically". Note that I do stay consistent with the numerical labeling of theorems (e.g. 2.12) in the book so that students can more easily reference the text to compare/study.</p> <p>I also sometimes use slightly different arguments than in the text, add in details that are left out of the book or leave out details that are explained in the book.</p> <p>E.g.:</p> <hr> <p><strong>2.8 Theorem.</strong> Every convergent sequence is bounded.</p> <p><strong><em>Proof.</em></strong> Assume $x_n\rightarrow a.$ Given $\epsilon=1,$ there is an $N\in\mathbb N$ such that $n\geq N$ implies $|x_n-a|&lt;1.$ Hence by the triangle inequality, $|x_n|&lt;1+|a|.$ On the other hand, if $1\leq n \leq N,$ then $$|x_n|\leq M:=\max\{|x_1|,|x_2|,\ldots,|x_N|\}.$$ Therefore, $\{x_n\}$ is dominated by $\max\{M,1+|a|\}. \quad \blacksquare$</p> <hr> <p>Now here is how I might actually write it on the board in class:</p> <hr> <p><strong>Thm 2.12</strong> $\quad x_n\rightarrow x \quad \Longrightarrow \quad x_n$ bdd.</p> <p><strong><em>Proof.</em></strong> Given $\epsilon&gt;0$ we can find $N$ s.t. $\forall n&gt;N,$ $|x_n-x|&lt;\epsilon.$ </p> <p>So $|x_n|\leq |x|+\epsilon$ when $n&gt;N.$</p> <p>Let $M=\max\{|x_k|; k\leq N\}.$ </p> <p>Therefore $|x_n|\leq\max\{M,|x|+\epsilon\}$ for any $n. \qquad \blacksquare$</p> <hr> <p>Of course, I also verbally explain each step of the work and will sometimes draw diagrams to help them informally/intuitively understand the argument, etc. </p> <p>My justification is that there is no need to copy it exactly as it is in the text as the students can simply read that, and that it might be beneficial for them to see it written up slightly differently so that they can become accustomed to different styles of mathematical writing.</p> https://matheducators.stackexchange.com/q/13464 5 Can some lovers of math truly never create something previously unseen? Greek - Area 51 Proposal https://matheducators.stackexchange.com/users/155 2018-01-14T18:04:37Z 2018-01-21T00:48:45Z <p>Can someone truly love math, and master and remember discovered calculations, counterexamples, proofs; but still fail to invent anything new (e.g. incapacity to prove anything unseen, calculate something that demands new tricks, or discover counterexamples)? I ask not about <a href="https://www.reddit.com/r/changemyview/comments/70tfpt/cmv_all_smart_people_are_capable_of_doing_well_in/dn6c10f/" rel="noreferrer">dyscalculia</a> that appears irrelevant, as I'm not asking about arithmetic. </p> <p>I bolded the relevant parts from <a href="https://www.reddit.com/r/changemyview/comments/70tfpt/cmv_all_smart_people_are_capable_of_doing_well_in/dn5vtav/" rel="noreferrer">the Reddit Post</a> beneath that's anecdotal: is there any evidence?</p> <blockquote> <p>Although I am usually not a full-time teacher, I've taught hundreds of students over the last 35 years.</p> <p>Students fall into five categories:</p> <ol> <li><p>A tiny number of brilliant students who will teach you something.</p></li> <li><p>A moderate number of pretty smart students who would probably do an OK job if you just handed them the textbook and left.</p></li> <li><p>A large number of average students for whom your technical class is serious work, but will get through with help from you and study.</p></li> <li><p>A moderate number of people who don't like or care about math and/or aren't particularly talented who might pass if they put in the work but probably won't.</p></li> <li><p>A small number of people who will never understand the material, no matter what.</p></li> </ol> <p>When I was young I refused to admit that that last category existed. I put a lot of time into a few people who worked really hard but tried to memorize and fake their way through, or just couldn't get it.</p> <p>And then something happened to convince me otherwise.</p> <p>I had a friend who started Math in University at the same time that I did, but he was about five years older. He'd already had two successful careers - he had been a journalist and then quit that to run a campaign for a politician who won his election - but he'd always loved math.</p> <p><strong>He did perfectly well in first year - not exceptionally but fine. But in second year, the trouble started. He could not create new proofs - no, not at all. He understood the material quite well, with some gaps, but writing proofs was a bridge he could not see how to cross.</strong></p> <p>I spent hours with him, but he started to do worse and worse.</p> <p>Eventually he vanished. I finally got in touch with his brother who said, "I really appreciate your having contacted us, but I'm not going to tell him you called. He had a nervous breakdown and his doctor said he should be kept away from any memory of what happened."</p> <p>Now don't get me wrong. My guess is that at least half the people who fail math do so because they just aren't interested enough to put the time in, and if they really wanted to, they would succeed.</p> <p><strong>But there's a core of people, often otherwise smart people - people who are even really interested - who simply don't think that way, and all the coaching in the world won't fix this problem.</strong></p> </blockquote> https://matheducators.stackexchange.com/q/13361 15 How to teach students the value of concrete counterexamples? Haudie https://matheducators.stackexchange.com/users/9240 2017-12-24T22:39:26Z 2017-12-29T03:25:25Z <p>I teach exercise sessions for a Linear Algebra course for 1st semester students in Europe. Students have to prepare some exercises at home. In class, I call on students to present their solutions.</p> <p>One big problem that I see is that in exercises of the form "prove or disprove", students' disproofs usually ends with an equation with variables instead of a concrete counterexample. For example, to prove that a certain statement does not hold true, a student proof may go like this: </p> <blockquote> <p>If the statement does hold, then [...] And in the end, we find that, for all real numbers $x,y$, that $x=y$. This is clearly not true, so the original statement is false.</p> </blockquote> <p>When I tell my students, "This proof is okay, but a concrete counterexample would be nicer", they usually say, "But from this equation I can see it just as clearly".</p> <p>I see three main reasons to advocate for concrete counterexamples instead:</p> <ol> <li>They force you to actually think about the existence of a counterexample, or to recognize that the resulting equation is "vacuous". (However, with some easy Linear Algebra examples, this does not seem to be an issue.)</li> <li>The proof looks simpler and more convincing for the reader. (But admittedly, not so for my students. It seems they would they have to derive the equation, find a concrete counterexample, and then write the disproof again with their newly discovered counterexample.)</li> <li>It is the usual, accepted style.</li> </ol> <p>However, I tell my students these reasons repeatedly and it seems to have no effect. A related issue is that they often "prove" a universally quantified statement via an example.</p> <p><strong>My questions:</strong></p> <ol> <li>Are there any easy examples where you could easily find an empty equation and no counterexample which I could show to my students?</li> <li>Has there been any reasons about why thinking in counterexamples is hard for fresh(wo)men?</li> <li>What can I do to improve the situation?</li> </ol> https://matheducators.stackexchange.com/q/13248 2 Strategies for learning proofs Björn Lindqvist https://matheducators.stackexchange.com/users/7837 2017-12-03T22:47:39Z 2017-12-04T18:59:27Z <p>What are the best methods for learning proofs? I'm tasked with learning two dozen proofs about the properties of continuous functions and real numbers in a week well enough to be able to present them. Like Taylor's Formula, the fundamental theorem of calculus, and Bolzano-Weierstrass theorem. What is the best way to do it? My main problem is that there are no exercises to practice on.</p> https://matheducators.stackexchange.com/q/13229 3 Is the Nomenclature of Triangle Congruency Proofs Consistent? Chaim https://matheducators.stackexchange.com/users/8258 2017-11-30T15:53:54Z 2017-11-30T19:47:27Z <p>My Geometry class is doing triangle congruency proofs these days. In general, we find three pairs of congruent parts (sides or angles) in two triangles; we show that these congruencies reveal that the triangles are either congruent or similar; and we conclude that further parts are congruent (in the case of congruent triangles) or proportional (in the case of similar triangles).</p> <p>In the column of justifications, therefore, the last two justifications are usually SSS (or SAS or ASA or AAS or AAA) and then either CPCTC (Corresponding Parts of Congruent Triangles are Congruent) or Definition of Similarity.</p> <p>I’ve done pretty much the same thing each time I’ve taught Geometry, but this year we’re using a text by Pearson in which this topic appears in Chapter 4. After doing this a zillion times, I have begun to wonder something about this nomenclature.</p> <p>Why is that where triangles are CONGRUENT, we DO refer to CPCTC, and we DO NOT refer to a definition; but where triangles are SIMILAR, we DO NOT refer to CSSTP, but we DO refer to a definition?</p> <p><a href="https://i.stack.imgur.com/t9iOC.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/t9iOC.png" alt="enter image description here"></a></p> https://matheducators.stackexchange.com/q/13198 12 Tasks that encourage argumentation MathGuy https://matheducators.stackexchange.com/users/7550 2017-11-24T18:22:27Z 2017-11-27T13:13:23Z <p>I am looking for resources that have tasks such as the one below that encourage argumentation. I want tasks that 8th graders could do but would also be appropriate for high school students. I want to do some kind of research on the mathematical practice "construct viable arguments and critique the reasoning of others". I've seen several good exploration tasks such as this one, but they are all scattered. Are there any good books or resources that include these types of tasks?</p> <p><a href="https://i.stack.imgur.com/8Gs9Z.png" rel="noreferrer"><img src="https://i.stack.imgur.com/8Gs9Z.png" alt="enter image description here"></a></p> <p>Fendel, D., Resek, D., Alper, L., &amp; Fraser, S. (1996). Interactive mathematics program year 1 - unit 2: The game of pig (p.99). Emeryville, CA: Key Curriculum Press.</p> https://matheducators.stackexchange.com/q/13075 5 Tutoring Discrete Mathematics Niels Kornerup https://matheducators.stackexchange.com/users/8922 2017-10-29T20:40:06Z 2017-10-31T04:29:53Z <p>A few weeks ago, I started tutoring a student in Discrete Mathematics (a subject I took a year ago). </p> <p>I have previously tutored both pre-calculus and calculus, but never a proof based class. I have been approaching the job by helping the student work through the homework and helping her figure out content that she couldn't follow during class. </p> <p>I am afraid that I might actually be doing the student more harm than good, as Discrete Mathematics is a proof based course (the first of this kind for her), and thus I am afraid that she will begin relying on me to get intuition for problem solving and writing proofs in general. </p> <p>I was thinking about transitioning to a structure where she proves everything that they proved in class again by herself (with help where needed). The idea is inspired by <a href="http://calnewport.com/blog/2008/11/25/case-study-how-i-got-the-highest-grade-in-my-discrete-math-class/" rel="noreferrer">this blog post</a> I found last night. Does anyone with experience tutoring a proof based math course have any advice for me?</p> https://matheducators.stackexchange.com/q/13060 0 Is it possible to have taken intro to proofs, calculus 3 and differential equations and still lack the ability to do proofs? user8788 https://matheducators.stackexchange.com/users/0 2017-10-28T05:14:35Z 2017-10-29T15:26:01Z <p><a href="https://matheducators.stackexchange.com/questions/2386/ideal-undergraduate-sequence">Ideal Undergraduate Sequence</a> <strong>Main question:</strong></p> <p>I looked above and what I'm interpreting out of it is that one should be able to do proofs after studying some intro to proofs class, calculus multivariable+vector calculus, and differential equations linear algebra. </p> <p>Is it possible that one won't be able to do proofs after finishing doing calc 3/vector calc and differential equations?</p> <p><strong>Background: (context)</strong></p> <p>I really want to be a pure mathematician and not some high school teacher in the future. I suck at other subjects. math's all I got left, i.e. I can't write, I can't code, manual labor or mcdonald's job sounds possible future. I'm taking calculus 3 and differential equations right now and I'm wondering whether there's no hope in me becoming a mathematician if I fail to be able to do proofs like the ones in algebra or analysis like weistrass M-test and something banach after finishing calculus 3 and differential equations.</p> <p>I cannot prove theorems at this point independently and can't study some class like real analysis or number theory right now.</p> <p><strong>Optional questions:</strong> (ignore if you're busy) Is what the link above says that if you can't do proofs after calc 3 and differential equations, you'll never be a mathematician?</p> <p>Does finishing taking calculus 3/vector calculus and differential equations transform your mind to being able to do proofs? I mean why did we suggest (in the hyperlink above) that people should take multivariable calculus and differential equations before a serious proof class? Are calc 3/ differential equations special? </p> <p>Are math teachers in high school the people who took calculus 3 and differential equations but just didn't have the intelligence or weren't transformed from such an experience to be able study proof writing? </p> https://matheducators.stackexchange.com/q/13034 10 Book request: teaching proving and reasoning at an American university shuhalo https://matheducators.stackexchange.com/users/8895 2017-10-22T23:42:56Z 2018-02-08T18:42:12Z <p>I am a European postdoc who recently teaching at a large public university in the United States. I will have to teach a course for undergraduate students that introduces them to proving and reasoning in mathematics.</p> <p>The students have possibly no exposure to mathematical reasoning in general. At the end of the course, they should be able to read and write proofs, and use the standard logical/set-theoretical notation. </p> <p>What books or lecture notes can you recommend (or dis-recommend) for such a course? I am particularly interested in material that gets them as close as possible to being able to read non-American textbooks. </p> https://matheducators.stackexchange.com/q/13032 13 Unique steps leading to a non-unique answer KCd https://matheducators.stackexchange.com/users/893 2017-10-22T18:52:32Z 2018-09-18T22:16:20Z <p>When asked to show a math problem has a unique solution, students sometimes think that if an algorithm leading to a solution has unambiguous instructions at each step (no need to make choices at any point) then the solution they find has to be the only solution. Loosely speaking, if each step in reaching the solution is "unique" then the end result has to be unique (the only possible solution).</p> <p>This is not true, and I think it would be nice to have a list of examples at different levels (of undergraduate mathematics) showing why this idea is mistaken.</p> <p>For example, if asked to solve $55x+32y = 1$ in integers then Euclid's algorithm for computing $\gcd(55,32)$ followed by back-substitution (reversing the steps of Euclid's algorithm) is a procedure where each step is completely determined by the previous ones and leads to the definite answer $(x,y) = (7,-12)$, but the original equation has infinitely many integral solutions: $(x,y) = (7+32t,-12-55t)$ for integers $t$.</p> <p>What other examples can people offer? I am not interested only in computational problems. </p>