Recent Questions - Mathematics Educators Stack Exchange most recent 30 from matheducators.stackexchange.com 2022-01-20T05:26:44Z https://matheducators.stackexchange.com/feeds https://creativecommons.org/licenses/by-sa/4.0/rdf https://matheducators.stackexchange.com/q/24820 4 How do I sketch a good gaussian curve freehanded, or by using only common sketching tools? Kiteration https://matheducators.stackexchange.com/users/19385 2022-01-19T04:22:10Z 2022-01-19T23:37:04Z <p>I'm a lousy artist. If I want my Gaussian curves to be accurately drawn when I use a whiteboard, or work with pen &amp; paper, what are my options?</p> <p>Is there a way to use a straight edge, or compass, or some other trick to getting accurate curves from the Gaussian family?</p> <p>I want to make more symmetrical sketches where the error of any given area under the curve is minimized.</p> <p><a href="https://i.stack.imgur.com/hnvdQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hnvdQ.png" alt="Normal distribution" /></a></p> https://matheducators.stackexchange.com/q/24819 2 Measures to quantify complexity of algebra equation Paul https://matheducators.stackexchange.com/users/5265 2022-01-18T22:12:12Z 2022-01-18T22:12:12Z <p>Like the title says, I am looking for ways to measure the complexity of an algebra equation. For now, I am focused on linear equations, but I would think any metrics could be generalized for quadratic, etc. I hoping the metrics can be useful for isolating student trouble spots and measuring progress in a rigorous way, as well as have a way to make progressively more complex equations to solve (this is specifically for U.S. Algebra I students).</p> <p>Some examples to point out what I am looking for:</p> <p>(1) <span class="math-container">$3x + 2 = 11$</span>. In this simple equation there are two operations, the moving the 2 over, and then dividing by 3. I would think these operations would have different measures?</p> <p>(2) <span class="math-container">$2(x+3)+1=13$</span>. Here there are two additional operations beyond (1), the distributive property and combining the numbers on the left. Already (1) and (2) could contain 4 quantitative measures.</p> <p>(3) <span class="math-container">$3-2(4-2x)=11$</span>. Here the working of negative signs add a different complexity.</p> <p>(4) <span class="math-container">$2x+3(x+3) = 4x+10$</span>. Here there are <span class="math-container">$x$</span>'s to combine, and an <span class="math-container">$x$</span> to move across the equals sign.</p> <p>(5) <span class="math-container">$2x+1=6$</span>. I am distinguishing this from the 1st equation because the answer is a fraction. Is this considered more complex?</p> <p>So I am hoping there can be a set of scores that I can assign to an equation to determine complexity. I've seen a lot of people talk implicitly about complexity, but I have not seen any explicit calling out of what makes equations complex.</p> https://matheducators.stackexchange.com/q/24816 2 Is copying working and explanation plagiarism in this context? mohan10216 https://matheducators.stackexchange.com/users/19265 2022-01-18T10:12:50Z 2022-01-18T21:31:38Z <p>In my institution I have a friend who is part of this Math ambassadors club where they write blog posts and share them online at medium.com. However, there is an issue with my friends post.After they shared it with me to go through, check for errors and edit, I looked online and found <a href="https://www.cantorsparadise.com/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793" rel="nofollow noreferrer">this</a>. My issue is that their working and explanations are all copied from the person who wrote this, my friend has only changed a few words(5-10). I am worried that if they post this they will get into a lot of trouble since plagiarism is dealt with very harshly in my institution. However, they denied copying from there. I don't mean to sound rude, but I do not believe them. How should I go about this? Should I report them? FYI: There were no citations in their article.</p> https://matheducators.stackexchange.com/q/24815 2 For 15 year olds, are there exercises — with full solutions — on the Fence Post or Off by One error? NNOX Apps https://matheducators.stackexchange.com/users/155 2022-01-18T07:58:05Z 2022-01-18T22:43:49Z <p>Which books contain practice questions — preferably with full solutions — to assist 15 year olds with the <a href="https://betterexplained.com/articles/learning-how-to-count-avoiding-the-fencepost-problem/" rel="nofollow noreferrer">Fence Post</a> or <a href="https://en.wikipedia.org/wiki/Off-by-one_error#Fencepost_error" rel="nofollow noreferrer">Off by One</a> error?</p> <p>Most students at my institution have not heard of this name, though some recognized the error. Some of them are hankering after more practice exercises, to assist them with preventing it! Even after I cover the concept in general, though without exercises, too many of them fail to spot and avoid it on exams and tests.</p> https://matheducators.stackexchange.com/q/24814 0 Mental Health in Mathematics lemniscate https://matheducators.stackexchange.com/users/19378 2022-01-17T18:44:17Z 2022-01-17T18:44:17Z <p>I am not sure if my question is relative to this meta but I still want to put forth my thoughts and concerns and questions because I think its not just me but others too who have similar issues.</p> <p>My concerns mostly revolve around students with mental and learning disabilities in academia and specifically from undergraduate level and upwards.</p> <p>What kind of learning paths they should adopt?<br /> How should teachers modify courses for them?<br /> What would be criteria for their scholarships?</p> <p>For it to be more clear and explicit I am going to use myself as an example.</p> <p>I got admitted into this graduate programme for theoretical computer science two years ago. In both of my semesters from the first year, I would find myself completely lost in whatever I was doing even in the topics and subjects which I was confident about. I showed my concerns to my professors after a couple of discussions with them I decided to take relevant Bachelors courses which would improve my basics. So I did that but I had to go through same difficulties in them so again after introspection and discussions with fellow academics, I enrolled myself officially into Bachelors but same problems exist even now.</p> <p>Recently I(27) have been diagnosed with dyslexia and ADHD I don't know how I am going to efficiently continue my academic career. My university is going to be helpful in terms of time required to finish my degree, but having to do Bachelors again from 1st world country after moving from 3rd world country is not economical at all and getting scholarships is not something you can be surely definite about. Most of the scholarships I have found they don't support my requirements (having to finish degree in longer time). So I want to know how to tackle such situations in European education systems because scholarships comes with responsibility of maintaining sufficient credits? So should there be a leniency in scholarships for such students?</p> <p>In my courses I have always felt that I need more time to finish assignments in order to have a good grasp of concepts but considering the disability I have and reports from my doctors I have only gotten to have extra time in the assignments not what I actually have proposed. So I want to ask that does my need justify my demand? Because I have found this in repetition that more time on exams seems not that of useful if I have not grasped concepts properly, since I would be spending time just to get bare minimum marks so that I have enough average before the exam for each course.</p> <p>Now I am at the point where I have passed two analysis(real) courses, one combinatorics course and I this loop of thinking is not stopping that as the courses get more advanced I don't know what to do because its just that I did not get to have a grasp of proofs properly. I have already revisited concepts and proofs from my combinatorics course twice with solving problems but whenever I look at same things after sometime I can't even recall definitions that I have to reread whole topic or sometimes chapter. I know I am still an amateur in this regard but blanking on definitions is a concern IMO after you have passed a course and repeated it. So such students should be given leniency according to their need? I ask this because a lot teachers put emphasis on time for homework when classes begin.</p> <p>I told and asked this because I do math for passion and I want to do it right(not perfect), perfect is what you strive for.</p> https://matheducators.stackexchange.com/q/24813 0 Problem solving approach to learning and psychology plants https://matheducators.stackexchange.com/users/17553 2022-01-17T10:36:55Z 2022-01-17T10:36:55Z <p>I try to have a problem solving approach to learning math. What i mean by this is if someone sets some questions or problems regarding the material i am reading how should i answer or what questions or problems should i make from the material i am reading to help me in learning,</p> <p>But, i confuse it with problem solving generally. Generally, i have tested my skills on problem solving and sometimes i fail to solve some problems. Probably i should not confuse these two ways or approaches?</p> <p>Also, what should i do to feel better about math and learning and not feel like having fear or sadness?I have read if i remember correctly that in the United States of America, many people have fear and sadness about learning and math and this affects me at least a little i think.</p> <p>Generally, i read undergraduate material in math and i try some papers which are uploaded on arxiv from researchers. Sometimes, i read free material uploaded from other sources.</p> https://matheducators.stackexchange.com/q/24807 3 How do you study subjects you're not that interested in Obamafish https://matheducators.stackexchange.com/users/19362 2022-01-14T01:13:46Z 2022-01-14T22:34:41Z <p>I'm an undergraduate who doesn't find analysis particularly interesting, but I'm taking a calculus on manifolds course next semester, so I'm reviewing measure and integration theory since my grasp on the subject should be stronger, but it's so difficult to actually sit down and learn the material when I'm uninterested in it. With subjects I like, I can sit down and read a textbook for hours every day without any problem.</p> <p>Does anybody have any advice? Topology was my favorite course so when I can think of problems topologically or have some topological motivation, it helps me want to learn these other subjects, but the textbook I use doesn't take a topological approach. If anybody has any advice (or textbook recommendations), I'd greatly appreciate it.</p> <p>Also, I can't just not learn this subject, either</p> https://matheducators.stackexchange.com/q/24801 2 Why is my 8th grade Algebra 1 tutoring student learning mean absolute deviation and standard deviation? blakedylanmusic https://matheducators.stackexchange.com/users/19352 2022-01-13T02:42:16Z 2022-01-13T09:25:56Z <p>I’m tutoring an 8th grade student in Algebra 1, and he showed me that their class learned how to find standard deviation and <strong>mean absolute deviation</strong> using the following formulas:</p> <p><span class="math-container">$SD=\sqrt{\displaystyle\frac{\Sigma (x_i-\mu)^2}{n}}$</span></p> <p><span class="math-container">$\textbf{Mean Absolute Deviation}=\displaystyle\frac{\Sigma |x_i-\mu|}{n}$</span></p> <p>I did not learn this when I was in Algebra 1 in 8th grade, and I was in the honors class. Is this because of Common Core? I know they’re trying to scatter more stats materials in the regular curriculum but I’m just shocked he was working on this at his level. He eventually understood it and was getting the correct answers but he definitely struggled a lot with it today before getting there. Just don’t understand why they’re covering this at this level — it seems a little advanced to me.</p> https://matheducators.stackexchange.com/q/24799 1 Math outside of undergraduate studies and proofs plants https://matheducators.stackexchange.com/users/17553 2022-01-12T11:01:10Z 2022-01-13T14:00:14Z <p>I read sometimes mathematical works of others outside my undergraduate studies. I think i can not follow the understanding of the proofs of theorems sometimes. What should i do? Should i read other things connected to the proofs i read to understand the proofs?</p> <p>I want to make my own theorems and proofs but i can not solve the open problems i read, although they are considered difficult from others.What should i do to make theorems and proofs? At least for them to be new and accepted from the scientific community.</p> <p>Could i make my own open problems and conjectures or questions and try to prove them? If i do it, what should i read and what do most of the mathematicians do to solve them?</p> <p>When reading theorems without the proofs and learning them, how will i know if a problem needs those theorems to be solved? About the proofs, how should i use what i learn from them on other possible solutions of problems?</p> <p>What does someone learn from proofs?</p> <p>Thank you.</p> https://matheducators.stackexchange.com/q/24798 6 Should proofs include a third “context” column? jackisquizzical https://matheducators.stackexchange.com/users/14430 2022-01-12T04:42:19Z 2022-01-15T12:39:17Z <p>Proofs, or any mathematical derivation, appearing in any real setting, such as a book or textbook or talk, or even when we're teaching it in class, includes a great deal of surrounding explanation. But when we ask students to regurgitate proofs, we ask for what is merely the skeletal core of the proof in this rib-like two column format. At the very least, it seems to me, proofs ought to have three columns, including a new multi-row lefthand column that describes (at least) the approach being taken in that section of the proof. By &quot;multi-row&quot; I mean that the proposed new lefthand &quot;row&quot; can encompass multiple rows of the basic two-column proof. The resulting format would look like this:</p> <pre><code> =================|===============|============== What we're | &lt;c=&lt;c | Reflective Property up to in | tABC~=tDEF | SAS this section | etc... | etc... =================|===============|============== What we're | etc... | etc... up to in | etc... | etc... this section | etc... | etc... </code></pre> <p>The context would, I think, largely reflect the reasoning and planning that went into (goes into) the proof, and would commonly, I think, end up representing lemmas that participate in the larger proof. One could say that these lemmas ought to be rolled off into prior proofs of their own, and I would agree. But we do not provide a way, in geometry, of naming and organizing proofs usefully so that prior short proofs (technically lemmas) can be looked up and referred to easily.</p> <p>Because we do not have a clear naming scheme for proofs, we cannot call upon them as one would functions in a programming language. Indeed, one might wonder why student of geometry aren't being taught geometry like one would teach a programming language: Here's a bunch of functions (lemmas) you can use, and here's how to use them. We do do this for some things, like the triangle congruence lemmas (SAS etc), and for some logical rationales (CPCTC, etc), but the dozen random theorems (lemmas) regarding parallelograms, mid segments, and so on aren't ready-to-hard functions with clear naming, so we end up re-deriving/proving them in the middle of other proofs, which makes the proofs into these long-winded, un-memorable and ultimately unwieldy things.</p> <p>The three-column format I'm proposing at least offers a way to internally organize proofs into logical segments so that even without addressing the problem of the previous paragraph, at least there is a a way of making the substructure explicit.</p> https://matheducators.stackexchange.com/q/24796 9 Should I upload slides before or after a class ablmf https://matheducators.stackexchange.com/users/11702 2022-01-10T22:49:22Z 2022-01-19T20:54:28Z <p>I used to post my slides before a class. But I noticed that many students simply read it while in class instead of listening . So I am thinking not doing it in the future. But they can still get it from students who have taken the class before.</p> https://matheducators.stackexchange.com/q/24795 1 Best books for mathematical statistics self-study? blakedylanmusic https://matheducators.stackexchange.com/users/19352 2022-01-10T21:02:49Z 2022-01-11T19:58:58Z <p>I'm hoping to start a masters in mathematics in the fall, and am hoping to find a good book on mathematical statistics to study so that I'll be able to take graduate level mathematical statistics once I start my degree. For context, my undergrad was in music, and I'm in the midst of taking prerequisites to qualify for the masters program. I'm taking finals for Linear Algebra and Calc III, and after that I will be taking Differential Equations, Methods of Proof, Real Variables, and Abstract Algebra. Are there any books you recommend for self-study in mathematical statistics? (preferably not too expensive if possible -- trying to save for school) Thank you!</p> https://matheducators.stackexchange.com/q/24790 2 How can 17 y.o. high school students intuit that P(n, r) stops at $n - (r - 1)$, not $n - r$? NNOX Apps https://matheducators.stackexchange.com/users/155 2022-01-10T00:31:26Z 2022-01-10T15:27:01Z <p>Every year, some 17 y.o. student makes the mistake of stopping <span class="math-container">$P(n, r)$</span> at <span class="math-container">$\color{darkorange}{(n - r)}$</span>, rather than <span class="math-container">$\color{forestgreen}{(n - (r - 1))}$</span>. Because they are in their last year of high school, they do understand why. But they goof up because this is counterintuitive, because <span class="math-container">$P(n, r)$</span> contains <span class="math-container">$r$</span>, but the last term contains <span class="math-container">$\color{forestgreen}{r - 1}$</span>.</p> <p>As lined in red below, I want students to understand this, not just memorize. How can they intuit this?</p> <blockquote> <p><a href="https://i.stack.imgur.com/0DZP7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0DZP7.jpg" alt="enter image description here" /></a></p> </blockquote> <p>David Patrick, <a href="https://artofproblemsolving.com/wiki/index.php/David_Patrick" rel="nofollow noreferrer">BS Math &amp; Computer Science, MS Math (Carnegie Mellon), PhD Math (MIT)</a>. <em>Introduction to Counting &amp; Probability</em> (2005), p 20.</p> https://matheducators.stackexchange.com/q/24786 0 What books teach the formula for the # of k-permutations of n objects, with x types, and $r_1,⋯,r_x$ = the number of each type of object? NNOX Apps https://matheducators.stackexchange.com/users/155 2022-01-08T23:40:01Z 2022-01-09T07:46:21Z <p>Some of <em><strong>my 16 year old students</strong></em> hanker after the formula for the # of k-permutations of n objects, with x types, where <span class="math-container">$r_1, ⋯, r_x$</span> = the number of each type of object. This is more generalized <a href="https://matheducators.stackexchange.com/a/7765/19348">than this question</a>.</p> <p>What books accessibly teach this formula? What books gently expound — fill in all gaps and steps in — <a href="https://mathoverflow.net/a/37218">this answer</a> by Prof. <a href="https://mathoverflow.net/users/972/suresh-venkat">Suresh Venkatasubramanian</a>?</p> <p>The book doesn't have to be written for 16 year olds. You can recommend books for undergraduates, but they must be readable and easygoing.</p> https://matheducators.stackexchange.com/q/24783 3 How long would it take someone to master the topics in the book like Book of Proof by Hammack and similar? a a https://matheducators.stackexchange.com/users/19230 2022-01-08T21:41:46Z 2022-01-09T22:23:11Z <p>If someone never had any experience with mathematical proofs and had only classes like Calc I-III (which he passed) without paying any attention to the proofs present in the textbooks, how long would it take this person to master the Book of Proofs by Richard H. Hammack?</p> <p>Now a few clarifications: This person would learn this book completely alone, at home, so no other students, professors or any other type of math enthusiasts. Let's say this person would learn 2 hours a day. And by &quot;mastering&quot; I mean this person would solve 90% of all exercises without looking at the solutions and understand all theorems.</p> https://matheducators.stackexchange.com/q/24782 4 Why has the chapter on second-order differential equations been moved to the website instead of being put in the book in Stewart Calculus 9th edition? HyperDimensionalBeing https://matheducators.stackexchange.com/users/17571 2022-01-08T16:50:05Z 2022-01-15T20:29:33Z <p>From the book</p> <blockquote> <p>The chapter on Second-Order Differential Equations, as well as the associated appendix section on complex numbers, has been moved to the website.</p> </blockquote> <p>It doesn't mention a reason in the book and I couldn't find an answer after searching the internet. Is there a reason related to pedagogy? Does the publisher profit from it?</p> https://matheducators.stackexchange.com/q/24777 6 How to convince a high school student that the $=$ symbol denotes identity? Vince Vickler https://matheducators.stackexchange.com/users/19341 2022-01-06T21:54:08Z 2022-01-08T02:23:27Z <p>In French language, arithmetic statements are often read, at the elementary school level, as , say, &quot; deux et deux font quatre&quot; , i.e. something like &quot; two and two make four&quot;.</p> <p>Out of this arises a belief according to which the <span class="math-container">$\Large=$</span> symbol expresses some sort of action , either an action performed by numbers themselves or by the person that operates the mental activity of computation which is supposed to be denoted by the <span class="math-container">$\Large +$</span> sign.</p> <p>This first belief may, in the head of older students, be replaced by the idea that <span class="math-container">$\Large =$</span> means &quot; has the same magnitude &quot; or &quot; has the same value as&quot;.</p> <p>I tried to show to high school students that the supposedly active meaning of <span class="math-container">$\Large =$</span> does not work anymore when the equality is reversed : <span class="math-container">$2+2$</span> may ( arguably) &quot; make&quot; <span class="math-container">$4$</span> , but would one say that <span class="math-container">$4$</span> &quot; makes &quot; <span class="math-container">$2+2$</span> ?</p> <p>But I did no manage to convince them that, at least in the case of arithmetic statements, the &quot; has the same magnitude &quot; interpretation is not correct.</p> <p>The identity meaning seems simply unbelievable to students.</p> https://matheducators.stackexchange.com/q/24770 3 Questions to help better understand the textbook David Steinberg https://matheducators.stackexchange.com/users/98 2022-01-03T14:17:59Z 2022-01-03T15:54:18Z <p>I am teaching a linear algebra class for math majors and non-majors out of the first 4 chapters of Lay's book. My plan is to have the students read a section prior to each class, have them answer a few questions about the reading (submitted online before class), and discuss their answers during class. (There will be a group work component too, but that's not why I am here.)</p> <p>I am trying to think of questions that would help them engage thoughtfully with the material. I feel good about asking some questions like &quot;find the row echelon form of the following matrix,&quot; but I think there are some more foundational questions that I might ask, like &quot;What new results/definitions are introduced in this section&quot; or &quot;how does this section relate to the previous section,&quot; or &quot;what new problem is introduced in this section&quot; etc. The kinds of question that experts ask themselves all the time (possibly without thinking about it).</p> <p>Can you suggest any such questions?</p> https://matheducators.stackexchange.com/q/24769 5 What do you think, is teaching on an actual board more efficient than using an online board? MathIsCool https://matheducators.stackexchange.com/users/19326 2022-01-03T00:25:05Z 2022-01-03T00:25:05Z <p>I am a sophomore math undergraduate and so far all of my university courses have been online due to the pandemic. I am really curious what you guys think about the efficiency of teaching mathematics via a drawing tablet compared to teaching it using the regular blackboards/whiteboards/whatever other types of boards exist that are found in most lecture halls.</p> <p>Let me give some motivation for this question since I think that it may seem odd for a student to ask this. Many of my professors claim that online math lecturing is not as good as in person math lecturing due to the limited space you have when you use a drawing tablet (Note: I am only going to focus on this point in this question. It is quite obvious that in person teaching is much better because we are social beings, but I want to discuss this particular aspect of online teaching). By this they mean that instead of having, say, 3 blackboards that you can write on, you just have as much space as your screen and that's basically it. Since I have only taken university math courses online, I can't really decide whether they are right about this and I don't want to judge based only on my high school experience. But the fact that this has been constantly brought up by many different people has inevitably got me thinking about whether my lectures would be much better (I have to mention that I believe that they are good in general) if classes were in person or if this is just something that stems from, say, a more canonical view towards lecturing. If you ask me, the biggest advantage of these drawing tablets is that the professor can save their notes and then send them to their students and in this way you basically get &quot;free&quot; lecture notes (by free I mean that the lecturer doesn't need to spend some extra time typesetting some notes, he just maybe has to polish a bit what he wrote). I find this to be superior to the traditional setting where you have to worry about getting the notes from someone in case you have some emergency and can't attent the lecture. In my view this advantage is pretty big and it seems to me that maybe after the pandemic is over and everyone resumes in person classes these drawing tablets could still be useful for some lectures if not all. But then again, my university education so far has been online, so I don't think that I am necessarily right. That's why I thought that this may be a good question to ask here.</p> https://matheducators.stackexchange.com/q/24765 13 Is there a measurable learning goal related to understanding proofs of important theorems? Duncan https://matheducators.stackexchange.com/users/19319 2022-01-01T23:26:26Z 2022-01-03T17:57:17Z <p>I believe that good math courses are structured around measurable learning goals. For example, &quot;can correctly replace a line integral with an equal double integral using Green's Theorem&quot; or &quot;can use the Sylow Subgroup Theorems to prove the smallest non-abelian simple group has order 60.&quot;</p> <p>I also believe that good math courses should give students a deeper understanding than &quot;follow these rules&quot;. In particular, at least the major level courses should certainly include proofs of main theorems, such as the Sylow Subgroup Theorems.</p> <p>But I cannot come up with a measurable learning goal (one that doesn't start with &quot;understands&quot;) that would motivate a class activity that includes proving important theorems.</p> <p>My example of using the Sylow Theorems is a good learning goal, but if that is the only goal, there is no reason to explain the proof of the Sylow Theorems. But I also feel that &quot;Can reproduce a proof of the Sylow Theorems&quot; is going too far.</p> <p>I hope you can understand the tension I'm getting at. Have you found any way to justify teaching well-known (beautiful!) proofs in a learning-goal motivated course design?</p> https://matheducators.stackexchange.com/q/24747 0 How can I visualize differential equations and Integration in real life? Ibrahim Omer https://matheducators.stackexchange.com/users/19303 2021-12-29T04:17:02Z 2022-01-14T00:21:48Z <p>How can we understand differential equations and Integration in real life so that we can understand calculus easily. All we do here, at university level is memorize calculus and get the answer. We cannot relate these beautiful equations to other physical phenomenon because we just memorize and didn't understand.</p> https://matheducators.stackexchange.com/q/17198 12 tutorial active learning AnyAD https://matheducators.stackexchange.com/users/10125 2019-09-29T08:53:12Z 2022-01-03T22:55:07Z <p>This is a question I asked on [Academia.se]. It did not get an answer, so I am re-posting it here.</p> <p>In the country where I live, university students studying mathematics usually attend lectures, consultation with their lecturers (if they have questions relating to the material beingtaught), and tutorial/practice classes.</p> <p>Years ago there was a change in the way tutorials are run. Now students work in groups of <span class="math-container">$2$</span>-<span class="math-container">$4$</span> in front of white boards, and the tutor walks around checking and commenting on the solutions written by students (on the white boards). (Previously the students would have asked questions, and the tutor would have solved the problems on a white board).</p> <p>The change was implemented on the basis of some research that suggested greater learning benefits for the students through more active participation and peer-consultation in problem solving. If anyone is familiar with, or can give a reference to, this research article/s, please feel free to provide that here.</p> <p>I'd be interested to learn how tutorials, or active participation in problem solving with peers in mathematics classics, are carried out in other countries.</p> <p>The benefit of the above practice class is that the student is `forced' to participate. Or at least one student from each group is, since there are always students who either don't attend or simply stand and contribute very little to the group discussion (even though they are encouraged to take turns and help each other understand, this does not necessarily occur).</p> <p>It is also hard to tell how much this helps an average student learn mathematics. Have there been any studies on this?</p> https://matheducators.stackexchange.com/q/11651 15 What are some recent, interesting, accessible pieces of mathematics NiloCK https://matheducators.stackexchange.com/users/308 2016-11-14T19:35:19Z 2022-01-13T21:09:05Z <p>Mathematics can come across as a sterile, dead subject - a catalogue of techniques long-ago decided, and forever relearned by each successive generation of students.</p> <p>This is <em>approximately</em> true for elementary and secondary mathematics, and for the standard progression of undergraduate courses (eg, Calculus 1,2,3, discrete math / combinatorics, ODE + Vector Calc, Analysis and Algebra).</p> <p>Of course, the subject is alive and kicking, with many thousands of active researchers learning, creating, refining, and publishing every day. But the vast majority of fresh research requires considerable expertise to understand, and are therefore inaccessible to younger students of the subject.</p> <p><strong>What, then, are some recent results that are interesting and accessible to students at (say) a secondary school level, which might exemplify that the subject remains active?</strong></p> <p>A couple of examples that come to mind (which could be fleshed out as answers) are the recent progress against the Twin Prime Conjecture, and the surprising observation that primes ending in $X$ 'favor' being followed by a prime ending in $Y$, for various $(X,Y)$ pairs.</p> <p>Where it's appropriate, please include links to any media treatment of the result.</p> <p>Let's roughly define 'recent' as being within the lifetime of some collection of students.</p> https://matheducators.stackexchange.com/q/10349 26 Why do students like proof by contradiction? Jessica B https://matheducators.stackexchange.com/users/4746 2016-01-07T15:46:58Z 2022-01-03T19:10:33Z <p>Every-so-often I come across proofs of the form</p> <ul> <li>Assume $X$ is false.</li> <li>Prove $X$ is true (without using that it is false).</li> <li>This contradicts that $X$ is false.</li> <li>Hence $X$ is true.</li> </ul> <p>I've seen students write such proofs, and I've seen them read this structure into a proof even when it isn't there.</p> <p><strong>Why do students seem to prefer this type of proof by contradiction to direct proof?</strong></p> <p>A couple of ideas I thought might be behind this:</p> <ul> <li>'proof by contradiction' is a thing they can give a name to, whereas direct proof doesn't have its own 'identity' in the same way;</li> <li>in a proof by contradiction you see at the start what you are aiming for, ie the opposite of your assumption (this won't always work, but is probably true of most cases they see);</li> <li>proof by contradiction changes the logic to something they find easier to think about (I came across this with a standard uniqueness proof - I think the idea that I had two objects that in the end turn out to actually be the same object was too confusing as an idea that was true, but it made more sense as a contradiction).</li> </ul> https://matheducators.stackexchange.com/q/10318 9 Advice on Proof-based Math Topics for High Schoolers Zaid Khalil https://matheducators.stackexchange.com/users/6185 2015-12-29T16:55:45Z 2022-01-10T04:18:36Z <p>I have a handful of high school students that are all prospective math/physics majors and have pooled their resources to hire me to teach them a proof based math course because it has become apparent to them in my physics class that proof and derivations are important. Basically I meet with them 2 hours a week and run it like a socratic method or students have to prove the theorems with my limited guidance and so far I have covered the following topics:</p> <ol> <li>GCD</li> <li>Euclid's Lemma</li> <li>Well Ordered Principle and Mathematical Induction</li> <li>Fundamental Theorem of Arithmetic</li> <li>Theorems of Elementary Arithmetic a*0=0, (-a)(-b)=ab etc.</li> <li>Arithmetic mean - Geometric Mean Inequality</li> <li>Pythagorean Theorem</li> <li>Cauchy Schwartz Inequality</li> <li>Irrationality of sqrt(p) where p is prime</li> </ol> <p>I will have only a limited amount of time with these students and I need to decide on what topics would be most valuable for them to be exposed to, here is the list of topics, which do you think would be most valuable:</p> <ol> <li>Conic Sections - going from the geometric definitions to the algebraic representations of conics</li> <li>Proofs of Archimedes: Areas of Circle, Quadrature of the Parabola, On the Sphere and Cylinder</li> <li>Exploring the completeness property of reals</li> <li>Sets, Nested Intervals and the Uncountability of the Reals</li> <li>Exploring sequences and series - in particular using telescoping series to derive Σi, Σi^2 , etc</li> <li>Area under curves Riemann Sums</li> <li>Counting and Binomial Theorem</li> <li>Sets and the Axioms of Probability</li> <li>Limits, Continuity (delta-epsilon proofs)</li> <li>Differentiability</li> <li>Properties of Exponential and Logarithmic Function using power series definition of the exponential function</li> <li>Vectors, Vector Spaces, Linear Operators</li> </ol> https://matheducators.stackexchange.com/q/10092 6 Aspiring HS Math Teacher: Textbooks for learning Algebra, Geometry, Trigonometry, and Calculus? SinaloaPaisa https://matheducators.stackexchange.com/users/5957 2015-11-27T02:21:35Z 2022-01-15T17:42:32Z <p>I plan to study for 6-12 months to take a high school mathematics teaching license exam. This one to be exact.</p> <p><a href="https://web.archive.org/web/20151123152128/http://www.fl.nesinc.com/pdfs/math6-12_tig_6thedition_051413_doe.pdf" rel="nofollow noreferrer">Florida Teacher Certification Examinations Test Information Guide for Mathematics 6–12</a></p> <p>Been out of school for roughly 8 years. Never took calculus before. Was never really the studious type in school, but managed out alright.</p> <p>Nonetheless, I am wanting to put a lot of time and effort into re-learning these subjects from the ground up not only to pass the test but also to help me be an effective teacher.</p> <p>Thus I am looking for the best textbooks to have to help aid me in this process.</p> <p>Thanks for the recommendations and any advice really. Cheers</p> https://matheducators.stackexchange.com/q/9847 27 Given a 3 4 5 triangle, how do you know that it is a right triangle? Mitch https://matheducators.stackexchange.com/users/2134 2015-10-27T19:20:55Z 2022-01-16T21:15:09Z <p>Without knowing the Pythagorean theorem, and in presenting reasons why the theorem might be true (without giving a full proof), is there any way to give examples of triangles that are intuitively understandable to be right triangles?</p> <p>For example, one can easily show (proof by <em>elementary</em> picture with elementary geometry) that two area 1 squares and one area 2 square (side <span class="math-container">$\sqrt{2}$</span> but no need to mention that) form an isosceles right triangle, by dissecting the unit squares into four smaller congruent isosceles right triangles and the area 2 squares into eight of the same triangles.</p> <p><a href="https://i.stack.imgur.com/LnnK6.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LnnK6.gif" alt="tiling showing one instance of PT: http://www.geom.uiuc.edu/~demo5337/Group3/tileproof.GIF"></a></p> <p>It is 'obvious that the abc triangle is right, and that the two smaller squares sum to the larger (with units the triangle tiles). </p> <p>But I don't know of a similar tiling for a 345 triangle. </p> <p><img src="https://i.stack.imgur.com/IlAGf.gif" width="375" height="400" alt="3 4 5 right triangle: jwilson.coe.uga.edu/emt668/EMAT6680.2001/Meyers/Emat%206690/Instructional%20Unit/image6.gif"></p> <p>The square on the hypotenuse is skew and impossible to visually validate. There's nothing to sum. It is simple calculation to note that <span class="math-container">$3^2 + 4^2 = 5^2$</span>, but its picture doesn't 'say' that the angle between the 3 and 4 sided square must be 90 degrees.</p> <p>So how do you know that a 3/4/5 triangle is right without proving the full force of the Pythagorean theorem?</p> https://matheducators.stackexchange.com/q/9837 11 Are there examples of countries where the use of CAS systems or graphing calculators was deemphasized or discontinued? Julia https://matheducators.stackexchange.com/users/1203 2015-10-26T19:35:26Z 2022-01-02T17:50:05Z <p>In the last 30 years more and more countries introduced graphing calculators and then CAS systems to their high-school students. But are there already any examples of a trend in the opposite direction? That is, examples where a state or country decided to row back and deemphasize or discontinue the use of CAS systems or graphical calculators in teaching mathematics (at the secondary level). If so, what were the (official) reasons for this decision? </p> <p>Please note that I don't want to discuss advantages or disadvantages of CAS systems in high school math, but just the very specific question above.</p> <p>An example of this phenomenon I am aware of is in a state of Germany, more specifically Baden Württemberg: from 2004 on at every Gymnasium [a type of high-school] graphical calculators were allowed (e.g. TI-83/84) in the final examination (Abitur) in one part of the exam. Yet, it has been decided that from the Abitur 2019 on only a simple scientific calculator will be allowed. Since this tool will not be allowed in the Abitur it changes the style of teaching. CAS tools or software like geogebra will be used from time to time on a PC/tablet/smartphone but most of the time, calculations will be done manually as well as in tests. </p> https://matheducators.stackexchange.com/q/2157 21 What are some good low-prerequisite examples for the heuristic advice "If you cannot prove it, prove something stronger."? user11235 https://matheducators.stackexchange.com/users/43 2014-05-05T07:19:15Z 2022-01-15T07:35:09Z <p>One useful trick in mathematics is to prove something stronger instead of the question asked.</p> <p>This works well in induction proofs (because strengthening the claim also strengthens the induction basis):</p> <blockquote> <p>Example: Prove $\frac1{1\cdot 2} + \frac1{2\cdot 3} + \dots +\frac1{(n-1)\cdot n} &lt; 1$. This admits a simple induction proof if one proves the exact formula instead.</p> </blockquote> <p>But there are also other examples</p> <blockquote> <p>Example: Proving that inscribed angles in a circle are equal becomes easier when we try to show that all of them are half the central angle.</p> </blockquote> <p>Of course, there are also lots of research examples that involve the trivialization of results as soon as new parameters are introduced or the whole problem is generalized to the "right" setting.</p> <p>Now, my question is the following:</p> <blockquote> <p>What are some good examples for the principle "Prove something stronger instead of the original statement" that would work well with first-year university students?</p> </blockquote> <p>Some calculus is ok, as long as it is introductory content. But generally, the more basic, the better, because the point is to illustrate the principle.</p> https://matheducators.stackexchange.com/q/22 27 What are some good examples to motivate the implicit function theorem? András Bátkai https://matheducators.stackexchange.com/users/61 2014-03-13T22:26:03Z 2022-01-04T18:32:45Z <p>I always had problems teaching the implicit function theorem in advanced analysis courses. This result is motivated by later applications, but it would be great to provide easily accessible examples to motivate the whole thing.</p> <p>I usually use Example 1.1.1 from <a href="http://books.google.de/books/about/The_Implicit_Function_Theorem.html?id=ya5yy5EPFD0C&amp;redir_esc=y" rel="nofollow noreferrer">Kranz, Parks: The implicit function theorem (Birkhäuser)</a>, which is <span class="math-container">$y^5+16y-32x^3+32x=0.$</span></p> <p>What else is there?</p> <p>ADDED in edit: Though I accepted a very nice answer, I would be happy for more answers and for more slist of possible examples.</p>