Most teachers, studies have shown, are able to distinguish between types of math problems as well as the primary strategies that children use to solve different types of problems. Most teachers can also predict what problems their students will be able to solve. However, for most teachers, this information about what individual children can do is not organized into a network of knowledge about children’s cognition in math. Since teachers’ knowledge of students’ problem-solving abilities is correlated with student achievement, an ongoing research and development project at the University of Wisconsin/Madison called Cognitively Guided Instruction has focused on organizing our understanding of children’s learning of mathematics into a coherent structure. Research has shown that students, in the classrooms of teachers who have studied Cognitively Guided Instruction, demonstrate better knowledge of both math facts and problem solving and have more confidence in their math ability.

Teachers interested in Cognitively Guided Instruction attend a four-week seminar during which they learn a sequential framework of math problems and study the way in which children think and learn about mathematics. They discuss instructional implications with other teachers, and design instructional programs according to their own beliefs and teaching styles. Teachers who use Cognitively Guided Instruction tend to rely more on problem solving and less on number facts than teachers without such training. These teachers use a wider variety of instructional strategies. They listen to their students describe their thinking and thus they know more about individual children’s problem-solving processes. Additionally, they are more likely to believe that instruction should be built on each student’s existing knowledge.

At this point, however, research has not been able to show how these experimental teachers successfully use knowledge about their student’s thinking to organize their class and develop a curriculum that leads to increased learning. In order to determine precisely how teachers changed their teaching to improve achievement, Elizabeth Fennema, Megan L. Franke and Thomas P. Carpenter, University of Wisconsin/Madison and Deborah A. Carey, University of Maryland, studied one teacher (Ms.J.) over several years, to document how she taught, how her teaching changed and developed following training, and the effect that this changed instruction had on her students’ achievement.

### An In-Depth Study of One Teacher’s Classroom

Ms. J. had been teaching first grade for 12 years in Madison, Wisconsin when she volunteered for a workshop on cognitively guided instruction. The ethnic and socioeconomic range of her students is wide. Her classes average 24 students, but many children enter and leave during each school year.

During the first year of their study, researchers gathered information about Ms. J.’s knowledge of children’s thinking, her beliefs about teaching and learning mathematics, and her cognitive perspective. That summer she attended a seminar on Cognitively Guided Instruction. In the following years, researchers observed Ms. J.’s math instruction and measured her students’ achievement each September and April. Ms. J. met monthly with other teachers participating in the study and with researchers to discuss ideas about teaching. In these meetings teachers talked about their instructional decisions and researchers shared what they knew from studies about how children think. The goal of these meetings was for teachers to help researchers understand how more in-depth knowledge of the way children think could be used to plan instruction. In the third year researchers continued to observe math instruction in Ms. J.’s class. They also talked with her and her students informally. At the end of the third year, a few children were randomly selected from each of her teaching groups (formed on the basis of problem-solving abilities). Each of these children was individually tested and interviewed to assess his or her number-fact knowledge and problem-solving ability.

### Ms. J.’s Teaching

In her knowledge of problem types, their relative difficulty and solutions for solving them, Ms. J. ranked near the top among teachers who had attended the CGI seminar. She was able to consistently predict which children would be able to solve which problems and which strategies they would use. She clearly understood the complexity of children’s thinking in math. Her knowledge was extensive, accurate, hierarchically organized and integrated in a complex way. Fennema et al. report that “interwoven with this knowledge was information about the use of counters, the relevance of the problem context to the children, the language used in the problems, the choice of number size and the selection of problems for which a variety of strategies could be used”.

Ms. J. expressed the belief that children should be taught to understand mathematics by building on what they already know. She learned what her students knew by listening carefully to them as they explained how they solved problems. In this way she learned what they were ready to do and at the same time, avoided drilling them on skills or ideas they had mastered. Compared with other teachers, Ms. J. questioned children more about their thinking. She also expected her students to use multiple strategies at a higher level. During math class she was almost always actively engaged with one child or group but was aware of what the other children were doing. The room was noisy, but observing the children revealed that they were working on and thinking about mathematics.

Although there was considerable variety in math activities, consistencies in her instruction became apparent. Children were expected to be actively engaged in mathematics, usually solving word problems written by Ms. J., their peers or themselves. They were expected to persist, and to be able to report how they had solved a problem and to reflect on their own thinking by comparing their solution with that of another student. They were also expected to listen to and try to understand the solutions of others.

Math class often began with students working together on an unfinished problem from the previous day, on a problem a student had been working on at home or on problems that Ms. J. wanted everyone to think about. After this, the children worked in three groups. Ms. J. would work with one group, the student teacher with another, and the third group would work individually or cooperatively. Ms. J. also used large sheets of newsprint to write story problems or number sentences for individual children to solve. She would listen to a child, assess and record that child1s thinking, and plan further instruction on these sheets. She constructed several sheets for each child each week. During “big sheet” time, Ms. J. was seated at a table and children would come to her with their sheets. Rarely did children wait in line to see her. They had learned to go on with their work until she was available. She asked each child how they solved the problem and if it had been too easy or too hard; children were always responsible for knowing how they arrived at an answer. She took notes on the big sheet itself and used these notes to create new sheets and as a record of the child1s progress. Ms. J. made instant instructional decisions that ensured that each child worked at his or her current skill level. She insisted that more advanced children solve problems in more advanced ways and would not accept easy answers from them. She made sure, however, that each child had successful experiences. Observers were surprised by the quantity and quality of cooperation among students. Ms. J. created an atmosphere in which mistakes were understood to be part of the process and in which children were very supportive of other children’s thinking. Ms. J. continually emphasized that everyone was responsible not only for his own learning, but for the learning of others as well. Children learned to respect the time it took to think, and they persisted at solving their problems. Ms. J. often said that no problem was too hard to solve. Problems unsolved by a child or group were often written on the board so that they could be discussed the next day. Persisting for several days was not unusual.

From almost the first day of school, children spent most of their time solving word problems constructed by Ms. J. She used what she knew about the children to vary the types of problems and the difficulty level. The context of the problems was derived from activities in other subject areas. Ms. J. stressed concepts over rote skills and math facts, in the belief that children would learn skills as they learned to solve problems. Teaching her students to invent and solve their own math problems was a primary goal. Her students especially enjoyed challenging themselves with large numbers. Always, however, the emphasis was on the solution strategy rather than on the answer.

Ms. J. introduced new ideas when she observed that children were ready for them. For example, when she noticed that some children demonstrated an understanding of grouping by 10, she gave those children problems with numbers higher than 20 to encourage regrouping. She also made sure the other children were aware of how groups of 10 had been used in the solutions of those problems. Before long, many children were regrouping by 10s and 100s. Although she did not teach specific lessons on how to solve particular problem types, she was always aware of the mathematical relationship of various ideas, and her comments stressed relationships that she hoped the children would learn. When children were ready, she added multiplication, division and fraction ideas to problems.

Ms. J. used a wide range of problems for assessment which she carried out in both written and oral form. The purpose of the assessment was to gain knowledge that could be used to make immediate decisions about how to structure the learning environment for each child. Ms. J. also used this knowledge to communicate with parents about their child’s progress.

### What Children Learned

The researchers assessed students’ knowledge at the beginning and end of each year. All the problems given to students as part of the assessment were more difficult than those typically found in first- or second-grade textbooks. The children were accurate in describing what they knew and did not know. Those children in the highest group were able to solve three-step word problems with more than one operation, division problems of 1-3 digits, fraction problems, problems with missing information and problems in which all the numbers exceeded 100. However, there were consistencies in the learning of all three groups. All the children demonstrated a strong sense of number. They knew readily which numbers came before and after a given number, and which numbers were greater or less than others. They were comfortable manipulating numbers, taking them apart or putting them together to make more than one number, counting by various numbers (2s, 3s, 5s, 7s, etc.). Although their strategies varied in maturity, all were able to solve addition and subtraction number facts correctly. Their understanding of place value was clearly evident in their solutions. Even a student with learning disabilities was confident that if he persisted, he was capable of solving any problem given to him by Ms. J.

### Conclusions

In summary, the children in Ms. J.’s classroom were significantly above average in math achievement at the end of the year. Her heterogeneous class of students clearly exceeded the standards proposed by the National Council of Teachers of Mathematics.

Unexpectedly, Fennema et al. found that Ms. J. used her knowledge of children’s thinking in surprising ways. She did not use the hierarchy of problem types taught in the seminar on Cognitively Guided Instruction to make sequential instructional plans. Instead of teaching the problems types in sequence, she exposed children to all types of problems almost from the first week of school. Her approach gave students opportunities to solve all types of problems using strategies at their current skill level. She sequenced instruction so that each student advanced from his or her current cognitive and skill level using problems that could be solved with a variety of strategies. This unorthodox approach provided an enriched curriculum that included experiences not usually offered to first-grade children. She encouraged them to try various strategies on the same problem and she found ways for them to explain their problem solutions. Ms. J. reported that through the workshop on Cognitively Guided Instruction she came to realize that her students could do more than she had ever expected and this enabled her to understand the complexity of children’s thinking and to “get a handle on all my children, from the lowest to the highest…”. From the moment she recognized this, she continually used knowledge about children’s thinking to expand her expectations of children and to structure their learning environments to enable them to do remarkable problem solving. “The more I challenged, the better they got.. it was the students who convinced me that Cognitively Guided Instruction works, and they went far beyond what I ever expected that they could do.” She says that teachers should have a thorough understanding of the content of problem solving, and of problem types and solution strategies before trying to impart them to students.

These researchers point out that the depth of knowledge about cognition in mathematics is greater than in some other subject areas. For this reason, Cognitively Guided Instruction may not be as easily applied to other subjects. Fennema et al. conclude that Ms. J. is clearly an exceptional teacher who has helped them to understand how knowledge obtained from research can be used in classrooms to enrich mathematical thinking and increase learning.*“Using Children’s Mathematical Knowledge in Instruction” by Elizabeth Fennema et al, American Educational Research Journal, v30 n3 p555-83 Fall 1993. *