Making connections in mathematics

• The call for making connections in mathematics is not a new idea, as it has been traced back in mathematics education literature to the 1930s and W.A. Brownell’s research on meaning in arithmetic (Hiebert and Carpenter, 1992).

• Though children use different strategies to solve mathematical problems in out of school contexts, they still develop a good understanding of the mathematical models and concepts they use as tools in their everyday activities (Carraher, Carraher and Schliemann, 1985; Nunes, Schliemann and Carraher, 1993).

• Mathematical meaning plays a vital role in student solutions of problems in everyday activities, especially compared to in-school problem solving activities that depend more on algorithmic rules. The strategies and solutions students construct to solve problems in real-world contexts are meaningful and correct, while the mathematical rules used by students in school are devoid of meaning and lead to errors undetected by the student (Schliemann, 1985; Schliemann and Nunes, 1990).

• Students need to build meaningful connections between their informal knowledge about mathematics and their use of number symbols, or they may end up building two distinct mathematics systems that are unconnected—one system for the classroom and one system for the real world (Carraher et al., 1987).

• Students need to discuss and reflect on connections between mathematical ideas, but this “does not imply that a teacher must have specific connections in mind; the connections can be generated by students” (p. 86). A mathematical connection that is explicitly taught by a teacher may actually not result in being meaningful or promoting understanding but rather be one more “piece of isolated knowledge” from the students’ point of view (Hiebert and Carpenter, 1992).

• Learning mathematics in a classroom differs from learning mathematics outside the school in these important ways:

1. Learning and performance in the classroom is primarily individual., while out of school activities that involve mathematics are usually group-based.

2. Student access to tools often is restricted in the classroom, while out-of-school activities allow students full access to tools such as books and calculators.

3. The majority of the mathematics activities in a classroom have no real-world context or connection, while out-of-school activities do by their very nature.

4. Classroom learning stresses the value of general knowledge, abstract relationships, and skills with broad applicability, while out-of-school activities require contextual knowledge and concrete skills that are specific to each situation (Resnick, 1987a).

• The skills and concepts learned in school mathematics differ significantly from the tasks actually confronted in the real world by either mathematicians or users of mathematics (Lampert, 1990).

• Students learn and master an operation and its associated algorithm (e.g., division), then seem to not associate it with their everyday experiences that prompt that operation (Marton and Neuman, 1996).

• Teachers need to choose instructional activities that integrate everyday uses of mathematics into the classroom learning process as they improve students’ interest and performance in mathematics (Fong et al., 1986).

• Students often can list real-world applications of mathematical concepts such as percents, but few are able to explain why these concepts are actually used in those applications (Lembke and Reys, 1994).

• Vocational educators claim that the continual lack of context in mathematics courses is one of the primary barriers to students’ learning of mathematics (Bailey, 1997; Hoachlander, 1997). Yet, no consistent research evidence exists to support their claim that students learn mathematical skills and concepts better in contextual environments (Bjork and Druckman, 1994).

• Hodgson (1995) demonstrated that the ability on the part of the student to establish connections within mathematical ideas could help students solve other mathematical problems. However, the mere establishment of connections does not imply that they will be used while solving new problems. Thus, teachers must give attention to both developing connections and the potential uses of these connections.