Monthly Archives: April 2011

This list of links was complied somewhere around 2005.  Ugh.  That seems like a long time ago.  I’m a lot better than this now.

Logic Puzzle

If someone can figure out this logic puzzle, please let me know.  I’m wondering why I saw SPARTAN brand hand soap at The University of Michigan.

Videos on Multiplying Fractions

A couple of students in my on-line classes have been asking questions about multiplying fractions.  Should you simplify before or after multiplying?  How should you simplify?  By cross-canceling directly, or by a prime factorization and cross-canceling combo?  Here are some videos I have made to help clarify the situation.

View on screencast.com »

Free Open Reference Interactive Geometry Book

Check out this website I found at mathopenref.com

I will definitely be using some of the applets here the next time that I teach geometry as part of a pre-algebra course. In fact, maybe I’ll have the students read through this and play with all of the applets as part of their homework (before) coming to class.

Thoughts on Assigning Homework

Proponents of assigning homework probably believe that students need regular practice outside of class.  While I’m totally not against students practicing math outside of class, I don’t believe that we should give homework simply so that the students can practice.  Let me clarify: I’m not saying that students do not need the practice at all.  And I do believe that some parts of mathematics need to be practiced until learned.  If a student needs to practice, that’s fine.  However, I am against giving homework simply for this purpose.  I sincerely believe that not all students benefit from this method of drill, drill, drill until you get it correct, or you memorize enough to be able to pass the upcoming test.  That’s not learning.  Homework should part of the learning experience.
My stance on homework is that the quality of the homework problems matters more than the quantity of the homework problems.   And I believe that this statement applies to both the instructors and the students.  Instructors need to design quality homework problems and students need to turn in quality solutions to the problems.  For some instructors and some groups of students, regular homework after every chapter can definitely be a good thing.  However, for some students, bi-weekly homework, or even one or two semester projects might be better.  I have used many formats for assigning homework, adapting them to each group of students.

Opponents of assigning homework might disagree with me with their massive claims that assigning homework hasn’t helped to improve test scores, so they just don’t assign it anymore.  Although, I’m not totally against this notion that there is no visible connection between homework and test scores, I think that the benefit of homework goes well beyond just whether the test scores are higher or not.  What I believe homework does is conditions students to see the problem, review the problem, and attempt to critically think and solve the problem.  Even if there are no numerical benefits from homework, what I have seen to be true is that homework helps to reduce math anxiety, test anxiety, and stress among the students.  So, even if the student gets the same score that he or she would have gotten without the homework, the student was saved plenty of hours of anxiety and stress.

Again, I would like to stress that I honestly believe that homework assignments need to be thoughtfully designed by the instructor and considered part of the learning experience.  Homework is essential.  It should take whatever form it needs to take in order to be beneficial for the students.

New Slope Game

I hope you enjoy this game!  I would love to have any feedback if you decide to use it.

Other Slope Resources

Applets
1.  Interactive Slope Applet – Although I ran out of time to actually use this with my own class, this is a wonderful resource that lets students click and drag points such that when the line between the point changes, the calculation of the slope of the line also changes on the screen as well.  Very useful!

2.  Slope-Intercept Equation Applet – This appears to be the exact same applet I introduced to you a few weeks ago in the Geogebra Tutorial video.  It’s a very simple resource that allows students to visualize the slope-intercept equation of a line by using sliders to change the slope and y-intercept.

Worksheets

MindMap for Basic Mathematics Project

I learned a lot from the class that I took on Online Learning and Teaching this semester.  We used the book Understanding by Design by Wiggins and McTighe.  This course has revolutionized the entire way that I think about teaching and learning.  As of late, when planning lessons, I keep finding myself needing to list the big idea, the misconceptions, and the essential questions.  I’ve also found this to be a good way to organize my own learning.  This book is truly a great read.
As part of this course, we had to design a demo course using the ideas in the book.  To present my course, I decided to use a mindmap, rather than a PowerPoint presentation.  I thought I would post it here as just a little sample of everything that I have been working so hard to do this semester.

http://www.mindomo.com/view?m=ed9c47bcafeb408495c2e52561fae659

Thoughts on Preparing Students for College

This is my response to this post in The Chronicle of Higher Education.

It is my understanding that there is still a major problem with equality in this country and that many students who go to inner-city or rural schools may not have the same opportunities as students who live in affluent suburban neighborhoods.  I completely understand that.  I completely agree that anyone who wants a chance at college should be given the opportunity to have a chance at college.  There are many arguments that can be made that college is not for everyone.  However, again, I sincerely believe that anyone who wants a shot at college should be given the opportunity.
But my problem with this article is that it seems to point out the fact that college is such a new opportunity for ‘black males and other underrepresented groups’.  What I think people tend to forget is that college is a new opportunity for any incoming freshmen.  Anytime a student enters college for the first time, there are going to be certain things that the student needs to adjust to, whether the student is in one of the underrepresented groups, or not.  I don’t think the fact that someone is from an underrepresented group should diminish the fact that other students who are not in these groups are probably struggling with many of the exact same adjustment issues.
Once you have entered college, you are officially on the same playing field.  Sure, you may have to take a few developmental classes to get you up to speed with some of the material that you may have missed due to the inequities in the high school education system, but in terms of adjustment, you are indeed dealing with many of the same issues that any incoming freshmen is dealing with.  I think that trying to find a commonality with the other freshmen, instead of feeling like you are a singleton in a sea of many is one of the most important things that you can do as a freshman.
Personally, I am a minority student who went to a inner-city high school.  Due to the lack of resources, there were many things that I didn’t get to do in high school that I would have loved to have done.  I didn’t wallow in my sorrow.  As soon as I graduated from high school, I took the first train out of town that I could find to a college that was as far away from being in the inner-city as I could at the time.  I was completely overwhelmed.  My grades weren’t nearly as high as they had been in high school.  I was very disappointed in my low grades my first semester in college compared to the high grades that I had in high school.
I could have become a statistic.  I could have fallen into a deep pit of depression and weighed my options of whether I should just drop out of college and move back home with my parents, or possibly out of embarrassment, not move back home and just live homeless on the street.  But I didn’t.  I realized that all of the students there, not just me, had to adjust to college.  I realized that the world didn’t revolve around me, my grades, and my ability or inability to graduate from college.  I didn’t need anyone to tell me this.  I didn’t need anyone to hold my hand throughout my entire college experience until the day I graduated.

My thought is this.  If a student goes to college and doesn’t realize what I realized on their own, if a student does not have the ability to independently think and weigh decisions on their own, and if a student wants to wallow in their sorrow or fall into their own personal pit of depression, I don’t think that we should play a role in helping the student become more dependent.  This will just create future workers who may have a college degree, but are not able to independently think and create ideas on their own.  I implore you, please, in preparing students for college, teach them about this.  Teach the students about the importance of independent thinking and putting their futures in their own hand.  But don’t help them walk their way into a future that they didn’t fully create on their own.

Thoughts on Critical Thinking in Mathematics

This is my response to this comment by a reader of The New York Times.

First of all, as a math instructor, I’m a little offended that this reader has called our education system pathetic without actually giving a solution to the problem.  Although, I understand where the reader is coming from in terms of needing a solid foundation in order to do higher mathematics, I want to share a share a story with you based on what I have learned from working with students whose first experience with mathematics was outside of the United States.
Last semester I had a student in my basic mathematics class who tried as hard as she could in the class.  She could do most of the arithmetic mentally, or by hand.  She was especially good at calculating fractions mentally because she followed a memorized formula for doing so.  She was used to being taught the formulas upfront and then shown some ways of how to apply the formulas.  I always discourage students from memorizing formulas for calculating fractions because it’s exactly that.  It’s nothing more than memorization that absolutely does nothing to help the person understand what a fraction actually is.
Throughout the entire semester, she was at the top of the class.  However, when the test came along, I had a question that made the students think about what a fraction really is. A question in which one actually had to know that a fraction of something, whatever it is, is a fraction of something that has been split up into a certain number of equal parts.  Somewhere in the process of the rote memorization that this student had done, she missed the point of equal parts, and tried several times to break an object into several unequal parts.  No one else missed this problem.  Because they took their learning experience beyond just simply memorizing a formula for calculating fractions that didn’t apply to every problem that involves fractions.
And that is where I think that this reader is missing the point.  I think that by having students think critically and develop the formulas for themselves, that certain learning occurs and details will be learned through the discovery of these formulas that might be missed otherwise.  I like to think of formulas as shortcuts for people who are too lazy to derive something for themselves.  Of course formulas are nice, but it’s always nice to know where the formula is coming from.  How can someone honestly say that they know where the formula is coming from if the only thing they know to be true about it is that someone told them that it holds true and therefore they should memorize it?

Discovery learning takes a long time, and you won’t get the answer right away.  But getting the answer right away is not the point of the whole thing.  And just because you don’t get the answer right away, doesn’t mean that math is difficult and impossible to understand.  It doesn’t mean that you should give up and blame the entire educational system.  You need to understand that true learning and true understanding takes longer to foster because the roots of it go deeper than any foundation that is set down using any other method of learning.

Remember the example of my student?  Her so-called solid foundation in the formulas that she memorized about fractions at a young age, her rock solid foundation, was crumbled in just seconds by one question asking her to actually interpret the meaning of what a fraction meant.  To me, this is sad, and I would never want this to happen to anyone that I know.