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.

Nice to know we’re not alone

I came across this editorial in the Manila (Philippines) Bulletin over the weekend. Of interest to me:

First, we must address the quality problem that plagues our basic and secondary education because this system is what feeds students into the college level. Our students’ performance in Math and Science are particularly worrisome. In the Trends in International Math and Science Study or TIMSS, our performance continues to be poor: Out of 45 countries, we ranked 41st in Science, and 42nd in Math. We are behind Tunisia and Morocco, and ahead of Ghana and Botswana.

I have two general reactions to this: First, I’m glad we’re not the Philippines. Second, it’s nice to know the same discussions are going on in other places.

…And the Counterpoint

A few days ago, I posted a link and some commentary about reform math in Washington. Today, I came across this post, which is specific to Everyday Math.

The author notes that, “Reform math has dominated our schools for more than 15 years. Over this period, our international ranking has plummeted.” It seems that the article in the Seattle paper directly refuted this claim. At any rate…

The author basically degrades Everyday Math, citing several states that have banned or failed to adopt the program for various reasons. Here’s what might be my favorite paragraph:

Everyday Math has been described as a “mile wide and an inch deep.” U.S. Secretary of Education Arne Duncan is calling for “more depth and less breadth” in education. States like Connecticut are heavily invested in reform programs like Everyday Math. The Hartford Courant newspaper recently reported that 40 percent of incoming college freshmen require non-credit “remedial” mathematics.

Mile Wide, Inch Deep: Show me a core basal program that isn’t. It’s a symptom of over 50 different sets of standards and a long-running debate over what students really need to know.

More Depth, Less Breadth: This should be the goal of every teacher. Figure out what your students know, what the “kinda” know, and what they don’t know, and then adjust your teaching to fit. I’m a big fan of Texas Instruments and what they are dong for education, but stories like the one I received in a TI email today send shivers up my spine: “Imagine having your whole year planned out before stepping foot in your classroom.”

Remedial Math: Only 40 percent? Seems low. Again, this is a symptom of more than the program. It’s about outdated standards, outdated teaching, and a refusal to move away from the teacher’s comfort zone.

So we’re back to the same place: It’s about instruction.

(Note that nashworld does a great job of highlighting the need for quality instruction-through his own experience-in a recent post.)

How many times do I have to tell you…

What did you expect?

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.