Math instruction: best practices and problems (1)

Below are different perspectives on math instructions submitted by our e-community about math instruction:

REVIEW OF TRADITIONAL AMERICAN MATH TEXTS:

A very interesting article reviews traditional American math textbooks and say these are the ones that work… read the whole article linked here

It Works for Me: An Exploration of Traditional Math Part II

Barry Garelick Columnist EdNews.org Part II of III The Rise of Test Scores

…While a complex mix of factors may be responsible for the general increase in
math achievement scores, there is also evidence that there were changes going on
with math textbooks during this period as well. The major players among math
education reformers of the 20’s through the 50’s include Leo Brueckner, Robert
Lee Morton, Foster Grossnickle, Arnold E. Moser, Guy T. Buswell and William A.
Brownell.Brownell, spoken well of by NCTM and various luminaries in today’s
reform movement, was the key reformer of the early twentieth century and
promoted what he called meaningful learning; i.e., teaching mathematics as a
process, rather than a series of end products of isolated facts and procedures
to be committed to memory. If this sounds like what the reformers are talking
about today, it is because the complaints levied against how mathematics is
taught, like the complaints about education in general through the years, have
been perennial. What is often not mentioned when these complaints are “replayed”
is that there have also been perennial solutions and some of these solutions
have actually been effective.
Brownell led the charge against the isolated, rote-memory type of math teaching
that came about in large part through the books and efforts of E. L. Thorndike,
another figure of education at that time.The reformers listed above, including
Brownell, all wrote math texts that were in use from the 30’s through the
60’s.The later books were written by Brownell (with Guy T. Buswell and Irene
Sauble) starting in the mid-50’s in a series called “Arithmetic We Need”.I’m
familiar with this series because they were the books I used when I was in
school. I have copies of these and other books in use by all the authors.All the
books give explanations of what is going on with specific mathematical
procedures, and topics were presented in a logical sequence that allowed
building upon previously learned and mastered material.But what is particularly
interesting are the explanations in the teacher’s manuals and prefaces to these
books:

From “Making Sure of Arithmetic” (Grade 6):”Each new process is explained in the
simplest terms, utilizing every graphic aid possible.From the beginning, meaning
and relationship are emphasized. As a result the pupil gains not only skill but
skill with understanding.” (Morton, et. al., 1946)
From “The New Curriculum Arithmetics” (Grade 7) “A program of mixed and
cumulative practice exercises insures mastery and retention of the processes and
topics studied.” (Brueckner, et al, 1941)
From “Growth in Arithmetic” (Grade 3): A comparison chart in the teacher’s
edition showing the difference between the older (Thorndike-derived) textbooks
and this one: “Older: Taught as facts, skills, and habits of procedure; Newer:
Taught to emphasize meanings, principles, and relationships. Facts and skills
developed after understanding.” (Clark, et al, 1952)
From “Teaching Arithmetic We Need” (Grade 5) “Each book in this series is built
upon a conception of arithmetic that involves two aims, the social aim and the
mathematical aim. Adherence to the latter aim requires that children see sense in
what they learn.”(Brownell, et. al., 1955a)
While the above sounds like they may have come from the introductions of various
reform texts, the main difference is that 1) there were actual textbooks that
students could read (as opposed to workbooks with problems but little to no
written explanations of what is going on) and 2) these textbooks contained
actual content that required mastery by the student.
You may well be wondering why the books are so heavy on drill if these reformers
came from the school of “progressivism” which embodied the notion of “teaching
the child” and which focused on a student-centered curriculum.The inclusion of
drills in these books is not inconsistent with the philosophy of the
progressives. The reformers believed in teaching the child, it’s true; but
through “meaningful learning” which in Brownell’s and others’ view, meant
teaching the concept with examples of how (and why) it worked, and then
providing the student opportunity to practice the procedure to ensure both
understanding and mastery of the skills involved in applying the concepts.
The drills in the books from the 40’s, 50’s and early 60’s are varied, mixed,
and cumulative (i.e., continually including problems that were addressed earlier
within new material). That they are cumulative is important, given recent
research showing the positive effects of such practice on problem solving.
(Mayfield and Chase, 2002). Also, there are drills that simply ask students to
identify the operations needed to solve the problems, to ensure that they know
when to apply, say, multiplication as opposed to division in solving a problem.
I would therefore add to the list of possible factors influencing the upward
trends in achievement scores from the 40’s through the mid-60’s the textbooks in
use, and the implementation of the theories behind them.This is not to say that
the traditional math of such time was perfect. If I had to compare the
“Arithmetic We Need” texts that I used with Saxon Math, or the math program used
in Singapore, I would say the latter two are superior with much more challenging
word problems. I can say, however, that the essentials of math were covered well,
which would include place value, why a particular algorithm worked, thorough
application of fractions and multiplication and division of fractions (similar
to Singapore’s approach) and application of procedures to solving word
problems. Here are two problems from the sixth grade book of “Arithmetic We
Need”:

“How many glass containers holding 3/16 quarts can be filled with water from a
quart bottle which is full?”
“If it takes 1 and hours to drive to the city,what part of the distance will Bill
and his father drive in ｾ hour?”
(Brownell, et.al., 1955b)
It cannot be denied that some teachers did not follow the texts and insisted on
an approach that relied onrote-memorization and math problems isolated from word
problems – an approach that Thorndike promoted and against which Brownell and
others rebelled. But neither Brownell, the other reformers of those times, nor
mathematicians, asked the teachers to teach math that way. Any lack of
continuity between textbooks used between grades was also not the fault of
authors or mathematicians. If by “traditional method” and “old way of teaching
math” people mean poor teaching and bad planning, it should be noted that such
was incidental to and independent of the textbooks used and the philosophy put
forth by the reformers.Barry Garelickis an analyst for the U.S. Environmental
Protection Agency in Washington, D.C. He is a national advisor to NYC HOLD, an
education advocacy organization that addresses mathematics education in schools
throughout the United States. Published November 6, 2007

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FUZZY MATHEMATICS

Texas Challenges City on Math Curriculum
By ELIZABETH GREEN The Sun Reporter of the
Sun November 20, 2007
The state of Texas has dropped a math curriculum that is mandated for use in New
York City schools, saying it was leaving public school graduates unprepared for
college.
The curriculum, called Everyday Mathematics, became the standard for elementary
students in New York City when Mayor Bloomberg took control of the public
schools in 2003.
About three million students across the country now use the program, including
students in 28 Texas school districts, and industry estimates show it holds the
greatest market share of any lower-grade math textbook, nearly 20%. But Texas
officials said districts from Dallas to El Paso will likely be forced to drop it
altogether after the Lone Star State’s Board of Education voted to stop
financing the third-grade textbook, which failed to teach students even basic
multiplication tables, a majority of members charged.
One board member, Terri Leo, who is also a Texas public school teacher, called
the textbook “the very worst book that we had submitted.” This year, the board
of education received 163 textbooks for consideration.
The board chairman, Don McLeroy, said the vote was part of a larger effort to
prepare more Texas students for college. “We’re paying millions of dollars to
the publishing industry,” Mr. McLeroy said. “We might as well get something
back.”

More on fuzzy maths:

Basic algebra is not covered in “integrated” math texts like Core. TERC in its first edition contains ABSOLUTELY NO ELEMENTARY ARITHMETIC, at least as it was taught to parents using efficient standard methods. Connected math did not contain the formula for computing an average. Now what is the possible purpose of a textbook which contains no standard contents whatsoever? — excerpted from Arthur Hu’s comments on the internet.

Watch the video “Math Edn: An Inconvenient Truth” at youtube.com which says essentially the same things but shows concrete examples of what’s wrong with the texts.

In another youtube.com video,  “Math Education: A University View“, an interesting and clear explanation was provided of why Washington State students have been failing math…coinciding with the introduction of the Reform and Constructivist method of math. The professor said he took his son’s own math edn into his hands, and enrolled them in Kumon tutoring as well as worked with them at night for remedial math work. He mentions that these textbooks use the method and perhaps some of us might want to review if we are using flawed math texts: Pearson’s Connected Math Program; McDougal Integrated Math; Interactive Math Program (IMP); Everyday
Math and TERC.

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A controversial viewpoint is offered below by an Oxford academic who advocates less than conventional methods of learning math. His ideas are not likely to catch on, but they are fun to read…

Who needs maths?
Compulsory lessons should go, says an Oxford academic. There are better ways to
get children to understand the key concepts – sudoku, for example. John Crace
reports Tuesday November 13, 2007 Guardian
Soon after the first sudoku puzzles began to appear in newspapers a couple of
years ago, there came hurried reassurances from worried editors. Sudoku might be
a number grid, they soothed, but don’t let all those nasty ones, twos and threes
frighten you, because you don’t need to be any good at maths to do it.
It was a message that summed up the national attitude to maths. Numbers are
something inherently difficult, to be feared and mistrusted. The subject carries
a lasting memory of childhood shame and frustration from which we never recover.
Maths is for geeks, nerds and misfits; the rest of us get by on a wing, a prayer
and a calculator.
Andrew Hodges, maths lecturer at Wadham College, Oxford, takes a different view
of the addictive puzzle. “Sudoku may not require long multiplication or
division,” he says, “but it is a very good puzzle that replicates the pattern of
thinking required to solve quite complex logical problems in maths. But no one
dares mention the association, for fear of putting off all those who like doing
it.”
Hodges has spent a lot of time thinking about these sorts of contradictions over
the past year while writing One to Nine: The Inner Life of Numbers, his
contribution to the growing catalogue of books that aim to make high-brow maths
user-friendly for middle-brow arty types. It has to be said that, like many
other authors before him, Hodges is only partially successful. You can be
reading quite happily for several paragraphs, enjoying the feeling of a new
world opening up, and then he loses you in a sentence. And no matter how often
you re-read that sentence, you’re still none the wiser. You have come up against
a barrier of understanding that language cannot easily transcend. In chapter
nine, for example, Hodges states: “By about 20 [years old], it is possible to
catch up with current knowledge in one small area – for instance, in the
heartland of mystery and discovery to which elliptic functions are the doorway.”
Hodges laughs: “Even when I was writing the book, I knew that it was part folly
because, for almost all readers, it was going to be an exercise in alienation
just to pick it up.”
Worst subject
He explains: “Most people seem to remember maths as their worst subject and have
developed a mental block about it. So there will be a lot of areas about which
they know nothing. Everyone who tries to popularise maths understands they are
going to come up against this, but most choose to ignore it and carry on
regardless. I felt it was important to address this barrier between writer and
reader head on.”
Which is easier said than done, when you realise just how limited many people’s
maths often is. Only last week, Camelot had to withdraw one of its scratchcards
when loads of punters complained that they couldn’t understand why -9 was a
lower number than -8.
For Hodges, the real battleground is the syllabus at key stages 3 and 4, where
the ante gets upped from relatively straightforward maths to something
altogether more complicated: an uneasy hybrid of the Athenian Euclidian abstract
logic that was so appealing to the Victorian gentry, and the relentless grind of
long calculation that has its roots in the pre-computer era when bosses needed
clerks to keep books and ledgers with metronomic accuracy.
“There still needs to be a syllabus that stretches the most able and provides
them with a route on to A-level and university,” he says. “But I think we should
consider abandoning it as a compulsory subject. What’s the point in a system
that brands all those who don’t get a maths GCSE as failures? All it does is
reinforce their sense that maths is boring and difficult, which is the last
thing we should be doing.”
He would like to see a change of approach. “We should be trying to find ways of
equipping children with the basic maths they will need to function adequately in
society. I’m sure there are wonderful examples of good teaching practice to be
found in schools, but the curriculum is very prescriptive and most teachers
don’t have the time to be creative. We should be looking at ways of teaching
maths skills through other media, such as electronic music and web design, that
are more relevant to most students.”
Hodges is by no means the first person to suggest a radical overhaul of maths at
secondary level. At the request of the government for a rethink of post-14
maths, Professor Adrian Smith, principal of Queen Mary, University of London,
published Making Mathematics Count in 2004, which branded the GCSE as “not fit
for purpose” and suggested doing away with the three-tier system of higher,
intermediate and foundation GCSEs. “It is crazy to have a system whereby the
highest grade in the foundation paper is a D,” he said. “What incentive is there
for anyone to try when they are guaranteed to fail?”
Smith also promoted the idea of functional maths for the less able – maths that
would equip students with basic numeric skills, such as percentages, measurement
and estimation – and called for more teachers and better continuing professional
development. Charles Clarke, then education secretary, accepted these proposals
wholeheartedly and the maths world sat back and waited for the promised land.

Barely a squeak
It hasn’t really come. There have been a few bright signs – the creation of a
National Centre for Excellence in the Teaching of Mathematics, and a scheduled
switch to a two-tier GCSE. But there’s been barely a squeak from Celia Hoyles,
the government’s maths tsar, in the past few years and the introduction of
functional maths, the key component to the reforms in most people’s opinion, has
been rather fudged. “It has all gone a bit quiet,” says David Benjamin, a maths
teacher at Folkestone academy in Kent. “We’ve seen a model that incorporates
elements of functional maths into the GCSE curriculum, but there’s been no sign
of a separate qualification.”
Inevitably, this means that some pupils are being forced to study ideas that are
beyond them, and teachers cannot do much about it. “A lot of kids have been
getting very negative messages about maths since they were born,” Benjamin says,
“from their parents who themselves struggled with maths, and from school. Maths
can seem quite unforgiving. You can have something explained several times and
still not really understand it, and when this happens, it’s easy to think you
are stupid. So our first aim is to build up the students’ confidence.”
But teachers have to get through a packed syllabus at a steady pace; they can’t
keep going over the material until everyone in the class has grasped it. What’s
more, some kids are never going to understand all the concepts, particularly in
algebra, not because they are stupid, but because they aren’t at the necessary
level of abstract cognitive development. Yet they have to be taught it
regardless. The consequences are all too predictable; failure breeds more
failure and disengagement generates greater disengagement.
Paul Ernest, a professor at Exeter University’s school of education and lifelong
learning, argues that traditional teaching methods disadvantage ethnic minority
pupils, girls, students with special needs and those from poor backgrounds, and
that considerations of social responsibility should be applied to maths
teaching.
“I disagree with people who think that mathematics is neutral and value-free,”
he says. “It is human made, therefore culturally influenced, and this makes
social justice central and relevant in mathematics.
“We need to think of different ways of contextualising maths to take
multi-culturalism, racism and sexism into account. Students need to see that
everyone owns maths, and that many countries have had their roles in the
development of the subject downplayed. We need to make maths more democratic and
discursive, so they are not afraid to suggest wrong answers.” As Ernest readily
accepts, this last suggestion demands a total rethink of teaching styles, as one
of the main problems students come across in maths is precisely that an answer
is simply either right or wrong. And it’s hard to get round that. Whereas an
essay can allow for shades of opinion and degrees of understanding, most school
– and even undergraduate – maths doesn’t throw the subject open to these
nuances.
A great deal to learn
“I think mathematicians have a great deal to learn from other disciplines,”
Hodges says, “and I would like to see more emphasis on making judgmental
assessments. This is a key skill, especially when dealing with human problems,
but it’s not one that mathematicians seem to learn, even at undergraduate
level.”
Maths remains a very particular animal, and one that politicians are reluctant
to tamper with. There’s no denying there have been improvements: the numbers
taking GCSE and A-level maths have risen and there are more teachers.
But the less able are still struggling, mainly because politicians worry it
would be electoral suicide to introduce a separate functional maths GCSE. They
can squeeze a bit of functional maths into the existing system and shave off a
bit of the more difficult stuff elsewhere, but going the whole hog is somehow
just too radical, inviting accusations of dumbing down. So for the time being,
instead of learning something functional, the less able will continue to learn
next to nothing.