Making meaning

Who is taking care of that?

The Hiebert & Grouws chapter I talked about last Friday pointed to two elements of instruction that cut across different pedagogical styles: explicit attention to concepts and struggling with important mathematics. In this post I want to concentrate on the first of these. We make things explicit by talking about them. So, somewhere in a mathematics classroom, there should be talk about meaning. I’m going to delve into two research papers, one co-authored by Grouws and one co-authored by Hiebert, and look for how meaning shows up in classroom talk.

Improving the traditional model

The Missouri Mathematics Effectiveness Project was a two-phase experimental study in 4th grade classrooms (Good & Grouws, 1979a, The Missouri Mathematics Effectiveness Project: An Experimental Study in Fourth-Grade Classrooms; 1979b, Experimental Study of Mathematics Instruction in Elementary Schools). The first phase studied 100 teachers in a relatively middle class district, and identified 9 teachers who were consistently effective and 9 who were consistently ineffective over two years, measured by gains for their students on the Iowa Test of Basic Skills. They developed a behavioral profile for each group through classroom observations, distilled a set of factors that distinguished the groups, and assembled them into a training program consisting of a 45-page training manual and two 90 minute sessions.

The second phase took place with 40 volunteer teachers in a different district with many schools in low socio-economic areas. Schools were randomly assigned to either receive the training and implement it, or to continue teaching as they had been. Students in the former group of schools generally outperformed students in the latter. “Treatment students moved from 26.57th to 57.58th percentile on national norms; controls moved from 29.80th to 48.81st.” The improvement in the control group suggests the possibility that teachers in that group sought to improve their teaching simply by virtue of being in the study, while continuing their traditional approach (the so-called Hawthorne effect)—they had been told that they would receive the training the following semester. In this case the possibility of a Hawthorne effect suggests that program was even more effective than the results show.

The program, called Active Mathematics Teaching, had the following characteristics:

(a) The program, in total, represents a system of instruction; (b) instructional activity is initiated and reviewed in the context of meaning; (c) students are prepared for each lesson stage so as to enhance involvement and to minimize student performance errors; (d) the principles of distributed and successful practice are built into the program; (e) teaching presentations and explanations are emphasized.

The lesson structure consisted of daily review, development, seatwork, and homework. The daily review and seatwork sections were fairly traditional: collecting homework, reviewing skills, some mental computation exercises for the daily review, and providing “uninterrupted successful practice” for the seatwork, with accountability by checking work at the end. The development section is where the meaning making comes in. In this section teachers should

1. briefly focus on prerequisite skills and concepts

2. focus on meaning and promoting student understanding by using lively explanations, demonstrations, process explanations, illustrations, etc.

3. assess student comprehension
   (1) using process/product questions (active interaction)
   (2) using controlled practice

4. repeat and elaborate on the meaning portion as necessary. [Emphasis added]

Product questions “ask students to provide an answer,” whereas process question ask student to “explain how the answer was or could be obtained.” Importantly for my main interest in this post, teachers are doing most of the explaining:

We feel that the presence of a few process questions in the development stage of a lesson are helpful (especially when a new principle is being introduced) because listening to a student’s explanation can help teachers diagnose inappropriate assumptions, etc., that students have made. However, we believe that most of the process development can be done through teacher modeling of process explanations rather than by asking students to respond to process questions. [Emphasis added]

The development section was the most novel element in the program; it was the one the teachers had the most difficulty implementing.

Developing an alternative model

Hiebert & Wearne (1993, Instructional Tasks, Classroom Discourse, and Students’ Learning in Second-Grade Arithmetic) followed second-grade classrooms for 12 weeks learning place value and multidigit addition/subtraction. Two of the classrooms were “alternative” classrooms that had already broken with the textbook tradition; the study observes and characterizes them precisely. They had fewer problems per lesson, more time per problem, more questions asking students to describe and explain strategies, longer student responses, and higher performance and greater gains by year’s end.

The difference between the classrooms is illustrated vividly by comparing transcripts. This is from Classroom A, one of the traditional classrooms.

Teacher: “Look at Row A. See where it says 580 take away 234? Okay, what are you going to ask yourself?”
Student: “Can I take 4 from 0?”
Teacher: “Can you?”
Student: “No.”
Teacher: “What do you do?”
Student: “Go to the 10s, cross out the 8 and put 7.”
Teacher: “Go to the 10s, cross out the 8 and put 7. Good. What are you going to do with the 10 that you took away?”

Here the talk is about working through the procedure. Contrast with this transcript from Classroom D, one of the alternative classrooms.

Teacher: “Jane, can you come up and tell us how you did this? I didn’t see any blocks out on your desk, so I want to see how you did this.”
Jane: “I counted in my head.”
Teacher: “Where did you start?”
Jane: “I put 145 down and then I counted 135.”
Teacher: “How did you count 135?”
Jane: “I put a hundred with the 135 to make 200.”
Teacher: “She combined the 100 here with the 100 here and got 200. Then what did you do?”
Jane: “I put the 4 and the 3 together.”
Teacher: “Why did you put the 4 and the 3 together?”
Jane: “Because they’re the ones.”
Teacher: Oh, so you’re combining the 4 and the 3 for a very specific reason. Can you tell us about that again?
Jane: Tens. They’re 10s. So I got 7.
Teacher: Why didn’t you write the 7 down?
Jane: Because its going to change.
Teacher: How do you know it’s going to change?
Jane: Because it’s 5 ones, and 5 and 5 is 10, and that makes it 8.
Teacher: What’s the answer then?
Jane: (Writes 280).
Teacher: Who has another way we can do this?

In this exchange, the meaning talk is coming from the student, with careful solicitation by the teacher. I loved the teacher move after Jane incorrectly referred to the 4 and the 3 as the ones. Not correcting, but not allowing it to pass either. The teacher instigated a pause and a reflection—”Oh, you had a specific reason.”

The authors summarize the difference between the two groups of classrooms:

Students receiving the conceptually based instruction spent a greater percentage of time working on the rationale for procedures and examining the legitimacy of invented procedures. Students receiving the traditional instruction spent a greater percentage of time practicing taught procedures.

Two different classrooms, a common element, and a way forward

These two papers had very different research methodologies and dealt with very different traditions of teaching. What interests me is the common principle that emerged from both: explicit attention to meaning. It sounds obvious, but as I have described here and here, it’s not easy.

I have presented a contrast that seems to be a dichotomy. The answer to the question in “Who is taking care of making the meaning?” seems to be either the teacher or the students. But in reality, in a classroom where talking is happening at all, a more complex mix emerges.

The 5 Practices for Orchestrating Productive Mathematics Discussions describe such a mix.¹ I’ll talk more about these in another post, but briefly, in a classroom where students are working on a problem, teachers anticipate what the students will do, monitor their work, select examples to discuss with the whole class, sequence the examples to build a coherent discussion, and then explicitly connect the different methods.

In these classrooms, teachers are conducting the orchestra and listening to the choir; teachers are bringing knowledge and seeing student thinking; teachers are making decisions on the fly every minute about which move to make next.

Many approaches to teaching are viable—explicit instruction, guided inquiry, some synthesis of both—if they share the common element of making meaning and attending to concepts.

1. Full disclosure: we make extensive use of this framework in the IM curriculum.