How Not to Teach Math

Kathy Liu Sun, a former high school math teacher who is now an assistant professor of education at Santa Clara University says too many teachers are having their students work on computers for the entire math lesson.

She writes:

Proponents argue that computer-based lessons allow students to go at their own pace and expose students to content they might not otherwise have an opportunity to see. But these benefits come at a high cost.

One of the most pressing problems is the content and focus of these digital lessons, which are often simply digital replications of traditional lecture based math lessons. (You remember these: teacher at board showing you example after example, followed by practicing a similar problem with different numbers twenty times over.)

Whether delivered digitally or in person, this type of instruction sends the wrong message about mathematics. It teaches students that mathematics is about mastering a set of procedures, rather than viewing mathematics as a creative subject that is about problem-solving and sense-making.

Research has shown that such an emphasis on mathematical procedures is not supportive of student learning and fails to help students to draw connections between key mathematical ideas, think critically, and problem-solve. These skills are particularly important for 21st Century citizenry and long-term achievement outcomes.

While your seventh grader engaging in digital math lessons might be learning pre-calculus procedures, she may not have any understanding of the underlying concepts that will be critical for future success.

Liu Sun may as well have been talking about my 13 year old self. Thanks to my Old School parents, meaning unending hours with flashcards, I had all the necessary math facts etched into my cerebral cortex. Consequently, I could solve equations with the best of them, but I didn’t understand the underlying concepts when the road kicked up. As a result, even my final math class, Algebra 2/Trig, proved difficult.

Liu Sun continues:

. . . consider how technology might genuinely support mathematical sense-making and problem-solving. A recent study conducted at Stanford University found that students who played a game that focused on the relationship between numbers, rather than memorized math facts, led to better learning outcomes.

Good educational technology, implemented at the appropriate time, can enhance math learning. Here are a few things to look for when examining technology to support mathematics learning:

  1. Exploration: The technology should provide opportunities for students to explore by conjecturing, testing out different ideas, and making mistakes. We should avoid digital learning programs that focus only on memorization or funnel students’ thinking

  2. Multiple Solution Strategies. Identify technology applications that have more than one way to solve the problems. For example, rather than using digital flashcards such as 3+4 = ?, we can identify apps that ask students to find pairs of numbers that add to 7. The latter question has many solutions such as 1 & 6, 2 & 5, 0 & 7 and supports students to understand how one whole number (in this case 7) can be broken into parts in multiple ways.

  3. Connections between concepts and procedures. Good educational technology supports students to focus on relationships, not discrete facts. Rather than choose a digital program that solely focuses on doing the same procedure over and over, identify a program that supports students to understand why the procedure works. For example, with regards to the earlier problem 3+4 = ?, a digital program that includes other representations, such as images of objects that students move around can better support to develop meaning of the procedure. Digital math games that focus solely on procedures should only be considered after students have strong understanding between concepts and procedures.

Like a wannabe Maggie Z, humble blog loyalist and middle school math teacher extraordinaire, I’m only going to assign Liu Sun partial credit for her blog post because she stops short of identifying any specific technology applications that check the three aforementioned boxes.

Thus proving blogging is harder than it looks.

Who Are You Drafting Off Of?

Like a lot of introverts, my need for solitude sometimes seems insatiable. Yet, I’m keenly aware we are social beings and that we need others to accomplish much of anything and to have any meaningful shot at genuine happiness.

Even though you won’t find me in the Nisqually Delta with binocs, DSLR camera, and ginormous lens dangling from my neck, or listening to bird calls on my iPad, I enjoy watching the birds we share our new spot with. Yesterday’s airshow was especially good. Two bald eagles took turns nipping each other (foreplay?) while a cormorant glided by obliviously . I also can’t get enough of watching geese and other migratory birds fly by in small, medium, and large “V’s”.

Migratory birds draft off of one other for the same reason cyclists do, to save about 25% of their energy. I wouldn’t be surprised if they also also benefit somehow from the social aspect of flying together.

Are you an investor, if so, are you “flying solo” or are you drafting off of someone more experienced, knowledgeable, and successful? Like this person. What about as a person, are you drafting off anyone to be a better human being? Or maybe, like me, as an educator, parent, and older person, you’re doing your best to, like Nairo Quintana in the picture below, “lead out” others in need of a positive example.

20175944-355979-800x531.jpgPhoto: Tim De Waele |

Postscript: I love this pic because even though Quintana is working at least 25% harder than Contador and company, he’s totally in control, meanwhile, everyone else is struggling mightily to hold his wheel. If only I had grown up Boyacense.

The Opioid Crisis

From Joel Achenbach in the Washington Post. “No longer ‘Mayberry’: A small Ohio city fights an epidemic of self-destruction”.

“Now you can get heroin quicker in these communities than you can get a pizza,” said Teri Minney, head of the Ross County Heroin Partnership Project. “They’re delivering.”

The addicts often shoot up in public places knowing that if they overdose they will be seen and potentially revived by police or paramedics carrying naloxone, the anti-overdose medication, also known by the brand name Narcan. One day in September, police and paramedics responded to 13 separate overdose calls, including one fatality: a man who died in an apartment right on Main Street. Meanwhile, a woman overdosed in her car as it idled at a Valero gas station with her 2-year-old daughter in the back seat. On that single day, seven children in the county were taken into government custody.

“It’s the Zombie Apocalypse,” says Gabis, the coroner.

The county’s health commissioner, Tim Angel, says he sees multiple generations of addicts now. He’ll ask a young patient who has come in for treatment, “How did you get involved in this?” and the answer will be, “My mother shot me up for my birthday when I was 14.”

Imagine if the President focused half as much energy trying to ameliorate the opioid crisis as he does immigrants.

Paragraph to Ponder

From “Understanding Teacher Retention and Mobility in Washington State” by Elfers, Plecki, and Windekens.

While the number of teachers in Washington state has increased by approximately 11,000 in the last twenty years, the racial and ethnic diversity of the teacher workforce made only minimal gains. In 1995-96, there were approximately 49,000 teachers, 94% of whom were White. In 2015-16, 90% of the state’s 60,000 teachers were White. The increase in diversity of the workforce was concentrated among Hispanics, with the percent of Hispanic teachers increasing from 1.7% to 3.9%, and the percent of Asian/Pacific Islander/Native Hawaiian teachers rising from 2.0% to 2.8%. The proportion of Black/African American teachers has declined from 1.6% in 1995-96 to 1.2% in 2015-16. The proportion of Native American teachers also declined slightly from 0.8% to 0.7% in the last twenty years.

This problem is almost always framed as “students of color need educated, professional role models that look like them” which is undeniable; much less appreciated though, is the need for White students to have excellent, ethnically diverse teachers.

Post-Truthism Explained

From “Why Facts Don’t Change Our Minds” by Elizabeth Kolbert.

Surveys on many. . . issues have yielded. . . dismaying results. “As a rule, strong feelings about issues do not emerge from deep understanding,” Sloman and Fernbach write. And here our dependence on other minds reinforces the problem. If your position on, say, the Affordable Care Act is baseless and I rely on it, then my opinion is also baseless. When I talk to Tom and he decides he agrees with me, his opinion is also baseless, but now that the three of us concur we feel that much more smug about our views. If we all now dismiss as unconvincing any information that contradicts our opinion, you get, well, the Trump Administration.

How to Fix the “Youth Sports Coaching” Problem?

A friend’s son, an 11th grader, is a very good SoCal basketball player. He wants a college scholarship, but his team is poorly coached, so attracting the attention of college coaches is more difficult. His team lost one game 70-62 after being up 60-45 after three! I do not know if their mascot is the Cougars.*

My friend really knows basketball and is exasperated at the coach’s incompetence. Granted, too often, overly subjective and meddlesome parents of young athletes are part and parcel of the problem, but that’s not my focus here. My focus is on the amazing discrepancy between what we require of beginning teachers versus coaches. Coaches are educators, for some student-athletes, even more influential ones than teachers. Yet all we tend to require from them is some CPR and child abuse training. We hope they “know” their sport inside and out and how to interact positively with their athletes. More generally, we hope they have the necessary dispositions to inspire their athletes to not just athletic, but life success.

Consider the economics of the problem. The aforementioned high school basketball coach gets $2,800 for the season, meaning way less per hour than his athletes earn at their weekend jobs. So the supply of coaches is severely limited. That means high school athletic directors are loathe to fire any coach that isn’t breaking the law. That is, as long as the parents’ protests are manageable, which they usually are since frustrated parents turn over every few years, meaning they never get sufficiently organized.

Given the paltry stipends coaches receive, it’s unrealistic to expect them to undergo any training remotely similar to student teachers. So what to do absent more incentives for outstanding educators to consider coaching? Could a more thorough and thoughtful interview process weed out incompetent and/or unkind coaches-to-be? Only if there’s more candidates to choose from right? And so we’re back to the stipends.

I really don’t know the answer to the question. A few outstanding high school coaches read the humble blog. Maybe they’ll enlighten us.


*inside Washington State joke; Scottie, love you as always