# Category Archives: Problem Solving

August 11, 2010, 8:35 am

# Student (mis)understanding of the equals sign

Interesting report here (via Reidar Mosvold) about American students’ misunderstanding of the “equals” sign and how that understanding might feed into a host of mathematical issues from elementary school all the way to calculus. According to researchers Robert M. Capraro and Mary Capraro at Texas A&M,

About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students.

Robert Capraro, in the video at the link above, makes an interesting point about the “=” sign being used as an operator. He makes a passing reference to calculators, and I wonder if calculators are partly to blame here. After all, if you want to calculate 3+5 on a typical modern calculator, what do…

August 8, 2010, 12:47 pm

# Calculus and conceptual frameworks

Image via http://www.flickr.com/photos/loopzilla/

I was having a conversation recently with a colleague who might be teaching a section of our intro programming course this fall. In sharing my experiences about teaching programming from the MATLAB course, I mentioned that the thing that is really hard about teaching programming is that students often lack a conceptual framework for what they’re learning. That is, they lack a mental structure into which they can place the topics and concepts they’re learning and then see those ideas in their proper place and relationship to each other. Expert learners — like some students who are taking an intro programming course but have been coding since they were 6 years old — have this framework, and the course is a breeze. Others, possibly a large majority of…

May 12, 2010, 12:52 pm

# Boxplots: Curiouser and curiouser

The calculus class took their third (and last) hour-long assessment yesterday. In the spirit of data analytics ala the previous post here, I made boxplots for the different sections of the test (Conceptual Knowledge (CK), Computation (C), and Problem Solving (PS)) as well as the overall scores. Here are the boxplots for this assessment — put side-by-side with the boxplots for the same sections on the previous assessments. “A2″ and “A3″ mean Assessments 2 and 3.

Obviously there is still a great deal of improvement to be had here — the fact that the class average is still below passing remains unacceptable to me — but there have been some definite gains, particularly in the conceptual knowledge department.

What changed between Assessment 2 and Assessment 3? At least three things:

• The content changed. Assessment 2 was over derivative rules and applications; Assessment 3 covered…

April 11, 2010, 2:56 pm

# MATLAB as a handout creator

One of the fringe benefits of having immersed myself in MATLAB for the last year (in preparation for teaching the Computer Tools for Problem Solving course) is that I’ve learned that MATLAB is an excellent all-purpose tool for preparing materials for my math classes. Here’s an example of something I just finished for a class tomorrow that I’m really pleased with.

I was needing to create a sequence of scatterplots of data for a handout in my Functions and Models class. The data are supposed to have varying degrees of linearity — some perfect/almost perfectly linear, some less so, some totally nonlinear — and having different directions, and the students are supposed to look at the data and rank the correlation coefficients in order of smallest to largest. (This is a standard activity in a statistics class as well.)

I could have just made up data with the right shape on Excel or…

April 8, 2010, 8:17 pm

# Analyze, hack, create

One of these days I’ll get back to blogging about the mathematics courses I teach, which make up the vast majority of my work, but the MATLAB course continues to be the place where I am working the hardest, struggling the most, learning the biggest lessons about teaching, and finally having the greatest sense of reward. This week was particularly rewarding because I think I finally figured out a winning formula for teaching a large portion of this stuff.

This was the last in a three-week series on introduction to programming. We had worked with FOR loops already. I had planned to look at WHILE loops in the same week as FOR loops, then have the students play around with branching structures in week 2, then have them apply it to writing programs to do numerical integration week 3 for use in their Calculus II class in which most of the class is currently enrolled. But the FOR loop stuff we…

March 17, 2010, 8:26 pm

# You can't become an expert in college

Cover of Outliers: The Story of Success

Here’s something of an epiphany I had at the ICTCM while listening to Dave Pritchard‘s keynote, which had a lot to do with the differences between novice and expert behaviors in problem-solving.

Malcolm Gladwell, in his book Outliers, puts forth a now-famous theory that it takes at least 10,000 hours to become a true expert in a particular area, at the top of one’s game in a particular pursuit. That’s 10,000 hours of concentrated work in studying, practicing, and performing in some particular area. When we talk about “expert behavior”, we mean the kinds of behaviors that people who have put in their 10,000 hours exercise as second nature.

Clearly high school or college students who are in an introductory course — even Dave Pritchard’s physics students at MIT, who are likely…

March 1, 2010, 9:21 pm

# MATLAB and critical thinking

My apologies for being a little behind the curve on the MATLAB-course-blogging. It’s been a very interesting last couple of weeks in the class, and there’s a lot to catch up on. The issues being brought up in this course that have to do with general thinking and learning are fascinating, deep, and complicated. It’s almost as if the course is becoming only secondarily a course on MATLAB and primarily a course on critical thinking and lifelong learning in a technological context.

This past week’s lab really brought that to the forefront. The lab was all about working with external data sets, and it involved students going to this web site and looking at this data set (XLS, 33 Kb) about electoral vote counts of the various states in the US (and the District of Columbia). One of the tasks asked students to make a scatterplot of the land area of the states versus their electoral vote count…

February 4, 2010, 10:14 pm

# 12 videos for getting LaTeX into the hands of students

There seem to be two pieces of technology that all mathematicians and other technical professionals use, regardless of how technophobic they might be: email, and $$\LaTeX$$. There are ways to typeset mathematical expressions out there that have a more shallow learning curve, but when it comes to flexibility, extendability, and just the sheer aesthetic quality of the result, $$\LaTeX$$ has no rival. Plus, it’s free and runs on every computing platform in existence. It even runs on WordPress.com blogs (as you can see here) and just made its entry into Google Documents in miniature form as Google Docs’ equation editor. $$\LaTeX$$ is not going anywhere anytime soon, and in fact it seems to be showing up in more and more places as the typesetting system of choice.

But $$\LaTeX$$ gets a bad rap as too complicated for normal people to use. It seems to be something people learn …

January 4, 2010, 7:00 am

# Wolfram|Alpha as a self-verification tool

Last week, I wrote about structuring class time to get students to self-verify their work. This means using tools, experiences, other people, and their own intelligence to gauge the validity of a solution or answer without uncritical reference an external authority — and being deliberate about it while teaching, resisting the urge to answer the many “Is this right?” questions that students will ask.

Among the many tools available to students for this purpose is Wolfram|Alpha, which has been blogged about extensively. (See also my YouTube video, “Wolfram|Alpha for Calculus Students”.) W|A’s ability to accept natural-language queries for calculations and other information and produce multiple representations of all information it has that is related to the query — and the fact that it’s free and readily accessible on the web — makes it perhaps the most powerful self-verification tool…

December 31, 2009, 5:30 am

# How to convert a "backwards" proof into a "forwards" proof

Dave Richeson at Division By Zero wrote recently about a “proof technique” for proving equalities or inequalities that is far too common: Starting with the equality to be proven and working backwards to end at a true statement. This is a technique that is almost a valid way to prove things, but it contains — and engenders — serious flaws in logic and the concept of proof that can really get students into trouble later on.

I left a comment there that spells out my  feelings about why this technique is bad. What I wanted to focus on here is something I also mentioned in the comments, which was that it’s so easy to take a “backwards” proof and turn it into a “forwards” one that there’s no reason not to do it.

Take the following problem: Prove that, for all natural numbers $$n$$,

$$1 + 2 + 2^2 + \cdots + 2^n = 2^{n+1} – 1$$

This is a standard exercise using mathematical…

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