Knights and Knaves

A big push at my school this year is to get kids to write and to present. You know... those "21st century skills." This has never been a problem for because I am asking my kids to write explanations and present proofs and such all the time (which is probably a reason no one else wants to teach geometry). One of the ways I get my kids to write and present, while at the same time applying some if/then thinking, is through the use of Knights and Knaves logic problems. An example is as follows:

A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie.You meet two inhabitants: Zoey and Mel. Zoey tells you that Mel is a knave. Mel says, `Neither Zoey nor I are knaves.'

So who is a knight and who is a knave?

When discussing a problem like this with my students - two inhabitants, each could be a Knight or a Knave - I like my students to examine the four possible cases. By ruling out those cases that lead to contradicting statements between the inhabitants, what is left must be true (an informal introduction to proof by contradiction?).

There does seem to be somewhat of a history to these types of puzzles. These puzzles, of course, lead some of my students to The Hardest Logic Puzzle Ever, as well as to an interesting variation on the theme which introduces a "Spy" that can tell the truth or tell a lie.

Different Learning Styles Bogus? What Did The Research Say?

This is a follow-up to my earlier post on this topic. I read the research article published a year ago (a YEAR ago? What does THAT tell you?) Learning Styles: Concepts and Evidence by Harold Pashler, Mark McDaniel, Doug Rohrer, and Robert Bjork in the journal Psychological Science in the Public Interest(PSPI), Volume 9, Number 3. You can read the article for yourself in a free preview.

As I stated in the aforementioned post, I do not care for the "data" that every administrator is ga-ga about, nor do I care much for educational research that treats students as widget coming off an assembly line. I do my best to find what works for each kid, and if educational research can inform what I do, great. But I feel that data is suspect...just too many variables.

In a nutshell, this was a literature review of studies concerning learning styles. It appears to be in response to the learning-styles industry that produces books, workshops, and learning-styles assessments. The rationale behind this industry is that to teach a student most effectively, you must first be able to accurately diagnose their learning style (hence, the need for the assessments). Once you have accurately diagnosed the learning styles of your students, the teacher can begin to implement a learning-styles approach (hence, the need for books and workshops).

If this is the case, what research is out there that demonstrates the effectiveness of a learning-styles approach? The authors were asked to find if learning-style practices were supported by scientific research. They looked for evidence that supported what they referred to as the meshing hypothesis; an instructional method shown to be most effective for students with a particular learning style is not the most effective instructional method for students with a different learning style. If we find evidence of an instructional method like this, then we have found evidence in support of the meshing hypothesis, and we would conclude that there is something to this learning-styles thing.

The authors found little evidence to support learning-style instructional methods. Some of the studies they reviewed did not use appropriate research methods to test the effectiveness of learning style instructional methods. The authors found that those studies that did use appropriate research methods at times contradicted the meshing hypothesis.

It should be noted that what the authors suggest a valid learning-styles study should entail would be along the lines of a students-as-widgets study. At least, that is my take on it.

The authors did acknowledge the anecdotal things that we all see every day in our classrooms. They did not argue the fact the people prefer to have information presented to them in a certain manner (visually, for example), nor did they argue that different people process information differently. They acknowledged that different disciplines call for different instructional methods (for example, content in Language Arts is presented differently than content in Mathematics). They also acknowledged that students may benefit from having content presented in a particular manner, and the teacher, noticing these patterns in their students, may plan for these student differences. The authors simply dispute that the current models of learning-styles assessments do not offer an instructional prescription for each student (which is what the learning-styles industry would want you to believe).

There were some interesting points made by the authors. One being how the popularity of the notion of learning-styles could be attributed to how it provides a means of sorting people into distinct groups. Schools LOVE this stuff. However, learning-styles does not have the negative connotation of tracking because is it a good thing to find out how I learn best, and that how I learn is different from how someone else learns. I am special, I am unique, and learning-styles tells me that (sarcasm).

Another point made reminds me of the point made in the article referred to in my first post. If a child is not succeeding in school, it must be because the child's learning styles are not being addressed.

This runs counter to the argument made in the Social Construction of Learning Disabilities, in that if a child is not succeeding in school, it must be because the child is imperfect in some way. Schools are perfect institutions, and there could never be something wrong with an institution created by us, for us. It had got to be the kid (more sarcasm). Nevertheless, a point made in this article seems to hold for both Learning Disabilities and Learning Styles. This is probably my favorite paragraph in the whole article. Wherever there is LD, you could replace it with LS (learning style) without changing much of the meaning:

"What is important to understand is that LD is intelligible only in the context of schooling. It is in schools, where children are routinely sorted and evaluated in terms of certain learning behaviors, that LD comes to life. Individual students cannot have LD on their own (McDermott & Varenne, 1999). The performance of LD requires an institutional framework that assigns particular meanings to students’ behaviors that, in other cultural contexts, do not carry the same significance. Learning disabilities defined in terms of a delay in acquiring specific (pre)reading skills, for example, make sense only in a context where these skills are salient. Similarly, a relationship between LD and attention demands a context that requires students to deploy their attention in particular ways. In short, the expression of LD requires the institution we know as school, although it is at least possible to imagine schools where students are not sorted by ability, where learning and not the rate of learning is most important, and where the absence of learning is not such a significant presence (pg 484)."

Changing the Culture

In an attempt to change the culture of my school, someone decided it would be good to spend money on large signs with motivational quotations from significant figures in our content areas. The motivational quotations are to go along with our school themes that were created during the summer. You know...things like "respect" or "cooperation." I am not sure what the seven themes are. I looked on our school webpage, but they are not listed any place that I can find them.


It is now the math department's turn to suggest some quotes to be placed on signs that will hang outside out room. Here are my tongue-in-cheek suggestions:


"I don't quite hear what you say, but I beg to differ entirely with you. " (De Morgan)


"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions." (Klein)


"Mathematicians are born, not made." (Poincare)


"To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls. " (Chebyshev)


"It is rare to find learned men who are clean, do not stink and have a sense of humour." (said about Leibniz)


http://www-history.mcs.st-and.ac.uk/Quotations/index.html