Lori, a 3rd grader, enrolled midyear at Gibson Elementary in Clark County, Nevada. She had been doing two- and three-digit problems involving place value at her former school, and her mathematics achievement test scores were above average. Yet when Lori was given an individual interview to assess her understanding of place value, these were the results: Teacher: [Indicating a pile of plastic snap cubes.] Can you show me a group of 10?Lori: Yes. [She snaps 10 cubes together to form a stick.]Teacher: Here's a card with a number on it [32].Can you read it?Lori: Thirty-two.Teacher: [Producing a place value board, a piece of cardboard that is blue on the left side, white on the right.] Here's a place value board. We can put tens on the blue side and ones on the white side. Can you show me what 32 means with the cubes on the place value board?Lori: [Places 3 pink cubes on the tens side and 2 white cubes on the ones side.]Teacher: Can you show me how you would do this? [Shows a card containing the problem 32 − 19.]Lori: You can't because there are only 5 cubes here.

To gain a better understanding of Lori's thinking level, the teacher gave her several more tasks. One task used toy animals: 9 cows and 4 pigs: Teacher: If we take all the cows away, will there be any animals left?Lori: Yes, 4 pigs.Teacher: If we take all the animals away, will there be any cows left?Lori: No, they'd all be gone.Teacher: Are there more cows or more animals?Lori: More cows because there are 9 cows and only 4 pigs.

Although Lori certainly understands that cows are animals, as indicated by her answer to the second question, she could not think of cows as “cows” and as “animals”

*at the same time*. Yet this is exactly what she was being asked to do with numbers. To comprehend the place value system, she needed to understand class inclusion—that 32 is 3 tens and 2 ones and at the same time 32 ones (Inhelder and Piaget 1964). Lori was missing a basic classification structure, one of several mental structures that undergird the child's construction of the number system. Piaget maintained that understanding elementary mathematics depends upon these structures and that failures are caused by moving students into symbols before they have had enough time to construct the relationships with objects (1948).Lori had evidently memorized enough to score well on the achievement test, but she lacked true understanding of the number system. Unfortunately, Lori was not an isolated case. Of the 45 third grade students who enrolled midyear at Gibson, only six students understood place value, even though their test scores indicated otherwise. A correct answer on a test does not assure that the student understands the material being taught. “How,” educators might ask, “can we ensure that students understand what they are taught?” Before we can teach for understanding, the question we need to ask is, “What do we understand about the students we are attempting to teach?”

## How Thinking Develops

At Gibson Elementary, the entire staff (including the principal) has been trained in the Developmental Activities Project (dap) in order to understand how students develop logical thinking, how to assess each student's developmental level, and how to create an environment to enhance development.

Begun in 1981 and used at Gibson since 1990, dap bases its framework on Piaget's 46 concrete operational mental structures. Mental structures are the processes we use to create relationships and to understand the data we take in through our senses. In other words, they are the things we think

*with*, whereas the content of our thought is what we think*about*. In the example of Lori, how the sets and subsets are related (cows belong to the category of animals) is the structure, and pigs/cows/animals are the content. There can be no understanding without the use of mental structures to relate one piece of information to another. “Learning” devoid of mental structuring is simply fact memorization, and memorized facts can be forgotten. In contrast, once structures are developed, they are wired into our circuits.Mental structures developed during the elementary years fall into several categories, including: classification, relations, number, spatial relations, time, and measurement. Within each of these categories, structures develop in a hierarchical sequence that is the same for all students. For each individual, however, development follows a unique timetable that is influenced by factors including social environment, maturation rate, experiences, and disposition to learn.

According to Piaget, students do not develop logical thinking structures from being told or shown what to do or from paper-and-pencil exercises but from acting upon objects. Gradually students internalize these actions, learning to perform them mentally and to represent them symbolically. For example, to develop classification, a student first needs to physically move objects (for example, a set of shapes) into designs or groups according to the student's own scheme. As the student decides where to put the various shapes, what to name the groups, and how to combine or separate groups, he or she builds a classification structure. Later the student might draw a picture of the arrangement. Still later, he or she would be able to classify mentally without needing objects to manipulate. Once the structure is developed, the student can apply it to any content.

## An Environment for Development

To create an environment that promotes the development of the various thinking structures, the dap approach requires sets of carefully designed materials. Sets are colorful and interesting and provide opportunities for many starting points and possible activities. Each set contains enough objects to keep a student interested, and each classroom has enough sets so that every student has a choice of sets. Materials are stored in clear bags or open tubs to facilitate selection; larger or messier materials are located at permanent centers. To spread out their materials to work, students need large spaces such as tables or open floor areas.

*for classification*: creepy crawlies, toy foods, Attribute Blocks, buttons;*for relations (ordering)*: Cuisenaire rods, bead necklaces of different lengths, paper dolls of different sizes, nesting cups;*for number*: Teddy Bear Counters, pennies, spray-painted lima beans, snap cubes;*for spatial relations*: Pattern Blocks, a model village, clothes on a clothesline, pegs and pegboards;*for time*: sand timers, sequencing cards and pictures;*for measurement*: Lots-o-Links, paper clip chains, strings, containers for water or beans.

Because students will be at different thinking levels and will progress at their own rates, each student chooses the activity he or she will do each day. Learning activities must fit each student's level to promote optimal development, and when students choose their activities, they find their own thinking levels. Motivation is built in, behavior problems disappear, and copying or comparisons among students are minimized. Teaching with objects is not chaotic and noisy when teachers manage the classroom with two basic rules: (1) Don't bother others, and (2) Take care of the materials.

## Understanding for Teaching

As students choose their sets, they are encouraged to construct their own arrangements, groupings, patterns, and relationships. The teacher interacts with individual students about their activities, beginning by diagnosing at what level the student is working. The teacher may gather information through informal questions and observations or may choose to give a formal assessment such as the class inclusion task used with Lori. Knowing the individual's developmental level lets the teacher ask questions tailored for each student.

If a student makes an error in thinking, the teacher asks questions, and, if ready, the student will self-correct. If the student is not ready to develop a structure, all of the explanations and reinforcements we might try will not increase understanding. For example, some adults become concerned because we leave Lori with a “wrong answer.” Telling Lori she was incorrect would not help her develop the class inclusion structure. It would do more harm than good, because she would not be constructing for herself. Children's errors are necessary rungs on the ladder of development. Because reinforcement for logical thinking makes students dependent upon the teacher, dap teachers avoid it.

The Development Activities Project approach uses a questioning cycle of open suggestions and further diagnosis. An important element of the cycle is the introduction of cognitive dissonance. When the teacher has ascertained the student's thinking level, he or she may ask a question that causes a slight disturbance in the student's intellectual equilibrium. For a student who is ready to develop a structure, the right question at the right time may initiate a process of mental reorganization.

Recently, a teacher described an interaction with a 5-year-old in which questioning stimulated a new level of thinking. The student, working with a set of toy animals, had grouped them in several ways and explained her groups. Toward the end of their 5-minute interview, the teacher pulled out a group of 10 horses and 3 dogs and asked: Teacher: Are there more horses or more animals?Student: More animals, of course!Teacher: How do you know?Student: Well, if you tried to count all the horses in the world it would take a real long time, but if you tried to count all the animals including the horses, you'd be dead first!

## Assessment for Understanding

The bottom line in teaching is always assessment: Do students understand what they are learning? We believe that the most valid assessment of student understanding is to provide a problem situation the student has never seen before. The more of the process the student supplies, the more we can be certain of student understanding.

Piagetian tasks for the 46 mental structures form our framework for assessment. By keeping the structural elements constant but varying the content and the context, new situations can be created. For example, the child's understanding of class inclusion could be gauged by changing the questions and materials, for example, “If we made a group of all of the students or a group of all of the people, which would be more?”

Other evidence of understanding may be collected—anecdotal records, checklists, photographs, and videotapes. Students also document their progress in the form of drawings, writings, charts, graphs, and constructions. Because organizing data is a logical process, students demonstrate more understanding if given access to the raw materials of record keeping—newsprint, colored paper, stampers and pads, crayons, glue—and encouraged to create their own record of their activities. Drawing helps children make the natural transition from actions on concrete objects to mental and symbolic representations of these actions.

## Successes with dap

Do students really develop understanding with the Developmental Activities Project? A carefully controlled, three-year longitudinal study in Iowa involving 340 students compared students in dap classrooms with students in classrooms using more traditional approaches to teach mathematics and science (Phillips 1989). Each child received an individual interview at the beginning of kindergarten, 1st, 2nd, and 3rd grades. Interviews included several Piagetian tasks testing for mental structures plus some performance math problems. Baseline data in kindergarten indicated that dap and control groups were essentially equivalent; however, the dap students scored significantly higher on all measures in each subsequent year. Along with the increased development of class inclusion (and several other structures), students showed a better grasp of place value!

At Gibson Elementary, further evidence of understanding is reflected in the comments of some of the parents, students, and teachers: Parent: My child sees so many things mathematically now.Teacher: Reteaching is virtually nonexistent because once a child has a structure in place, it is not forgotten.Parent: My daughter finds so many ways to solve a problem.Student: This is the most I have learned in such a short time in all the schools I've been to. I learned so, so, so much math this year.

The words of one teacher summarize our experience with the dap approach in developing understanding: To be present when a child develops a structure and to know that I set the stage by asking a question means more than having a child memorize my words. If they do it themselves, I know they have something that will last the rest of their lives. Teaching within the Developmental Activities Project framework is teachingwithunderstandingforunderstanding.