The“math wars” of recent years could be framed as a conflict between those who advocate for direct instruction on basic facts and algorithms vs. those who advocate for a more constructivist approach through which students learn their own ways of solving problems. This representation is oversimplified, but that’s what happens in a war of words.

So let’s not get stuck in the conflict. Instead, let’s focus on areas of agreement.

1. Can we agree that all math textbooks have strengths and weaknesses, and that more important than the book is the ability of the teacher to strategically use the resources of that curriculum?
2. Can we agree that the quality of interaction between teacher and student is more important than whatever latest reform is being touted?

In order to see what is really going on between teacher and student, we need to look deeper than the structure or the room, the type of materials that are being used, or the way in which students are grouped. Liping Ma, in Knowing and Teaching Elementary Mathematics, reflected on how people may misinterpret the situation when comparing traditional classroom settings in China to so-called reform-minded classrooms in the U.S.“If you look carefully at the kind of mathematics that the Chinese students are doing and the kind of thinking they have been encouraged to engage in, and the way in which the teacher’s interactions with them foster that kind of mental and mathematics process, the two kinds of classrooms are actually much more similar than they appear.”

Conventional Versus High-Achievement

In a report to the California State Board of Education Review of High Quality Experimental Mathematics Research, the authors described two types of lessons: conventional lessons and high-achievement lessons. Conventional lessons essentially are comprised of two parts. “First, students sit passively and watch as the teacher demonstrates how to do something new... (and) then abruptly, students are expected to know and independently apply the information newly introduced moments earlier.” Meanwhile, high-achievement lessons have three parts:

• In the first phase, teachers not only demonstrated, but explained, and asked many questions, checked for understanding, or conducted discussions.
• In the second phase students gradually transition from teacher regulation to self-regulation... (which may vary) from students helping one another collaboratively to high levels of teacher help with feedback and correctives.
• In the third phase students demonstrate their ability to independently recall and/or generalize and transfer (what was taught).

In high-achievement lessons, the majority of the instructional time is spent in phase two so that students get substantial opportunity to “own” the concept being taught.

In another report, Cognitively Guided Instruction: A Research-Based Teacher Professional Development Program for Elementary School Mathematics, the authors shared the results of a three-year longitudinal study examining the relationship between teachers’ beliefs and instruction. They used “levels” to describe the beliefs and instructional practices of categories of teachers.

Level 1 teachers believe that children need to be explicitly taught how to do mathematics... (and these) teachers generally demonstrate the steps and procedures.... and the children practice applying the procedures... and there is little or no discussion of alternative solutions.
Level 2 teachers begin to question whether children need explicit instruction... and the teachers alternately provide opportunities for children to solve problems using their own strategies and show the children specific methods.
Level 3 teachers believe that children can solve problems without having a strategy provided for them.... and children spend most of the mathematics class solving and reporting their solutions to a variety of problems.
Level 4 teachers use what they learn from listening to students to make instructional decisions... these teachers continually reflect back on, modify, adapt, and expand their (instructional) models based their understanding of their students’ thinking.

Building on a Framework

Following is a lesson plan framework you can use to develop or adapt lessons that improve the quality of interaction between you and your students. The framework is essentially a set of heuristics that can help you construct a more powerful lesson. It may help you fill in the gaps from an incomplete textbook lesson, assure you that the textbook lesson is complete, or help you come up with a lesson on your own. Here is the framework with examples from actual textbook lessons.

1. What is the stated objective of the lesson you are to teach? How could this be turned into a more meaningful objective? Meaningful objectives include why we are learning this concept and more specifically state related and underlying concepts that aid in true comprehension.

Meaningful objectives are rarely provided by textbooks. The teacher, like the Level 4 teachers previously described, must reflect on the students’ thinking, the related and underlying concepts, and their prior experience with this concept in order to come up with the meaningful objective. Why learn this concept? Why should I care about this? Does learning this help me in daily life? And what related and underlying concepts will really be needed to fully comprehend this concept?

While this may seem like an excruciating process, it is necessary in order to come up with the meaningful objective which will then drive your lesson. If the stated objective is:

Students will calculate the area of a rectangle.

then the meaningful objective might be:

Students will understand that any area can be broken up into equal parts and added to help determine the size of that area for some real purpose such as finding out how long it would take to do something or finding out how many materials may be needed to build something. Students will also understand that if the area is a rectangle and if these equal parts are squares, you can create an array and multiply to find the area.

This is long, but it helps clarify what we should teach.

If this step were avoided, as it often is, the Level 1 teacher might just go ahead and teach a conventional lesson demonstrating the length x width formula without really finding out if students really understand the underlying and related concepts. Students may learn a formula, but it would become a wasted opportunity to allow them to explore the interconnectedness of concepts much less transfer their knowledge to new situations.

2. How can I find out what the students already know about these concepts in a way that also helps them understand why we are learning these concepts?

This step merges two best practices—activating prior knowledge and providing an anticipatory set (from Madeline Hunter’s Elements of Instruction). Often the best way to activate prior knowledge is simply to ask the students questions directly related to the meaningful objective. For example:

• How do you find out how big an area is?
• What do you know about squares and rectangles?
• When we say, “the area of the football field,” what do we mean?
• What is meant by the phrase “equal parts”?

The anticipatory set is intended to motivate the students. Once again, a question related to the reason why we are learning this concept is one of the best ways to do this. You could ask the students, “Would you rather mow the lawn at the park or three of your neighbors’ lawns?” You could also ask, “What are some ways to figure that out?”

3. How can I model or demonstrate the related and underlying concepts and gradually move from teacher-regulation to student self-regulation and—like the Level 4 teacher—utilize the students’ thinking to adapt and guide my instruction?

One way to move consistently from teacher-regulation to student self-regulation and to utilize student thinking to guide the lesson is the Model and Gradual Release method. Here’s how it might work in the lesson on area with suggestions for how much teacher vs. student regulation there should be:

A. Model completely all of the underlying concepts in figuring out the area of something students can relate to, see, or experience. (100 percent teacher regulation/0 percent student self-regulation)
B. Partially model some of the concepts using another example. Have selected students model whatever you don’t. (50/50)
C. Provide a third example, and ask the students to completely model that solution. (10/90)
D. Provide a fourth example, have pairs or small groups find the solution, and ask for volunteers to model. (5/95)
E. Provide a fifth example, and have students find the solution independently. (5/95)
F. Provide a sixth example, have students come up with alternative ways to solve the problem, and model them for the other students. (5/95)
G. Have students come up with an example and then come up with alternative ways to solve the problem and/or new problems that extend the learning to new concepts. (100% student regulation)

Depending on your students’ developmental level, you can skip steps in the process and move more quickly to student self-regulation.

Regardless of the math curriculum that is used or the math reforms being touted, it is important to remember the role of the teacher. Teachers control how textbook resources are used and control their interaction with the students. How teachers handle those responsibilities probably has a greater impact on student achievement than almost anything else.

Questions

1. How do your beliefs about learning math impact your instruction?
2. What are the strengths and weaknesses of your math textbook series?
3. How can you make math objectives more meaningful to students?
4. What are some ways you can find out how your students think about math concepts?
5. How can you change how you teach based on your students’ thinking?
6. How can you move more quickly from 100% teacher regulation to 100% student self regulation?

Resources

Carpenter, Thomas P.; Fennema, Elizabeth; Franke, Megan Loef; Levi, Linda; Empson, Susan B. Cognitively Guided Instruction: A Research-Based Teacher Professional Development Program for Elementary School Mathematics, National Center for Improving Student Learning and Achievement in Mathematics and Science; September, 2000.
Dixon, Robert C.; Carnine, Douglas W.; Lee, Dae-Sik; Wallin, Joshua. Review of High Quality Experimental Mathematics Research, National Center to Improve the Tools of Educators, University of Oregon. March, 1998.
Ma, Lipping. Knowing and Teaching Elementary Mathematics. Lawrence Erlbaum Associates, Publishers, 1999.

Rod Haenke has been guest editor of this series of Best Practice articles for Today’s Catholic Teacher. Rod will continue to serve as guest editor for this series and is actively seeking authors who would be willing to share their knowledge and expertise in next year’s series. Stay posted for topics for next year’s articles. Rod is a former Catholic school teacher and has taught Master’s level courses for St. Thomas University and St. Mary’s University in Minnesota. He is founder and director of Instructional Designs, Inc., and can be reached at rhaenke@instructionaldesigns.net.