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…


The Support of Others

It is amazingly empowering to have the support of a strong, motivated, and inspirational group of people. – Susan Jeffers, author

One thing that was both encouraging and discouraging to me when I was teaching was the support or lack of support for students to learn math. With just a few encouraging words, teachers, mentors, counselors, administrators, friends, and parents can inspire and promote a love of (or at least interest in) mathematics in a young person. A few discouraging words can have an equal, if not greater, effect in the opposite direction.

We had a guidance counselor at the high school where I last taught who was notorious for telling students that it was OK if they weren’t good at math, because she wasn’t either. This leads, in some cases, to apathy and a general dislike of mathematics. I was reminded of this today when I read an advice column. In part, the mother writes, “I told her that, like most women, I wasn’t good in math so if she got a D, that was OK.”

What was the first piece of advice for this mother? “Shift your attitude.” I think there is a need for a general attitude shift about mathematics for all stakeholders. If you are involved in the education of children, please use encouraging words that support, rather than tear down, a child’s confidence in their own ability to do mathematics. It’s just one more thing we can all do to make mathematics better for everyone.

Purposeful Assessment

Like changing for the sake of change, assessing for the sake of assessing has some major downsides. Meaningful assessment can take many forms (try a search for meaningful assessment), but it should always have one outcome: improved opportunity to for students to learn.

This from a recent post on the Rational Mathematics Education blog: “Creating ANY good test item is challenging, but creating test items that actually tell us what we need to know to improve teaching, learning, and parenting when it comes to academic subjects is a major challenge.”

Ask Kermit. It’s not easy being green. And it’s not easy to write a good assessment that gives meaningful data.

Raising the Bar; or, Change for the Sake of Change

I picked up this bit of news out of Delaware. The state should be applauded for their efforts to raise the bar on student achievement. However, it doesn’t seem like they’re looking forward to scores increasing at all. They have assumed that raising the expectations on the test will result in an increase in the number of students that do not pass the test.

Change for the sake of change is not really responsible, yet we do it all the time. An increase in student achievement won’t come from changing a test or adopting a new textbook. Change comes when we design and implement a meaningful curriculum based on student needs.

Factors for Student Achievement

I saw a nice article (from the March 23, 2010 Hechinger Report) that is ultimately about factors that impact student achievement in math. after noting that “Among the top-performing countries, no pattern in pedagogy emerges. There is, in fact, wide variety in mathematics teaching practices worldwide,” the report goes on to identify three issues that impact student achievement in math.

First, curriculum. This is a symptom of well-intentioned standards that make teachers and administrators feel lie they have to teach everything, every year, or the kids just won’t learn. Included in the article is this table, comparing grade 3 assessments:

Assuming (safely) that the assessment is reflective of the intended curriculum, it is easy to see why curriculum plays a role in student learning.

Second, assessment. Not the summative assessments that are still so prevalent in classrooms, or the faux-formative assessments that teachers (including me) use to help them feel better about themselves. It’s about real, ongoing, meaningful contextual assessment that informs instruction and helps all kids learn. The article specifically points to the overuse of multiple-choice assessments, popular because they are easy to score but notoriously bad at providing information about the process students use to solve the problem.

Third, teachers. This part of the article took me back an earlier post that addressed some of the problems with the way we do business. The report notes that, “It’s no secret that American elementary and middle school teachers often have weak math skills,” and then goes on to cite Deborah Ball’s comment: “This is to be expected because most teachers – like most other adults in this country – are graduates of the very system that we seek to improve.”

Improving math education for all students remains a work in progress. When we realize that these factors, among others, are all part of the big picture, then we can begin to work toward the improvement we need.

Another high school math debate

In a follow-up to the last post, I came across an article in the Salt Lake Tribune (math education in Utah is a particularly fascinating topic). It seems that a district superintendent sparked a debate with the state superintendent when he sent a tweet that called the state’s position on high school math standards, “curious.”

The disagreement comes not from whether students should take more math in high school, but rather from what math they should take. The state superintendent believes that all students should take math through Algebra 2, and then have options for further study. His critic believes that all students should take calculus.

I agree with the first idea, for a few reasons. Calculus has been inappropriately crowned the king of math. Calculus is merely a doorway to further studies in math or a related field. Students considering a career that is rich in mathematics (pure math, math education, engineering, physics, etc.) should plan to take calculus, preferably in high school.

Many college-bound students will benefit more from a statistics course (required if they choose to attend graduate school) than a calculus course. Most students, regardless of their career plans, would benefit from a course in discrete math, although most schools and districts are slow to consider this path.

The danger of the argument is that these options are being labeled “tracks,” a negative term that implies that students that take statistics are not as smart or capable as students that take calculus. The responsibility lies with the schools and teachers to ensure that this ability grouping doesn’t happen, and that students are given every opportunity to follow the path of their choosing beyond Algebra 2.

Rethinking high school math

I came across an interesting article in the Delaware Cape Gazette regarding high school math. There were two things that attracted my attention:

High school Principal John Yore said teaching geometry at ninth grade is ideal. “The top scores come from students who’ve had geometry or better. Students who take geometry at ninth grade do better at upper-level high school courses and on any standardized test, including the SAT.”

In response, I would say, “Of course they do!” If you take geometry in ninth grade, you are far more likely to take an upper-level course, let alone do well at it. It is difficult – nearly impossible – to take a Calculus course if you don’t take geometry in ninth grade. One result of taking an upper-level course is a higher score on standardized tests; most college entrance exams  assess content through precalculus.

[One board member] asked if the district needed to address math at the middle schools, as well. Robert Fulton, high school education supervisor, said both middle schools already have math specialists. The priority is the high school, he said, which needs support.

So what do the math specialists do? Ideally, the specialists’ time is spent working with teachers, focused on effective instruction. But this isn’t enough. The board needs to be asking what systems have been put in place to address students that fall behind in middle school. Waiting to address problems in high school doesn’t work (I’ve been there).

In this case it’s about instruction, but it’s also about effective (and early) intervention.