…And stop belaboring the concept of the equals sign

My long-time math crony Barry Garelick has recently alerted me to claims about how students don’t understand the “equals” sign and therefore require lessons on the underlying concept of mathematical equality. 

(These claims are based on studies that potentially confuse conceptual understanding of the equals sign with the ability to do basic arithmetic or involve additional confounds like the effects of tutoring.)

The notion that teachers should be devoting class time to conceptual understanding of the equals sign reminds me of this old post from Out in Left Field.

Stop belaboring the concepts: the limits of “conceptual understanding”

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The Question of Authorship in Autism

The latest news about colleges and universities falling for facilitated communication reminds me of this old Out in Left Field post–from 2015. I had forgot about how long some of these issues have been on my mind.

Autism diaries: the question of authorship

“I just assumed you were coming back to the kitchen,” says J. “In fact, I did not even think of trying to leave the fan on all night.”

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Yes, but is it teachable?

A recent study (still just a pre-print) cited by supporters of “social-emotional learning” (SEL) reminds me of all the fanfare about “grit” (remember grit?) and of an old post of mine from Out In Left Field.

But first, a bit about this latest study, entitled “The Roles of Social-Emotional Skills in Students’ Academic and Life Success: A Multi-Informant, Multi-Cohort Perspective.” Its methodology appears rigorous: 

a multi-informant (self, teacher, and parent) and multi-cohort (ages 10 and 15 from Finland, N = 5,533) perspective to study the association between 15 social-emotional skills and 20 educational (e.g., school grades), social (e.g., relationships with teachers) psychological health (e.g., life satisfaction), and physical health outcomes (e.g., sleep trouble).

But what does it conclude?

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Reasonable accommodations for moderate autism?

As I noted in a recent comment, the most helpful accommodations J. got in college–and perhaps the most helpful accommodations he’s ever had–were:

1. A written transcript of the entire class.

2. The option to take certain classes online.

These two accommodations, increasing the quantity of text-based information and text-based exchanges, made it easier for him to review things would otherwise go in one ear and disappear out the other. 

And the second accommodation eliminated the greatest challenges of group work: those spontaneous, in-person exchanges involving fast moving conversations and non-verbal communication.

But what about what is often the greatest challenge of all in moderate autism: comprehension, especially reading comprehension. Here, from Out in Left Field, is a follow-up post to the last one.

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Facilitating people by “providing access”

The more I think about the kinds of notions I discussed in this old post from Out in Left Field, the more I think about how all this is part of a broader phenomenon that includes facilitated communication.

The broad assumption is that children have already picked up various academic skills on their own and that all the adults need to do is to “provide access.”  “Provide access”, in turn, seems to mean whatever support is necessary for students to “demonstrate understanding.”

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Isn’t it pretty to think so?

The increasing demands for the full inclusion in regular education classes of profoundly autistic students who are purportedly unlocked by facilitated communication, along with Greg Ashman’s recent post on full inclusion, have reminded me of this old Out in Left Field post of mine:

The Full Inclusion illusion

Even as J winds down his final year in the Philadelphia public school system, I’m still learning new things about how the district operates. For example, I recently learned that the one-size-fits-all approach to high school English predates the Common Core. For at least the last dozen years, all Philadelphia high school English teachers have had to choose from a specific list of literary works/authors, which include Steinbeck, Edgar Allen Poe, the Canterbury Tales, and Shakespeare plays. It doesn’t matter that, according to a 2013 report, only 53.4% of the school district’s 11th graders scored proficient or advanced on the Literature section of the state’s new Keystone test, and that only 10.1% and 10.6%, respectively, of English Language Learners and special ed students with IEPs scored proficient or advanced in Literature.

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Is diminished Joint Attention really not a problem for word learning in autism?

(Cross-posted at FacilitatedCommunicaton.org.)

The standard view of autism as a socio-cognitive disorder accounts for the language delays/limitations in autism by:

  1. Observing that Joint Attention (JA) is diminished in autism and
  2. Pointing to causal connections between JA and word learning.

In my last post on the autism-related research of Morton Gernbacher, I critiqued Gernsbacher’s claim that JA is impaired in autism and concluded by saying that:

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The tutoring fallacy–Facilitated Learning?

I’ve lately been thinking about how some of what goes on in Facilitated Communication relates to some of the ways in which tutoring can go awry. Tutors have a natural urge for their tutees to succeed. They naturally tutor by asking questions, some of which help lead towards the answers. Unless they closely monitor themselves closely, they may end up failing to bring their tutee to the level of mastery they think they have. 

Here’s an old Out In Left Field post on this phenomenon–and a few more.

The Tutoring Fallacy: where clear explanations fall short

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.