Connecting Manipulatives and Symbols Promotes Mathematics Learning

• In Event: 2-082 - External Representations in Mathematical Thinking and Learning

Integrative Statement

Researchers in education and psychology have long supported the use of concrete objects to facilitate children’s mathematics learning (e.g., Burns, 1996; Montessori, 1964). Yet, the use of concrete objects does not guarantee success (McNeil & Jarvin, 2007). To increase their effectiveness, many researchers recommend drawing more explicit links between concrete objects and abstract symbols. For example, a practice guide focused on mathematics suggests that instructors should “help children link formal math vocabulary, symbols, and procedures” to their informal experiences with concrete objects (Frye et al., 2013, p. 44). The argument is that instructional practices that highlight these connections (via comparison, gesture, etc.) will facilitate children’s knowledge of the connections, which in turn will promote deep conceptual understanding. The goal of the current study was to assess the empirical relation between children’s knowledge of these connections and their overall learning from a math lesson.

Eighty-two children (M age = 6.7 years) learned about place value in a one-on-one tutoring session. The lesson focused on identifying the value of each digit in a series of three-digit numbers using base-ten blocks (e.g., the 3 in 134 represents 3 tens). After the lesson, children completed a 3-item Block Task, which assessed their knowledge of the connections between the blocks and symbolic numbers. Children were shown a symbolic number and asked to show that number using blocks (Figure 1). Each child also completed a pretest to assess prior knowledge, a posttest to assess learning, and a transfer test with four-digit numbers.

Children exhibited low prior knowledge on the pretest (M = 4.8 out of 12 [40%], SD = 1.8) and often misunderstood symbolic three-digit numbers (e.g., selecting 90057 as the way to write “nine hundred fifty-seven”). The lesson helped some children understand the connections between base-ten blocks and symbolic numbers. The average Block Task score was 1.8 out of 3.0 (SD = 1.2) and 40% of children solved all three items using the most efficient, correct strategy. Importantly, it was the children who understood the connections that learned the most. We ran an ANCOVA with Block Task mastery as the independent variable (yes vs. no) and posttest scores (percent correct) as the dependent variable. We included pretest scores, age, gender, and ethnicity as covariates. There was a significant positive effect of Block Task mastery, F(1, 76) = 4.59, p = .03 (Figure 2). Indeed, children at mastery on the Block Task improved nearly twice as much from pretest to posttest compared to children who were not at mastery. We ran a similar ANCOVA analysis with transfer scores as the dependent variable and found a similar main effect of Block Task mastery, F(1, 76) = 6.36, p = .01 (Figure 2).

Thus, children’s ability to connect concrete objects and math symbols predicted their learning from a lesson. These results provide empirical support for the idea that concrete objects can be effective, but only to the extent that they “aid students in building, strengthening, and connecting various representations of mathematical ideas” (Clements, 1999, p. 49).

Authors

• Emily R. Fyfe, Indiana University
  Presenting Author

• Andrea Marquardt Donovan, University of Wisconsin-Madison
  Non-Presenting Author