Author Archives: Mr. Chase

Patient problem solving

Posted on by
1

There are lots of qualities that make someone “good” at math. Knowledge and skills are important, and so is creativity. But perhaps the most important qualities are patience and persistence.

I was inspired to write this because of this post by Alexander Bogomolny.

So many of my students shut down as soon as they see a problem, especially those students who have had a bad relationship with math over the years. Some students even give up before really understanding (or reading!) a question. This is even more true when it comes to ‘word problems.’ If you haven’t yet seen this now famous TED talk from Dan Meyer, I encourage you to watch it now (and visit his great blog!). I think he might be the first person I’ve heard use the phrase “patient problem solving” so I’ll give him the credit for that :-).

The importance of patient problem solving has broader application than just math, of course. In so many areas of life, we give up too easily when faced with a problem. We don’t realize, that if we just looked a the problem a little longer, if we came back to it a few more times, if we dove a little deeper, the problem would crack.

When I do my graduate class homework, I find great value in starting the homework problems as soon as possible. Sometimes the problems just need to percolate in my brain!

For those of us who teach, it’s important to keep ourselves fresh and engaged in mathematical problem solving on a regular basis so that we can (1) remain familiar with what real mathematicians actually do, and (2) relate to (and empathize with) our students who are being faced constantly with fresh problems.

The folks over at Math Fail recently encouraged us in that direction, saying “If you didn’t focus on mathematics today, you should at least keep the gears in motion by trying to solve this mathematical puzzle”  and then they proceeded to give us a good puzzle. Go try it now! (I solved it this morning with my wife as we were driving to church!)

I want my students to experience the immense satisfaction that comes from having solved a stubborn problem. I want them to know that if you bang your head enough against the problem, eventually it will crack! Alexander Bogomolny made this point in the the inspiring blog post I mentioned above, saying

Still, there is great satisfaction in having solved a problem – even a simple one, and extra satisfaction in being able to appreciate an elegant proof; this kind of satisfaction is multiplied manifold after you devised a solution on your own. Yes, it all may start with inspiration, but to keep the flame burning involves hard work…the more you sweat, the greater the satisfaction.

Some of my students are starting a big math paper this week, in which they choose their own topic. One of my hopes for them is that they get to experience the deep satisfaction that comes from actually doing mathematical thinking and solving hard problems. There’s also great satisfaction in coming up with good mathematical questions! And they’ll have a chance to do that too.

On a related note, when reading math textbooks, students sometimes don’t understand that reading math is very different than reading other subjects. In other subjects you might be willing to devote 5 minutes per page. But in math, a reader shouldn’t be discouraged if it takes 20 minutes or more to understand a page of text. Math is dense!

Pictures with equations

Posted on by
Reply

Check out this awesome blog post by Richard Clark on the Alpha Blog.

 

Follow the link to see lots of great pictures made with equations. These pictures are so complicated it makes you wonder, is there any picture we can’t make with equations? My first answer is NO.

Think about vector-based graphics. Vector graphics, for those who’ve never heard the term, are pictures/graphics that are stored as a set of instructions for redrawing the picture rather than as a large array of pixels. You’ve used vector graphics if you have ever used clip-art or used the drawing tools in Microsoft Office, or if you’ve ever used Adobe Illustrator, or Inkscape. The advantages of vector graphics include very small files and infinite loss-less resizeability. How can vector graphics achieve this? Well, like I said, vector graphics are stored as rules not pixels. And by rules, we could just as easily say equations.

So the answer is certainly YES we can make any picture using equations. I think the harder question is can we make any picture using ONE equation? Or one set of parametric equations? Or one implicit equation?

What constraints do we want to impose? Do fractals/iterative/recursive rules count?

I am curious to find out how the creators of these picture-equations came up with them. It seems infeasible to do this by trial and error, given the massive size of these equations.

Oh, and if you haven’t yet seen the Batman Curve, you better go check that out too.

 

Teaching and Grace

Posted on by
Reply

Thanks to my dad for the hat tip. If you haven’t yet seen it, this talk (available in text & audio) is a MUST READ for all teachers. Thank you, Francis Su, for sharing your thoughts on teaching and grace!

Your accomplishments are NOT what make you a worthy human being.

I highly recommend, over and above any other teaching book I’ve read, Teaching with Love and Logic. The message of that book is similar to the message Professor Su shares. Let kids know you like them for who they are, not for how they perform in your class!

Cubic polynomials and tangent lines

Posted on by
2

Just read an article in the most recent NCTM Mathematics Teacher magazine called “Students’ Exploratory Thinking about a Nonroutine Calculus Task” by Keith Nabb. I really, really enjoyed this article. Maybe for some this isn’t new, but I didn’t know this fact:

Average two of the roots of a cubic polynomial. Draw a tangent line to the cubic at this point. Did you know it will always pass through the third zero?? Incredible!

Here’s a nice site that I just googled that goes through one proof. However, the charm of the article mentioned above is that there are many interesting proofs that students came up with, some of which are more or less elegant (brute force algebra with CAS, Newton’s Method, just to name two of the four strategies mentioned in the article).

I wish I could give you the whole article, but you have to have an NCTM membership to see it. Here’s the link, but you’ll have to log in to actually see it.

“Japanese” Multiplication

Posted on by
1

My brother sent me a link to this video that teaches “Japanese” multiplication (thanks Tim!):

I learned about this technique in my History of Math class, and Vi Hart talked about it in a video back in 2011:

She does a nice job showing why there’s nothing particularly special about this Japanese “visual” multiplication. Here are a few reasons why it’s not better, as far as I’m concerned:

  1. It’s not faster (sometimes it is, but most of the time not). As Vi points out, counting the number of dots in a rectangle by hand is ridiculous.
  2. It’s painful when the numbers are bigger than 1, 2, or 3 and when there are more than 2 digits in the numbers (just try multiplying 976 x 8937 for example).
  3. Zeros make things difficult (use dashed lines?)
  4. Carrying is still required.
  5. It’s perhaps more error prone, since it relies on your counting all the intersections.

In the end, to multiply two numbers you still have to multiply all their digits by each other and deal with carries, no matter which method you choose. I think it’s still worth teaching various methods of multiplication to students in an effort to make the abstract more concrete.