Saturday sharing: scary research findings

[Maria Droujkova]

This week's topic is math anxiety. Reading the responses to the Supreme Ruler task, especially the scary Evil Overlord lists, made me think that certain bravery is required to teach mathematics. We need to be brave ourselves to model bravery for our anxious students. While we don't feed poor students to grizzly bears (as Laura suggested for maximizing anxiety), they may feel like they are facing a hungry grizzly, in some learning situations.

http://p2pu.org/en/groups/ed218-developing-mathematics-the-early-years/content/week-11-math-anxiety-the-supreme-ruler-march-26-april-1/ 

Toward this goal, I want to invite you to share a piece of research that you find scary.

It can be from the area of mathematics education - there are a lot of scary studies there! Or it can be a research finding from another social science, or any area that relates to learning mathematics - and everything does in some way, because mathematics is everywhere. If you have ways of coping with that fear, share those. If not, maybe another course participant will reply to your email with coping strategies.

~*~*~*~*~*

I will go first. One of the scariest findings for me is the series of Milgram experiments. Essentially, the study concludes that as a teacher, I am likely to do things that belong in an Evil Overlord list - if ordered to do so by an authority. Here is the description of one of the experiments from Wikipedia: http://en.wikipedia.org/wiki/Milgram_experiment

"The volunteer subject was given the role of teacher, and the confederate, the role of learner. The participants drew slips of paper to determine their roles, but unknown to the subject, both slips said "teacher", and the actor claimed to have the slip that read "learner", thus guaranteeing that the participant would always be the "teacher". At this point, the "teacher" and "learner" were separated into different rooms where they could communicate but not see each other. In one version of the experiment, the confederate was sure to mention to the participant that he had a heart condition.

The "teacher" was given an electric shock from the electro-shock generator as a sample of the shock that the "learner" would supposedly receive during the experiment. The "teacher" was then given a list of word pairs which he was to teach the learner. The teacher began by reading the list of word pairs to the learner. The teacher would then read the first word of each pair and read four possible answers. The learner would press a button to indicate his response. If the answer was incorrect, the teacher would administer a shock to the learner, with the voltage increasing in 15-volt increments for each wrong answer. If correct, the teacher would read the next word pair.

Before conducting the experiment, Milgram polled fourteen Yale University senior-year psychology majors to predict the behavior of 100 hypothetical teachers. All of the poll respondents believed that only a very small fraction of teachers (the range was from zero to 3 out of 100, with an average of 1.2) would be prepared to inflict the maximum voltage. Milgram also informally polled his colleagues and found that they, too, believed very few subjects would progress beyond a very strong shock."

Psychology majors and professors were wrong. 

I don't have 100% reliable ways of coping with this fear - it still bothers me a lot. My best hope, I think, is the clarity about what is and isn't good for students - like lists you made this week. This clarity, together with the knowledge that the pressure from an authority can override moral reasoning, does help. More ideas would be appreciated.

Cheers,
Maria Droujkova
919-388-1721

Make math your own, to make your own math

 

[Garrett, Sandra]

I found an intersting piece about math called A Mathematician's Lament and you can read the whole thing here-   http://www.maa.org/devlin/LockhartsLament.pdf  It's long, but extremely interesting!  The peice ends with a description of math curriculum.  What is scary to me is to realize all of these years later that maybe math isn't a foreign language to me.  Maybe simply hearing the word "math" wouldn't bring about instant cringing in me if things had been done differently. When I logged int his morning and read this assignment out loud, my husband who was nearby replied that I would obviously write that anything about math is scary to me... and he wasn't joking! Scary is now knowing things can be different for my kids, and knowing they may not be.

 

The Standard School Mathematics Curriculum

LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not

something you do, but something that is done to you. Emphasis is placed on sitting still, filling

out worksheets, and following directions. Children are expected to master a complex set of

algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part,

and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables

are stressed, as are parents, teachers, and the kids themselves.

MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures,

akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are

handed out, and the students learn to address the church elders as “they” (as in “What do they

want here? Do they want me to divide?”) Contrived and artificial “word problems” will be

introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.

Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’

and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent

preparation for Algebra I.

ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this

course instead focuses on symbols and rules for their manipulation. The smooth narrative thread

that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance

algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no

characters, plot, or theme. The insistence that all numbers and expressions be put into various

standard forms will provide additional confusion as to the meaning of identity and equality.

Students must also memorize the quadratic formula for some reason.

GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of

students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy

and distracting notation will be introduced, and no pains will be spared to make the simple seem

complicated. This goal of this course is to eradicate any last remaining vestiges of natural

mathematical intuition, in preparation for Algebra II.

ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate

geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic

simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety

of standard formats for no reason whatsoever. Exponential and logarithmic functions are also

introduced in Algebra II, despite not being algebraic objects, simply because they have to be

stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder

mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.

TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory

definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of

a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and

obsolete notational conventions, in order to prevent students from forming any clear idea as to

what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All

Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and

symmetry. The measurement of triangles will be discussed without mention of the

transcendental nature of the trigonometric functions, or the consequent linguistic and

philosophical problems inherent in making such measurements. Calculator required, so as to

further blur these issues.

PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked

attempt to introduce late nineteenth-century analytic methods into settings where they are neither

necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to

obscure the intuitively clear notion of smooth change. As the name suggests, this course

prepares the student for Calculus, where the final phase in the systematic obfuscation of any

natural ideas related to shape and motion will be completed.

CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it

under a mountain of unnecessary formalism. Despite being an introduction to both the

differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be

discarded in favor of the more sophisticated function-based approach developed as a response to

various analytic crises which do not really apply in this setting, and which will of course not be

mentioned. To be taken again in college, verbatim.

***

And there you have it. A complete prescription for permanently disabling young minds— a

proven cure for curiosity. What have they done to mathematics!

There is such breathtaking depth and heartbreaking beauty in this ancient art form. How

ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an

art form older than any book, more profound than any poem, and more abstract than any abstract.

And it is school that has done this! What a sad endless cycle of innocent teachers inflicting

damage upon innocent students. We could all be having so much more fun.

 

 

 

 

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[Denise G]

I think this is the scariest education research I've read recently: Benny's Conception of Rules and Answers in IPI Mathematics

"The IPI program has been one of the most comprehensive attempts at developing an individualized instructional technology. As such it has been a valuable and promising experiment in education. However, Benny’s case appears to indicate that there are inherent weaknesses in the IPI mathematics program.

"Benny is a 12 year old sixth grade pupil with an IQ of 110-l15. He has been using IPI mathematics since second grade. He appeared to his teachers to be making good progress in mathematics, but it was discovered later that he understood incorrectly some aspects of his work. He had also developed learning habits and views about mathematics that would impede his progress in the future. Although there are probably many factors that contribute to his difficulties in mathematics, his case suggests that the effect of IPI mathematics on the understanding and perception of the subject by pupils of other backgrounds and abilities should be investigated."

Benny seemed to be one of the better students, cruising through the math program and passing tests with relative ease, yet when the researcher talked to him about what the math meant, it became clear that he didn't understand hardly anything. He confidently asserted "rules" that made no sense at all, yet seemed unconcerned about the contradictory results (for example, that 4/11 and 11/4 were equal to the same number, namely 1.5).

How many Bennys are there in our schools? And how can we identify them and help them?

_______________________

Here is some other research I find disturbing: Preservice Teachers’ Understandings of Relational and Instrumental Understanding

"Although the preservice teachers’ course had included teaching about understanding a number of misconceptions about the meanings of relational and instrumental understanding were evident in the responses of a sizeable minority, along with evidence that many held beliefs that were likely to result in them teaching instrumentally."

(In order to understand the article, one first must read Relational and Instrumental Understanding by Richard Skemp.)

Among the things the preservice teachers believed:

• It's unrealistic to expect relational understanding.
• Relational understanding was a topic you could maybe get to, if you organized your class time well.
• Relational understanding meant seeing how math related to other things.

In all, nearly a third of the students misunderstood relational understanding. Now, I wouldn't find that scary if it was just them misunderstanding the words "relational" and "instrumental", because those are rather weird terms anyway. But what I'm afraid of is, even when they were in a class about learning to teach math, with professors who made a point of teaching about the difference between a conceptual, interrelated, deep understanding of math (relational) and a follow-the-recipes understanding (instrumental), the students just didn't get it.

I try to help homeschoolers learn to see math as a rich web of relationships and ideas -- a sort of mental adventure. Most of them grew up with an instrumental understanding, and that's the only vision of math they've ever heard of. If it's THIS hard to change preservice teacher's understanding of math, what hope do I have that a few small blog posts and forum comments (or even my short little books, whenever I manage to finish them) can change anyone's view?