Excerpts from a Variety of Sources: Tools and Representations Virtual Spring 2016 SIM K-12 Article One

Excerpts from a Variety of Sources: Tools and Representations Virtual Spring 2016 SIM K-12 Article One Concreteness Fading in mathematics and Science Instruction: A Systematic Review Emily R. FyFe, Nicole M. McNeil, Ji Y. Son & Robert L. Goldstone Published online: January 2014 Concrete materials, which include physical, virtual, and pictorial objects, are widely used in Western classrooms (Bryan et al. 2007), and this practice has support in both psychology and education (e.g., Bruner 1966; Piaget 1970). There are at least four potential benefits to using concrete materials. First, they provide a practical context that can activate real-world knowledge during learning (Schliemann and Carraher 2002). Second, they can induce physical or imagined action, which has been shown to enhance memory and understanding (Glenberg et al. 2004). Third, they enable learners to construct their own knowledge of abstract concepts (Brown et al. 2009). Fourth, they recruit brain regions associated with perceptual processing, and it is estimated that 25 40 % of the human cortex is dedicated to visual information processing (Evans-Martin 2005). Despite these benefits, there are several reasons to caution against the use of concrete materials during learning. Specifically, they often contain extraneous perceptual details, which can distract the learner from relevant information (e.g., Belenky and Schalk 2014; Kaminski et al. 2008), draw attention to themselves rather than their referents (e.g., Uttal et al. 1997), and constrain transfer of knowledge to novel problems (e.g., Goldstone and Sakamoto 2003; Sloutsky et al. 2005). A number of researchers recommend avoiding concrete materials in favor of abstract materials, which eliminate extraneous perceptual details. Abstract materials offer increased portability and generalizability to multiple contexts (Kaminski et al. 2009; Son et al. 2008). They also focus learners attention on structure and representational aspects, rather than on surface features (Kaminski et al. 2009; Uttal et al. 2009). However, abstract materials are not without shortcomings. For example, solving problems in abstract form often leads to inefficient solution strategies (Koedinger and Nathan 2004), inflexible application of learned procedures (McNeil and Alibali 2005), and illogical errors (Carraher and Schliemann 1985; Stigler et al. 2010). In general, abstract materials run the risk of leading learners to manipulate meaningless symbols without conceptual understanding (Nathan 2012). 10 Given that both concrete and abstract materials have advantages and disadvantages, we propose a solution that combines their advantages and mitigates their disadvantages. Specifically, we argue for an approach that begins with concrete materials and gradually and explicitly fades toward more abstract ones. This concreteness fading technique exploits the continuum from concreteness to abstractness and allows learners to initially benefit from the grounded, concrete context while still encouraging them to generalize beyond it. Concreteness fading was originally recommended by Bruner (1966). He proposed that new concepts and procedures should be presented in three progressive forms: (1) an enactive form, which is a physical, concrete model of the concept; (2) an iconic form, which is a graphic or pictorial model; and finally (3) a symbolic form, which is an abstract model of the concept. For example, in mathematics, the quantity two could first be represented by two physical apples, next by a picture of two dots representing those apples, and finally by the Arabic numeral 2. The idea is to start with a concrete, recognizable form and gradually strip away irrelevant details to end with the most economic, abstract form. We use the term concreteness fading to refer to the three-step progression by which the physical instantiation of a concept becomes increasingly abstract over time. Since Bruner s time, several researchers have adopted similar approaches to the concrete versus abstract debate, advocating the use of concrete materials that are eventually decontextualized or faded to more abstract materials (Goldstone and Son 2005; Gravemeijer 2002; Lehrer and Schauble 2002; Lesh 1979). 10/11 1

Article One (continued) Concreteness Fading in mathematics and Science Instruction: A Systematic Review Emily R. FyFe, Nicole M. McNeil, Ji Y. Son & Robert L. Goldstone Published online: January 2014 On the surface, the benefits of concrete and abstract materials seem to be in opposition. It appears we have to choose grounded, contextualized knowledge or portable, abstract knowledge. However, we argue that both goals can be achieved simultaneously. Specifically, we suggest that concreteness fading allows concepts to be both grounded in easily understood concrete contexts and also generalized in a manner that promotes transfer. In this way, concreteness fading offers the advantages of concrete and abstract materials considered separately, the advantages of presenting concrete and abstract materials together, and finally, the advantages of presenting them together in a specific gradual sequence from concrete to abstract. Here, we explicate the unique theoretical benefits of this progressive fading sequence for both learning and transfer. See Fig. 1 for a schematic theoretical model of concreteness fading and its potential benefits 11/12 12 2

Article Two Magical Hopes: Manipulatives and the Reform of Math Education Deborah Loewenberg Ball American Educator Summer 1992 My main concern about the enormous faith in the power of manipulatives, in their almost magical ability to enlighten, is that we will be misled into thinking that mathematical knowledge will automatically arise from their use. Would that it were so! Unfortunately, creating effective vehicles for learning mathematics requires more than just a catalog of promising manipulatives. The context in which any vehicle - concrete or pictorial - is used is as important as the material itself. By context, I mean the ways in which students work with the material, toward what purposes, with what kinds of talk and interaction. The creation of shared learning context is a joint enterprise between teacher and students and evolves during the course of instruction. Developing this broader context is a crucial part of working with any manipulative. The manipulative itself cannot on its own carry the intended meanings and uses. 18 Although kinesthetic experience can enhance perception and thinking, understanding does not travel through the fingertips and up the arm. And children also clearly learn from many other sources - even from highly verbal and abstract, imaginary contexts. Although concrete materials can offer students contexts and tools for making sense of the content, mathematical ideas really do not reside in the cardboard and plastic materials. 47 More opportunities for talk and exchange - not just of techniques, but of students thinking, of the pitfalls and advantages of alternative models, and of ways of assessing what students are learning - are needed. If manipulatives are to find their appropriate and fruitful place among the many possible improvements to mathematics education, there will have to be more opportunities for individual reflection and professional discourse. 47 3

Article Three The Effects of the Concrete Representational Abstract Integration Strategy on the Ability of Students With Learning Disabilities to Multiply Linear Expressions Within Area Problems Tricia K. Strickland, PhD1 and Paula Maccini, PhD2 Remedial and Special Education 2012: 34(3), 142 153. the ability of secondary students with learning disabilities to multiply linear algebraic expressions embedded within contextualized area problems... Abstract The special education literature suggests several research-supported instructional strategies to assist students with mathematics LD or difficulties in accessing the algebra curriculum. Specifically, strategies such as the Concrete Representational Abstract (CRA) graduated instructional sequence, graphic organizers, and explicit instruction have been identified as having positive effects for teaching algebra content to students with LD (Strickland & Maccini, 2010). In the CRA sequence, the algebra concept or skill is taught first through concrete manipulatives. After students have mastered the algebra task using concrete manipulatives, they use representations or pictures of the physical manipulatives until mastery is demonstrated. Results of the current study indicate that secondary students with LD can (a) learn to multiply linear expressions to form a quadratic expression when provided with the CRA-I strategy and the expansion box and (b) develop a conceptual understanding of the quadratic expression as a generalizing statement that is a representation of a contextualized area problem with tabular data. 143 143 143 4

Article Four The Effect of Manipulative Materials on Mathematics Achievement of First Grade Students Bobby Ojose and Lindsey Sexton The Mathematics Educator 2009: Vol. 12, No.1, 3-14. Suydam and Higgins (1977) performed a meta-analysis of 40 research studies into the use and effectiveness of manipulatives on students achievement in mathematics. 60% of the studies indicated that manipulative had positive effect on student learning; 30% showed no effect on achievement; and 10% showed significant differences favoring the use of more traditional (nonmanipulative) instructional approaches. In similar work, Sowell (1989) performed meta-analysis of 60 additional research studies into the effectiveness of various types of manipulatives with kindergarten through post-secondary students. On the basis of this research, she concluded that achievement in mathematics could be increased through the long-term use of manipulatives. 10 Marsh and Cooke (1996) analyzed the effect of using manipulative (Cuisenaire rods) in teaching third-grade LD students to identify the correct operations to use when solving math word problems. After using the manipulatives, students showed statistically significant improvements in their ability to identify and use the correct operations to solve the problems. Also, in a study of 1,600 fourth-and fifth graders, Cramer, Post, & delmas (2002) compared the achievement of students using a commercial curriculum for learning fractions with the achievement of students exposed to specialized curriculum that placed great emphasis on the use of manipulatives. Students using the manipulative-based curriculum had statistically higher mean scores on posttests and retention tests. 11 It is also important to note that children cannot learn mathematics simply by manipulating physical objects. When using manipulatives, teachers should closely monitor students to help them discover and focus on the mathematical concepts involved and help them build bridges from concrete work to corresponding work with symbols. 12 While children can remember information taught through books and lectures, studies show that deep understanding and the ability to transfer and apply knowledge to new situations requires learning that is founded on direct, concrete experience. 12 5

Article Five A Meta-Analysis of the Efficacy of Teaching Mathematics With Concrete Manipulatives Kira J. Carbonneau, Scott C. Marley, and James P. Selig Journal of Educational Psychology 2013: Vol. 105, No.2, 308-400. Potential instructional moderators of the efficacy of teaching with manipulatives can be derived from contemporary human development and cognitive theories (McNeil & Jarvin, 2007). According to these theoretical explanations, concrete manipulatives facilitate learning by (a) supporting the development of abstract reasoning (Bruner, 1964; Montessori, 1964; Piaget, 1962), (b) stimulating learners real-world knowledge (Baranes, Perry, & Stigler, 1989; Rittle-Johnson & Koedinger, 2005), (c) providing the learner with an opportunity to enact the concept for improved encoding (Kormi-Nouri, Nyberg, & Nilsson, 1994), and (d) affording opportunities for learners to discover mathematical concepts through learner-driven exploration (Bruner, 1961; Papert, 1980;Piaget & Coltman, 1974). Each of these theoretical explanations provides instructional characteristics that may reduce or increase the effectiveness of math manipulatives. 381 A recent synthesis of the instructional guidance literature indicates the provision of instructional guidance results in greater performance on learning outcomes relative to pure discovery (Alfieri, Brooks, Aldrich, & Tenenbaum, 2011). Reading and listening strategy research further supports the importance of instructional guidance when using concrete manipulatives (Glenberg, Brown, & Levin, 2007; Marley, Levin, & Glenberg, 2007, 2010; Marley et al., 2011). However, Martin (2009) warned that too much instructional guidance with concrete manipulatives can impede learning by confining students to interpretations that do not transfer to novel circumstances. If this so, it is expected that the provision of high instructional guidance with manipulatives will result in lower performance on outcomes related to transfer of learning. 382 A potential problem with physical enactment has been identified by several authors (Martin, 2009; Sarama & Clements, 2009). The simple act of moving manipulatives is likely not sufficient for promoting learning. Without explicit instruction, children may not move objects in a manner that appropriately represents the mathematics concept being taught. 382 According to Dean and Kuhn (2007), in order for student-controlled strategies to be effective, students must engage in instruction over an extended period. In addition, Sowell s (1989) metaanalysis provides evidence that extended use of manipulatives had a positive effect on measures of retention. Results from the moderator analysis of the present study contradict these findings. Studies that were less than 45 days had a higher mean effect on student learning within the aggregated data. 385 Results from this meta-analysis begin to focus the inconsistencies seen within the manipulationbased literature. Findings indicate that using manipulatives in mathematics instruction produces a small- to medium-sized effect on student learning when compared with instruction that uses abstract symbols alone. Additionally, results revealed that the strength of this effect is dependent upon other instructional variables. Instructional variables such as the perceptual richness of an object, level of guidance offered to students during the learning process, and the development status of the learner moderate the efficacy of manipulatives. 398 6

Article Six Effective Pedagogy in Mathematics/Pàngarau Best Evidence Synthesis Iteration [BES] Glenda Anthony and Margaret Walshaw, Massey University Published online: February 2007 https://www.educationcounts.govt.nz/ data/assets/pdf_file/0007/7693/bes_maths07_complete.pdf In mathematics education, artefacts offer thinking spaces they are tools that help to organise mathematical thinking (Askew, 2004; Meyer et al., 2001). Symbolic artefacts or inscriptions characteristic of mathematics include the number system, algebraic symbolism, graphs, diagrams, models, equations, notations for fractions, functions, and calculus, and so on. (English, 2002). Other tools include pictorial imageries, analogies, metaphors, models (such as pizzas, chocolate bars, and tens frames), examples, stories, illustrations, textbooks, rulers, clocks, calendars, technology, and problem contexts (Presmeg, 1992). 126 The type of artefacts that teachers make available to students affects students mathematical reasoning and performance. In a study involving the learning of measurement, Nunes, Light and Mason (1993) explored the extent to which students thinking could be attributed to different artefacts. They found that the choice of tools students can access does make a difference to their achievement. Blanton and Kaput s (2005) description of the tools that supported algebraic reasoning in a third grade classroom included objects such as in/out charts for organising data and concrete or visual artefacts such as number lines, diagrams, and line graphs for building and making written and oral arguments. When used effectively to support learning, these objects became referents around which students reasoned mathematically. 126 Inscriptions are not limited to representations of the number system. Telling stories with graphical tools is a core element of the statistics curriculum. A critical component in the development of students thinking and reasoning is transnumerative thinking; that is, changing representations of data to engender an understanding of observed phenomena (Chick, Pfannkuch, & Weston, 2005, p. 87). In order to develop statistical literacy, Chick et al. urge that students from the middle years and up need more exposure to multivariate sets, as opposed to univariate data. These researchers argue that students should be given opportunities to create their own representations before being introduced to conventional ones, claiming that the effectiveness of standard forms may be more apparent if the students first grapple with their own representations. 127 Bremingan s (2005) study of 600 students use of diagrams in an Advanced Placement Calculus examination found that the modification or construction of a diagram as part of students problemsolving attempts was related to problem-solving success. More than 60% of students who achieved success in set up and over 50% of students who achieved partial success in set up either modified or constructed diagrams (p. 271). Moreover, the study found that errors identified in the solutions of students who achieved partial success appeared to be, at times, related to their diagrams. 129 7

Article Six (continued) Effective Pedagogy in Mathematics/Pàngarau Best Evidence Synthesis Iteration [BES] Glenda Anthony and Margaret Walshaw, Massey University Published online: February 2007 https://www.educationcounts.govt.nz/ data/assets/pdf_file/0007/7693/bes_maths07_complete.pdf Pape, Bell, and Yetkin (2003) found that one of the features of classroom instruction that emerged as critical to students learning was the use of multiple representations. In their seventh grade classroom study, they found that, when students were engaged in solving rich problems or creating complex representations, they were motivated and accomplished significant mathematical thinking. Multiple representations lighten the cognitive load of the learner by providing conceptual tools for thinking. 129 Baxter, Woodward, and Olson s (2001) study in five elementary schools in reform mathematics programmes with the U.S. noted differences among the classes in terms of the mathematical role that manipulatives played. Observations of 16 low-achieving target students revealed that whilst in some classes manipulatives were a distracter, in others they provided a conceptual scaffold. In three of five classrooms, manipulatives became the focus rather than the means for thinking about mathematical ideas. 133 Research has found that tools can provide effective compensatory support for students with learning disabilities. Jones et al. s study (1996) illustrates how a nine-year old learning-disabled student, Jana, used the 100s chart to move beyond pencil-and-paper computation. The episode took place within a series of lessons structured around calculations relating to garage sale purchases. 133 Jana spoke for herself and her partner: We picked the picture frame for 38c and the poster for 15c and we just have 7c change. When asked to explain how they know they would have 7c change, Jana said, We just thought about the 100s chart. We started with 38 and went down to 48 and then counted 5 more. So we paid 53c that gives us 7c back because we had 60c to spend. During early instruction with this graphic aid, Jana was encouraged to move a finger along the chart as she counted. With practice, she developed the skills to visualise the counting-on process just by thinking of the 100s chart using the chart as a compensatory tool to compute two-digit sums mentally. 8