Recent Questions - Mathematics Educators Stack Exchange most recent 30 from matheducators.stackexchange.com 2020-11-23T16:44:12Z https://matheducators.stackexchange.com/feeds https://creativecommons.org/licenses/by-sa/4.0/rdf https://matheducators.stackexchange.com/q/19195 2 Start drawing a graph david https://matheducators.stackexchange.com/users/13795 2020-11-22T09:40:15Z 2020-11-22T12:59:12Z <p>I want to give to my students instructions/advises for drawing (directed) graphs (with few nodes, less than 15) in draft mode (before drawing final one on the official test paper).</p> <p>But I really have no idea what to say that would be relevant... Myself, I do it quite randomly before finalize.</p> <p>Is there some relevant instructions that could be given?</p> https://matheducators.stackexchange.com/q/19183 2 Interesting Trigonometry problems athos https://matheducators.stackexchange.com/users/9059 2020-11-19T21:24:56Z 2020-11-22T16:13:22Z <p>After explaining some basic trigonometry to my kid, such as <span class="math-container">$\sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$</span>, Law of sines, Law of cosines, I wonder if there are some interesting quizs for him to work on? Even better if it's a book.</p> <p>Example for &quot;interesting&quot;, <span class="math-container">$\sin 0 = \frac{\sqrt{0}}{2}$</span>, <span class="math-container">$\sin \frac\pi 6 = \frac{\sqrt{1}}{2}$</span>, <span class="math-container">$\sin \frac\pi 4 = \frac{\sqrt{2}}{2}$</span>, <span class="math-container">$\sin \frac \pi 3 = \frac{\sqrt{3}}{2}$</span>, while <span class="math-container">$\sin \frac\pi 2 = \frac{\sqrt{4}}{2}$</span>. Or more details in wiki page <a href="http://Trigonometric%20constants%20expressed%20in%20real%20radicals" rel="nofollow noreferrer">Trigonometric constants expressed in real radicals</a>.</p> <p>Explanation in searching of interesting problems -- I believe exercises are necessary for one to really get the ideas and tricks of trigonometry, but they might also be boring, so if there are a bunch of interesting problems, that'll be perfect.</p> https://matheducators.stackexchange.com/q/19179 5 Why do we write numbers with decreasing place values? Jasper https://matheducators.stackexchange.com/users/667 2020-11-19T17:26:12Z 2020-11-19T17:51:21Z <p>This question came up while teaching ~16 year olds binary numbers. Why do place values increase to the left and not the other way round?</p> https://matheducators.stackexchange.com/q/19169 0 Weekly Problem Sets for Lang's Undergraduate Algebra and Undergraduate Analysis [closed] John Clever https://matheducators.stackexchange.com/users/13911 2020-11-19T02:19:46Z 2020-11-19T02:19:46Z <p>For Lang's Undergraduate Algebra and Undergraduate Analysis, I cannot find any weekly problem sets online. I've found that doing every problem in every section takes far too long, and without problem sets, I don't know which problems to do without doing all of them.</p> <p>Does anyone know where I can find online weekly problem sets for these textbooks?</p> https://matheducators.stackexchange.com/q/19164 28 How to answer questions about the purpose of learning math? BKE https://matheducators.stackexchange.com/users/14091 2020-11-18T22:00:36Z 2020-11-22T05:50:08Z <p>What are some good answers to questions eg. &quot;why do we need to study square roots&quot;? Of course the answer depends highly on who is asking. For the scope of my question, I have a student in mind, who is - given their interests for further education - likely never going to use square roots in adult life, but still has to learn them. (Of course my question is not about square roots specifically. It can be any topic which, realistically, will not be used &quot;in real life&quot; by the student.)</p> <p>I have some answers, but I feel that they all fall short of being a satisfactory answer:</p> <ol> <li><p>&quot;Because you have to know it for your grades, so later you will have more choice of what you want to do in life&quot; - this is my honest answer, but I fear it might be a source of resentment for the student against the education system, which I don't want.</p> </li> <li><p>Bring some forced example from &quot;real life&quot;, eg. &quot;John has a son whose age is the square root of his age. John is 49 years old. How old is his son?&quot; - I want to avoid these at any cost, any student will see through these and will be enforced in his view, that &quot;math is nonsense&quot;. Even if the example is better and <em>I think</em> more relevant &quot;in real life&quot;, if it fails to resonate deeply with the student, then it will backfire as an example.</p> </li> <li><p>&quot;It teaches you to think abstractly&quot; - I am deeply skeptical of some generic problem solving ability. I think we are good at solving problems that we practice solving, and it's not obvious how one transfers to another problem. And even if generic problem solving activity is a thing, it is hard to see why practicing square roots is the best thing to improve it, instead of something else that the student is more interested in. I think, hearing this answer, most students will nod along and still think to themselves that math is nonsense.</p> </li> <li><p>&quot;Distract from the question&quot;. The &quot;why learn this&quot; question usually arises from a thought process, that is just distracting the student. He finds square roots hard to understand, so he is looking for an excuse to think about something else. He never seems to ask the &quot;why do this&quot; question for activities that he otherwise just enjoys eg. reading a novel, playing music, drawing etc. and if asked &quot;why&quot;, he will rationalize why it is a good thing to do. So anything that distracts from the &quot;why question&quot; and manages to bring focus back to thinking about the problem itself is good. (If you have any concrete strategies for this, I appreciate if you share it.) However: I still think, that the &quot;why learn this&quot; question is important and I would not like to ignore it completely.</p> </li> <li><p>I am an applied mathematician myself, so I can bring real life examples where math (beyond basic arithmetic) is useful. But the student will likely not have the knowledge to fully appreciate it, so while he might be impressed momentarily, I don't think I can expect any consistent success. It certainly makes sense to try connecting the applications of math with school curriculum every once in a while, but honestly these are often very far away from each other.</p> </li> </ol> <p>Is there any good strategy to answer to the &quot;why learn this&quot; question, that I missed?</p> <p>Did I make a conclusion above, which is absolutely wrong?</p> <p>If possible, I am looking for answers which are based on at least some empirical evidence, but I am also interested in concrete examples where something has worked really well.</p> <p><strong>EDIT:</strong> thanks for all the contributions. Some further clarification about my question:</p> <ol> <li><p>I look for direct answers and not analogies. Math is not sports, art, or music. To me, such answers are essentially distraction (see point 4 above).</p> </li> <li><p>Several answers are in the direction of &quot;it's part of general knowledge&quot;, I see this as appeal to authority and as such, the nicer version of appealing to grades (point 1 above).</p> </li> <li><p>Some answers are quite specific, eg. lots of suggestions are about finance (which btw I personally find really boring). Some topics in math connect better with finance than others and having one &quot;go-to example&quot; isn't a really useful mindset when dealing with a concrete student and a concrete topic.</p> </li> </ol> https://matheducators.stackexchange.com/q/19158 2 Logarithms chronologically before algebra Michael Hardy https://matheducators.stackexchange.com/users/205 2020-11-15T23:31:36Z 2020-11-22T01:53:47Z <p>Do any textbooks or (somewhat?) standard curricula introduce logarithms and their applications in arithmetic without assuming the students know any algebra?</p> https://matheducators.stackexchange.com/q/19154 8 Question about function notation Ferenc Beleznay https://matheducators.stackexchange.com/users/14957 2020-11-15T16:19:06Z 2020-11-18T16:44:33Z <p>In the textbook I am using to teach mathematics to high school students I found the following illustration about composition of functions.</p> <p><a href="https://i.stack.imgur.com/14J94.png" rel="noreferrer"><img src="https://i.stack.imgur.com/14J94.png" alt="enter image description here" /></a></p> <p>I do not agree with this illustration. For me <span class="math-container">$g$</span> is the slicer, <span class="math-container">$g(x)$</span> is the sliced potato, <span class="math-container">$f$</span> is the fryer and <span class="math-container">$f(g(x))$</span> is the bowl of french fries. I would introduce the notation (not on the diagram though) of <span class="math-container">$f\circ g$</span> for the process of slicing and then frying.</p> <p>Am I being overly pedantic?</p> https://matheducators.stackexchange.com/q/19144 3 What is an algebraic explanation of why the product of the slopes of perpendicular lines is $-1$? [duplicate] Joseph O'Rourke https://matheducators.stackexchange.com/users/511 2020-11-14T01:22:06Z 2020-11-15T14:29:03Z <blockquote> <p><strong>Q:</strong> What is a succinct, clear and purely <em>algebraic</em> explanation of why the product of the slopes of perpendicular lines is <span class="math-container">$-1$</span>?</p> </blockquote> <p>Here I am aiming for high-school students (in the U.S.). I have a purely geometric explanation (below), but I would like to supplement it with a purely algebraic explanation.</p> <hr /> <img src="https://i.stack.imgur.com/wPnqn.jpg" /> <p>Rotating a slope triangle by <span class="math-container">$90^\circ$</span>.</p> https://matheducators.stackexchange.com/q/19140 8 How do you teach students about the concept of a proof? iYOA https://matheducators.stackexchange.com/users/14497 2020-11-10T19:43:26Z 2020-11-15T00:01:01Z <p>I get this question a lot from new students who are taking their first proof-based math class. They are struggling because they don't have that fluency with proofs, to begin with. They don't know what constitutes valid proof or they have trouble approaching questions that ask them to prove something*.</p> <p>*Note this is <em>different</em> from asking an experienced student to prove a hard theorem, where the difficulty arises in the actual content of the proof, not the various methods of how to go about proving it.</p> <p>When I think back on how I learned proofs, I grew up spending many years reading proofs and solving hard problems, and then eventually I started constructing my own proofs. But I would say it happened organically. It would be the same way a native speaker grows up in their home country, hearing the language being spoken around them and gradually assimilating it themselves. This is NOT useful to a student taking their proof-based class who has to immediately learn to be 'fluent' in the language of proofs.</p> <p>What do you tell a student in this position? Or what would you recommend to be the most efficient way to get that understanding?</p> <p>Edit: I realized there's really two main questions going on here:</p> <ol> <li>How do you go about teaching proofs in general?</li> <li>Let's just say they're in an introductory class that already assumes some basic fluency of proofs. (Like a first-year course in real analysis or something). How can they go about acquiring this fluency on their own?</li> </ol> https://matheducators.stackexchange.com/q/19137 1 How to create an online examination in a small class that ensures certain academic integrity Field Aussie https://matheducators.stackexchange.com/users/14865 2020-11-10T14:39:42Z 2020-11-10T14:49:22Z <p>I am teaching a Calculus class with around 30 students. The classroom is about 100 m^2.</p> <p>So far I have been using myopenmath.com to assign homework problems to my students. Now I would like to deliver the Mid-term examination via myopenmath.com.</p> <p>The set-up in my mind is as follows: the students will bring their own devices into class and access myopenmath.com to get the exam problems. The exam is an open-book exam, but the students are not allowed to receive support from anyone else. I’ll wander around the class and supervise the exam.</p> <p><strong>My question</strong>: do we have any kind of technology to support this exam so that it minimises the chance of trying to cheat from the students?</p> <p>My problem is that I have to run the exam on my own, without any support from the university. Of course, the traditional examination form (written exam) might help to prevent academic misconduct. But I love the advantage of myopenmath.com in providing immediate feedback and creating randomized questions.</p> https://matheducators.stackexchange.com/q/19136 5 A video game for teaching the concept of a mathematical proof Erel Segal-Halevi https://matheducators.stackexchange.com/users/13986 2020-11-10T09:00:59Z 2020-11-10T09:00:59Z <p>Two students in my game development course would like, as a final project, to develop a video game for teaching mathematics. In contrast to the <a href="https://matheducators.stackexchange.com/q/4208/13986">many other games for teaching maths</a>, which focus mainly on problem solving, they want to focus on the concept of a mathematical proof, in particular: proofs of classic theorems in number theory, such as Euclid's theorem or the fundamental theorem of arithmetics. Students often find such proofs hard to learn due to their &quot;infinite&quot; nature: while any finite number of examples is not sufficient, there is a finitely-written proof that covers <em>all</em> infinitely-many cases. The difficulty is both in understanding the proofs and in writing their own proofs.</p> <p>As an example in-game idea that might be used to illustrate these abstract concepts, we thought of a hero that must defeat a monster. The hero can shoot at the monster some special cases of the theorem (e.g. prime numbers), but this only slows the monster down temporarily. In order to defeat it completely, the hero must build an &quot;induction machine&quot; that can construct cases endlessly. There are many other proof concepts that should be considered.</p> <p>My questions are:</p> <ul> <li>Are there video games that aim to teach the concepts of mathematical proofs (rather than just problem solving)?</li> <li>What are some research papers regarding the difficulties encountered by students in learning mathematical proofs?</li> </ul> https://matheducators.stackexchange.com/q/19124 14 Replacement for the Pac-Man grid analogy Misha Lavrov https://matheducators.stackexchange.com/users/9609 2020-11-07T16:22:40Z 2020-11-10T16:58:05Z <p>To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and &quot;identify opposite sides&quot;. We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Provided that our material is stretchy enough, which isn't an issue topologists concern themselves with.)</p> <p>In the past, I've described this description of the torus by analogy with <a href="https://en.wikipedia.org/wiki/Pac-Man" rel="noreferrer">Pac-Man</a>. In this video game, Pac-Man can leave the screen on one edge, and come back onto the screen from the same position on the other side.</p> <p>But I want to abandon this analogy, and come up with something better, because:</p> <ul> <li>If you haven't played Pac-Man, it's not very helpful - and how many people these days have? I think I've played Pac-Man on a TI calculator a total of once or twice in high school.</li> <li>If you <em>have</em> played Pac-Man, it's not very helpful, because in a typical Pac-Man maze, the &quot;tunnels&quot; that allow this wrapping-around behavior only go one way: from left to right. So a Pac-Man level is more like a cylinder than a torus.</li> </ul> <p>Are there better analogies?</p> https://matheducators.stackexchange.com/q/19117 2 Are differential equations considered calculus and included in a calculus class or is it its own class? Luke Justin https://matheducators.stackexchange.com/users/14909 2020-11-06T01:44:44Z 2020-11-18T05:44:21Z <p>Are differential equations considered calculus and included in a calculus class or is it its own class? Also, if it is its own class then what calculus classes does it come after?</p> https://matheducators.stackexchange.com/q/19092 1 When are students taught implicit and parametric representations of curves? Joseph O'Rourke https://matheducators.stackexchange.com/users/511 2020-11-02T21:01:43Z 2020-11-20T09:02:30Z <p>Do students learn implicit equations (such as <span class="math-container">$x^2+y^2-r^2 = 0$</span>) and parametric equations (e.g., <span class="math-container">$x=a t^2,\;y= 2 a t$</span>) in a first course in algebra, which in the US would be early high school, maybe 9th grade? Or not until pre-calculus or calculus, late high school or early college? Perhaps parametric equations are not part of standard curricula?</p> <p>I'm trying to gauge how much it is reasonable for me to assume when writing for high-school students.</p> https://matheducators.stackexchange.com/q/19089 9 Is there a name for paths that follow gridlines? David Elm https://matheducators.stackexchange.com/users/8642 2020-11-02T17:02:42Z 2020-11-10T02:48:14Z <p>I'm writing up an activity where students are looking at pathlengths that follow along gridlines. <a href="https://i.stack.imgur.com/miFJ5.png" rel="noreferrer"><img src="https://i.stack.imgur.com/miFJ5.png" alt="enter image description here" /></a></p> <p><a href="https://i.stack.imgur.com/AqVSd.png" rel="noreferrer"><img src="https://i.stack.imgur.com/AqVSd.png" alt="enter image description here" /></a></p> <p>Is there a word or phrase that is commonly used to describe those paths, but doesn't include diagonals?</p> <p>I'll probably call them 'grid line paths', but if there is a common term, I'd like to align with that.</p> https://matheducators.stackexchange.com/q/19088 4 Writing mathematics in real time for lectures using Latex N.B. https://matheducators.stackexchange.com/users/14894 2020-11-02T16:59:51Z 2020-11-15T16:27:41Z <p>I am supposed to hold tutorial sessions for an undergraduate course in calculus. But the software provided by the university is not good at all. I saw the question suggesting to use Ziteborad but it is not what I am looking for.</p> <p>I am searching for some real time Latex compiler which would allow me to write what I am explaining on the fly without waiting to compile each time I add a sentence or a formula. Stack Exchange has an incorporated MathJax compiler whose functionality is similar to what I would like. Is there not a program or browser page which visualizes what I write in Latex code on a big page easy to present via screen sharing?</p> https://matheducators.stackexchange.com/q/19085 3 Clearest verb phrases for operations Nick C https://matheducators.stackexchange.com/users/470 2020-10-31T14:04:51Z 2020-11-08T01:30:42Z <p>What is the clearest way to indicate various operations along the following lines:</p> <p><span class="math-container">$f(x) = 3x$</span>: The function <span class="math-container">$f$</span> multiplies its input by 3.</p> <p><span class="math-container">$g(x) = x-5$</span>: The function <span class="math-container">$g$</span> decreases its input by 5.</p> <p><span class="math-container">$h(x) = 2^x$</span>: The function <span class="math-container">$h$</span>...Raises 2 to the power of its input? Exponentiates its input on a base of 2? Takes 2 to its input?</p> <p>In the first two cases, I made a choice of whether to refer to the operation's name (&quot;multiply&quot;) or a (hopefully) plain language action being performed (&quot;decrease&quot;). For the function <span class="math-container">$f$</span>, I could have said that it <em>triples its input</em> or <em>increases its input by 200%</em>, but these don't seem to generalize well for the purposes of communicating, and the latter is almost never obvious to students. Similarly, I could have said that the function <span class="math-container">$g$</span> subtracts 5 from its input, but I am not convinced this is any simpler than <em>&quot;decreases its input by&quot;</em>.</p> <p>Is there a simplest verb phrase for exponents? What do you think is clearest for students if I want to maintain the form &quot;The function <span class="math-container">$h$</span> ____________&quot;?</p> https://matheducators.stackexchange.com/q/19044 5 How important is it to come up with or learn an elementary solution? Ma Joad https://matheducators.stackexchange.com/users/9339 2020-10-22T08:47:41Z 2020-11-16T10:54:56Z <p>Note: by &quot;elementary&quot; I mean &quot;without using more advanced theory and tools&quot;.</p> <p>Students are sometimes required or encouraged to solve very difficult problems using limited number of tools and machinery. It is not uncommon in competition-style exams that problems in algebraic/analytic number theory are solved using elementary number theory (i.e. without analysis or abstract algebra). Various other exams in the world also have a limit on the number of tools that can be used. For example, sometimes, people are required to prove something similar to mean value theorem for a specific given function (with an explicit expression) solely by very complicated algebraic computations, without Calculus, because the rigorous <span class="math-container">$\epsilon$</span>-<span class="math-container">$\delta$</span> definition of derivatives and limits are not yet taught.</p> <p><strong>What are the reasons why we wish to tackle hard problems with elementary methods? And is it beneficial or not?</strong></p> <p>If we are building a theory from axioms, when we write proofs, we must only use things that are already proven. But in the situation I describe above, we are not building up a theory; instead, we are <strong>applying</strong> some theory to solve a problem, so in this situation, what are the reasons why we sometimes limit our range of tools? What are some pros/cons of this?</p> <p>This might be, to some extent, a matter of taste, but it is still interesting to know reasons for this.</p> https://matheducators.stackexchange.com/q/19017 43 How to explain that winning the lottery is not a 50/50 distribution? WoJ https://matheducators.stackexchange.com/users/8076 2020-10-19T18:41:58Z 2020-11-15T02:38:13Z <p>When casually discussing with my 13 yo child about probabilities, he told me</p> <blockquote> <p>there is a 50% chance to win at the lottery</p> </blockquote> <p>To what I said</p> <blockquote> <p>no, there is a 1 chance over 90 million</p> </blockquote> <p>(I roughly estimated <span class="math-container">$_{7}^{49}\text{C}$</span> which I think is more or less the lottery here)</p> <p>To what he replied</p> <blockquote> <p>no: either you win, or you don't. That's the probability of the fact to win.</p> </blockquote> <p>He is obviously wrong, me being the educated father and him the child with silly ideas. I am now sitting and thinking about a counter-argument.</p> <p>On the serious side, I am trying to quantify his answer from a mathematical perspective but I believe that the whole premise of his reasoning is wrong (but I am not sure where).</p> <p><strong>Note</strong>: I am asking the question here and not on Math SE because it is in my opinion more a matter of how to explain math to children (and accessorily, to their parents), more than a question about probabilities.</p> <p><strong>Note 2</strong>: I should have made it clear that we are there after many discussions on probability so he understands the &quot;number of positive outcomes&quot; / &quot;number of all possible outcomes&quot;. What he said was more like an invitation for discussion about the concept of &quot;winning or not, as a single event&quot;.</p> <p>All the answers are really interesting, I will have a hard time picking something up for the chosen one (but will upvote all)</p> https://matheducators.stackexchange.com/q/18524 8 Logic and proofs in secondary school Rusty Core https://matheducators.stackexchange.com/users/7930 2020-07-02T20:11:11Z 2020-11-13T17:37:10Z <p>Inspired by the question <a href="https://matheducators.stackexchange.com/questions/18494/when-do-college-students-learn-rigorous-proofs">When do college students learn rigorous proofs?</a>, I became curious when pupils in secondary schools learn about proofs, what kinds of proofs they are, how rigorously they are taught, do they learn any formal logic before or simultaneously with learning proofs?</p> <p>I am not focussed on American schools, I would like to learn how proofs are tackled around the world.</p> https://matheducators.stackexchange.com/q/18351 3 Is there a 'statistics theory' course plan for practitioners? avgvstvs https://matheducators.stackexchange.com/users/676 2020-05-18T16:02:36Z 2020-11-15T08:02:05Z <p>My job is starting to have me delve into categories that require things like regression analyses on data sets, essentially I'm being introduced to &quot;Data Science&quot; type material. Coming from a computer science background however, I'm aware of how easy it is to misapply statistics.</p> <p>My only course on statistics was over a decade ago and it didn't include any of the theoretical underpinnings that actually explained <em>why</em> the formulas worked, which would help me understand when to use method X vs method Y. I'm dreadfully afraid of creating false findings.</p> <p>I'm looking for teaching resources that would hopefully bridge the gap from &quot;I can do X, Y, or Z&quot; but would give enough math to help me understand when and where to use what. If there's anything like Roger Penrose's &quot;The Road to Reality&quot; for statistics, that would help. Otherwise any suggestions of what a good course plan for self-study would look like would be welcome.</p> <p>[EDIT] Clarifications:</p> <p>Having a comp sci background, I know for example that if the problem before me seems like a graph-theoretic problem, I can reach for my copy of &quot;Graph Theory and its Applications&quot; from Gross/Yellen.</p> <p>Physics isn't my forte, but when I want to leverage the mathematics I do have, I can reach for &quot;The Road to Reality&quot; by Roger Penrose.</p> <p>And if it's dealing with Data structures, I can check out &quot;The Art of Computer Programming&quot; from Donald Knuth, and for algorithms--although it's a textbook--there's also Cormen's &quot;Introduction to Algorithms,&quot; or even Skeina's book &quot;The Algorithm Design Manual.&quot; All of these works provide enough theory and proofs to make it pretty clear <em>why</em> these tools work. I'm fishing for items in this same category for Statistics, but judging by the comments, there isn't quite this level of organization and unity in statistics?</p> https://matheducators.stackexchange.com/q/17951 2 Teaching Quantifiers Before Logical Connectives 10understanding https://matheducators.stackexchange.com/users/13576 2020-02-23T14:38:53Z 2020-11-22T16:07:30Z <p>In this short question, I would like to ask whether it is possibly good to teach quantifier before logical connectives in a logic introduction lecture?</p> <p>I know there is a relationship between them but my question is based on my observation that students usually already get the concept of existential quantifier. Before introducing them to more complex forms of statement, maybe they can understand basic form of quantifier statements first (of course, after introducing open and closed sentence). This is just an idea. Let me know if anyone has done this because I have never seen one.</p> <p>Thank you very much!</p> https://matheducators.stackexchange.com/q/17161 30 Quote to show students don't have to fear making mistakes dietervdf https://matheducators.stackexchange.com/users/5898 2019-09-25T21:14:19Z 2020-11-21T00:01:21Z <p>I have some high school students which seem to be <em>afraid</em> of making mistakes. They are hesitant to make exercises in class because they want their course notes to be super clean, without any mistakes. The following has <em>often</em> happened in my class. A student writes (when <span class="math-container">$a$</span> is a positive real) <span class="math-container">$\sqrt{16a^2+9a^2} = 4a+3a = 7a$</span>. When the correct solution is shown and the mistake the student made is discussed, the student erases his mistake and writes down the correct solution. However, when studying the contents, he is not reminded of the mistake. (Which would be very valuable)</p> <p>Does anyone know of a nice quote which shows that there is great learning potential in making mistakes (and figuring out why!). (Especially in math)</p> <p>I know of the following quote by prof. Francis Su. A nice quote, but it's more about the value of persistence.</p> <blockquote> <p>Struggling is a good thing… it’s where learning happens, it’s what we professors are always doing in our research… the struggle is the most interesting place to be.</p> </blockquote> https://matheducators.stackexchange.com/q/14919 7 In teaching mathematics, should one always follow some international standards such as ISO 80000-2? Zuriel https://matheducators.stackexchange.com/users/6166 2018-12-15T14:39:12Z 2020-11-15T05:36:02Z <p><a href="https://en.wikipedia.org/wiki/ISO_80000-2" rel="nofollow noreferrer">ISO 80000</a>-2:2009 is a standard describing mathematical signs and symbols developed by the <a href="https://en.wikipedia.org/wiki/International_Organization_for_Standardization" rel="nofollow noreferrer">International Organization for Standardization</a> (ISO). In teaching mathematics, should one always follow this standard? </p> <p>As an example, there is no universal agreement on <a href="https://math.stackexchange.com/questions/283/is-0-a-natural-number">whether zero should be considered as a natural number or not</a>. But since there is an international standard which considers zero as a natural number (though the teacher may not like this convention), should one teach base on this standard only in order to avoid any confusion?</p> <p>Another example is, people use <span class="math-container">$\log x$</span> for both natural and common logarithm. In my teaching I usually spend 10 minutes explaining the difference and the fact that in most calculators, <span class="math-container">$\log$</span> means common logarithm and in <a href="https://www.wolframalpha.com/input/?i=log(x)" rel="nofollow noreferrer">WolframAlpha</a>, <span class="math-container">$\log$</span> means natural logarithm. My personal rule of thumb is, <span class="math-container">$\log$</span> means natural logarithm starting at pre-calculus and means common logarithm before pre-calculus. Based on ISO 80000-2, one should use <span class="math-container">$\lg$</span> for common logarithm and <span class="math-container">$\ln$</span> for natural logarithm. Under this convention, ambiguity ceased to exist. </p> <p><strong>Edit</strong>: The Chinese goverment published a <a href="https://www.google.com/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=3&amp;ved=2ahUKEwj96emrt6PfAhXrlOAKHYLrBHMQFjACegQIBBAC&amp;url=http%3A%2F%2Flxbwk.njournal.sdu.edu.cn%2FCN%2Fitem%2FdownloadFile.jsp%3Ffiledisplay%3D20170223092530.pdf&amp;usg=AOvVaw2UBIegpOqbr391oAYjugsx" rel="nofollow noreferrer">standard</a> in 1993, requiring all institutions in China teach that <span class="math-container">$0$</span> is a natural number. So at least in China, the dispute is forcefully solved.</p> https://matheducators.stackexchange.com/q/12377 10 What is the difference between "numeracy" and "number sense"? Dag Oskar Madsen https://matheducators.stackexchange.com/users/39 2017-05-27T12:22:59Z 2020-11-16T01:43:31Z <p>Is there a difference between <em>numeracy</em> and <em>number sense</em>, or are they synonymous? In my language they are often both translated to the same word (<em>tallforståelse</em>).</p> <p>I'm thinking that perhaps numeracy describes a competency, while number sense is more about having a "feel" for numbers or understanding relations between numbers. Is this how the terms are used in education literature? </p> https://matheducators.stackexchange.com/q/9689 14 How do you explain why perpendicular lines have negative reciprocated slopes? David Steinberg https://matheducators.stackexchange.com/users/98 2015-09-30T21:41:03Z 2020-11-16T18:26:01Z <p>For my purposes, I am interested mostly in a medium-sized liberal art college setting. My students have mostly seen this before, but it is not something they understand. When discussing parallel lines, I have them try to find a point of intersection, fail to find such a point, and conclude that the lines never meet, which I think is convincing. </p> <p>I am trying to find ways that they can convince themselves that two lines with negative reciprocated slopes are perpendicular. Do you have a method that you like?</p> <p>(also: not sure how to tag this question)</p> https://matheducators.stackexchange.com/q/7528 42 Real-world examples of more "obscure" geometric figures celeriko https://matheducators.stackexchange.com/users/3237 2015-03-04T17:39:35Z 2020-11-09T11:12:27Z <p>As part of my secondary geometry class I like to hook students by presenting real-world examples (usually images I find online or have taken myself) of different geometric shapes from real life. For instance, a lesson on the area of a circle might start out with a picture of a pizza pie or a lesson on the midsegments of triangles might start out with a picture of the Triforce. However, there are some geometric figures that I have had a hard time finding interesting, real-world examples of. Those figures (and I know I am forgetting a bunch..) are:</p> <ul> <li>Segment of a circle</li> <li>Secant line</li> <li>Trapezoid (Isosceles or not)</li> <li>Inscribed angle</li> <li>Parallel lines cut by a transversal</li> </ul> <p>I was wondering if anyone had any ideas for these geometric figures of interesting, real world examples? Also, I think it would be great that if people are aware of really cool real world examples for the more "standard" geometric figures to post those as answers as well. For instance, the Dockland Building at the Port of Hamburg is an astoundingly perfect parallelogram :) <img src="https://i.stack.imgur.com/tFbU0.jpg" alt="enter image description here"> Having a collection would be very helpful for teachers because I have not found a better way to get my students right into the groove by starting class off with a brief discussion about an interesting picture!</p> https://matheducators.stackexchange.com/q/7281 16 How to explain the difference between the fraction a / b and the ratio a : b? Abdallah Abusharekh https://matheducators.stackexchange.com/users/3306 2015-01-24T05:54:16Z 2020-11-20T14:54:17Z <p>I found it difficult to explain the difference between the fraction a / b and the ratio a : b. This subject is for pupils of grade 5. So is there a real difference between them and how to explain the difference in simple way ?!</p> https://matheducators.stackexchange.com/q/362 14 What is a good physical example of Stokes' Theorem? mirams https://matheducators.stackexchange.com/users/25 2014-03-17T11:02:09Z 2020-11-08T15:11:55Z <p>I find it useful to give physical examples of theorems, especially in vector calculus - for example $\nabla f$ being the direction of maximum ascent on a surface $f$.</p> <p>What is a good example for <a href="http://en.wikipedia.org/wiki/Stokes%27_theorem">Stokes' Theorem</a>?</p> https://matheducators.stackexchange.com/q/130 33 Examples why university education is important for future high school teachers Markus Klein https://matheducators.stackexchange.com/users/114 2014-03-14T16:28:14Z 2020-11-19T00:28:48Z <p>At my university, the students in math are mixed up (1/3-1/2 are bachelor/master students, the rest are future high school teachers). A problem arising very often is the discussion dramatically summarized by "I don't need this in high school/I already know enough, don't bother me with this abstract stuff". </p> <p><strong>Are there good examples/showcases/questions/arguments/etc. illustrating how important math education at university is for their life as high school teachers?</strong> </p> <p>(Expect from general discussion like "You need to know more than the school kids/What if they change the content of high school education/What will <em>you</em> tell the school kids when the say that they wanted to study something completely unrelated to math?" - If you want to, you can also add general issues, but I am more interetesed in concrete examples)</p>