The story of Billy, a first grader who started school at a disadvantage.
This story, a true one from a case study of a first-grade teacher’s classroom, shows the power of giving a struggling child problems that he can solve and explain.
Billy started first grade six weeks late, having never been to kindergarten. He couldn’t count or recognize numerals. His teacher, Ms. J, helped him learn to count and as soon as he could count, she gave him word problems to solve and asked him “How did you figure that out?” He spent the majority of the year solving and discussing word problems and, in the process, learning a great deal about himself as a mathematician. Here’s how it happened:
Ms. J. and the other children helped Billy learn to count objects, first to five and then to ten. …When he continued to have great difficulty recognizing numerals, Ms. J. gave him a number line with each number clearly identified. Billy carried the number line with him continuously, and if he needed to know what a numeral looked like, he would count the marks on the number line and know that the numeral written beside the appropriate mark was the numeral he needed. As soon as Billy could count, Ms. J. began giving him simple word problems to solve. On a sheet of paper, she would write a word problem such as, ‘If Billy had two pennies, and Maria gave him three more, how many would he have then?’ … Either Ms. J. or another child would then read the problem to Billy, who would get some counters and patiently model the problem. In this problem, Billy made a set of two cubes and a set of three cubes and then counted all the cubes. Ms. J. would then ask Billy to explain how he got his answer. He would tell her what each set meant, and how he had counted them all and gotten five and then counted up his number line to know what five looked like.
During mathematics class, Billy might solve only two or three of these simple problems, but he knew what he was doing, and he was able to report his thinking so that Ms. J. could understand what he had done. When Ms. J. was sure he understood the simple problems, such as the joining and separating result-unknown problems, she moved on to somewhat harder problems and to somewhat larger numbers. She encouraged Billy to make up his own problems to solve and to give to other children. Almost all of Billy’s time in mathematics class during the year was spent in solving problems by direct modeling or in making up problems for other children to solve.
When we interviewed Billy near the end of the year, he was solving problems more difficult than those typically included in most first-grade textbooks. Billy had become less reliant on his number line, and he could solve result-unknown and change-unknown problems with numbers up to twenty. Although at that point Billy was not yet able to recall basic arithmetic facts, he nonetheless understood conceptually what addition and subtraction meant, and he could directly model problems to find the answer. Billy was no less proud of himself or excited about mathematics than any other child in the classroom. As he said to the school principal, ‘Do you know those kids in Ms. J.’s class who love math? Well I’m one of them.’ (Peterson, Fennema, & Carpenter, 1991, pp. 89-91)
Excerpted from: Peterson, P., Fennema, E., & Carpenter, T. P. (1991). Using children’s mathematical knowledge. In B. Means (Ed.) Teaching advanced skills to educationally disadvantaged students (pp. 68-111). Menlo Park, CA: SRI International.
Update: Sometimes people want to know what happened to Billy after his first-grade year. Ms. J, who stays in touch with her former students as much as possible, reports that he moved away the next year and she lost track of him.
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