Speaking of ideas that contemporary individuals or school of thought have claimed to have invented, consider the "growth mindset." As front-page article in this week's Education Week explains:

The concept of growth mindsets has gained a foothold in many schools, where teachers emphasize that the brain can grow and change and that students don't enter school with a set of unchangeable strengths and weaknesses. In general, that means praising effort over personal traits and encouraging students to learn from mistakes by developing new strategies to approach problems.A brain that grows and changes and learns new skills? Praising effort rather than personality traits? Encouraging students to learn from mistakes? To try new strategies when other strategies fail? What revolutionary ideas!

Equalling groundbreaking is "research that shows that student mindsets and persistence are linked to academic success." In other words, if you believe that you can overcome weaknesses, and if you work persistently, then you're more likely to succeed! However counter-intuitive this may be, Edweek notes that more schools are buying into this idea that ever before:

Some teachers are also making efforts on their own to learn about the mindset concept. Stanford University's Project for Education Research That Scales, or PERTS, released a series of online courses about mindsets for parents and teachers last month.(Yet another revolutionary idea: PERTS!).

Some of the courses' key concepts are: encouraging students to ask questions that they may be afraid to ask; telling students that they can learn from their mistakes; and presenting specific mistakes and discussing them with the class.

Also key is the concept of a "sweet spot": a point where you find yourself struggling. Apparently this is where you need to persevere (and "unpack").

What is reasonable isn't new, and what is new isn't reasonable. Aside from confusing labels with concepts and dressing up old concepts in new labels, the growth mindset crowd has some new advice on math instruction. Here Edweek turns to Philip Uri Treisman, "a mathematics professor and director of the University of Texas' Charles A. Dana Center, which focuses on math and science education":

Merely challenging students to change their mindsets without also changing the way math is taught can be "dangerous," Treisman said. Without a grasp on math skills and opportunities to apply those skills and develop strategies, students will receive the message that even effort can't help them improve, he said.Educators, it turns out, are undermining the growth mindset (and presumably also the "unpacking" of the "sweet spot") by prescribing particular algorithms for particular problems. In order to promote a growth mindset in math, you need to teach through "open problems":

which challenge students to explain a concept rather than quickly identify one solution. This gives them a chance to explore strategies for solving a problem and recognize there is often more than one way to make sense of it rather than judging their own math skills by whether or not they get the initial answer correct.If math were music, Treisman adds, mastering the basic concepts would be like learning scales and leading students through discussions of open problems would be like playing songs. Inspiring this insight, perhaps, are equally insightful insights from Paul Lockhart and Keith Devlin.

One of the PERTS instructors is the eminently ethical and trustworthy Jo Boaler. Boaler notes that:

In a traditional problem, a teacher may give students the dimensions of a rectangle and ask them to find its perimeter. In an open problem, a teacher may ask students to draw three rectangles with a certain perimeter and explain their work.The Edweek article provides the following illustration:

Interestingly, a 2006 paper by Cornelia S. Große (University of Bremen) and Alexander Renkl (University of Freiburg) has reached a slightly different conclusion about how multiple strategies/solutions affects student success. (Many thanks to Barry Garelick for sending me this paper!). Entitled "Effects of multiple solution methods in mathematics learning" and published in Learning and Instruction (Volume 16, pp. 122-138), it discusses how:

we found that learning with multiple solutions reduces the learners’ effort to spontaneously solve parts of the problems on their own... In addition, the learners’ insight into analogies between examples seemed to have been hindered. It can be assumed that dealing with multiple solutions is so demanding that the learners do not have enough ‘‘free’’ cognitive capacity available to compare examples or to try to solve parts of them on their own.The multiple solutions approach of "open problems," it seems, might actually

*undermine*growth mindsets.

Will this 9-year-old finding ever permeate the American edworld? Or, when it comes to fostering growth mindsets, is there but a single, pre-prescribed strategy for everyone?

## 6 comments:

The open problem doesn't strike me as a bad problem, exactly, but it does seem much more difficult than the traditional problem. The traditional problem looks like a reasonable next step after you've explained what a perimeter is. The open problem would be completely baffling if it was the next step after explaining perimeter.

If you explained the definition of perimeter, then did a bunch of traditional problems, then did the open problem it could be a reasonable progression.

The trouble as I see it in Younger Daughter's homework is that they go right to the open problems, with little or no practice via traditional problems. It's too much to digest all at once.

How about an infinite number of solutions in the form y = 13 - x. I wonder what the grader would do if the student drew a line on a coordinate plane representing all possible answers.

I strongly object to the idea that kids should be given opportunities to struggle. This is so demoralizing for kids and, in my experience, causes them to give up rather than try harder.

My favorite ed quote:

“You know what’s the worst kind of instruction? The kind of instruction that makes kids feel stupid. And that’s what a lot of that discovery stuff does; their working memory gets overloaded, they’re confused. That’s bad instruction,” said Anna Stokke, an associate professor in the University of Winnipeg’s department of mathematics and statistics, who wrote the C.D. Howe Institute report. [ Decline of Canadian students’ math skills the fault of ‘discovery learning’: C.D. Howe Institute ]

From Ch 19 of Conversations on the Rifle Range published at OILF

In all my classes, I required my students to answer warm-up questions at the beginning of class. I used two types of questions:. One was a review-type question to apply what they recently learned. The other required them to some apply their prior knowledge –or what was familiar—in a new or unfamiliar situation. Some may view this as an inquiry-based approach, or an application of the “struggle is good” philosophy that adherents of Common Core seem to say is necessary to develop perseverance in problem solving, as well as the all-important and frequently undefined “grit”. I view a short amount of struggle as appropriate provided that explanation is provided shortly after. That way, even if students do not succeed in solving a problem, most are receptive to explanations that they might otherwise tune out.Barry's approach is the one that my math-loving kid prefers. Unfortunately, it has been awhile since he's had a teacher at school use it - four years, I think - but math outside of school does. The teacher 4 years ago also had kids volunteer to explain their approach(es) at the board.

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