Mathematics in the City:
Cathy Fosnot, Director
This study reports the results of student achievement data (N= 7821) in grades 3, 4, and 5 in New York City classrooms of teachers involved in a five-year inservice project ( Mathematics in the City at City College of New York) comprised of a two-week summer institute, a year long course investigating the developmental progression of children’s mathematical ideas, and co-teaching on-site. Coupled with the inservice, teachers were provided with Investigations in Space, Number, and Data and Maths in Context , two of the reform-based NSF-funded curricula . The teachers participating came from CSD #2, 3, 4, 5, and 6.
The difference in test scores after only one year of the program produced significant gains in all three grades in comparison to scores of same-grade children in same schools, same districts, but not involved in the program and still using traditional curricula. The mean of the experimental children was solidly within level three, whereas the control group mean was in level two. A covariance analysis was done on grade 4 and 5 data, controlling for entering level by using the prior year’s scores, to ensure that the gains were due to the program. A subtest analysis was also run to look at number relations, computation, operations, measurement, geometry, data and probability, patterns and functions, and problem solving. Differences were significant for the program children in all categories except for computation, operations, and algebra. Here the program children still scored higher than the control group children, but the differences were not significant. The gains on problem solving were highly significant, once again in favor of the program children. Results also showed that the longer the teacher stayed involved in the inservice program, the higher the scores.
Assessing Number Sense and Computation Strategies
Critics of the reform have argued that children using reformed-based curricula do not learn to compute efficiently. To examine computation specifically (and number sense in general) a test was designed to measure the level of computation strategies used. Nine classrooms that were deemed exemplars of the reform were chosen from across three school districts. “Exemplar” was defined as implementing consistently and well what was being taught in the inservice. Matched controls were found (characterized by traditional practice but controlling for socioeconomic levels and teacher experience). A significant difference in strategies was found. Children in the experimental group were found to have a better understanding of number relations. They often solved problems mentally through decomposition and tinkering appropriately with number relations. In comparison, the control group children often rewrote every problem in column fashion, ignored number relations, and performed algorithms regardless of the numbers. For example, experimental children often solved 38 + 39 + 40 + 41 + 42 by simplifying to 80 + 80 + 40, or to 5 groups of 40, and 60% of the third graders tested got the correct answer. In contrast the majority of the control group children rewrote the problem in column fashion. Several made place value errors, and only 48% got the correct answer. Answers from control group children on the problem 147 – 28 ranged from 14 to 966 with 20% of the total answers given not even within a more reasonable range of 100 to 147! No experimental child gave an answer above 147, and 98% of the total answers were within a reasonable range. By far the most interesting finding however, was that the effect of sense-making in classrooms was cumulative. In grade three the difference in unattempted problems was not significant. On grade four it was, and by grade five the difference was highly significant. A learned helplessness was developing in control classrooms. These findings support the argument that when mathematics is taught with realistic contexts, children will build their own ideas and make sense of problems mathematically in their own ways. They will trust in their ability and will at least make attempts to solve difficult problems. In contrast students who are not encouraged to mathematize in their own ways develop a learned helplessness. Math anxiety sets in and when faced with difficult mathematical problems, they give up.
Follow-up Studies to look at Low-performing Schools
For the last four years we have been following the test score data on a targeted SURR school in Harlem. In 1999 it was one of the lowest performing schools in mathematics in the whole city with only 13% of the children in the school achieving levels three and four (proficient). Trailblazers and TERC were introduced and used erratically in 2000. In 2000-2001 we (MITC) were asked to provide staff development, the administrator and teachers attended our workshops at the college, and a decision was made to adopt TERC consistently. Scores on the 2000 tests showed that 20% of the children in the school were on levels three and four. The difference in 2001 was not significantly higher, but by 2002 scores showed a dramatic change reaching nearly 30%. The third grade scores partially explain the rise as they moved in these years (1999 – 2002) from 9% to nearly 40%. These data are especially revealing because 2001-2 was the first school year that the third graders would have had three consistent years of a reform curriculum.