Please excuse this LONG post. TO AVOID CROWDING HARD-DRIVES PLEASE
DON'T HIT THE REPLY BUTTON! Because Benezet's work should be of
interest to mathematicians, physicists, and educators generally, I
hope subscribers will forgive my cross-posting to discussion lists
with archives at:
I have set off most quotes by bracketing them with the authors NAME, as"
NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME
Name wrote "Blah, blah, blah."
NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME-NAME
In the recent Math-Learn thread "Arithmetic and Algebra," initiated
by Michael Paul Goldenberg's 29 Jan 2002 00:10:11-0500 post regarding
Andre Dehaene's (1997) conclusion that algebraic calculation and
arithmetic are processed in different brain regions, there have
recently been several interesting comments regarding the ground
breaking work of Louis Paul Benezet (1935/36). [Some of the relevant
posts may be accessed by typing "Benezet" in the "Search Archive"
slot at the primitive Math-Learn archives
<http://groups.yahoo.com/group/math-learn/>, for 10 hits as of 5
February 2002 8:00:00-0800.] The following are my responses (R) to
four Math-Learn comments(C) on Benezet:
111111111111111111111111111111111111111111111111111111111111
C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1-C1
REX BOOGS on 31 Jan 2002 18:15:34+1000, wrote:
BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS
"Another study, conducted over 70 years ago, came to a similar conclusion."
Here "conclusion" refers to one given by Victor Steinbok in his post
of 29 Jan 2002 23:28:39-0500 regarding a Russian topologist who
"assembled a small group of four- and five-year olds to whom he
started 'lecturing' in higher math concepts. . . (and thereby). . .
realized that the kids actually not only could understand the
concepts but were demonstrating insights far superior to some of the
insights we've had from some of the math PhDs on this list . . . he
left the standard mathematical sequence for the school to take care
of. . ."
"Over 70 years ago in Manchester, New Hampshire, children learnt no
formal arithmetic until grade 6 (about age 11). The program's
creator, Superintendent Louis Benezet, describes it like this:
'In the fall of 1929 I made up my mind to try the experiment of abandoning
all formal instruction in arithmetic below the seventh grade and
concentrating on teaching the children to read, to reason, and to recite -
my new Three R's. And by reciting I did not mean giving back, verbatim, the
words of the teacher or of the textbook. I meant speaking the English
language. I picked out five rooms - three third grades, one combining the
third and fourth grades, and one fifth grade.'
And later...
'The distance from Boston to Portland by water is 120 miles. Three steamers
leave Boston, simultaneously, for Portland. One makes the trip in 10 hours,
one in 12, and one in 15. How long will it be before all 3 reach Portland?
In the ninth-grade students in Manchester, traditionally taught, 6 out of 29
gave the right answer; the experimental second grade 'had an almost perfect
score.' "
According to a later post by Rex - see C2 - Rex is recommending ONLY
the Benezet (1935/36) articles.
BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS
R1-R1-R1-R1-R1-R1-R1-R1-R1-R1-R1-R1-R1-R1-R1-R1-R1.
I agree completely with the perceptive Rex Boogs, except that, IMHO,
many subscribers might also be interested in the eight fascinating
"Related Articles" at the Benezet Centre
<http://www.inference.phy.cam.ac.uk/sanjoy/benezet/>, regarding work
related to Benezet by:
1. Alan Schoenfeld (University of California - Berkeley) and Kurt
Reusser & Rita Stebler (University of Bern) on the baleful effects of
rote learning,
2. The late Hassler Whitney, formerly of the Princeton Institute of
Advanced Study,
3. Andrew Gleason of Harvard,
4. David Hammer of the University of Maryland,
5. John Clement & Jack Lochhead (Univ. of Massachusetts), & George S.
Monk (University of Washington),
6. Sanjoy Mahajan (University of Cambridge) & Richard Hake
(University of Indiana),
7. Jack Lochhead (DeLiberate Thinking) & Arthur Whimbey (Whimbey.com), and
8. National Research Council - Mathematical Sciences Education Board.
2222222222222222222222222222222222222222222222222222222222222222
C2-C2-C2-C2- C2-C2-C2-C2-C2-C2-C2-C2-C2-C2-C2-C2-C2-C2-C2
REX BOGGS wrote on 1 Feb 2002 17:02:43+1000:
BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS
"The articles that I found fascinating . . . (at the Benezet Centre
<http://www.inference.phy.cam.ac.uk/sanjoy/benezet/>. . . . and I
suggested folks read if they were still following this thread, were
the three written by Benezet (1935/36) himself. Number 7 has NOTHING
to do with Benezet's articles. It . . . ("Related Article #7") . . .
relates to books by someone called Arthur Whimbey, probably written
after Benezet was dead to What is his connection to the three Benezet
articles. . .(Benezet 1935/36)?
BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS-BOOGS
R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2-R2
R2. I disagree that "Related Article #7" has NOTHING to do with
Benezet's articles." Furthermore, that "someone called Arthur
Whimbey" is relatively well known in the U.S. math education
community and Whimbey's connection to Benezet (1935/36) is explained
below.
Article #7 reads (CAPS indicate hot-linking):
"Material at <HTTP://WWW.WHIMBEY.COM>, especially the ALGEBRA-FREE
CHALLENGE. Just as calculators can turn mathematics into button
pushing, algebra can turn it into symbol pushing . . . [see, e.g.,
the classic Clement et al. (1981)]. . . . Solving problems without
algebra encourages graphical, visual methods of solution -- methods
that require thought."
Clicking on the above hot-linked <HTTP://WWW.WHIMBEY.COM>, yields the
web page of "Whimbey.com" where it is stated that: "We plan to post
here, from time to time, various items of interest to the users of
the many different books. . .
(<http://www.whimbey.com/Books/books.htm>). . . by Arthur Whimbey
and his vast cast of co-authors . . . [see e.g., Whimbey & Lochhead
(1999), Whimbey et al. (1999), Lochhead (2000)]."
Whimbey and Lochhead are relatively well known in the US for, among
other things, their contributions - very much in the spirit of
Benezet - to the heralded program SOAR (Stress On Analytical
Reasoning) <http://www.xupremed.com/a.SSA/soar.html> at Xavier, a
small historically black University in New Orleans [see, e.g.,
Lochhead (2000), pp. 3-5].
"If you send us a postal address we will mail you worked solutions to
these problems. These solutions are completely free of ALGEBRA
CONTAMINATION!" (My EMPHASIS.)]
What is the meaning of "ALGEBRA CONTAMINATION"? According to Jack
Lochhead (2001):
LOCHHEAD-LOCHHEAD-LOCHHEAD-LOCHHEAD-LOCHHEAD-LOCHHEAD
"The term 'algebra contamination' is not intended to be technical
jargon.It simply refers to the fact that the solutions (which can be
requested -though no one ever has) do not use algebra. They use
other, I think more transparent, means of mathematical thinking. . .
. diSessa (2000) . . . points out that Galileo did not know algebra
(it hadn't yet been invented) and really had to struggle to make his
arguments. The things Galileo had to work through are probably
exactly what we should be putting our introductory. physics students
through. Only with algebra (like with calculators) the students can
get answers without really thinking. ACTUALLY THIS ALL TIES IN VERY
NICELY TO BENEZET (BUT THEN EVERYTHING IMPORTANT DOES).
LOCHHEAD-LOCHHEAD-LOCHHEAD-LOCHHEAD-LOCHHEAD-LOCHHEAD
33333333333333333333333333333333333333333333333333333333333333
C3-C3-C3-C3-C3-C3-C3-C3-C3-C3-C3-C3-C3-C3-C3-C3-C3-C3
In response to the above Boggs post of "C1" above, WAYNE BISHOP wrote
on 31 Jan 2002 07:15:15-0800(my EMPHASIS):
BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP
C3a. "All this. . . (work by Benezet). . . really shows is that
there's nothing new under the sun. At least not in education over
the last century."
C3b. "REFUSAL TO DIRECTLY TEACH FOR TESTABLE READING AND WRITING
COMPETENCE, TO DIRECTLY TEACH FOR TESTABLE MATHEMATICS COMPETENCE HAS
BEEN GOING ON FOR A VERY LONG TIME."
C3c. "THE PEOPLE IT. . .(does "IT" mean failure to teach for testable
results, or Benezet's delay of arithmetic algorithms??). . . HURTS
MOST ARE THE LOW SES. . .(Socioeconomic Status). . . KIDS WHOSE
PARENTS ARE NOT WELL EDUCATED; those with no compensating mechanisms.
Their only access to upward societal mobility (barring the lottery,
exceptional talent in sports or music, or some such) is through
education but we keep trying, decade after decade, to deny them that
opportunity. Amazing."
C3d. "It . . .(the material at the Benezet Centre). . . IS. . .
(fascinating). . . Especially number 7. . .(Related Article #7
regarding). . . SOLVING STANDARD ALGEBRA PROBLEMS WITHOUT ALGEBRA.
Algebra competence appears to correlate with future academic success
even more consistently than formal reading competence (mostly because
algebra competence implies a level of reading competence but not vice
versa) so what to do? Avoid developing algebra competence.
Especially among ignorant po' folk. Amazing."
BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP
R3-R3-R3-R3-R3-R3-R3-R3-R3-R3-R3-R3-R3-R3-R3-R3-R3-R3
R3a. Bishop seems to imply that the Benezet-Berman experiment was
(is) the same old educational stuff of the past century. I wonder if
Bishop could amplify and/or document this assertion?
R3b. As an antidote to Bishop's erroneous implication that the
Benezet experiment involved no testing for mathematical competence, I
quote the abstract of Mahajan & Hake (2000):
MAHAJAN&HAKE-MAHAJAN&HAKE-MAHAJAN&HAKE-MAHAJAN&HAKE
"Abstract. Should teachers concentrate on critical thinking,
estimation, measurement, and graphing rather than college-clone
algorithmic physics in grades K-12? [For non-US readers: Children
begin kindergarten (grade K) roughly at age 5 and finish the highest
grade (12) roughly at age 18.]
Thus far physics education research offers little substantive
guidance. Mathematics education research addressed the mathematics
analogue of this question in the 1930's. Students in Manchester, New
Hampshire were not subjected to arithmetic algorithms until grade 6.
In earlier grades they read, invented, and discussed stories and
problems; estimated lengths, heights, and areas; and enjoyed finding
and interpreting numbers relevant to their lives. In grade 6, with 4
months of formal training, they caught up to the regular students in
algorithmic ability, and were far ahead in general
numeracy and in the verbal, semantic, and problem solving skills they
had practiced for the five years before. ASSESSMENT WAS BOTH
QUALITATIVE [e.g., ASKING 8TH GRADE STUDENTS TO RELATE IN THEIR OWN
WORDS WHY IT IS `THAT
IF YOU HAVE TWO FRACTIONS WITH THE SAME NUMERATOR, THE ONE WITH THE
SMALLER DENOMINATOR IS THE LARGER'; AND QUANTITATIVE [e.g.,
ADMINISTRATION OF STANDARDIZED ARITHMETIC EXAMINATIONS TO TEST AND
CONTROL GROUPS IN THE
6TH GRADE. . .(see Berman 1935). . . Is it finally time for a
science counterpart of the Benezet/Berman Manchester experiment of
the 1930's?"
MAHAJAN&HAKE-MAHAJAN&HAKE-MAHAJAN&HAKE-MAHAJAN&HAKE
R3c. In my opinion, Bishop's comment "C2c" above is ambiguous. If
Bishop's "IT" means failure to teach for testable results, then it
certainly does NOT apply to the Benezet-Berman experiment as
indicated in R3b above. If Bishop's "IT" refers to Benezet's delay
of arithmetic algorithms until the sixth grade, then the delay did
NOT work to disadvantage of low SES kids. In this regard, I quote
Sherman Stein (1996):
STEIN-STEIN-STEIN-STEIN-STEIN-STEIN-STEIN-STEIN
". . .Responding to this challenge . . .(to cut the curriculum). . .
L.P. Benezet, the superintendent in Manchester, New Hampshire, wrote
back 'it is nonsense to take eight years to get children through
ordinary arithmetic. the whole subject could be postponed until the
seventh grade, and could be mastered in two years by any normal
student.' . . . Over a period of several years he then conducted
experiments . . . (Benezet 1935/36). . . He knew he could count on
the cooperation of children and teachers. But what about the parents?
How many parents would let their children serve as guinea pigs in
such a risky experiment? Though he sent out notices to all parents
telling them what he planned, he got no protests. Luckily for him,
'NOT ONE PARENT IN TEN IN THE DISTRICTS SPOKE ENGLISH AS THEIR MOTHER
TONGUE. HAD I GONE INTO SCHOOLS WHERE THE PARENTS WERE HIGH SCHOOL OR
COLLEGE GRADUATES, I WOULD HAVE HAD A STORM OF PROTEST - AND NO
EXPERIMENT.
Here is what happened. In Benezet's words, 'The 6th graders were
divided into two groups. The experimental group had no arithmetic
until beginning the sixth grade and the traditional group had it
starting in the third grade. At the beginning the traditional group
excelled. By April the two groups were on a par. IN LESS THAN A YEAR
THE EXPERIMENTAL GROUP HAD BEEN ABLE TO ATTAIN THE LEVEL OF
ACCOMPLISHMENT WHICH THE TRADITIONALLY TAUGHT CHILDREN HAD REACHED
AFTER THREE AND A HALF YEARS.' Moreover, the pupils were shown the
reasons behind the processes, such as, "why a correct answer is
obtained in the division of fractions by inverting the divisor and
multiplying.
What did he do with the time saved? He put it into 'reading, reason,
and recite.' THE "EXPERIMENTAL" CHILDREN DEVELOPED MORE INTEREST IN
READING, A BETTER VOCABULARY, AND GREATER FLUENCY THAN THE PUPILS WHO
CAME FROM HOMES WHERE ENGLISH WAS SPOKEN. . . .
I know that it will be impossible to spread his message. Too many
parents speak English. AND SO IT IS THAT HIS REFORM DISAPPEARED FROM
THE STAGE, LEAVING SCARCELY A TRACE. In home schooling or small
private schools where pupils proceed at there own pace, not an
imposed one, one can see Benezet's theory confirmed." (My CAPS.)
STEIN-STEIN-STEIN-STEIN-STEIN-STEIN-STEIN-STEIN
R3d. I wonder if Bishop could document his claim that "algebra
competence appears to correlate with future academic success even
more consistently than formal reading competence." And even IF that
were the case, Lochhead is NOT advocating the avoidance of
algebra-competence development. Instead he is suggesting that
beginning students use "more transparent means of mathematical
thinking" - see the Lochhead quote in "R2" above. Lochhead's
suggestion is consistent with physics education research
demonstrating that the effectiveness of introductory mechanics
instruction can be markedly improved when interactive engagement
methods based on verbal, graphical, pictorial, and kinesthetic
experiences are substituted for the traditional plug-and-chug
substitution of numbers into rote-memorized algebraic mechanics
equations (see, e.g., Hake 2002).
In any case, Lou Talman's comment of 31 Jan 2002 13:03:07-0700 on
Bishop's dismissal of SOLVING STANDARD ALGEBRA PROBLEMS WITHOUT
ALGEBRA gives added weight to the case against Bishop's point "C3d"
(My CAPS):
TALMAN-TALMAN-TALMAN-TALMAN-TALMAN-TALMAN-TALMAN-TALMAN
"And yet I recall that Andre Toom spoke approvingly, even glowingly,
of the practice of asking kids to solve 'standard algebra problems'
without algebra--a practice he said was very common in the old Soviet
Union. I don't recall that anyone ever suggested that the Soviet
Union, for all its flaws, was weak at teaching kids to do
mathematics. MIGHT IT BE
POSSIBLE THAT THERE ARE MORE WAYS THAN ONE TO DEVELOP COMPETENCE IN
MATHEMATICAL THINKING?"
TALMAN-TALMAN-TALMAN-TALMAN-TALMAN-TALMAN-TALMAN-TALMAN
So Toom appears to be in agreement with diSessa (2000), Benezet
(1935-36), and Lochhead (see quote in "R2" above), and disagreement
with Bishop. I wonder if Lou Talman has the relevant Toom reference.
4444444444444444444444444444444444444444444444444444444444444
C4-C4-C4-C4-C4-C4-C4-C4-C4-C4-C4-C4-C4-C4-C4-C4-C4-C4
C4. On 31 Jan 2002 23:37:32 -0800, Wayne Bishop wrote [in response
to a 1 Feb 2002 17:02:43+1000 post by Rex Boggs stating that "Benezet
carried out comparative studies between his reform method and
traditional methods. While I am sure folks could find minor flaws in
his methodology, the weight of evidence strongly suggests that his
radical approach resulted in students with above average ability to
reason mathematically and to express themselves, and with minimal
formal instruction in arithmetic in Years 1 to 6."]
BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP (My CAPS)
C4a. "Unless it . . . (the Benezet study) . . . was done on a fairly
large scale, with ordinary teachers and ordinary students, IT SHOULD
BE VIEWED EXACTLY AS IT HAS BEEN FOR ALMOST THREE-QUARTERS OF A
CENTURY. IRRELEVANT IN THE REAL WORLD."
C4b. "No, the MINUSCULE evidence strongly suggests that his radical approach
might warrant further study to see if it might be applicable in MORE NORMAL
SETTINGS. Sure. Why not?"
C4c. "But. . .(using). . . controlled studies using REAL STUDENTS AND
REAL TEACHERS AND TRADITIONAL ASSESSMENT DEVICES, not new ones
invented for
the occasion."
C4d. "And with informed, signed, parental consent. Then we might
have something useful. . ."
C4e. "In the meantime, do what the Singaporeans do. At last report,
they are first in the world. Attempting to emulate successful
programs is my recommendation for educational improvement, not
creating new ones out of whole cloth and ELIMINATING STANDARD
ASSESSMENTS OF COMPETENCE to keep parents from storming the castle
(which is exactly what happened in
here California in the mid-nineties) . . . .for a somewhat different
interpretation of the California mid-nineties experience see Becker &
Jacob( 2000)."
BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP-BISHOP
R4-R4-R4-R4-R4-R4-R4-R4-R4-R4-R4-R4-R4-R4-R4-R4-R4-R4
R4a. While it is true that the Benezet experiment has been virtually
ignored for three-quarters of a century, that is NOT because the
study (a) had any of the faults alleged by Bishop: small scale,
unusual teachers, unusual students (these faults were NOT present,
see "R4c" below); or (b) was judged to be "irrelevant to the real
world." Rather, I think the Benezet study has been ignored because
most people in the mathematics education community have been unaware
of it.
R4b. "Miniscule evidence"? Granted one experiment cannot suffice to
definitely demonstrate the efficacy of any pedagogical strategy, but
the Benezet (1935/36) evidence in more than miniscule and DOES
warrant further study (National Research Council 1990).
Unfortunately, it is virtually forgotten in the mathematics
educational community - just as in physics education the wheel (or
more often the flat tire) must be reinvented over and over again. As
for "more normal settings," what could be more "normal" for the
1930's in the USA than a school in an industrial mill town
(Manchester, New Hampshire) catering to low SES students?
R4c. If Wayne would take time from his doubtless busy schedule to
actually read Benezet (1935/36) he would discover that Benezet DID
perform controlled experiments and did use "REAL STUDENTS," and "REAL
TEACHERS," and (for the time) "TRADITIONAL ASSESSMENT DEVICES," (on
the latter point, see Berman 1935). Benezet did NOT use new students,
new teachers, or new assessments "invented for the occasion."
R4d. As indicated in the Stein quote in "R3c" above, Benezet DID
receive signed consent from the parents but, in many cases, the
consent was not "informed" because the parents of the SES children
COULD NOT READ ENGLISH! Had they read English then the Benezet's
experiment would probably never have taken place - and as Stein
points out, it will probably not take place today in the public
schools because most parents CAN read English and fervently believe
(urged on by the likes of Bishop his Mathematically Correct
<http://ourworld.compuserve.com/homepages/mathman/> colleagues) that
if 'reading & riting & rithmetic" in the days of yore was good enough
for them then it's good enough for their kids. I'll leave it to
ethicians to balance Benezet's sin of obtaining uninformed consent
(why didn't Benezet send out consent forms written in the parents'
native language?), against his contribution of performing (arguably)
one of the most significant educational experiments in the history of
American education.
R4e. I think Michael Paul Goldenberg's 01 Feb 2002 08:15:42-0500 post
constitutes a sensible rebuttal to Bishop's point "C4e":
GOLDENBERG-GOLDENBERG-GOLDENBERG-GOLDENBERG-GOLDENBERG
"As to emulating 'successful' programs. . .(such as allegedly occur
in Singapore). . . , this is yet more question-begging. Is
it a slam dunk that if something works in country A that it will work in
country B? Could there be complexities that make a method which is fine for
one place unworkable in another? Is it reasonable to assume that cultural
differences, to name but one consideration, might make a method
inappropriate for ALL countries? And this assumes that we all accept that
the tests putting Singapore on top are actually meaningful and that they
measure what we generally believe should be measured to evaluate
mathematical achievement.
Finally, the claim that parents in California 'stormed the castle' is
disingenuous at best. . . It's a classic partial truth. Some parents
did indeed loathe specific programs implemented in California during
the '90's. Some non-parents as well. And some folks were terribly
upset by programs that preceded that swing of the pendulum. Some are
enraged by the programs currently approved by the current powers that
be in California. Parents and
non-parents do have the right to get upset, but their being upset does not
present a prima facie case against the things they are upset about. Even if
one accepted the notion that the 'will of the people' were the most
important consideration (as opposed to what people whose job it is to
conduct and/or respond to the best available research believe should be
done), the final question being begged is whether even a simple majority of
parents in California actually stormed the castle or wanted to. I can't help
but wonder if this were not simply a case of the squeaky wheel getting the
grease."
GOLDENBERG-GOLDENBERG-GOLDENBERG-GOLDENBERG-GOLDENBERG
For yet more commentary on the work of Benezet, see Mahajan & Hake
(2000) and Hake (2000, 2001a, 2001b, 2001c, 2001d).
Richard Hake, Emeritus Professor of Physics, Indiana University
24245 Hatteras Street, Woodland Hills, CA 91367
<rrhake@earthlink.net>
<http://www.physics.indiana.edu/~hake>
REFERENCES
Becker, J.P. and B. Jacob. 2000. "The Politics of California School
Mathematics: The Anti-Reform of 1997-99,"Phi Delta Kappan,
March; <http://www.pdkintl.org/kappan/kbec0003.htm>. "The authors
tell the story of a powerful group of parents and mathematicians in
California who manipulated information and played off of the public's
perception of our 'failing schools' to acquire political clout.
Through this telling, they hope that other states will be able to
adopt a more rational course as they reconsider their policies."
Benezet, L.P. 1935/36. "The teaching of arithmetic I, II, III: The
story of an experiment," Journal of the National Education
Association 24(8), 241-244 (1935); 24(9), 301-303 (1935); 25(1), 7-8
(1936). The articles were
(a) reprinted in the Humanistic Mathematics Newsletter #6: 2-14 (May
1991); (b) placed on the web along with other Benezetia at the
Benezet Centre
<http://wol.ra.phy.cam.ac.uk/sanjoy/benezet/>. See also Mahajan & Hake
(2000).
Berman, Etta. 1935. "The Result of Deferring Systematic Teaching of
Arithmetic to Grade Six as Disclosed by the Deferred Formal
Arithmetic Plan at Manchester, New Hampshire." Masters Thesis, Boston
University, USA.
Butterworth, B. 1999. "What Counts: How every brain is hardwired for
math." Free Press; especially Chapter 4 "Numbers in the Brain."
Clement, J., J. Lochhead, & G. Monk. 1981. "Translation difficulties
in learning mathematics," Am. Math. Monthly 88(4), 286-290 (1981);
online as "Related article #5" at the Benezet center
<http://wol.ra.phy.cam.ac.uk/sanjoy/benezet/> as a pdf
<http://www.inference.phy.cam.ac.uk/sanjoy/benezet/clement-high.pdf>:
"To investigate the source of the errors we had observed . . .(in
translating practical situations into mathematical notation). . . .
we collected data on the following simpler problem. . . "Write an
equation for the following statement: 'There are six times as many
students as professors at this university.' Use S for the number of
students and P for the number of professors. On a written test with
150 calculus-level students, 37 percent missed this problem, and
two-thirds of the errors took the form of a reversal of variables
such as 6S = P. In a sample of 47 non-science majors taking college
algebra, the error rate was 57 percent."
Dehaene, S. 1997. "The Number Sense: How the Mind Creates
Mathematics" (Oxford University Press). See also Butterworth (1999).
Hake, R.R. 2001d. Response (online at
<http://www.pkal.org/events/roundtable2002/paper-madison-response-hake.html>
to Bernie Madison's essay on assessment "Assessment: The Burden of a
Name"
<http://www.pkal.org/events/roundtable2002/index.html>. Quoting from
the former paper: "An expert but forgotten practitioner of such
assessment . . . (done to enhance teaching, increase learning, and
improve programs because it is a part of those processes). . . . was
Louis Paul Benezet (1935/36), a man far ahead of his (and our) time."
Hake, R.R. 2002. "Lessons from the physics education reform effort."
Conservation Ecology 5(2): 28; online at
<http://www.consecol.org/vol5/iss2/art28>. "Conservation Ecology," is
a FREE "peer-reviewed journal of integrative science and fundamental
policy research" with 11,000 subscribers in 108 countries. Volume 5,
issue 2 <http://www.consecol.org/Journal/vol5/iss2/index.html>
contains a special feature on "Interactive Science Education."
National Research Council. 1990. "Reshaping School Mathematics: A
Philosophy and Framework for Curriculum" (Mathematical Sciences
Education Board (that in 1990 included Jack Lochhead); pp. 30-31;
online at <http://books.nap.edu/catalog/1498.html>: "THERE IS SOME
EVIDENCE TO SUGGEST THAT PAPER AND PENCIL CALCULATION INVOLVING
FRACTIONS, DECIMAL LONG DIVISION, AND POSSIBLY MULTIPLICATION ARE
INTRODUCED FAR TOO SOON IN THE PRESENT CURRICULUM. Under currently
prevalent teaching practice, a very high percentage of high school
students worldwide never masters these topics - just what one would
expect in a case where routinized skills are blocking semantic
learning (e.g., BENEZET, 1935/36. The challenge for curriculum
development (and research) is to determine when routinized rules
should come first and when they should not, as well as to investigate
newer whole-language strategies for teaching that may be more
effective than traditional methods. THIS IS AN AREA WHERE MORE
RESEARCH NEEDS TO BE DONE." (Our EMPHASIS.)
Stein, S.K. 1996. "Strength in Numbers: Discovering the Joy and Power
of Mathematics in Everyday Life" (Wiley, 1996).
Whimbey, A., J. Lochhead, and P.B. Potter (1999). "Thinking Through
Math Word Problems: Strategies for Intermediate Elementary School
Students." Erlbaum. See also at
<http://www.whimbey.com/Books/thinking/thinking.htm>.