tag:blogger.com,1999:blog-25024739247844423642018-05-21T23:59:40.403-07:00Introduction to Mathematical ProofsA place to share my interests in the Introduction to Proofs course and to gather information about free resources for this course.Ted Sundstromnoreply@blogger.comBlogger16125tag:blogger.com,1999:blog-2502473924784442364.post-57968034403388089852014-01-24T06:20:00.003-08:002014-01-24T17:51:24.849-08:00Portfolio Project in a Proofs Course<span style="font-size: large;">In a post on this blog dated August 13, 2013, I expressed my opinions about the importance of writing in the introduction to proofs course. At Grand Valley State University, our introduction to proofs course, MTH 210 Communicating in Mathematics, is in the university's Supplemental Writing Skills (SWS) Program. Following is a description of this program that I include in my course syllabus.</span><br /><span style="font-size: large;"></span><br /><a name='more'></a><span style="font-size: large;"><br /></span><span style="font-size: large;">MTH 210 is a designated SWS (Supplemental Writing Skills) as described in the GVSU catalog. Completion of Writing 150 with a grade of C or better (not C-) is a prerequisite. SWS credit will not be given to a student who completes this course before completing Writing 150. SWS courses adhere to certain guidelines. Students turn it a total of at least 3000 words of writing during the semester. Part of that total may be essay exams, but a substantial amount of it is made up of finished essays or reports or research papers. The instructor works with the students on revising drafts of their papers, rather than simply grading the finished pieces of writing. At least four hours of class time are devoted to writing instruction. At least one-third of the final grade is based on the writing assignments.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The primary assignment that the professors at Grand Valley use to satisfy this requirement is the so-called Portfolio Project. My version of the portfolio project this semester consists</span><span style="font-size: large;"> of ten mathematical problems and one expository essay that will require a proof of at least one mathematical result.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Instead of describing this project here, you can click on the following links to download the three documents that I am using for this project.</span><br /><span style="font-size: large;"><br /></span><br /><ul><li><span style="font-size: large;"><a href="https://dl.dropboxusercontent.com/u/10471758/portfolio-guidlelines-2014-winter.pdf" target="_blank">Guidelines for the Portfolio Project</a></span></li><li><span style="font-size: large;"><a href="https://dl.dropboxusercontent.com/u/10471758/portfolioprobs-2014-winter.pdf" target="_blank">The Portfolio Problems</a></span></li><li><span style="font-size: large;"><a href="https://dl.dropboxusercontent.com/u/10471758/essay-2014-winter.pdf" target="_blank">The Portfolio Essay</a></span></li></ul><br /><span style="font-size: large;">I also plan to make these documents available on the website for Mathematical Reasoning: Writing and Proof. If you are an instructor and are interested in obtaining the LaTeX files for these documents, please contact me at <a href="mailto:mathreasoning@gmail.com">mathreasoning@gmail.com</a>.</span><br /><div><br /></div>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-26264898978482713472014-01-23T13:02:00.001-08:002014-01-23T13:02:17.630-08:00A Couple of Requests<span style="font-size: large;">As many of you know, I am the author of a book for the Introduction to Proofs course. My book is <i>Mathematical Reasoning: Writing and Proof</i>. I have made this book available to download for free using a Creative Commons License. You can download the book at the <a href="https://sites.google.com/site/mathematicalreasoning3ed/" target="_blank">website for the book</a>. A soft cover version of this book can also be purchased for less than $20 at <a href="http://www.amazon.com/Mathematical-Reasoning-Writing-Ted-Sundstrom/dp/1492103853/ref=sr_1_1?ie=UTF8&qid=1388771063&sr=8-1&keywords=ted+sundstrom" target="_blank">Amazon.com</a>.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;"><b>Request #1</b></span><br /><span style="font-size: large;">Advertising is one of the difficulties with an open-source book. There are websites that give lists of free textbooks, but there is usually no information from users of the book. So my request to those of you who have used (or are using) the book to send me a short quote about the book or a longer review of the book. I would like to include these quotes and reviews on the website for the book. You can make the quote or or review as a response to this post or you can send it to me at </span><a href="mailto:mathreasoning@gmail.com" style="font-size: x-large;">mathreasoning@gmail.com</a><span style="font-size: large;">. If you do so, please include your name and affiliation so that I can include that with the quote or review. If you prefer to have it be anonymous, just tell me so and I will post it that way.</span><br /><span style="font-size: large;"></span><br /><a name='more'></a><span style="font-size: large;"><br /></span><br /><span style="font-size: large;"><b>Request #2</b></span><br /><span style="font-size: large;">I would like to have the website for the book be a place where people can go to obtain or discover resources for the book. This is already done on the website but besides the link to screencasts for the book, I have written all of the resources that are available on the website. So I would like you to send me copies of resources that you have developed and are willing to share. You may send these to me at </span><a href="mailto:mathreasoning@gmail.com" style="font-size: x-large;">mathreasoning@gmail.com</a>. <span style="font-size: large;">I will create a resources page associated with the website and include things (with credit) from other people. Again, please let me know your name and affiliation so that I can give you proper credit.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">I am hoping that the addition of these resources will make the website for the book better and more importantly, will make the textbook for useful for instructors and students.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Thank you in advance for your help.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-40225106658126047732014-01-19T15:38:00.002-08:002014-01-19T15:38:18.878-08:00Updates for Mathematical Reasoning: Writing and Proof<span style="font-size: large;">Before I describe a few small updates to the materials that are available for <i><a href="https://sites.google.com/site/mathematicalreasoning3ed/" target="_blank">Mathematical Reasoning: Writing and Proof</a></i>, I would like to remind people to check out the <a href="http://aimath.org/textbooks/approved-textbooks/" target="_blank">List of Approved Open-Source Textbooks</a> that are available through the <a href="http://aimath.org/" target="_blank">American Institute of Mathematics</a>.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">In preparing for class this semester, I revised the study guides that are available for <i>Mathematical Reasoning: Writing and Proof</i>. These guides are available of the <a href="https://sites.google.com/site/mathematicalreasoning3ed/" target="_blank">web site for the book</a>. In addition, because "flipping" a proofs course can be a difficult and time-consuming thing to do, I have written short fact-based quizzes for most sections of the textbook. I usually give these quizzes at the start of class. (Students are supposed to study the section of the textbook along with the screencasts that are available for the text.) Instructors who would like to obtain a copy of these quizzes and their solutions, should contact me at <a href="mailto:mathreasoning@gmail.com">mathreasoning@gmail.com</a>.</span><br /><span style="font-size: large;"></span><br /><a name='more'></a><span style="font-size: large;"><br /></span><br /><span style="font-size: large;">I have also written problem sets for most of the sections of the textbook that I have students work on during class. Again, I have solutions for these problems and instructors can learn how to obtain these problem sets and their solutions by sending me a message using the email address above.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">I have found these quizzes and problem sets to be very useful for a flipped-class model using the textbook.</span>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-20414962090092305562013-10-25T08:52:00.001-07:002013-10-25T08:52:29.006-07:00Too Much Content?<span style="font-size: large;">A friend of mine recently posted a link on Facebook to the following blog post:</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;"><a href="http://garyrubinstein.teachforus.org/2013/10/19/death-of-math/" target="_blank">http://garyrubinstein.teachforus.org/2013/10/19/death-of-math/</a></span><br /><br /><span style="font-size: large;">This has the provocative title "The Death of Math." Side note: The use of the word "math" tends to bug me. In formal writing and public writing, I always try to use the term "mathematics."</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">This is a fairly long post and what I want to focus on now is one of the two recommendations Mr. Rubenstein makes to "fix mathematics." This one is:<span style="font-family: inherit;"> Greatly reduce the number of required topics and to expand the topics that remained so they can covered more deeply with thought provoking lessons and activities. (The second recommendation is to make mathematics beyond the 8th grade into electives.)</span></span><br /><span style="font-size: large;"><span style="font-family: inherit;"></span></span><br /><a name='more'></a><span style="font-size: large;"><span style="font-family: inherit;"><br /></span></span><span style="font-size: large;"><span style="font-family: inherit;">While the second recommendation could make for some interesting discussions, it was the first one that I started thinking about in terms of the introduction to proofs course. My first thought was that I agree with this recommendation but it is quite difficult to implement. This seems to be a fairly constant struggle that I have to balance the work students do to start "thinking like a mathematician" to the content they may need in future courses.</span></span><br /><span style="font-size: large;"><span style="font-family: inherit;"><br /></span></span><span style="font-size: large;"><span style="font-family: inherit;">At Grand Valley State University, our introduction to proofs course is MTH 210 - Communicating in Mathematics, and I believe our first priority in the course is to develop students' abilities to construct mathematical proofs and then to write them in a coherent fashion according to guidelines established by the mathematical community. Since we need content to "learn to think like a mathematician," what content do we use? Using content from previous course can be difficult because not every student has the same mathematical background, and it is quite likely that the "previous" content that we would select will have been forgotten by many of the students. So here is what we do at Grand Valley. I welcome comments, suggestions, and criticisms.</span></span><br /><span style="font-size: large;"><span style="font-family: inherit;"><br /></span></span><span style="font-size: large;">Elementary number theory dealing with properties of even/odd integers, divisors and multiples, and congruence. This is the material that is used to introduce the various proof techniques studied in the course.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">After dealing with the proof techniques, we continue to use the methods of proof while introducing the following content:</span><br /><span style="font-size: large;"><br /></span><br /><ul><li><span style="font-size: large;">Elementary set theory.</span></li><li><span style="font-size: large;">Functions.</span></li><li><span style="font-size: large;">Relations and equivalence relations.</span></li></ul><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">This may not seem like enough content for a 4-credit course, but content is not the main goal of the course. In fact, I often skip some sections of the text that I have written. For example, the chapter on set theory includes a section on the Cartesian product of two sets and a section on indexed family of sets. I skip both of these except for introducing the definition of the Cartesian product because it is used in the chapter on functions.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The last thing studied in the course is equivalence relations and when we had a 3-credit course, we barely had time to introduce the important idea of equivalence classes. I am hoping that we will now have some more time to spend with this now that we have a 4-credit course.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The textbook, Mathematical Reasoning: Writing and Proof, also includes a chapter on more number theory (basically greatest common divisors and prime factorization) and a chapter on finite and infinite sets. We do not cover these chapters in our course at Grand Valley but I know others like to include these topics in their course. So they remain in the textbook.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">One good thing about writing this is that it reinforced my opinion that our approach for the introduction to proofs course is reasonable and that we really do not have too much content in the course.</span><br /><span style="font-size: large;"><span style="font-family: inherit;"><br /></span></span><span style="font-size: large;"><span style="font-family: inherit;"><br /></span></span><span style="font-size: large;"><strong style="background-color: white; border: 0px; color: #333333; line-height: 18px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="font-family: inherit;"><br /></span></strong></span><br /><br />Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-83643107309432325152013-09-30T12:43:00.003-07:002013-09-30T12:43:32.388-07:00Version 1.1 of Mathematical Reasoning: Writing and Proof<span style="font-size: large;">With the fall semester now in full swing, I am finding it very difficult to find time to think about topics for this blog and to write posts for this blog. So this one is short.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">I have released Version 1.1 of the textbook Mathematical Reasoning: Writing and Proof. There is no difference in content between this version and Version 1.0. I have made only two changes:</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">I have added a <a href="https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxtYXRoZW1hdGljYWxyZWFzb25pbmczZWR8Z3g6MTk2OTM3Yzc4Y2ZiMWU0Yw" target="_blank">"Note to Students"</a> that comes before the preface. I do not know why this took me so long, but it is quite clear that students do not read the preface. So I wrote a short note that explains the features of the textbook to the students and what they can do to effectively use the book. Please use the link above to download this note and let me know what you think.</span><br /><span style="font-size: large;"></span><br /><a name='more'></a><span style="font-size: large;"><br /></span><br /><span style="font-size: large;">I have changed the Creative Commons License for the book to BY-NC-SA. This is the <a href="http://creativecommons.org/licenses/by-nc-sa/3.0/" target="_blank">Creative Commons Attribution-NonCommerical-ShareAlike 3.0 Unported License</a>. For more information, click on the link.</span>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-83273252895189704852013-09-10T15:56:00.001-07:002013-09-10T15:56:16.446-07:00Study Guides for Mathematical Reasoning: Writing and Proof<span style="font-size: large;">In a post on August 25, I indicated that I was working on study guides for each section of Mathematical Reasoning: Writing and Proof. I have now completed study guides for the book through Chapter 5. These can be downloaded on the website for the book at <a href="https://sites.google.com/site/mathematicalreasoning3ed/">https://sites.google.com/site/mathematicalreasoning3ed/</a>.</span><div><span style="font-size: large;"><br /></span></div><div><span style="font-size: large;">I hope students will find these useful tools to help them with their study.</span><div><span style="font-size: large;"><br /></span></div><div><br /></div></div>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-4800130689448991362013-09-05T11:26:00.002-07:002013-09-05T11:26:59.194-07:00LaTeX Workshop in Class<div class="MsoNormal"><span style="font-size: large;">In a post on this blog on August 14, I indicated that I was going to require my students in the introduction to proofs course to use LaTeX. This is the first time I have done this. I am quite nervous about requiring the use of LaTeX because I am not sure how the students will react to this. Students seem to be quite comfortable using a word processor and it is usually not too difficult for them to then incorporate the use the equation editor. Most students have used MS Word and some used Open Office. I am not fond of the equation editors in these two word processors, and I often encourage my students to use MathType, especially with MS Word.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-size: large;"></span><br /><a name='more'></a><span style="font-size: large;"><br /></span></div><div class="MsoNormal"><span style="font-size: large;">Yesterday, I introduced my students to LaTeX. Students were asked to bring their laptops to class and to prepare for class by installing LaTeX or signing up for a WriteLaTeX account before class. We then had a 50-minue LaTeX workshop in class. I supplied them with a .tex document that will serve as the template for their work in the class. I also supplied them with a reference document (<a href="https://dl.dropboxusercontent.com/u/10471758/latex-examples.pdf" target="_blank">latex-examples.pdf</a>) and an exercise for them to work on during class (<a href="https://dl.dropboxusercontent.com/u/10471758/latex-exercise.pdf" target="_blank">latex-exercise.pdf</a>). I was very pleased with the results. Most students only progressed through about the first page of the exercise document, but I was very happy with their enthusiasm and willingness to try to learn LaTeX. All of the students were able to do something and obtained a successfully compiled LaTeX document. I am very hopeful that having students use LaTeX will be a success. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-size: large;"><br /></span></div><br /><div class="MsoNormal"><span style="font-size: large;">I am hoping to have a couple more LaTeX workshops during the semester, but we do have other resources for students at Grand Valley as well. First, there students may bring their laptop and their LaTeX issues to my office. We also have a tutor in our Mathematics Center that can help students with LaTeX. Finally, Robert Talbert, a colleague of mine at Grand Valley, has produced about ten screencasts about LaTeX to help students learn LaTeX. These screencasts are available on the <a href="http://www.youtube.com/playlist?list=PLF975D9D3C9B50FF7" target="_blank">GrandValley Math Channel on YouTube</a>. Feel free to have your students use these screencasts.</span><o:p></o:p></div>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-40217106807776097262013-08-28T20:45:00.000-07:002013-08-28T17:48:58.656-07:00Flipped Classroom Documents<span style="font-size: large;">In a few of my recent posts, I have made reference to using a flipped classroom model for teaching the introduction to proofs course. To get a sense of what this means, following are links to some documents I use. These documents were intended for use in class on Wednesday August 28 for a 2-hour session.</span><br /><span style="font-size: large;"><br /></span><br /><ul><li><a href="https://dl.dropboxusercontent.com/u/10471758/2013-08-28-guide.pdf" style="font-size: x-large;" target="_blank">Study Guide</a><span style="font-size: large;"> (distributed on Monday August 26).</span></li><li><a href="https://dl.dropboxusercontent.com/u/10471758/quiz%231-sec11-12.pdf" style="font-size: x-large;" target="_blank">Quiz</a><span style="font-size: large;"> (taken at the start of class on Wednesday August 28).</span></li><li><a href="https://dl.dropboxusercontent.com/u/10471758/2013-08-28-problems.pdf" style="font-size: x-large;" target="_blank">Classroom Problems</a><span style="font-size: large;"> (Students worked on these in class in groups of 3 students on Wednesday August 28).</span></li></ul><br /><span style="font-size: large;"></span><br /><a name='more'></a><span style="font-size: large;"><br /></span><span style="font-size: large;">As you can see, the study guide includes a list of the screencasts that are relevant to Section 1.2. Students are expected to use the text and these screencasts and come to class prepared to work on the problems. The quiz is generally a simple quiz that mainly tests to see if the students have picked up the information presented in the text and screencasts. As you consider what happened in class, please keep in mind that my section of the course meets two times a week for 100-minute class sessions.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">When the quiz was</span><span style="font-size: large;"> completed, we will spent a few minutes discussing the quiz and had some time for questions about the section. The students then worked in groups of 3 on the problems while I circulated around the room checking on their progress and responding to questions. Toward the end of class, I will had each team turn in their work on one of the problems. I will basically grade them on a done or not done basis. We then had about 15 minutes left to respond to questions and perhaps set the stage for the next class.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-12890420919182863752013-08-25T14:35:00.003-07:002013-08-25T16:36:20.440-07:00Study Guides for Mathematical Reasoning<span style="font-size: large;">Classes start tomorrow. I will be teaching one section of MTH 210 - Communicating in Mathematics (and two sections of Trigonometry). As I have indicated, I used a flipped or inverted classroom model last winter for MTH 210 and I will do so again. To help students with this method, I wrote study guides for each section of Mathematical Reasoning: Writing and Proof. I wrote them specifically for my class including due dates, etc. For some reason, I wrote them using Word. I have now started to rewrite these using LaTeX in a more generic form that other people (including students) can use. So if you are interested, you can <a href="https://dl.dropboxusercontent.com/u/10471758/guides.zip" target="_blank">download a pdf file with study guides</a> for the first six sections of the book. I will soon be adding these to the <a href="https://sites.google.com/site/mathematicalreasoning3ed/" target="_blank">web site for the book</a> and will be able to make the LaTeX source files available. Please contact me at mathreasoning@gmail.com if you are interested.</span>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-33286892336311589892013-08-22T16:09:00.002-07:002013-08-23T11:53:21.578-07:00Compassion and Imagination<span style="font-size: large;">"To be a good teacher, exercise your compassion and imagination; to be a good pupil, exercise your capacity to learn independently."</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">As we get close to the start of classes, I realize that my time on the golf course will significantly decrease. Now what does that have to do with the quote above? I wish this was my quote but it is by M. Scott Peck from his book <i>Golf and the Spirit: Lessons from the Journey</i>. With classes and golf on my mind, I recalled that one chapter from the book was on teaching and learning. I have not read the book for several years and so I decided to take it out and this was the first chapter I read. The book deals mainly with teaching golf professionals and their students but also relates this to psychotherapists and their patients. (M. Scott Peck, M.D., is a psychiatrist.) </span><br /><span style="font-size: large;"></span><br /><a name='more'></a><span style="font-size: large;"><br /></span><span style="font-size: large;">One of the points he makes is that many golf teaching pros are not very good teachers and that the problem is inherent in their role, which is to teach golf. Being well-motivated professionals, they want to make sure their students get their money's worth and consequently, they overteach. Does this sound familiar? I know that I am often guilty of overteaching. Peck's point is that such overteaching can create not only an aversion to golf lessons but aversions to golf itself.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">Although I have not had much experience with the flipped classroom model, I have found it to be helpful in keeping myself from overteaching. A lot of responsibility is placed on the student but their work gives us a good place from which to start. In my proofs class, I will have 5 to 7 groups collaboratively working on problems and as I go around the room, I cannot spend too much time talking with one particular group. I have to develop my ability to give them ideas and suggestions that may help them get back on track and to be able to do the work themselves and developing a capacity to learn independently is important.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">This is one of the big differences Peck describes between experienced golfers (learners) and inexperienced ones. Even the best golfer (mathematician) will make a truly horrible shot. However, the professional golfer will then immediately analyze what he/she did wrong to help make sure it does not happen again. Novice golfers look to an authority (the golf instructor) to tell them what to do. As mathematics teachers, I think we can take some lessons from this chapter of <i>Golf and the Spirit</i>.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-33503665191856591662013-08-06T13:48:00.000-07:002013-08-06T13:48:15.465-07:00Alice in Wonderland<span style="font-family: Times, Times New Roman, serif; font-size: large;">The following quote from <i>Alice in Wonderland</i> by Lewis Carroll is pertinent to the so-called forward-backward method for direct proofs. I thought you might find it interesting and perhaps useful.<br /><br />One day Alice came to a fork in the road and saw a<br />C h e s h i re cat in a tree.<br />"Which road do I take?" She asked.<br />His response was a question: "Where do you want to go?"<br />"I don't know," Alice answered.<br />"Then," said the cat, "it doesn't matter."</span><br /><!--[if !supportLineBreakNewLine]--><br /><!--[endif]-->Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-61225662017409883002013-08-06T13:19:00.001-07:002013-08-06T13:26:18.041-07:00Interesting Methods of Proof<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-UEWylLUmTYk/UgFZsI5tmyI/AAAAAAAAACI/eECkN-BT1Lk/s1600/HarrisMiracle1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-UEWylLUmTYk/UgFZsI5tmyI/AAAAAAAAACI/eECkN-BT1Lk/s1600/HarrisMiracle1.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: Times, Times New Roman, serif; font-size: large;">This is a cartoon by Sydney Harris and shows a proof technique that I do not accept in my courses, even though I wish I could use it from time to time. Check out a <a href="http://www.sciencecartoonsplus.com/gallery/math/index.php" target="_blank">collection of cartoons related to mathematics</a> by Sydney Harris.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: Times, Times New Roman, serif; font-size: large;"></span></div><a name='more'></a><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><br /><div class="separator" style="clear: both; text-align: left;"><span style="font-family: Times, Times New Roman, serif; font-size: large;">Here is another cartoon with a proof technique that I hope I have never used.</span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-I2RVUoilRpA/UgFbeNFUpYI/AAAAAAAAACg/eMsU4Wlt850/s1600/images+(1).jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-I2RVUoilRpA/UgFbeNFUpYI/AAAAAAAAACg/eMsU4Wlt850/s1600/images+(1).jpg" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: left;"><br /></div>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-84313067192755021472013-08-06T06:30:00.003-07:002013-08-06T06:33:34.523-07:00Screencasts and the Inverted Classroom<div class="MsoNormal"><span style="font-family: Times, Times New Roman, serif; font-size: large;">A very important supplement to my textbook, <i><a href="https://sites.google.com/site/mathematicalreasoning3ed/" target="_blank">Mathematical Reasoning: Writing and Proof</a></i>, is an extensive collection of screencasts developed by Robert Talbert, a colleague of mine at Grand Valley State University. Although these screencasts are structured around this book, they can be used in conjunction with any introduction to proofs course. Please review these screencasts and see if they are suitable for use in your course. Most of the screencasts deal with one topic or example and are approximately 10 minutes long. They are a great supplement to a course and give the students a chance to have examples done outside of the classroom. You can find the complete collection of screencasts in the <a href="http://www.youtube.com/playlist?list=PL2419488168AE7001" target="_blank">MTH 210 Playlist</a> in the <a href="http://www.youtube.com/user/GVSUmath" target="_blank">GVSU Math YouTube Channel</a>.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: Times, Times New Roman, serif; font-size: large;"></span><br /><a name='more'></a><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span></div><div class="MsoNormal"><span style="font-family: Times, Times New Roman, serif; font-size: large;">These screencasts are very valuable if you want to design your course using an inverted classroom model. I did that last winter semester and I was very pleased with the results. Students were more engaged with the material than in a more traditional classroom model, and we had some very interesting discussions during class. This is because students have access to me while they are doing some of the most difficult work for the course. I am no expert on the inverted classroom, but I will continue to use this model in MTH 210 – Communicating in Mathematics.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span></div><div class="MsoNormal"><span style="font-family: Times, Times New Roman, serif; font-size: large;">If you would like to learn more about the inverted classroom, you should read the series on the inverted classroom by Robert Talbert in his blog <a href="http://chronicle.com/blognetwork/castingoutnines/category/inverted-classroom/">Casting Out Nines</a>. Although it is not in this series, the post on “<a href="http://chronicle.com/blognetwork/castingoutnines/2012/07/10/inverting-the-transition-to-proofs-class/">Inverting the transitions-to-proof class</a>” is very informative.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span></div><div class="MsoNormal"><span style="font-family: Times, Times New Roman, serif; font-size: large;">Robert and I and some other colleagues at Grand Valley have now used the inverted classroom model for our transitions-to-proof course, and I am pleased at how well my text, <i>Mathematical Reasoning: Writing and Proof</i> works with this model. It was not specifically written for an inverted classroom, but in reality, that term was not being used when I started writing it over 12 years ago. However, I did design the book to be studied by students and have incorporated active learning strategies into the book. The most evident of these are the preview activities that begin each section. The book also has several progress checks for students in each section, and these seem to help the students quite a bit. For me, the combination of the book and the screencasts are about as good of a combination for an inverted classroom as there is.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><br />Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-2108870594851517312013-08-05T06:48:00.004-07:002013-08-06T06:34:20.837-07:00Writing Proofs<span style="font-family: Times, Times New Roman, serif; font-size: large;">Issues dealing with the writing of mathematical proofs should be addressed throughout an introduction to proofs course. Completing a proof does bring personal satisfaction, but mathematicians also have the responsibility to be able to communicate their results to others. For a long time now, I have emphasized the importance of writing in courses I teach for mathematics majors, and I have been actively involved in helping students develop their writing abilities in the introduction to proofs course. </span><br /><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><span style="font-family: Times, Times New Roman, serif; font-size: large;">This is also important to others in the Department of Mathematics at Grand Valley State University as is shown the following two paragraphs, which are based on a document developed by the Department of Mathematics at Grand Valley State University for use in the Writing Center at the university.</span><br /><span style="font-family: Times, Times New Roman, serif; font-size: large;"></span><br /><a name='more'></a><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><br /><div class="MsoNormal"><span style="font-family: Times, Times New Roman, serif; font-size: large;">At the risk of oversimplification, doing mathematics has two distinct stages. The first stage is to convince yourself that you have solved the problem or proved a conjecture. This stage is a creative one and is quite often how mathematics is actually done. The second equally important stage is to convince other people that you have solved the problem or proved the conjecture. This second stage often has little in common with the first stage in the sense that it does not really communicate the process by which you solved the problem or proved the conjecture. However, it is an important part of the process of communicating mathematical results to a wider audience.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><span style="font-family: Times, Times New Roman, serif; font-size: large;">A mathematical proof is a convincing argument (within the accepted standards of the mathematical community) that a certain mathematical statement is necessarily true. A proof generally uses deductive reasoning and logic but also contains some amount of ordinary language (such as English). A mathematical proof that you write should demonstrate that you have gained a deep understanding of the mathematical concepts involved and should convince an appropriate audience that the result you are proving is in fact true. </span><br /><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><span style="font-family: Times, Times New Roman, serif; font-size: large;">It should be clear from this that we consider writing to be an important part of the mathematical process. This is one reason the title of our introduction to proofs course is "Communicating in Mathematics." Issues dealing with the writing of mathematical proofs are addressed throughout the course. This was also one of the reasons that I wrote the book <a href="https://sites.google.com/site/mathematicalreasoning3ed/" target="_blank"><i><b>Mathematical Reasoning: Writing and Proof</b></i>. </a> As I search the Internet now, it is great to find many documents dealing with the writing of proofs in mathematics course. The Department of Mathematics at GVSU has also developed such a document, which can be accessed <a href="https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxtYXRoZW1hdGljYWxyZWFzb25pbmczZWR8Z3g6MjVkNDhkODQxOWYxYmJmMw" target="_blank">at this link</a>. Because of such documents, I am sure that having a textbook dealing with these issues is not as important as it used to be. However, I still like the idea of having a text that supports what I try to have my students learn.</span><br /><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><span style="font-family: Times, Times New Roman, serif; font-size: large;">Feel free to make comments describing what you feel should be part of the writing guidelines for the introduction to proofs course. One nice things about having an open-access text is that I can include revisions much more quickly than I could before.</span><br /><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><br /><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-28195933687079320552013-08-03T14:42:00.003-07:002013-08-06T06:34:52.262-07:00Free Textbooks<div style="margin-bottom: .0001pt; margin: 0in;"><span style="font-family: Times, Times New Roman, serif; font-size: large;">Perhaps one of the more important resources for a course is a textbook. One difficulty with choosing a textbook for an introduction to proofs course is that there is no "standard" syllabus for the course. Many colleges and universities develop their own version of this course. That being said, there are several textbooks from which to choose and there are a growing number of free or low-cost textbooks.<o:p></o:p></span><br /><span style="font-family: Times, Times New Roman, serif; font-size: large;"></span><br /><a name='more'></a><span style="font-family: Times, Times New Roman, serif; font-size: large;"><br /></span></div><span style="font-family: Times, Times New Roman, serif; font-size: large;">A very good place to find free textbooks is the<span class="apple-converted-space"> </span><a href="http://www.aimath.org/textbooks/textbooklist.html" target="_blank">Open Textbook Initiative<span class="apple-converted-space"> </span></a>of the<span class="apple-converted-space"> </span><a href="http://www.aimath.org/" target="_blank">American Institute of Mathematics.</a> One of the books in the Introduction to Proofs section is the text that I have written,<span class="apple-converted-space"> </span><b><i><a href="https://sites.google.com/site/mathematicalreasoning3ed/" target="_blank">Mathematical Reasoning: Writing and Proof</a></i></b>. The first edition of this book was published in 2003 by Pearson Eduction, Inc. and was initially developed as a textbook for the proofs course at Grand Valley State University, MTH 210 - Communicating in Mathematics. The second edition was published in 2007, and I started developing a third edition sometime in 2009. However, Pearson decided not to publish the third edition and so the copyright for the text has been returned to me. I have decided to distribute the book through a Creative Commons License. Anyone can dowload a free copy of the book (pdf) file through <a href="http://scholarworks.gvsu.edu/books/7/" target="_blank">Grand Valley State University ScholarWorks</a>.</span>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0tag:blogger.com,1999:blog-2502473924784442364.post-56739801771770421992013-08-03T14:20:00.005-07:002013-08-05T08:04:56.465-07:00Welcome<div><span style="font-family: Times, Times New Roman, serif; font-size: large;">In this blog, I plan to share:</span></div><ul><span style="font-family: Times, Times New Roman, serif; font-size: large;"><li>My thoughts about introduction to proofs courses for the mathematics majors.</li><li>Information about the text that I have written for this course, <b><i><a href="https://sites.google.com/site/mathematicalreasoning3ed/" target="_blank">Mathematical Reasoning: Writing and Proof</a></i></b>.</li><li>Information about other free resources for an introduction to proofs course including other free textbooks for the course.</li></span></ul><ul></ul><span style="font-family: Times, Times New Roman, serif;"><span style="font-size: large;">Please feel free make suggestions for </span><span style="font-size: large;">blog in the comments, or y</span></span><span style="font-family: Times, 'Times New Roman', serif; font-size: large;">ou can send a messageto me at mathreasoning@gmail.com.</span>Ted Sundstromhttps://plus.google.com/109237311726662407916noreply@blogger.com0