From the New York State
Common Core Sample Questions
1. Is a piece of string mathematically comparable to a number line?
2. Should alternative answers for A and C be, respectively, 3/4 and 1/4?
For left-brainers and kin: thoughts on education, left-brainedness, autism, and right-brain biases.
From the New York State
Common Core Sample Questions
Barry Garelick, who wrote various letters under the name Huck Finn, published here, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number 12:
Barry Garelick's recent piece Education News, Undoing the ‘Rote Understanding’ Approach to Common Core Math Standards, got me thinking about one of the things that Reform Math has backwards.
Barry talks about the emphasis by today's math educators on ad hoc methods like "making tens" at the expense of traditional algorithms like borrowing and carrying. And so we see more and more worksheets like this one:
Two problems from the Spring 2013 North Carolina Measures of Student Learning Common Core Algebra II Exam:
Last week, CNN ran an opinion piece by Pasi Sahlberg, the former director general in the Finnish Ministry of Education and Culture (and now a visiting professor at Harvard's Graduate School of Education) explaining “Why Finland’s schools are top-notch.”
This is not the first piece that Stahlberg has written for American readers, and one likely reason he’s been getting so much attention is that what he tells us so exactly matches what so many of us want to hear. What makes Finnish schools so great, it turns out, is that they focus more on funding, educational equity, child-centered play, and “the whole child,” and less on testing and “narrow academic achievement”:
There are three things that have positively affected the quality of Finnish schools that are absent in American schools. First, Finland has built a school system that has over time strengthened educational equity. This means early childhood education for all children, funding all schools so they can better serve those with special educational needs, access to health and well-being services for all children in all schools, and a national curriculum that insists that schools focus on the whole child rather than narrow academic achievement.Stahlberg doesn’t mention that the U.S. spends more per pupil than Finland does, and, in particular, a great deal on special education. Nor does he reconcile the claim that Finland has early childhood education for all children with the fact that Finns famously don’t start school till age 7. As for the implication that U.S. schools are, by comparison, narrowly focused on achievement, he doesn’t mention that Finnish schools lack sports teams, marching bands, and proms.
Second, teachers in Finland have time to work together with their colleagues during the school day. According to the most recent data provided by the OECD the average teaching load of junior high school teachers in Finland is about half what it is in the United States. That enables teachers to build professional networks, share ideas and best practices. This is an important condition to enhancing teaching quality.But is the only factor? Is networking even the most important factor in teacher quality? Stahlberg doesn’t mention here that Finland recruits its teachers from the top 10% of college graduates, while only 23% of U.S. teachers come from even the top third of college graduates.
Finally, play constitutes a significant part of individual growth and learning in Finnish schools. Every class must be followed by a 15-minute recess break so children can spend time outside on their own activities. Schooldays are also shorter in Finland than in the United States, and primary schools keep the homework load to a minimum so students have time for their own hobbies and friends when school is over.I agree with Stahlberg that American kids need many more 15-minute outdoor recess breaks, and that our primary schools should assign much less homework. But there are a couple of important distinctions he omits. First, if you include indoor recess, in-class games, and indoor shows and movies—much more common in U.S. schools than elsewhere—American students are getting many more breaks than it first appears. The Finns send their kids outdoors in all kinds of weather; so should we. And if we simply reduce the passive, couch-potato breaks from learning, we can increase the time available for true recess without reducing the time available for true learning.
In the toughest math classes I took in college, it happened a couple of times that a certain classmate of mine would ask me to explain what was going on. He seemed to have more trouble understanding the material than I did, and my verbal explanations seemed to help him understand it better. So much better that he, now an accomplished engineer, went on to score higher than I did on all three class tests.
For many Reform Math acolytes, the ability to communicate your reasoning to others, and to talk about math more generally, is the apotheosis of mathematical understanding. It’s much higher level, supposedly, than “just” being able to get the right answer.
But does saying intelligent things about math necessarily mean that you can actually do math? One situation that often ends up suggesting otherwise is tutoring.
When you tutor someone one-on-one, at length, over a long enough period of time, it’s easy to think that you are comprehensively probing the breadth and depth of their understanding--simply by conversing with them broadly, and by asking the right questions and follow-up questions. Surely what your tutee says reflects what s/he knows. Surely it’s not possible for him or her to carry on articulately about something they don’t fully understand. And surely it’s not necessary to test their ability to do tasks independently in the more traditional, detached testing format of a written examination.
But some of the same things that make tutorials so great—their fluidity and flexibility, and the apparent close-up they provide into students’ understanding—are also their biggest downside. It’s not hard for tutors to accidentally provide more guidance than they intend to; to lead the tutee towards the correct answer; to fail to create situations, complete with awkward silences, in which tutees have to figure things out completely on their own.
Furthermore, when it comes to math in particular--symbolic, quantitative, and visual as it is--verbal discussion only captures so much. One can converse quite intelligently about limits, for example, without actually being able to actually find a limit, or about the properties of functions without being able to construct a formal proof of any of those properties.
Too often I’ve seen tutors grossly overestimate the ability of their more verbally articulate tutees to do the actual math—until they find independent testing turning out results much lower than they expected.
To put it in terms that are only semi-mathematical, clear verbal explanations are neither a necessary nor a sufficient condition for mathematical mastery. And it’s only the latter that correlates significantly with true mathematical understanding.
In a recent Op-Ed in the New York Times, Adam Grant, a professor of management and psychology at Wharton, shares his wisdom about America’s college admissions system:
The college admissions system is broken. When students submit applications, colleges learn a great deal about their competence from grades and test scores, but remain in the dark about their creativity and character. Essays, recommendation letters and alumni interviews provide incomplete information about students’ values, social and emotional skills, and capacities for developing and discovering new ideas.
This leaves many colleges favoring achievement robots who excel at the memorization of rote knowledge, and overlooking talented C students. Those with less than perfect grades might go on to dream up blockbuster films like George Lucas and Steven Spielberg or become entrepreneurs like Steve Jobs, Barbara Corcoran and Richard Branson.Apparently robots who mindlessly memorize things do better on their SATs than humans who read passages carefully, recognize grammatical errors, and know algebra backwards and forwards. And apparently those mindless robots also get better grades than humans who produce careful, thoughtful work—or who are socially savvy enough to know how to please their teachers and motivate them to grade them generously.
Assessment centers give nontraditional students a better chance to display their strengths. For example, imagine that a college wants to focus less on book smarts and more on wisdom and practical intelligence. Rigorous studies demonstrate that we can assess wisdom by asking applicants to give advice on moral dilemmas: What would you say to a friend who is considering suicide? How should a single parent juggle family and work? The answers offer a window into how well students balance different interests and values.Sure, I’m a big fan of wisdom, but should colleges (as opposed to, say, trade schools) really be favoring practical intelligence over more abstract, theoretical capacities? As for book smarts, when it comes to the street or the lobby or the conference room or the corporate ladder, these may, indeed, be a bug rather than a feature. But isn't academia, even now, still largely about… books?
Similarly, we can identify candidates with strong interpersonal and emotional skills by watching students teach a lesson to a challenging audience — as Teach for America does when assessing applicants. And tests have already been developed to measure creativity and street smarts, which predict college grades over and above high school grades and SAT scores, while reducing differences among ethnic groups. By broadening the range of criteria, assessment centers make it possible to spot diamonds in the rough.Assessing college applicants for their teaching skills might be a good idea—but not as a tool for personality discrimination (which, have I mentioned this already?, is a bad idea—and unethical to boot). A student's teaching ability indicates, more directly than it indicates his/her interpersonal and emotional skills, how well s/he can articulate to fellow classmates his or her ideas, insights, perspectives, backgrounds, viewpoints, and values—the diversity of which are a huge part of the college experience.
Third, when students submit essays and creative portfolios in the current application system, it is impossible to know how much help they have received from parents and mentors. In an assessment center, we can verify that students are personally responsible for the work they produce.Right. But there’s a cheaper alternative that doesn’t involve Grant’s assessment centers—cheaper, because it involves processes already in place. These would be those standardized tests that Grant is so eager to jetison. They include, in particular, the SAT critical writing test. This test, via the much-loathed 5-paragraph essay, is the one source of proctored, unaided student writing that college admissions officers have at their disposal (however rarely they actually read these particular essays). When the new SAT is rolled out next year and the essay becomes optional, we will, indeed, need some sort of a replacement.