MATH
Effective teaching
why must we do the same thing to both
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The teacher’s modeling of new material teaching, and traditional versus reform-
sides? and must be clear and concise; and based teaching. Although these categories
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Lessons frequently involve solving
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Once students are ready to succeed, and labels might have been useful for some
problems using different procedures signifi cant time should be allotted for purposes in the past, the instructional
and then examining the similarities error-free practice. features that facilitate conceptual
and differences between them. How is Lessons should progress from the understanding and skill effi ciency do not
Jack’s procedure different from Martha’s teacher’s presentation and the modeling of fall neatly into these categories. They no
procedure? Note: It is not necessary for mathematical material to practice time for longer capture the distinctions suggested
students to practice multiple procedures students. During the modeling part of the by the data. Attending explicitly to key
for solving similar kinds of problems, lesson, the teacher presents material clearly mathematical relationships, for example,
but comparing different procedures is and in a meaningful, organized way. The can be done within any of these methods.
benefi cial. teacher also asks questions throughout this These fi ndings suggest a weighted
part of the lesson. These questions are mostly balance between the two instructional
Work and wrestle product questions requiring students to approaches might be appropriate, with a
Teachers should provide opportunities for provide only answers and are asked publicly heavier emphasis on the features related to
students to wrestle with key mathematical for the entire class to hear. The transition conceptual understanding.
ideas and ensure that students do some from the whole-group portion of the lesson
of the important mathematical work in the to individual student practice occurs only What we know
lesson. after the students are well prepared. The fast
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To improve conceptual understanding,
Allowing students to work hard to pace of the lesson keeps students’ attention
make explicit the important
make sense of mathematics does not while ensuring that students move from
mathematical relationships and ask
mean standing by while they become one task to another successfully. Lessons
students to “work and wrestle.”
unnecessarily frustrated and confused, nor are content-focused, so that class time is
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To improve skill effi ciency, use rapid
does it mean presenting problems that are devoted to mathematical tasks as opposed
pacing, modeling, and moving to error-
well beyond their reach. But it does mean to managerial tasks, such as handing back
free practice.
providing time during the lesson when papers or discipline.
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Balance these two approaches, with
students are allowed to work on problems
a heavier emphasis on conceptual
they don’t immediately know how to solve, Not a simple correspondence
understanding.
and it does mean resisting the temptation The apparent disparity between one set
to jump in and tell students how to do of instructional features for conceptual
something at the fi rst sign of uncertainty. understanding and a completely different About the authors
At least some time during each math set of features for skill effi ciency, breaks James Hiebert is the Robert J Barkley
lesson, teachers should engage in the down when examining the results of studies Professor of Education at the University of
following activities: on conceptual understanding that also Delaware where he teaches prospective
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Pose mathematics problems to students report a signifi cant increase in students’ teachers and doctoral students. He conducts
that are just beyond what they currently skills. Apparently, it is not the case that only research in classroom learning and teaching,
know how to handle. Useful problems one set of instructional features facilitates and teacher preparation.
are those for which students have many conceptual learning and another set Douglas A Grouws is Research Professor and
of the prerequisite skills but that require facilitates skill effi ciency. Two quite different William T Kemper Fellow at the University
something more or different. kinds of features both appear to promote of Missouri where he works with doctoral
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Ask students to present their solution skill learning. students and conducts research. His
strategies for a challenging problem Perhaps the nature of skill learning research focuses on mathematics teaching
and engage the class in examining the is somewhat different under the two and curriculum evaluation.
mathematical validity of the strategies. instructional approaches. The measures
It is well known that understanding develops used to distinguish different kinds of skill
Further reading
as people try to resolve perplexities or competencies have not been sensitive
dilemmas that cannot be immediately sorted enough to confi rm this. But we can say that
Good TL & Grouws DA, (1977) A
out. Wrestling with perplexing situations instructional methods are likely to facilitate
Process-product Study in Fourth-grade
often results in rethinking ideas and creating more than one kind of learning. So, when
Mathematics Classrooms. Journal of
new and better explanations for how things choosing methods it is wise to consider the
Teacher Education, 28 (3), 49–54.
work. full set of learning goals that are valued. If Hiebert J & Grouws DA, (2007) The Effects
both conceptual understanding and skill of Classroom Mathematics Teaching on
Skill effi ciency effi ciency are desired, the evidence would Students’ Learning. In FK Lester Jr (Ed.),
How can effective instruction help recommend using the instructional features Second Handbook of Research on
students develop effi ciency in executing for conceptual understanding, or using a Mathematics Teaching and Learning
mathematical procedures? Patterns in the balance between the two approaches. 371–404. Charlotte, NC: Information Age
results of research point to a constellation Publishing.
of instructional features that facilitate the Old dichotomies are not helpful
development of students’ skill effi ciency: A number of categories have been frequently
Hiebert J et al. (2005). Mathematics
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Classes should be well organized, fast used to contrast methods of teaching,
Teaching in the United States Today (and
paced, and focused on mathematics; such as didactic versus discovery, direct
Tomorrow): Results from the TIMSS 1999
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Teachers should set the speed, organize instruction versus inquiry-based teaching,
Video Study. Educational Evaluation and
the lesson, and present material; student-centered versus teacher-centered
Policy Analysis, 27, 111–132.
fall 2009 Better: Evidence-based Education 11
Better(US) Fall09.indb 11 14/10/09 13:05:01
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