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Marsha Ratzel on Teaching Math

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The Great Fire Wall of China

As my regular readers know, I am writing from China these days, and have been doing so four years so far. Sometimes the blog becomes inaccessible to me, making it impossible to post regularly. In fact, starting in late September 2014, China began interfering with many Google-owned entities of which Blogspot is one. If the blog seems to go dark for a while, please know I will be back as soon as I can get in again. I am sometimes blocked for many weeks at a time. I hope to have a new post up soon if I can gain access. Thank you for your understanding and loyalty.

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Tuesday, April 24, 2012

Parents In Contempt

As long as schools hold the public (who pays their bills, by the way) and parents in utter contempt, I seriously doubt that they will be able to build the relational trust necessary to academic achievement. Even worse is the contempt demonstrated toward those parents and members of the public who are education colleagues who might actually have a message worth listening to. I am just going to tell the story. A friend's child brought home a math assignment on perimeter and area. One of the problems was unsolvable as presented. It looked something like this (I do not have the actual diagram since child already turned her homework in):

The problem is unsolvable because there is not enough information. There is no way to know whether the angles that “look” 90 degrees are in fact 90 degrees. There is also no way to know whether the vertical segment that “looks” like it bisects the base in fact does so. The child made these points in class, but the teacher shot her down. So thinking I was being helpful to a young teacher, I sent her an email. After some brief introductory remarks, I wrote,
... the last problem of a recent homework assignment on perimeter and area had insufficient data to solve the problem.  One of the principles of geometric diagrams is that we never go by appearance.  We solve using givens and proven facts.  Since that particular diagram gave no indicators of equivalent length, bisection, 90-degree angles, etc, no conclusions could be drawn regarding the length of the side opposite the one measuring 15 units, or any other non-given length, without making unsubstantiated assumptions.  We are miseducating children if we teach them, even indirectly, bad thinking habits.  One of the purposes of math instruction is the logical and critical thinking skills it cultivates. And if we are as worried about high-stakes tests as we say we are, we will teach students to think properly.  A favorite trick of bubble tests is to lead the students down a primrose path of faulty assumptions. If you would like discuss ways to help students learn to think, please let me know.  I spent a lifetime teaching math, first to junior high and high school, and later, college students.
The teacher sent a reply that seemed quite understanding, but showed that she somehow missed the point. She replied,
I completely understand what you are saying in the email. I also concur with what you said, however, I did verbally tell the students that angles that appear to be right are. ..However, if the student drew it out on the grid paper, then that student could find an area for the figure they drew. We are at an entry level with these problems, and so I looked at how the student drew the problem out, and determined if their area and perimeter matched that drawing.
Am I the only one who found the reply disturbing? So I wrote back,
Thank you for your reply. "I did verbally tell the students that angles that appear to be right are."  Angles that appear to be right are most decidedly not right just because of appearance.  I really think that if the student's math level is lower, then it is all the more critical to precisely teach thinking skills.  Drawing on grid paper does not really help, because it only pushes students harder to make unwarranted assumptions.
I closed by repeating my offer to help. Then I got this curt dismissal:
Hi again, Thank you for your response, it seems that on this particular problem I did not satisfy your criterion and I am sorry to have let you down. Have a nice day.
Okay, so she really does not want any help, I guess. And apparently she is happy to miseducate kids. I figured I would just move on, until....the school's guidance counselor contacted me, and let me know that she considers my communication with the teacher inappropriate and unprofessional. Seriously? She even challenged my right to have any conversation with the teacher by pretending that I was discussing a student, not a math problem. Now when I was teaching, the only time I passed communications on to administration was if it contained a personal insult of some kind, of course, a very rare occasion. Maybe it is unfair to extrapolate from one experience, but I assure you, schools routinely show contempt for parents, the public, and even the education-savvy members of the public. We must be careful that we as teachers refrain from thinking we get to define the terms of parental involvement. I know plenty of teachers who actually resent fully involved parents if those parents dare to challenge the teacher or the school. Then teachers vilify the parent as a "helicopter parent" in an ad hominem attempt to dismiss with contempt the parent's concerns. A lot of schools want to limit parent involvement to conferences and making cupcakes.

Wednesday, April 18, 2012

Unresolvable “Science” Debate?

Will the controversy between evolution and creationism ever end? Is it destined to swing forever on the pendulum of public opinion? The entire controversy is sustained on both sides by too much emotional investment in unexamined assumptions. The latest pretext for acrimony is a Tennessee bill intended to permit teachers “to help students understand, analyze, critique, and review in an objective manner the scientific strengths and scientific weaknesses of existing scientific theories covered in the course being taught.” In case that wasn't clear enough, the bill repeats its intention from the other way round. No teacher shall be prohibited from “helping students understand, analyze, critique, and review in an objective manner the scientific strengths and scientific weaknesses of existing scientific theories covered in the course being taught.”

I read the bill. It is only two pages of plainly-worded text. It mentions creationism not at all. It does refer to scientific theories, of which there are many, each with its own strengths and weaknesses. However, creationism is not one of those theories because it is not science.

You see, science is all about collecting only evidence that can be perceived with just the five senses. The sense may be amplified as when we use a telescope or other instrument. Science is concerned with explaining data collected only with the five senses. Other data is not considered.

An explanation that tries to account for extra-sensual data is, by definition, not a scientific theory. Nevertheless, due to public confusion and the desire of some that creationism be recognized as a scientific theory, it will be in science class that students ask their questions. Teachers need to be prepared to answer them while respecting deeply held religious beliefs.

The bitter acrimony is really unnecessary. It is easy and reasonable for students to accept that science attempts to explain only sense-based data. Most of the problem stems from a widespread misunderstanding of what science is.

As far as evolution goes, it suffers from historical bar-lowering, as it has weaknesses that do not adequately account for the scientific facts. Even within my lifetime, scientists have weakened the definition so much as to create a near tautology: evolution is change over time. Many science texts state it just like that. Others pretty it up a little, “evolution can be precisely defined as any change in the frequency of alleles within a gene pool from one generation to the next.” However, such a definition in non-controversial. Organisms do change over time. The biblical Jacob realized it thousands of years ago when he made a deal to work for Laban, receiving only the spotted sheep as his wage. Laban promptly removed all the spotted sheep from the herd. Nevertheless, by careful breeding, Jacob was able to create a herd of mostly spotted sheep from a herd of un-spotted sheep.

Years ago the National Association of Biology Teachers (NABT) used a fairly stringent definition: "The diversity of life on earth is the outcome of evolution: an unsupervised, impersonal, unpredictable and natural process of temporal descent with genetic modification that is affected by natural selection, chance, historical contingencies and changing environments." Unwritten, but understood and unquestioned, was the additional idea that mechanisms of descent were robust enough to account for the change from say, sponge to zebra.

In fact, this unstated implication is the root of the controversy. Pro-evolutionists (as distinguished from scientists) believe the implication; Anti-evolutionists (again, as distinguished from scientists) do not. For many, the implication goes directly to deeply-held belief systems. Later, the NABT deleted the words “unsupervised” and “impersonal.” Today, there is no definition on the website at all. One of the reasons that the definition of evolution has gotten weaker and weaker is that the data, especially as regards speciation, is inconclusive, and fails to support the more robust definition. There are lots of instances where it is not at all clear whether two organisms are members of different species. A high-quality university level biology book addresses the speciation continuum and other issues, but it can be a tough read.

As inconceivable as it may be to some, it is possible to discuss the weaknesses of evolutionary theory without smuggling in creationism. Only ideologues would consider the mere mention of evolution's weaknesses as an attack upon evolution. For critical thinkers, it is the grist of intelligence-making. As F. Scott Fitzgerald said, “The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still retain the ability to function.”