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.

The Discursive Practice of Learning Disability

One of the things I do when studying a research article is to go through the references of the article and see what catches my eye. When reading the Social Construction of Learning Disabilities, I was drawn to the work of Ray McDermott and Herve Varenne and in particular, their article Culture as Disability. What is culture as disability? The following passage provides a definition:

"The world is not a set of tasks, at least not of the type learned, or systematically not learned, at school, but made to look that way as part of political arrangements that keep people documenting each other as failures. Over the past forty years, school performance has become an exaggerated part of established political arrangements, and, by pitting all against all in a race for measurable academic achievement on arbitrary tasks, school has become a primary site for the reproduction of inequality in access to resources. The use of the term LD to describe, explain, and remediate children caught in a system of everyone having to do better than everyone else is a case in point. Even if used sensitively by people trying to do the right thing for the children apparently disabled, the term has a political life that involves millions of people operating on little information about the consequences of their work."

This, in turn, led me to the essay The Discursive Practice of Learning Disability: Implications for Instruction and Parent-School Relations by D. Kim Reid and Jan Weatherly Valle. The following excerpt describes things perfectly (my emphasis added):

"In current practice, teachers assign academic tasks deemed “grade appropriate” and hold expectations for a specified range of responses that represent mastery. The child who responds consistently outside this specified range eventually will, in most cases, be noticed as “a person with qualities to be discovered by agents of the school” (Varenne & McDermott, 1998, p. 215). The child then becomes the object of intense observation and documentation, a process reserved only for children who perform outside of the anticipated range of response but whose capacity for learning is suspected to be “normal.” In order to confirm or rule out the possibility of LD, a knowledgeable teacher makes a referral so that a psychologist (and perhaps other experts) can administer a battery of psychoeducational tests to the child to generate an individualized psychoeducational report based largely on the results. Soon, a special education committee meets to discuss the test results and to determine the child’s eligibility for individualized special education services. If the child is deemed eligible on the basis of “really being” a special education student, an Individualized Education Program (IEP) is developed. The transformation from ordinary schoolchild to disabled individual is now complete.(page 469)"

Why does it appear that I am so bitter about special education and LD?  I don't know. I think it started a dozen or so years ago. During a conference with the parents of one of my students, it was suggested by someone in the room that the parents should explore different medication options for their child. The parents never questioned it! I was dumbfounded. There is obviously something wrong with my school that would require a student to be medicated just to make it through the school day. 

It was at roughly the same time I stumbled upon John Taylor Gatto's Dumbing us down: The hidden curriculum of compulsory schooling and in it, The 7-Lesson School Teacher. This sort of started it all, and I have been trying to fight it from the inside ever since.

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)."

Different Learning Styles Bogus?

Found this on Farnam Street, a blog ran across when searching for stuff about the Prisoner's Dilemma for a Mathematical Games class I teach (sort or an applied discrete mathematics course). I am a bit wary of educational research anyway. I just feel that there are too many variables that can't be controlled. Nevertheless, I am looking forward to reading the research.

Different Learning Styles Bogus?

Beauty and Facial Symmetry

After seeing this article earlier this morning on Yahoo, I was reminded about the article Fostering Mathematical Inquiry with Explorations of Facial Symmetry written by Michael Todd Edwards, a good friend of mine at Miami University up in Oxford, Ohio. If you are an NCTM member who subscribes to Mathematics Teacher, login and check it out.

Poorly Drawn Quadrilaterals

I got this idea from HERE. The idea is this...you are presented with a quadrilateral that is drawn poorly, but which is marked correctly. Your job is to determine the best possible name for the quadrilateral (if it can be determined) or if it is a quadrilateral at all. Some examples are below...




So...what IS the best name for these quadrilaterals?

I have created a GeoGebra applet that generates up to 24 different poorly drawn quadrilaterals for anyone who is interested. You can find the applet HERE.

Semester Exams

Finished my semester exams today, all on moodle, which means my grading is done.

I have been a moodle user for the past couple of years, and have used moodle to incorporate a weekly spiral review of algebra and geometry topics (among other things I use moodle for). This year, I decided to use it for my semester exams.

Here's how it worked, along with some background.

These weekly moodle assignments are my student's only homework for me...twenty questions a week. I use the calculated question feature to create a hundred versions of the same problem, so different students will have the same questions, but in a different order, and with different numbers. My hope is that my students will talk about the problems (and they do), and not just copy answers (which they can't).

When they submit an answer, they know immediately if their answer is correct. If it is correct, they receive full credit for the problem. If it is incorrect, they receive a 20% penalty. They can attempt the assignment as many times as they want with the understanding that each attempt will mean a new version of the assignment with the same questions but with different numbers. I simply record their best score for the week.

Grading this way, I would hope that every student gets a perfect score. Even with an unlimited number of attempts each week and regular moodle moots I hold for students to stop in and ask questions, this does not happen. Nevertheless, I feel this is an effective way to get the kids to stay sharp on old content.

Over the course of the semester, I have created almost 200 questions like this, categorized by content (Triangles, Circular Areas, Trigonometry, etc). Many of these questions assume the use of GeoGebra or NspireCAS. Other questions require kids to make a choice: Should I complete this by hand or use GeoGebra?

Their exams were constructed the same way, from the same questions, with just a couple of exceptions.

(1) Every student had a different exam. I finally figured out how to randomly include any number of questions from a question category. For example, every student had three right triangle questions selected from 20 possible right triangle questions.

(2) The students could attempt the exam as many times as they wanted in the 2-hour time period. Just as on the weekly assignments, students knew immediately if their answer was correct or incorrect. However, when they attempted the exam again, their correctly answered questions were on the new attempt, along with their answers.

(3) Students could know their grade at any time. Because they could make repeated attempts on the exam, when they finished one attempt, they knew what their grade was on that attempt. This, of course, meant that no one needed to get a perfect score on the exam, because they only needed to get 90% of the questions correct.


Of course, this is not perfect. To use the calculated questions feature in moodle means I am asking a question that can be solved by applying an algebraic algorithm. I guess this could be good or bad. If I ask a question like

Quadrilateral ABCD is a square. Point A has coordinates (8,19). The diagonals intersect at point M, which has coordinates (1,-3). The vertices are labeled in a counter-clockwise direction. What is the x-coordinate of point B?

students need to know something about a square, how to construct it, that the diagonals are congruent and perpendicular, that is possesses 90 degree rotational symmetry.


I am not concerned that my students had seen the exam questions before on their weekly assignments. I am concerned more about the two hour exam period (we do this, the argument goes, to get our kids ready for college. Why, then, don't we have class just three days a week?) than if the kids have seen the questions. Besides, I'm not sure the kids even realized this until they took the exam, even though I had told them this numerous times before.

Anyway, this will work for now. My grading is done.

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

My Favorite Curriculum

Phillips Exeter Academy's mathematics curriculum. I do not know how they do it. I think if a student were to work their way through this curriculum, starting in Math1 and ending in Math4, they would be able to do some good mathematics.

I have used Math2 with my (so called) honors geometry students for three years. I can't convince anyone else in my department to go along with me.

The Social Construction of Learning Disabilites

I was reminded of the article The Social Construction of Learning Disabilites by Curt Dudley-Marling after sitting through an IEP re-evaluation meeting the other day. After all, if you have attended an IEP meeting, that is all the regular education teachers do is sit through them.

This student's recognized disability is in the area of mathematics. I call it "recognized" because I am not sure I believe there is a disability. The results of the test administered by the special educators to support this claim looked to me like the results for any one of my students. The test ranked her below average, average, or above average on a dozen or so different content strands.

Let's see...on the handful of strands that concerned mathematical operations, this student was ranked as "below average," estimated to be performing somewhere in the 7th to 9th grade level (this student is an 11th grader). Nothing out of the ordinary there. I would reckon that the majority of my students would fall in this range on these strands. Somewhere earlier in their schooling, someone decided that an appropriate modification for this student would be to allow them to use a calculator. These sections of this test were completed without a calculator. No wonder.

Let's see...on the handful of strands that involved operations with fractions and real numbers, this student scored below average, again estimated performing in the 7th to 9th grade range. Again, par for the course when it come to the majority of my students. For some of these sections, the student was permitted to use a calculator. I guess who ever decided on the calculator modification forgot to teach how to use a calculator. I wonder what calculator they used? Did they use their TI-84? Did they use one of my NspireCAS handhelds? Was it one of the Ohio Graduation Test calculators?

Let's see...on the handful of strands that are actually addressed in my classes - Algebra, Measurement, Geometry, Probability, Shapes - this student was ranked in the average to above average range on all, performing in the 10th grade to 11th grade range. I would imagine and expect that the majority of my students would be here.

I guess I do not understand how this disability works, or how it is constructed.