Re: Meaningless problems in algebra texts

[John Clement <clement@HAL-PC.ORG>]

> -----Original Message-----
> From: Forum for Physics Educators [mailto:PHYS-L@list1.ucc.nau.edu] On
> Behalf Of John Denker
> Sent: Monday, November 08, 2004 11:05 AM
> To: PHYS-L@LISTS.NAU.EDU
> Subject: Re: Meaningless problems in algebra texts
>
>  > John C wrote in part:
>  > |
>  > | Consider that less than 10% of HS graduates can apply formal
>  > | logic according to a paper by Lawson et al.  This means that
>  > | students will have great difficulty really understanding
>  > | proof.  Now it is possible to get them to do proof, but it is
>  > | not possible to do it by most of the conventional methods.
>  > | Consider that the majority of middle school children are
>  > | concrete operational so that the conventional methods of
>  > | teaching algebra are doomed to failure.  This means they are
>  > | incapable of hypothetico-deductive logic, but they can use
>  > | empirical inductive logic.  Actually only about 20% of HS
>  > | graduates are fully able to use hypothetico-deductive
>  > | deductive logic and some are able to use it part of the time.
>
> That's pretty much meaningless.
>   -- First, it switches back and forth between high-school and
>    middle-school.  Alas what's true for one is not true for the other.
>   -- Also, it switches back and forth between Piagetian theory (which
>    speaks about what _can_ be learned) and numerically-precise yet
>    vaguely-defined assertions about _has_ been learned.

Ok, maybe I should put some of what I said in perspective.  There is a large
body of research published in JRST and elsewhere which bears on this
subject.  Essentially the ideas of Piaget have been used by a number of
researchers who I would label either Piagetians or neo-Piagetians.  Shayer
and Adey fall into the former camp, while Lawson is in the latter.  Lawson
also worked with Karplus, Renner and others.

Piagetian theory actually does not say what can be learned, but the stage
theory says that individuals must go through various stages in cognitive
development.  The age at which a particular stage has been reached varies a
lot, and this was recognized by Piaget.  Lawson et al has carried these
ideas on and has deomonstrated that attainment of various stages develops
gradually and not all at once.  In other words they are developmental
meaning that they develop and can not just be taught in a conventional
course.  That does not mean, however, that it is not possible to promote
development.  At various stages there are limitations on what can be
understood, but teaching that promotes passage to the next stage will also
improve the learning of concepts that are difficult at the current stage.
Meanwhile the original clinical Piagetian tests have been turned into more
easily scorable paper and pencil tasks that have remarkable reliability.
This was originally done by Shayer, with an even easier version by Lawson.

As to middle school vs high school, there is apparently a dividing line
about age 10+ at which it is possible to promote development of "formal
operational" reasoning.  This is actually misnamed because it does not mean
that students can use formal logic, but rather that they can reliably use
proportional, conservation, statistical, correlational, sequencing ...
reasoning.  As a result middle school to college form a continuum in which
students are capable of developing "formal operational" reasoning, so these
educational levels should be discussed together.  Lawson showed that there
is pre-frontal lobe development which is the reason for the 10+ dividing
line.

Further evidence comes from the work of Shayer and Adey, and Lawson.
Karplus is credited with inventing "The Learning Cycle", which Lawson and
others have shown promotes transition to formal operational thinking.
Shayer and Adey have used a variant of the learning cycle called Thinking
Science which in 2 years during middle school is very successful in pushing
up students.  However tests of Thinking Science before age 10 have shown no
ability to push up students to formal operational thinking.  It does however
push students up to the higher levels of concrete operational.

I have used the Lawson Piagetian test which uses a scale of 0-12.  I have
found that students below 10 can not get 100% Hake gain on the FMCE.  Indeed
the maximum gain seems to be proportional to the Lawson test score below 10
and would seem to be zero when the Lawson test is zero.

Shayer and Adey who have published a number of papers and books on the
subject claim that many aspects of algebra are impossible to understand
below the formal operational level.  From my own personal observations, this
is probably correct.  As a result the primary task needs to be using
appropriate tasks and methods to push the students up in their thinking.  At
present there are two very good methods.  The first is Thinking Science
which has been designed solely for this purpose and has 30 70 min lessons
which must be spaced over about 2 years, preferable in middle school.
Another method which has been used with low performers is Feuerstein's
Instrumental Enrichment, but also is helpful with average students.  The
original large scale test of IE pushed students in a residential school up
to average IQ.  Meanwhile their cohorts (other students in the school) who
received an equivalent time in teacher designed enrichment were about 30%
lower.  All students were about 2 years behind in schooling when they
started the program.  This test was in Israel, so when the students were all
drafted it was easy to get IQ scores on all of them.

Now as to the comments about empirical-inductive vs hypothetico-deductive.
That is straight from Lawson.  He has taken the original Piagetian labels
and essentially renamed the original two higher levels.  He also has
convincing evidence that the basis for formal operational thinking is
hypothetico-deductive or the ability to reliably ask what if questions.

How does this relate to proof?  Well, if students can not understand formal
logic, then formal proof becomes almost impossible.  As Lawson has shown
only about 10% of HS graduates can understand and apply formal logic.  In
addition this percentage is much lower in 9th and 10th grade where proof is
someimes introduced.  However, Lawson has also argued with some good
evidence that most logic is not based on formal reasoning.  Essentially
students need to use reasoning, but not necessarily formal proof.

Incidentally there have been studies which showed that courses in which
formal logic is studied, do not actually improve student ability to use such
logic.  They merely learn about it.  They memorize the various categories of
logic and can talk about it, but when given a task which requires logic,
they fail to use it.  BTW conventional computer science courses also fail to
develop this ability.  There is actually a test of this called the PLT,
Propositional Logic Test.

Getting back to Lawson, he has hypothesized that the reason why formal
logical ability is much slower to develop is that it is simply not needed in
most tasks confronting students.  So if you wish to push the ability to do
proof, you need curriculum which confronts students with tasks that require
formal logic to be solved.

As a result of this evidence I am very skeptical of attempts to push formal
proof in secondary school.  It is always done in such a manner that it does
not promote logical thinking.  However, it should be possible to do it in a
properly constructed setting in which student perceive the necessity to
prove things.  Proof should flow naturally out of an attempt to make sense
of the material and not just be inserted as a formally taught subject.  At
first it could be just getting students to do compelte explataions and by
asking the question "How do you know?".

Incidentally Lawson has also shown that there is a higher level of thinking
which he dubs Theoretical.  This is necessary for easy acquisition of
concepts where there are unobservable things.  This includes evolution, and
virtually all physics concepts.  I would recommend that anyone who wishes to
pursue some of these ideas pick up the books by Shayer and Adey, and also
read Lawson's "Science Teaching and the Development of Thinking".  There are
also any number of articles in JRST (Jour of Reseach in Sci Teaching)

As educators our primary task should be to push up student thinking, and
then secondarily to promote curricula.

John M. Clement
Houston, TX