The third quiz in Proofs Class

This term I am teaching our introduction to proofs class IBL style with some great notes from Coe College.  My twist on the class is to have oral quizzes.  So far each has been a bit different.

1. First quiz, I gave the students a choice of three problems that we had already discussed and let them choose their favorite and present a solution to me in my office.  
2. Second quiz, I choose a longer proof of a proposition from their notes, removed all the punctuation, printed it out on cards with magnets on the back and had the students put the proof in order.  I chose a dandy because half of the proof had two cases and nearly every student wanted to start their proof with the line "there are two cases".
3. Third quiz . . . So, now it is time for the third quiz.  I want to give the students an incorrect proof and have them find (and hopefully correct) the errors.  In this chapter we went over functions and cardinality, so hopefully the proof will have something to do with that.  However, I am drawing a blank.

Here are my basic learning objectives for every quiz which I snagged from a variety of web sites:

A successful student will be able to:

• Determine what is exactly given by the hypothesis of a statement, and what exactly is
• required to be proven.
• Identify and use definitions correctly.
• Grasp the sequence of implications and quanti fications of the statement the student has to
• prove
• Justify a proof passage by citing the correct previously known result
• Identify and distinguish
• conclusions from supporting statements,
• relevant from extraneous material,
• necessary steps from corollaries,
• factual from normative statements,
• logical fallacies in a proof,
• questionable term usages,
• unstated assumptions,
• the purpose of a proof passage or a proposition.
• relate analogies and di fferences among proofs and proof techniques.
• identify the fundamental part of a proof from the technical details and grasp the structure
• of a proof.
• outline the main points of a proof and correlate them.

How Do I Assess This?

This term I am teaching our Foundations of Analysis course to 15 students IBL style.  Third time may be the charm, but I am finding it to be going quite well.  However, I don't want to give them a traditional exam.  So . . .

I am experimenting with oral quizzes.

• Students come to my office in their time.
• They are given a selection of problems to work on that are identical to, or very similar to, ones the class has worked through.  The student chooses a problem.
• They work the problem on the white board in front of me.
• I have them explain their work and why they think what they have done solves the problem.

I have a process rubric that I am still refining that includes things like

- Student identifies and uses definitions correctly.

- Student can justify a proof passage by citing the correct previously known result.

This way, I can correct a student who mis-remembers a definition, but still note that he was able to use all other definitions correctly.

So far, the process is going well and, since it is very similar to how class works, the students are comfortable with the situation and able to perform at their best.

I'm just not any good at math

This post is inspired by a comment by a Senator who was worried about pornography being injected in the Common Core because they are talking about fuzzy math and replacing numbers with letters.

Everyone who teaches or studies math has heard the phrase "I can't do math" too many times to count.  It is really frustrating for many reasons and it is really holding our educated citizenry back.  I do believe that some people really struggle with math just like some people really struggle with reading, but those numbers are probably on the same order.

Sometimes I try to just laugh it off.  One of the lines I use is, "If everyone was good at math, then I would be out of a job."  Now, this is just as true as the statement it is a response to, but everyone can laugh and move on with the conversation.  However, what is most frustrating is that what most people mean by math isn't MATH!  Some of them struggle with fractions, or rational functions, or fractional exponents (see the trend?).

When my math majors start talking about not being good at math, it is much more important to listen.  Teaching #IBL Advanced Calculus, the students are doing math, many of them for the first time.  This feeling I remember.  The bane of my existence was Basic Algebra in graduate school and I am pretty sure I still don't have a good grasp of how Algebraists think.

One of my students was talking in class about how much she LOVED her differential equations class, especially when juxtaposed with her struggles in advanced calculus.  She is a math major because she really enjoyed Calculus in high school and college.  For her, differential equations is that capstone experience where the math she fell in love with is coming around again in her life and it just all makes sense.  I worked to explain that she loves the application of mathematics to problems of physics, chemistry and engineering.  This is a wonderful thing in, and of, itself and is worthy of celebration.  Meanwhile, advanced calculus, that class that contains the word she enjoys, is not asking the questions she wants to struggle with.  She can do it, she just doesn't want to.

My student is a math major, but she isn't a mathematician and she doesn't want to be. She finds the application of math that most people struggle with easy, but she isn't drawn to the creation of math.  This is much like my relationship with art.  I celebrate the art which is created, but I am not drawn to create it myself.  I enjoy singing, but I know I will never be a professional singer.  In a moment of personal honesty, this student may say that she is not any good at math, but not in the way most people mean.  She means that she is not drawn to being a professional mathematician, but instead is content to use the mathematics to make the world a better place.

In our department, the upper level mathematics courses require students to earn a certain number of math culture points as a way of indoctrinating the students into the culture of mathematics.  We ask students to go to talks and conferences, participate in contests, read articles about mathematics and solve math problems.  It isn't IBL, but it does help students learn how to answer, What is math?

Now, to return to replacing numbers with letters.  The algebra we teach in high school, even with the changes proposed by the Common Core, is not mathematics.  It is working on problems that, realistic, or unrealistic, can be solved by application of mathematical conclusions.  NEARLY ANYONE can do this.  It takes patience, generalization and pattern recognition, but not some deep talent.

The most frustrating part of the statement I'm just not any good at math is that the people saying it don't have any idea what mathematics is!  When I teach an #IBL course, part of what I do is demonstrate real mathematics to the students.  And this is what makes all the difference.

One Good Idea: WriteLatex

I don't know how the rest of the bloggers keep up, but here is my next post about my Advanced Calculus class this term.

One new thing I am trying this term is to have the students write up solutions to the questions they solve on WriteLatex.  I have a free account and I have shared the file with the class.  This is working well with the 8 students in my class, but I am not sure how well it would work with a much  larger class.  It has also completely changed the class dynamics.

Past: Students would volunteer to present solutions to problems in class.  They would write up their solutions on the board (chalk or white) and the other students would review their work.  A lot of time was taken writing on the board since they don't really know how to talk and write at the same time.  Not much work was actually written up and turned in.

Present:  Students are going on the web site and writing up the problems they can do.  Then, in class, they present their solution and talk through it.  I think the most disconcerting part is when they discover an error and they correct it using their own laptop.  Meanwhile words just appear and disappear on the projected document.  The biggest problem I am having is getting the students who feel behind to submit proofs.

Then, I can go into the document and write comments about their proofs.  I have even been known to substitute more elegant LaTeX code as an example of the power of the environment (did you know about the \tag command?).

We did have one day when the WriteLatex server went down in class, but the students all have printed copies of the questions and we just proceeded using the boards.  Another problem is that the chapters are starting to get unwieldy.  I went through and chopped up Chapter 1 with each section having its own file because it would take a while to find where we were in the document.  I have heard that Sage Math Cloud has backwards and forwards searching which would be nice, but I hadn't heard about that project until over a week into the term.  Boy, it can be hard staying on the bleeding edge of things.

Conclusions: I love that the students are learning to communicate mathematics in writing and that we do not need to take time for writing.  This was not something I was expecting at all.  The original plan had them presenting at the board and only writing up their work afterwards.  However, I do need to find a way to "reserve" problems for students who are behind.

#IBL #WriteLatex

Week 1 - What, exactly, is a real number

Advanced Calculus came to a screaming halt today.

First, the students asked for some more information about sets and help developing intuition about upper bounds and least upper bounds.  We also discussed the difference between something being an element of a set and something being a subset and/or a proper subset.  Heads were nodding, so we continued on - for two questions.  What was the stumper?

• If x squared is 2, prove or disprove that x is a real number.

After a lot of head scratching, the class broke up and worked together on the boards for the rest of the time.

Everyone else was taking pictures of the board, so I decided to record a few myself.

Week 1 - Students are a fruit salad

My advanced calculus class of students is a fruit salad of experiences.  I have sophomores through seniors who have taken no proofs classes through either our abstract algebra course or our introduction to proofs class.  Why do I think of this as a fruit salad?  Because they are all different flavors and textures, but are all welcome in the class.  I have one banana who is pretty sure he is in over his head, but having had him in other classes, I know he is capable.

Before the term begins

I am teaching MA 305, Advanced Calculus, at Wartburg IBL style for the second time starting Monday.  I have the course on my mind when I had dinner at one of our Chinese restaurants last night.  Here are the appropriate fortune cookie sayings that I collected for my students:

• Only the person who risks is truly free.
• Good timber does not grow with ease; the stronger the wind, the stronger the trees.
• Never despair, but if you do, work on in despair.
• Only the educated are free.

The first couple of weeks of the class are going to feel to my students that someone has taken them out to the middle of the lake, thrown them in, and asked them to learn how to swim.  I know they can be successful, but they are in for a tough time for a bit.