## Learning and Mathematics

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### Conceptually based instruction - Hiebert and Wearne (1992)

James Hiebert and Diana Wearne, of the University of Delaware, describe in this article an experiment they conducted in which they compared text-based instruction with conceptually based instruction in a series of lessons on place value and related concepts. Their findings indicate that students need to be able to make links between different representations (or forms) of the same concept.

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.

"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