Jan Mokros, Mary Berle-Carman, Andee Rubin, and Kim O'Neil
TERC, Cambridge, MA
During the 1994-95 school year, several second grade classrooms in the Boston area implemented the Investigations in Number, Data, and Space curriculum, which is a K-5 curriculum program emphasizing number, data, space, and the mathematics of change. We were particularly interested in examining how children using a quite different approach to learning number operations would make sense of a variety of problems involving number operations. How would Investigations children, who had not learned any standard algorithms, perform on number tasks relative to children who had learned standard algorithms and had also invented some of their own strategies? The Investigations curriculum emphasizes children's construction of their own strategies for solving problems. At the second grade level, there are three, month-long units that focus on numerical operations. Students are not taught algorithms at all, but are encouraged to develop their own strategies based on sound number sense and understanding of the meaning of each problem.
For example, consider this problem:
Yesterday at the park, I counted 39 pigeons. When a big dog walked by, 17 of them flew away. How many were still there?
The problem can be approached in many effective ways, including counting up by ones, tens, or a combination of tens and ones; counting down in a similar fashion; or by moving between "landmark" numbers, such as 40, 30, or 20. Other strategies involve the decomposition of one or both numbers into "friendlier" numbers, with subsequent adjustments made to handle the leftover pieces of the numbers. In the curriculum, students experience inventing and applying their own ways of solving problems, making sense of other people's strategies, and keeping track of the processes involved in arriving at solutions.
Method
The assessment of the effectiveness of this approach to teaching/learning operations consisted of end-of-year individual interviews, lasting approximately 25 to 45 minutes, with 50 students. Thirty students from three different classrooms made up the Investigations group, and twenty students from two classrooms using another curriculum made up the Comparison group. The Comparison group was using a curriculum that emphasized both invented procedures and learning traditional algorithms. Both groups were from suburban schools, with the Comparison students from a somewhat wealthier community with a smaller class size and more Caucasian students.
The interviewer posed a series of word problems tapping children's understanding of subtraction, combining, and comparing, as well as their ability to decompose numbers and find factors. One problem was to distribute 52 fish among three fish tanks (and subsequently do the task with the stipulation that one fish tank has 13 fish.) Another involved finding how many more beans 33 is than 17. Part two of this problem involved combining 33, 17, and 5 beans. Another problem asked how many pencils are left when you start with 66 and sell 29. The final problem involved identifying factors, or numbers that you could "jump" by on a game board and land exactly on 24, and then exactly on 100. Children were encouraged to use manipulatives and representations, and to explain how they were thinking as they solved the problems. They were asked to solve the problem using a different method if they were having trouble with the first strategy. The interviewer recorded children's actions, use of paper and pencil, use of manipulatives, and their verbalizations.
Scoring of the problems involved the use of an analytic scoring rubric, with scorers kept blind concerning the identity of students. We addressed these questions:
- Was the child's answer accurate or inaccurate?
- If inaccurate, were they off by 1 or 2, off by 10 or a group of 10s, or off by some other number?
- What was the child's primary strategy? (e.g., "chunking" and counting by tens, counting solely by ones, use standard algorithms)
- Did the child use manipulatives, mental strategies, standard algorithms, or a combination?
- Was the child able to explain his/her thinking clearly through talking, writing, or drawing?
The rubrics were developed by four TERC researchers and curriculum developers, and were highly reliable: Between 80% and 100% reliability was achieved on all problems on all five of the questions raised above (accuracy, type of inaccuracy, strategy, use of materials, and explanation.)
Results and Discussion
Students in the Investigations group achieved a higher accuracy level on the problems than students in the Comparison group. Out of 7 points total, the mean for Investigations students was 4.6 items correct, compared with 3.25 for the Comparison Group (t = 2.13, p <.05). Investigations students were significantly more accurate (p < .05) on three of the addition/subtraction problems, including the subtracting pencils problem, the comparing beans problem, and the combining beans problem. On the problem distributing fish, the results were suggestive but non-significant. On factoring problems, there were not significant differences between the two groups. Further exploration revealed that children in neither group had much experience with factoring problems.
|
Problem |
Investigations |
Comp. |
Chi square |
p |
|---|---|---|---|---|
|
66-29 pencils |
60% |
32% |
3.76 |
.05 |
|
comparing 33 & 17 |
69% |
40% |
4.06 |
.04 |
|
combining 33 +17+ 7 |
>81% |
45% |
6.80 |
.009 |
|
distributing 52 fish |
86% |
65% |
3.06 |
.08 |
|
distributing 52 fish, 13 in one tank |
71% |
55% |
1.38 |
NS |
|
factors of 24 |
59% |
40% |
1.64 |
NS |
|
factors of 100 |
62% |
58% |
.24 |
NS |
Use of the standard algorithm, on problems where it could have been useful, was higher among students in the Comparison group, however neither group was particularly successful using it. On the subtraction problem (66-29 pencils), 35% of the Comparison group used the traditional algorithm involving borrowing, while only 7% of the Investigations group used it. One of the two Investigations students who used the algorithm was accurate, while only two out of seven (less than 30%) of the Comparison using the algorithm were accurate. On the problem involving combining 33 + 17 + 7, 20% of the Comparison group used the algorithm whereas only 3.3% of the Investigations group did.
The Investigations students seemed more selective or discriminating in their choice of strategies. For example, 33% of Investigations students versus 50% of Comparison group students relied primarily on mental arithmetic to solve the problems. However, the Investigations students were accurate with these mental strategies 80% of the time, while the Comparison group achieved accuracy only 30% of the time. Another finding was that 33% of Investigations students chose to use manipulatives as their predominant strategy, and 90% of these students achieved a medium or high level of accuracy. On the other hand, a lower proportion ---20%--of Comparison group children relied predominantly on manipulatives, and of those who did so, only 50% achieved a medium or high degree of accuracy. The findings also suggested that for the Investigations group, more than the Comparison group, different problems elicited different strategies from students.
The process students went through to match a strategy to the problem, and to their capabilities is fascinating. Many children, particularly those in the Investigations group, would start with a strategy that involved going up or down by tens, realize they were not quite sure this would work, then switch to a predominately concrete strategy that involved using counters or manipulatives (in some cases, grouping these manipulatives by tens). The concrete strategies were more conservative, and in some ways more primitive, but tended to work well. The fact that children often used an amalgam of concrete strategies and mental arithmetic, and used manipulatives on more difficult parts of problems, suggest that they are on their way to learning more sophisticated strategies but need a more familiar strategy in order to ensure accuracy.
These data suggest that Investigations students are aware of who they are as mathematical learners. They choose strategies that work for them, that make sense to them, and with which they can be successful. As a group, Investigations students showed greater flexibility than Comparison group students in being able to choose from a wide range of strategies. Perhaps as a result, they are more accurate with whatever strategies they choose. The process of inventing strategies that make sense for the learner within the context of a given problem is an emphasis of the Investigations curriculum, and this emphasis was clearly reflected in the findings.
Acknowledgements
The research described in this paper was supported, in part, by the National Science Foundation Grant No. ESI-9050210. Opinions expressed are those of the authors and not necessarily those of the Foundation.