**James Hiebert** and **Douglas Grouws** reveal which elements of math instruction have been shown to help students’ conceptual understanding and their skill efficiency ** **

**DECIDING WHICH INSTRUCTIONAL METHODS **are most effective for increasing students’ learning continues to be one of the great challenges for educational research. Should teachers use Method A or Method B? Which one will show the best results?

What we know |
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● To improve conceptual understanding, make explicit the important mathematical relationships and ask students to “work and wrestle.” ● To improve skill efficiency, use rapid pacing, modeling, and moving to error-free practice. ● Balance these two approaches, with a heavier emphasis on conceptual understanding. |

An important truth about the effectiveness of instructional methods is that particular methods are not, in general, effective or ineffective. Instructional methods are effective for something. Educators always need to be clear about what this something is when they talk about the effectiveness of instructional methods.

Focusing on the following two learning goals, we ask which instructional methods are most effective: conceptual understanding – the construction of meaningful relationships among mathematical facts, procedures, and ideas; and skill efficiency – the rapid, smooth, and accurate execution of mathematical procedures. These two learning goals are central to mathematics learning and have often competed for attention.

**Conceptual understanding **

Research conducted over the past 75 years has spanned a wide range of mathematics topics, age groups, and class settings. The results point to two important features of teaching that can help the development of students’ mathematical understanding.

**Work and talk **

Teachers and students should intentionally and explicitly talk about, and work on, important mathematical relationships.

At least some time during each lesson should be spent on the following activities:

- Examining relationships among facts, procedures, and ideas within a lesson and across lessons. Is one problem a special case of the preceding problem? How the problems solved today are similar to and different from the problems considered yesterday? How do linear graphs, tables of ordered pairs, and linear equations all represent the same linear function?
- Exploring the reasons why procedures work as they do. In addition to practicing procedures, students should examine and discuss why the procedures work, especially when new procedures are being introduced. Why do we usually add from right to left? When we solve an equation, why must we do the same thing to both sides? and
- Lessons frequently involve solving problems using different procedures and then examining the similarities and differences between them. How is Jack’s procedure different from Martha’s procedure? Note: It is not necessary for students to practice multiple procedures for solving similar kinds of problems, but comparing different procedures is beneficial.

**Work and wrestle **

Teachers should provide opportunities for students to wrestle with key mathematical ideas and ensure that students do some of the important mathematical work in the lesson.

Allowing students to work hard to make sense of mathematics does not mean standing by while they become unnecessarily frustrated and confused, nor does it mean presenting problems that are well beyond their reach. But it does mean providing time during the lesson when students are allowed to work on problems they don’t immediately know how to solve, and it does mean resisting the temptation to jump in and tell students how to do something at the first sign of uncertainty.

At least some time during each math lesson, teachers should engage in the following activities:

- Pose mathematics problems to students that are just beyond what they currently know how to handle. Useful problems are those for which students have many of the prerequisite skills but that require something more or different.
- Ask students to present their solution strategies for a challenging problem and engage the class in examining the mathematical validity of the strategies.

It is well known that understanding develops as people try to resolve perplexities or dilemmas that cannot be immediately sorted out. Wrestling with perplexing situations often results in rethinking ideas and creating new and better explanations for how things work.

**Skill efficiency **

How can effective instruction help students develop efficiency in executing mathematical procedures? Patterns in the results of research point to a constellation of instructional features that facilitate the development of students’ skill efficiency:

- Classes should be well organized, fast paced, and focused on mathematics;
- Teachers should set the speed, organize the lesson, and present material;
- The teacher’s modeling of new material must be clear and concise; and
- Once students are ready to succeed, significant time should be allotted for error-free practice.

Lessons should progress from the teacher’s presentation and the modeling of mathematical material to practice time for students. During the modeling part of the lesson, the teacher presents material clearly and in a meaningful, organized way. The teacher also asks questions throughout this part of the lesson. These questions are mostly product questions requiring students to provide only answers and are asked publicly for the entire class to hear. The transition from the whole-group portion of the lesson to individual student practice occurs only after the students are well prepared. The fast pace of the lesson keeps students’ attention while ensuring that students move from one task to another successfully. Lessons are content-focused, so that class time is devoted to mathematical tasks as opposed to managerial tasks, such as handing back papers or discipline.

**Not a simple correspondence **

The apparent disparity between one set of instructional features for conceptual understanding and a completely different set of features for skill efficiency, breaks down when examining the results of studies on conceptual understanding that also report a significant increase in students’ skills. Apparently, it is not the case that only one set of instructional features facilitates conceptual learning and another set facilitates skill efficiency. Two quite different kinds of features both appear to promote skill learning.

Perhaps the nature of skill learning is somewhat different under the two instructional approaches. The measures used to distinguish different kinds of skill competencies have not been sensitive enough to confirm this. But we can say that instructional methods are likely to facilitate more than one kind of learning. So, when choosing methods it is wise to consider the full set of learning goals that are valued. If both conceptual understanding and skill efficiency are desired, the evidence would recommend using the instructional features for conceptual understanding, or using a balance between the two approaches.

**Old dichotomies are not helpful **

A number of categories have been frequently used to contrast methods of teaching, such as didactic versus discovery, direct instruction versus inquiry-based teaching, student-centered versus teacher-centered teaching, and traditional versus reform-based teaching. Although these categories and labels might have been useful for some purposes in the past, the instructional features that facilitate conceptual understanding and skill efficiency do not fall neatly into these categories. They no longer capture the distinctions suggested by the data. Attending explicitly to key mathematical relationships, for example, can be done within any of these methods.

These findings suggest a weighted balance between the two instructional approaches might be appropriate, with a heavier emphasis on the features related to conceptual understanding.

**About the authors **

**James Hiebert** is the Robert J Barkley Professor of Education at the University of Delaware where he teaches prospective teachers and doctoral students. He conducts research in classroom learning and teaching, and teacher preparation.

**Douglas A Grouws** is Research Professor and William T Kemper Fellow at the University of Missouri where he works with doctoral students and conducts research. His research focuses on mathematics teaching and curriculum evaluation.

**Further reading **

Good TL & Grouws DA, (1977) A Process-product Study in Fourth-grade Mathematics Classrooms. *Journal of Teacher Education*, 28 (3), 49–54.

Hiebert J & Grouws DA, (2007) The Effects of Classroom Mathematics Teaching on Students’ Learning. In FK Lester Jr (Ed.), *Second Handbook of Research on Mathematics Teaching and Learning *371–404. Charlotte, NC: Information Age Publishing.

Hiebert J et al. (2005). *Mathematics Teaching in the United States Today (and Tomorrow): Results from the TIMSS 1999 Video Study. Educational Evaluation and Policy Analysis*, 27, 111–132.