Friday, October 25, 2013

Too Much Content?

A friend of mine recently posted a link on Facebook to the following blog post:

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."

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:  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.)

Monday, September 30, 2013

Version 1.1 of Mathematical Reasoning: Writing and Proof

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.

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:

I have added a "Note to Students" 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.

Thursday, September 12, 2013

Some Thoughts and Observations about My Flipped Classroom

So far, my introduction to proofs class has met 5 times.  We meet two times per week for 100 minutes, and I am continuing to use the flipped classroom model that I started last semester.  The big difference is that this course is now a four-credit course and last semester it was a three-credit course.  So last semester, we met two times per week for 75 minutes.

This is a great development but it has also created some challenges.  Our goal in increasing this to a 4-credit course was not to add more content but to be able to spend more time working with the students as they attempt to do proofs and to spend more time on issues dealing with writing in mathematics.  The basic model (times are approximate) for the flipped classroom last semester was:

Tuesday, September 10, 2013

Study Guides for Mathematical Reasoning: Writing and Proof

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

I hope students will find these useful tools to help them with their study.

Thursday, September 5, 2013

LaTeX Workshop in Class

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.

Wednesday, August 28, 2013

Flipped Classroom Documents

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.

  • Study Guide (distributed on Monday August 26).
  • Quiz (taken at the start of class on Wednesday August 28).
  • Classroom Problems (Students worked on these in class in groups of 3 students on Wednesday August 28).

Sunday, August 25, 2013

Study Guides for Mathematical Reasoning

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 download a pdf file with study guides for the first six sections of the book.  I will soon be adding these to the web site for the book and will be able to make the LaTeX source files available.  Please contact me at if you are interested.

Thursday, August 22, 2013

Compassion and Imagination

"To be a good teacher, exercise your compassion and imagination; to be a good pupil, exercise your capacity to learn independently."

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 Golf and the Spirit: Lessons from the Journey.  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.)  

Sunday, August 18, 2013

Doing Mathematics is a Slow Process

Over the past few years, I have of course been teaching the introduction to proofs class at Grand Valley (MTH 210 – Communicating in Mathematics) and have taught a few sections of a 100 level trigonometry course.  In both of these courses, one thing that has struck me has been the impatience of students and their desire to get a quick answer or simply to get things done quickly.  One of my tasks now seems to be to try to slow down the students and instill in them the idea that doing mathematics is a slow process.  I am not sure why this has been on my mind lately.  Perhaps it is because our life in general seems to be continually getting faster and students today have become use to being able to get information more quickly than ever before.  (Another reason may be that I am trying to slow down as I am approaching retirement.)

Wednesday, August 14, 2013

Doubt and Proof in Mathematics

Here is a link to a blog post written by Robert Talbert, a colleague of mine at Grand Valley State University.

LaTeX in the Proofs Course

Let me first say that I am interested in hearing from anyone who has required students to use LaTeX in their introduction to proofs course as I will be requiring my students to use LaTeX for the first time this fall.

For over ten years, faculty at Grand Valley have been requiring students to submit a “portfolio of proofs” as part of the requirements for MTH 210 – Communicating in Mathematics.  (This is our introduction to proofs course.)  These problems are often posed in the form of a conjecture, which the students have to either prove or disprove.  MTH 210 is a so-called Supplemental Writing Skills (SWS) Course and the portfolio project is used to satisfy the requirements for such a course.  Basically, a designated SWS course give students an opportunity to submit their work for review before submitting it for a final grade.

Tuesday, August 6, 2013

Alice in Wonderland

The following quote from Alice in Wonderland 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.

One day Alice came to a fork in the road and saw a
C h e s h i re cat in a tree.
"Which road do I take?" She asked.
His response was a question: "Where do you want to go?"
"I don't know," Alice answered.
"Then," said the cat, "it doesn't matter."

Interesting Methods of Proof

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 collection of cartoons related to mathematics by Sydney Harris.

Screencasts and the Inverted Classroom

A very important supplement to my textbook, Mathematical Reasoning: Writing and Proof, 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 MTH 210 Playlist in the GVSU Math YouTube Channel.

Monday, August 5, 2013

Writing Proofs

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.  

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.

Saturday, August 3, 2013

Free Textbooks

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.


In this blog, I plan to share:
  • My thoughts about introduction to proofs courses for the mathematics majors.
  • Information about the text that I have written for this course, Mathematical Reasoning: Writing and Proof.
  • Information about other free resources for an introduction to proofs course including other free textbooks for the course.
Please feel free make suggestions for blog in the comments, or you can send a messageto me at