http://mathematicallysane.com/analysis/reformvsbasics.htm
John
A. Van de Walle Ten years after NCTM�s release
of the
In order to understand how
we arrived at this state of war it seems useful to offer some definitions of
- Counting accurately to 100 or more. - Mastery of basic facts for all four operations. - Pencil and paper computation skills with whole numbers, decimals and fractions. - Solving percent problems. - Knowing and using formulas for area and perimeter of basic shapes. - Mastery of measurement conversions (2 pints in a quart, 3 feet in a yard, etc.). Of course the above is an obvious over simplification.� Regardless of your specific list, the position of the basics camp is that kids should know some �basic stuff.�� Basics are about what children should know � content.� It is trivial pursuit mathematics. Reform mathematics is a
bit more difficult to pin down but the principal positions can be found in the
1989 These two positions, reform and the basics, are not opposite ends of the same continuum.� On one hand, the basics tend to be about content, specifically about the content that was common when today�s adults were in school.� On the other hand, reform is much more about how children learn and how to achieve the content goals one desires. Again, this is an oversimplification.�
The basics camp has also taken some extreme positions.� In some states, California
being a prime example, the skills recommended are not always appropriate for
the grade levels suggested nor are they reflective of today�s societal needs.�
Many skills such as multiplication by a 3-digit decimal number are archaic in
a world of omnipresent calculators.� And most onerous is the demand in some
quarters for nothing other than direct instruction, what Tom O�Brien (1999)
so aptly labeled �parrot math.�� This approach suggests that children mimic
mindlessly what teachers model with the hope that somehow the mimicry will lead
to learning.� Do parrots understand?� Ernst von Glassersfeld (1995) talks of
50 years of domination by behaviorists.� �The behaviorists succeeded in eliminating
the distinction between The reformers have also made some mistakes and are guilty of misguided emphases.� By praising the values of calculators and continually harping about tedious computations, we in the reform camp failed to emphasize many of the valid content objectives in mom and dad�s typical list of basics.� We became obsessed with all sorts of interesting problems that ate time in the mathematics period.� (Nineteen pigs and chickens with a total of 54 legs.� How many pigs and how many chickens?)� Detailed applications and �alternative assessments� became so involved with the context that often the mathematics was difficult to find (Rainforest math).� Manipulatives, cooperative groups and calculators became hallmarks of reform no matter how superficially or inappropriately these materials and methods were used.� In short, we forgot that there is much of what is basic that is still important.� Basic facts are essential!� All children must be able to compute!� Mom and dad think we are pushing �fuzzy math� and have forgotten what is important. The reformers feel as though they are being returned to the dark ages by state mandates.
We simply cannot go backwards!� The behaviorists� parrot math, direct-instruction approaches are doomed to failure.� Evidence of this can be found in any gathering of adults, even educated adults, where most will gladly proclaim that to them mathematics is a mystery.� Most adults believe that mathematics is composed of mindless rules and while it is certainly important, they were �never any good at math.�� Direct, rule-oriented methods failed them miserably and will fail today�s children in exactly the same manner. � Today we know a lot about how children learn and how they learn mathematics.� There is abundant evidence that instruction reflective of a constructivist theory learning is more effective than teaching by telling.� The NSF supported reform curricula all reflect, though somewhat imperfectly, a constructivist view of learning.� Most reform-movement advocates espouse constructivism as the best explanation for how children learn.� Unfortunately, we have not moved very far in this direction if measured by the typical classroom in the United States.� The basic tenet of constructivism
is simply this: Since ideas are used to develop new knowledge, those existing ideas will necessarily be connected to the new idea.� The result is a network of meaningful, related, useful ideas.� The more ideas and the more interconnected they are, the better all of the ideas are understood.� Things make more sense when they can be related to a lot of the ideas already understood. Constructivism is a theory
about how we learn.� If it is correct then that is how On the other hand, when mathematical ideas are used to create new mathematical ideas, useful cognitive networks are formed.� Return to 7x8 and imagine a class where children discuss and share clever ways to figure out the product.� One child might think of 5 eight�s and then 2 more eight�s.� Another may have learned 7x7 and noted that its just one more seven.� Still another might look at a list of 8 seven�s and take half of them (4x7) and double that.� This may lead to the notion that double 7 is 14, and double that is 28 and double that is 56.� Most of these methods suggest ways to think about other facts where doubling or five�s can be useful.� The result is a meaningful network of ideas.� Meaningful networks of ideas mean fewer details to retain, easier recall of ideas after extended periods of time, better application of ideas to novel problems, and a feeling that mathematics makes sense.� Apply this same idea to any area of study in your experience.� When you are able to or helped to connect new ideas to those you already have, the new information seems to make sense; you understand it and most likely will remember it easily.� Conversely, when facts do not connect with things you know they seem foreign or mysterious and they are almost certainly forgotten or confused.�
The main thesis of this paper is the following:
The term � for which the solution has not been explained, � that begins where kids are (with their ideas), � that is challenging mathematically, and � for which justification and explanations for answers, methods, and results are understood to be the responsibility of the students. A problem � may be posed for individual students, pairs, or groups, � may involve hands on materials or drawings, or involve only mental work, and � may involve calculators or not. A problem should always include the expectation of a report be it oral, written or some other product such as a poster or graph. The intent of the above
definition is to broaden the concept of problem to include any task that causes
children to wrestle or struggle with the mathematics that we want children to
learn.� It is important to understand that mathematics is to be taught A few examples are in order: Grades K-1:
Grades 2-3:
Grades 3-5:
It is important to notice that these examples involve very traditional content.� Children engaged in these problems are likely to �bump into� and develop or construct the desired ideas.� It is not a matter of discovery but rather the personal development of the expected curriculum, based on the students� existing ideas.� In the first example above, students will find ways to think about a number in two parts.� Some children will be very systematic.� Others will be quite haphazard, starting each combination anew.� When these ideas are shared, the thoughts from the class as a whole will aid in everyone�s development.� Part/whole concepts are at the very foundation of addition and subtraction facts.� In the third example, students have to wrestle with the size of fractions.� They will need to use what they understand about the meanings of top and bottom number and the relative size of fractional parts.� Some may learn from others in their group since all students in a class rarely master basic fraction concepts.�� Some may use a physical model but the explanations that develop at the end of the lesson will go beyond simple showing of fraction pieces.� Computation and estimation are built much more easily on a foundation of good fraction concepts such as these.
Admittedly, teaching through problem solving is a difficult task when compared with teacher-directed instruction guided by a traditional basal textbook. Why should anyone want to go to this trouble?� Here are some good reasons: 1.
Problem solving places the focus of the students� attention on 2.
Solving problems develops the belief in students that they are capable
of 3. Problem solving provides ongoing assessment data that can be used to make instructional decisions, help students succeed, and inform parents. 4. It is a lot of fun!
A problem-solving lesson should always have three components: before, during and after.
�Depending on the task you might do a related or simpler task.� For computational situations you may want them to do the task mentally or make and share estimates before working with paper and pencil.�
Be certain to plan ample time for this portion.� Often this is when the best learning will take place.� Twenty minutes or more is not at all unreasonable for a good class discussion and sharing of ideas. Examples: Grades 3-5:
In the Before portion of the lesson, have students supply the missing part of 100 after you supply one part.� Try numbers like 80 or 30 at first then try 47 or 62.� You might also ask students if the answer to the problem is more or less than 300.� Now let go.� Students here will develop an add-on approach to subtraction.
Grades 3-5:
This last example is designed to help students develop a method for multiplying two 2-digit numbers.� Earlier tasks may have been rectangles such as 30 by 8 or 40 by 60.� The discussion or after portion of the lesson may take place the second day with as much as a whole period devoted to developing written schemes for solving this problem.� Grades 1-2:
Here students working alone or in pairs will come up with a wide variety of strategies for this fact.� In the after portion of the lesson, focus on those strategies that are most efficient. If no one uses a make-ten strategy (3 on to 7 is 10 and 2 more is 5), you will know to plan a subsequent lesson involving ten frames and adding on to make numbers in the teens.
The following ideas have been learned from elementary teachers who have been working hard at developing a problem-solving approach in their classrooms. 1.�� 2.�� 3.�� 4.�� 5.�� 6.��
1.�� ����� Number concepts, basic facts and even computational procedures can be developed as examples in this paper have demonstrated.� Even a standard algorithm can be explored as a problem after children have invented a variety of meaningful methods of their own.� Show how a standard procedure works and pose the problem of figuring out why it works.� Even traditional mathematics makes sense and children can be expected to make sense of it. 2.��
3.��
4.�� ����� There never will be
a perfect set of tasks for your class; every class is different.� Here are places
to begin:� 1)� Look at the lessons in your textbook.� Many of the teach or explain
sections on the pupil pages can be transformed into good problems.� Try it!�
2)� Read the NCTM journals regularly!� Copy and file articles to go with each
chapter in your text or each section of your curriculum.� Don�t wait until you
need an idea to begin flipping through journals.� 3)� Check out NCTM publications.�
The 5.�� ����� Absolutely!� The error
is to believe that drill is a method of developing ideas.� Drill is appropriate
when a) the desired concepts have been meaningfully developed, b) flexible and
useful procedures have been developed, 6.�� Do not give in to the temptation to �tell �em.�� Set it aside for the moment.� Ask yourself why it bombed.� Did the students have the ideas they needed?� If not, that tells you where to go next.� Was the task too advanced?� Often we need to regroup and offer students a simpler related task that gets them prepared for the one that proved difficult.� Was the task getting at the ideas you wanted to develop?� When you sense a task is not going anywhere, regroup!� Don�t spend days just hoping that something wonderful might happen.� If you listen to your students you will know where to go next.
The most fundamental or
basic thing to keep in mind is this:�
Commission on Standards
for School Mathematics.� (1989).� O�Brien, T. C. (1999). Parrot
math.� von Glassersfeld, E. (1995).�
A constructivist approach to teaching.� In L. P. Steffe & J. Gale (Eds.),
Battista, M. T. (1999).�
The mathematical miseducation of America�s youth:� Ignoring research and scientific
study in education. Hiebert, J. (1999). Relationships
between research and the NCTM Standards.� Schoen, H. L., Fey, J. T.,
Hirsch, C. R., & Coxford, A. F. (1999).� Issues and options in the math
wars. Return to Reform vs. the Basics |
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