Article:
Hiebert, J. and Wearne, D. (1992). Links Between Teaching and
Learning Place Value With Understanding in First Grade, Journal for
Research in Mathematics Education, 23 (2) 98-122.
Overview of Experiment:
Hiebert and Wearne conducted their experiment with 151 first-grade
children distributed through six classes. They focused on a
twelve-day unit on two-digit place value in January, and a ten-day
unit on two-digit addition and subtraction without regrouping in
late April and early May.
Two teachers taught their classes the regular text-based lessons,
while each of the authors taught alternative lessons to two of the
remaining classes. The students received a pretest on basic place
value concepts in December, and, after each unit, a test that
included sections on place value and two-digit addition and
subtraction with and without regrouping. Seventy-two children were
interviewed after each of the units to evaluate strategy use. An
observer was also present in each classroom for twenty lessons and
was later asked to assess the differences between the two classroom
formats.
Some definitions:
"Links" are the connections between different classroom elements:
"questions of teaching and questions of learning and the potential
links between them" and "links between instruction, understanding,
and performance" (p. 98).
"Representations" are means used by the teacher to transmit
mathematical knowledge to the students. These can include
manipulatives such as base-10 blocks and Unifix cubes, and written
symbols such as formulas or problems.
"Conceptually based instruction" is used here to mean the kind of
teaching that fosters understanding of the reasoning and logic
behind mathematical ideas. Place-value concepts include grouping
issues (10s, 1s, etc.) and the use of units for writing and using
numbers. The authors maintain that while in text-based instruction
these ideas may not be transitted to or understood by the students,
in conceptually based instruction the focus is on learning why we
use place value and the ability to transfer this knowledge to
mathematical achievement.
Quotes and Comments:
"Our objectives in this report are to contribute to the literature by
describing what one form of conceptually-based instruction looks like
as it is operationalized in classrooms, by considering its effects in
learning, and by searching for links between instruction, understanding,
and performance" (p. 99).
"Several principles guided the development of instruction that would
support students' efforts to make connections. First, external
representations (physical, pictorial, verbal, symbolic) were used as tools
for demonstrating and recording quantities, acting on quantities, and
communicating about quantities. Second, once a particular
representation was introduced (e.g. base-10 blocks) it was used
consistently to allow students to practice using it as a tool and to become
familiar with the uses it afforded. Third, the representations were used
to solve problems as well as being analyzed as interesting artifacts in
their own right. Fourth, class discussions focused on how the
representations could be used and on how they were similar and
different. The general aim of these principles was to help students
become comfortable with different forms of representation and to build
relationships between them.
"Several additional guidelines were used to design instruction. Physical
materials and verbal stories were used as the initial representations for
quantities and actions on quantities. Pictures of the physical materials
that had been manipulated by the students were then used for
convenience and for focusing class discussions. Finally, written
symbols were introduced as efficient ways of recording the quantities
and actions that had been explored and discussed using the other
representations. Once a particular form of representation was
introduced, it was used continually and interchangeably with previous
forms" (pp. 100-101).
For example, "The January lessons began by posing problems of
finding how many objects there were in large sets, mostly sets between
50 and 100. Class discussions and suggested strategies began with
counting by ones and shifted to counting more efficiently by grouping
and counting by twos, fives, and eventually, tens. One kind of object
investigated was Unifix cubes. These were eventually grouped into
quasi-permanent bars of 10. Two-digit numerals were introduced as
efficient ways of recording the size of sets. Discussions frequently
included the two ways of interpreting the written number -- as ones and
as tens and ones. For example, 57 was interpreted as 57 individual
objects and as 5 groups of 10 objects with seven left-over objects. Base-
10 blocks were introduced as tools, like Unifix cubes but already
hooked together. Most tasks at this point were story situations that
involved dealing with large sets by, for example, packaging them
together for sale in groups of 10. Students used base-10 blocks or
pictures of the blocks or written numbers to help them solve the
problems and then described the strategies they used" (p. 100).
From the observations, three factors were identified that differed
consistently between the text-based and alternative forms of instruction:
coherence within lessons, time spent on individual problems, and the
way in which physical materials were used. Coherence within lessons
was defined as the types of changes that occurred between activities.
These changes could involve any of the following: topic of discussion;
material being used; the representation form used by the teacher (spoken
language, physical materials, pictures, written symbols, and oral or
written stories or description); and the representation form requested of
the students.
In general, students in the alternative classrooms spent more time using
the physical materials, more time was spent in the alternative classrooms
on individual problems, and the alternative classes made fewer shifts in
topic and use of physical material, but more frequent changes in
representational forms requested of the students (which could potentially
help build connections between representations, especially if the topic
remains consistent).
The authors also identify other differences between the types of
instruction, including: "(a) an encouragement in the alternative-
instruction classrooms to consider alternative responses and
procedures compared with a tendency in the textbook-based classrooms
to guide students through a single step-by-step procedure; (b) using the
connections between representations throughout the lesson compared
with introducing one representation and then moving to another
representation without exploiting the connections; and (c) laying the
groundwork in one lesson for future ideas compared with developing
ideas in a somewhat inconsistent way" (p. 116).
Results of the tests and interviews showed many individual differences
within and between classes, but also produced some significant
results. Specifically, conceptually based "instruction seemed to
facilitate somewhat higher levels of understanding, and this seemed to
translate into higher levels of performance on tasks that exploited such
understanding" (p. 113). Students who did not have a conceptual
understanding of place value and related procedures could not ultimately
modify and transfer their knowledge to new skills (namely addition and
subtraction with regrouping).
"We conclude that students' understanding and performance are
connected in complicated ways and that instruction influences the way in
which the interaction plays out. The data reported here suggest that
understanding, as measured by the place-value tasks, does not translate
directly into procedures [i.e. addition/subtraction with regrouping] but
that it does interact with procedures to yield increased flexibility and
power. However, this interaction is influenced by the instructional
environment, and, in this case, flourished more when instruction
attempted to facilitate students' understanding rather than procedural
proficiency" (p. 121).
-- summarized by Jane Ehrenfeld
Learning and Mathematics
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