Proportionality in Graphs

As I was looking through the results of the 2016 Presidential election, I found the graph below. The colors aren’t as important to me as the design: the use of proportion to show the electoral college votes for each state helps to illustrate why candidates spend their time in some states as opposed to others.


Source: The Telegraph 9 November 2016

So, how would you use this with students? Share your ideas in the comments…


Wait. I Can MAKE Kids Learn?

I came across something interesting today. I ventured (for the first time) onto Google Books to see what they had to offer. I made my way to K-12 Mathematics – no big surprise there. And I came across this:

The Equation for Excellence: How to Make Your Child Excel at Math by Arvin Vohra.

Perhaps I’m naive, but I’m not familiar with Arvin or this book. I’m not going to talk about the author – I’ll let you form your own opinions based on his website.

I looked at the table of contents, and Chapter 11: The Calculator Fallacy caught my eye. So I started reading. I will admit that some of the points are valid and made me stop to think, but there is a general theme of “calculators make students lazy” and “teachers are misinformed.”

Then we get to this: “A student solving a complicated problem spends very little time doing actual calculations. Most of the time is spent examining relationships and determining what concepts apply.”

Wait. Didn’t he just make the case for calculators? I used graphing calculators to help students examine relationships and link concepts. If they used the calculator to multiply six and four, so be it.

The author then supports his statement: “The student who does math by hand has these concepts ingrained in his mind, and is adept at using them.”

Again, wait. Did he just tell us how students gain conceptual knowledge? Wow. We’ve been trying to figure that out for a while, and here was the answer all along. Make them do the work by hand. (Nobody’s ever tried that one before.)

Doing math by hand does not build a solid conceptual foundation for learning. Models help students build this foundation. Rich activities that apply learning help build this foundation. Regurgitating facts and working everything out by hand do not build conceptual understanding.

Finally, this assumption: “Thus, he rapidly sees relationships between various formulas and concepts, and can quickly figure out how to do the problem.”

I can count on one hand the number of students who made connections between formulas and concepts by simply doing problems by hand. I agree with the idea that a calculator in the hands of a less effective teacher is a dangerous thing. But the author discounts the role that a calculator can play in discovering patterns and understanding relationships, and the role of an effective teacher in promoting this kind of calculator use.

Washington’s Miniature War

If you follow the ongoing saga of the “math wars” at all, you are likely familiar with the long-running debate in the state of Washington. Many school districts in Washington were early adopters of NSF-funded “reform” mathematics curricula, and much of the debate surrounding these programs has come out of Washington. (If you don’t believe me, do a search on YouTube for math.)

Even given this background, I was a bit surprised to see this guest editorial in the Seattle Times regarding discovery-based math. Of particular interest to me were the comments.

I think we’d like to believe the math wars are over. This article, and the related comments, bring us back into reality. It begs the question, “Will the math wars ever end?”

Why do I care?

You might ask, “Why does he care about math education so much?”

And then I come across something like this:

If you see what’s wrong with this, you get my point.  If you don’t see what’s wrong with this, you’ve made my point.

If you need a hint, let me know.  I’m still a teacher at heart.

(Image from

How many times do I have to tell you…?

Take a look at this article in the Salt Lake Tribune.

Yet another example of people missing the point: it’s about instruction.

The State of Utah would be much better off spending their money on competitive grants to districts to provide training for their teachers in concept-based mathematics.

I won’t rant about this anymore.  I think this says it all.

Foundations for Success

File this under, “Stuff I meant to post last week and didn’t.”

The National Mathematics Advisory Panel released its final report last week, titled Foundations for Success. You can find the report and sub-reports in a variety of file formats at the NMAP home page.

The report is heavily grounded in “high-quality” research and includes six key elements that I would summarize as follows:

  1. Curriculum Focal Points;
  2. How students learn;
  3. Teacher content knowledge and pedagogical content knowledge;
  4. Quality first instruction;
  5. Quality, focused assessment; and
  6. Education research.

Remember, these are my summaries, not theirs. I would suggest looking at the report if you’re at all interested in what the group had to say, as they are much more verbose than I.

Starting Something New

(cross-posted at Looking for r)

I like my blog.  It gives me an opportunity to talk about things related to math education that are important to me.  But sometimes I’ve got more to say, and it’s not always about math, and sometimes it’s not even about education.

Enter the new blog.  I’m excited about this, and hopefully it will keep my creativity flowing a bit more evenly.  Look around and comment.  I’m looking forward to this!  Of course, if you prefer the linearity of the old blog, it will still be around.  I’ve got a lot more to say about math education…