Concept Rich Mathematics Instruction Pdf

Instructional Interventions for Students with Mathematics Learning Disabilities

Margo A. Mastropieri , ... Dannette Allen-Bronaugh , in Learning About Learning Disabilities (Fourth Edition), 2012

Discussion and Future Directions

A review of research on mathematics instruction involving students with learning disabilities revealed a variety of behavioral, cognitive, and metacognitive approaches, which have been found to be effective in improving both mathematical computation and mathematical problem solving in students with LD. Most metacognitive instruction has been combined with very explicit instructional strategies during which students are taught, and then provided ample opportunities of practice.

Research in metacognitive strategy training has revealed the effectiveness of such training across an expanding area of tasks, as well as types of training. Of particular interest in recent years is the research on explicit problem solving, including the expansion to higher-level math including algebra and research on affectively oriented self-instruction. This research has extended our knowledge of the potential breadth of metacognitive training, with respect to types of intervention as well as content area.

Similar to earlier reviews (Mastropieri, Scruggs, & Shiah, 1991), a number of investigations in the present review were concerned with calculation performance. More recently instructional packages have included practice on number combinations within instructional packages that also contained problem solving (see RtI section). This finding appears to contrast strongly with the expressed views of the National Research Council (1989) and the National Council of Teachers of Mathematics (1989), who have repeatedly argued against the emphasis on computation over conceptual development. However, it has been clearly demonstrated that students with LD and MLD frequently exhibit persistent difficulties mastering basic number facts and computational skills (Geary, 2004), as well as in simple verbal problem solving (Lerner & Johns, 2012).

Early word problem-solving interventions involved relatively simple and straightforward problems of the sort typically found in math workbooks (for an exception see Hutchinson, 1993). Such problems do not generally correspond to the NCTM (1989) emphasis on "word problems of varying structures" (p. 20), such as problems that require analysis of the unknown, problems that provide insufficient or incorrect data, problems that can be solved in more than one way, or that have more than one correct answer (see Baroody, 1987; Parmar & Cawley, 1996). However, more recent research with students with LD and MLD and in the areas of RtI appears to meet more recent NCTM (2000) standards. These research packages are multifaceted and integrate metacognitive problem-solving strategies, including computational practice and strategies, concrete to abstract teaching sequences using manipulatives, explicit instruction and multiple practice opportunities that use a range of exemplars to facilitate generalized learning (e.g., Bryant et al., 2011).

Nevertheless, finding evidence-based practices in math remains challenging for teachers. Currently few studies meet established criteria by the What Works Clearing House which places teachers in the awkward position of searching for math programs. This is especially true for teachers of students with LD and MLD. Future research efforts would do well to conduct rigorous studies which could yield more available evidence-based practices and programs for students with LD and MLD. Overall, it can be stated that research in mathematics education for students with LD and MLD is progressing steadily. Future researchers and practitioners will be able to benefit greatly from the insights gained from the present research and look forward to more research in the future.

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Catching Math Problems Early: Findings From the Number Sense Intervention Project

Nancy C. Jordan , Nancy Dyson , in Continuous Issues in Numerical Cognition, 2016

3.1.1.3 Number Operations

Addition and subtraction operations are a focus of primary school mathematics instruction ( Jordan et al., 2009), although these abilities start to develop much earlier. Many preschoolers successfully solve nonverbal calculations with object representations (eg, the child is shown 3 dots that are then hidden with a cover; two more dots are slid under the cover, and the child indicates that 5 objects are now under the cover) (Levine, Jordan, & Huttenlocher, 1992; Huttenlocher, Jordan, & Levine, 1994). Early facility with nonverbal calculations predicts later conventional addition and subtraction skills (Levine et al., 1992). In primary school, children learn to solve arithmetic combinations with totals to 10 and then to 20. They also solve story problems that vary in semantic complexity (Riley, Greeno, & Heller, 1983; Hanich, Jordan, Kaplan, & Dick, 2001).

Children make use of a variety of calculation strategies (Carpenter, Hiebert, & Moser, 1983). Counting is the predominant method for solving simple addition and subtraction problems. Counting allows children to calculate exactly how much less or how much more one number is from another (National Research Council, 2009). For combinations with totals of 10 or less, young children can represent each part of the problem on their fingers and then count all of the fingers to get the sum. By the end of kindergarten, some children see they can count on from the first or the larger addend to get a sum (eg, for 5 + 2, the child counts 6, 7 on two fingers to get 7), a more efficient method than counting out both addends (Baroody, Lai, & Mix, 2006). Children who count on their fingers in kindergarten develop fluency with number combinations earlier than children who do not count on their fingers (Jordan, Kaplan, Ramineni, & Locuniak, 2008). Children also learn that numbers can be broken into smaller sets of numbers, sometimes referred to as conceptual subitizing (Clements, 1999). For example, the number 4 can be seen as 1 and 3, or 2 and 2. Thinking about different "partners" for sums and relating them to the inverse operation of subtraction (1 + 3 = 4 and 4 − 1 = 3) encourages meaningful mental problem solving (Fuson, Grandau, & Sugiyama, 2001). As children become proficient with some arithmetic combinations, they can use this knowledge to derive solutions to other combinations (eg, 3 + 3 = 6, so 3 + 4 = 7). Children with mathematics difficulties do not make good use of adaptive counting calculation strategies and thus are more error prone (Geary, 1994; Jordan, Kaplan, Oláh, & Locuniak, 2006).

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Development of Mathematical Language in Preschool and Its Role in Learning Numeracy Skills

David J. Purpura , ... Yemimah King , in Cognitive Foundations for Improving Mathematical Learning, 2019

Interventions to Improve Mathematical Language

A strong emphasis on mathematical language is believed to be a critical component of the success of early mathematics instruction ( NCTM, 2006), and researchers have emphasized the importance of including a mathematical language component in general mathematics instruction (Chard et al., 2008; Clements & Sarama, 2011). However, to date only a few intervention studies have focused on directly improving children's mathematical language skills (Hassinger-Das, Jordan, & Dyson, 2015; Jennings, Jennings, Richey, & Dixon-Krauss, 1992; Powell & Driver, 2015), and these studies have provided strong evidence that children's mathematical language can be improved through intervention. For example, Hassinger-Das et al. (2015) randomly assigned kindergarten children to a number sense plus mathematical language intervention, a number sense only intervention, or a business-as-usual control group. They found that children in the mathematical language group outperformed the other groups on immediate posttest measures of mathematical language, but not general mathematics measures. Yet, at a delayed posttest, the mathematical language group showed significant improvement on mathematical skills in comparison to the other conditions. These findings suggest that although some benefits of mathematical language instruction may not have been immediately found, children may have been provided with access to the content of instruction by gaining a better understanding of the language components of mathematics which allowed them to further develop their mathematics skills. Importantly, and similar to research conducted by Jennings et al. (1992) and Powell and Driver (2015), the mathematical language intervention targeted both mathematical knowledge and numeracy skills. Though these studies provide evidence of the malleability of mathematical language skills, they cannot be used to disentangle the effects of mathematical language specifically, without the additive component of mathematical knowledge. Critically, despite prior correlational evidence that mathematical language is predictive of mathematical development, these correlational and intervention studies cannot be used to make causal assertions regarding this relation.

To address the issue of whether or not mathematical language is causally related to numeracy development, Purpura, Napoli, Wehrspann, and Gold (2017) conducted an intervention study that provided mathematical language instruction with no additional numeracy instruction. Forty-seven preschool children from Head Start centers were randomly assigned to a storybook reading intervention or a business-as-usual control group. Children were pre- and posttested on measures of numeracy, mathematical language, and general vocabulary. The mathematical language storybook intervention was administered for approximately 15 min per day, 2–3 days per week, for 7 weeks (M sessions = 14.5). For the first 6 weeks, interventionists read one book per week to small groups of 2–3 children at a time. The foci of books alternated weekly between quantitative and spatial language, though both quantitative and spatial terms were highlighted when possible. During the seventh week, children were allowed to choose one of the six books for each session to review content and to ensure prolonged engagement. An eighth week was used to make up sessions with students who missed earlier sessions. A key aspect of this intervention was that it was designed to include only mathematical language terms (e.g., more, less, near, far) and not numeracy skill content (e.g., number names, counting, addition) to assess the causal relation between mathematical language and numeracy skills. Over the course of the intervention, children were exposed to 56 different mathematical language terms either as part of the story or as questions asked by the interventionists.

Dialogic reading (Arnold & Whitehurst, 1994; Lonigan, Anthony, Bloomfield, Dyer, & Samwel, 1999) was used as the primary instructional framework of this intervention in order to promote children's own use of mathematical language and more fully engage them with the content. The utility of dialogic reading for improving children's language skills is likely derived from the emphasis on eliciting verbal responses and descriptions from children (Lever & Sénéchal, 2011). It was hypothesized that this framework would be effective for improving mathematical language as mathematical curricula rich in interaction are effective for improving both mathematics and language skills (Sarama, Lange, Clements, & Wolfe, 2012; see also Hassinger-Das et al., 2015). After the intervention was completed, the results of the study indicated that the intervention had significant effects on both mathematical language skills (Hedge's g = 0.42) and numeracy skills (Hedge's g = 0.32), but not general vocabulary. These findings indicate that promoting children's mathematical language skills also positively affects their general numeracy skills (e.g., a measure that broadly assesses children's skills in one-to-one correspondence, numeral identification, set comparison, numeral comparison, number order, ordinal number knowledge, story problems, and formal addition). It is likely that exposure to mathematical language terms in an interactive context where children repeatedly heard the words and were also encouraged to use the words themselves allowed children to form an understanding of the terms. As such, it appears that mathematical language may causally underlie numeracy development.

Despite the positive findings from the Purpura, Napoli, et al. (2017) intervention study, it is unclear which aspect of mathematical language—quantitative or spatial—was driving the positive transfer to numeracy skills. Our work only examined effects of the intervention on the broad measure of mathematical language. However, this question can begin to be answered by examining the intervention's impact on the quantitative and spatial mathematical language terms separately. When the mathematical language measure was separated into quantitative (six items) and spatial domains (10 items), these new exploratory analyses indicated that the effects on mathematical language were primarily found for quantitative mathematical language (Hedge's g = 0.80; Fig. 1) and not spatial language (Hedge's g = 0.20; Fig. 2). These findings suggest that instruction on quantitative mathematical language, but not spatial mathematical language, underlies the early development of numeracy skills.

Furthermore, by examining the pretest and posttest scores of each group in Figs. 1 and 2, it can be seen that for spatial mathematical language, both groups improve by about two points from pre- to posttest. In contrast, for quantitative mathematical language, only the intervention group improved, suggesting that children may not have acquired these quantitative terms over the course of normal schooling. Ultimately, it is likely that quantitative language affects numeracy development because it provides approximate benchmarks of quantitative relations (Gunderson et al., 2015) by which children can then develop more exact knowledge of specific numbers and relations about numbers (Shusterman, Slusser, Halberda, & Odic, 2016). However, given that both quantitative and spatial language were taught in an integrative manner, the more specific claim of whether it is instruction in quantitative language or spatial language that underlies this relation cannot yet be answered fully. A number of critical questions regarding impacts of mathematical language instruction and mechanisms by which it improves children's mathematics skills need to be evaluated through future research.

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The Power of Comparison in Mathematics Instruction: Experimental Evidence From Classrooms

Bethany Rittle-Johnson , ... Kelley Durkin , in Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts, 2017

Abstract

Comparison is a fundamental cognitive process that supports learning in a variety of domains. To leverage comparison in mathematics instruction, evidence-based guidelines are needed for how to use comparison effectively. In this chapter, we review our classroom-based research on using comparison to help students learn mathematics. In five short-term experimental, classroom-based studies, we evaluated two types of comparison for supporting the acquisition of mathematics knowledge and tested whether prior knowledge moderated their effectiveness. Comparing different solution methods for solving the same problem was particularly effective for supporting procedural flexibility across students and for supporting conceptual and procedural knowledge among students with some prior knowledge of one of the methods. We next developed a supplemental Algebra 1 curriculum to foster comparison and evaluated its effectiveness in a randomized-control trial. Teachers used our supplemental materials much less often than expected, and student learning was not greater in classrooms that had been assigned to use our materials. Students' procedural knowledge was positively related to greater implementation of the intervention, suggesting the approach has promise when used sufficiently often. This study suggests that teachers may need additional support in deciding what to compare and when to use comparison.

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The Development of Basic Counting, Number, and Arithmetic Knowledge among Children Classified as Mentally Handicapped

Arthur J. Baroody , in International Review of Research in Mental Retardation, 1999

2 TEACHING

Analogous to the argument over how NMH children should be taught mathematics, there is currently heated debate over how to best teach mathematics to children with special needs, including those with mental retardation. On the one hand, some proponents of reform argue that the traditional direct-instruction approach should be entirely replaced by a child-centered approach that embodies developmentally appropriate practices (e.g., Gestwicki, 1995). On the other hand, others have countered that developmentally appropriate practices may not be effective with children with special needs and that special educators should rely on direct instruction (e.g., the teacher modeling correct procedures and providing special feedback) (see, e.g., Atwater, Carta, Schwartz, & McConnell, 1994; Carta, 1995; Carta, Schwartz, Atwater, & McConnell, 1991).

Yet others who basically support the philosophy of developmentally appropriate practices—including the NCTM (1989, 1991)—take a more moderate view. Rowen and Cetorelli (1990), for example, noted that there is no one correct way to teach mathematics. According to the National Research Council (1987), effective teachers use a variety of methods to enhance the mathematical power of their students. In other words, effective teachers may implement the investigative approach in various ways: from teacher-guided inquiry or discovery learning to open-ended projects (Baroody, 1998). Indeed, although instruction should, for the most part, be meaningful, purposeful, and inquiry-based, there may be times another approach is needed (Baroody, 1998).

In the past, there have been sporadic reports that children with mental retardation can benefit from a purposeful, meaningful, and inquiry-based approach to mathematics instruction (see, e.g., Beattie & Algozzine, 1982; Ross, 1970). For instance, the editor's note to Foti (1959) pointed out:

Some twenty years ago, the editor experimented with a combination of shop work and mathematics with some retarded boys ages ten to thirteen. Our outgrowth of the experiment was the "discovery" by pupils of non-conventional methods of obtaining answers to simple computational exercises. The carpenters' steel square became a most useful device. (p. 158)

Recent research (Ezawa, 1996; personal communication, August 27, 1997) further raises the hope that children with mental retardation can profit from an investigative approach that focuses on patiently helping them construct basic counting, number, and arithmetic concepts and skills. For example, teachers can foster the development of informal addition strategies by taking advantage of everyday situations and introducing projects, games, or simple word problems. As children invent more sophisticated strategies, they can encourage them to share their inventions with others. Moreover, teachers can prompt their classes to discuss these new procedures. While it may take several years before children with mental retardation invent advanced strategies such as counting-on, they may learn, for instance, the value of sharing or discussing ideas or strategies and that there are different ways to doing things (Ezawa, personal communication, August 27, 1997). These "lessons" and the construction of meaningful knowledge may help to promote greater flexibility—a hallmark of adaptive expertise.

Even so, to date there is little hard evidence that the investigative approach recommended by the NCTM (1989)—and proponents of "developmentally appropriate practices" (Bredekamp, 1993)—is effective with most, or even many, children with mental retardation. Still unanswered, then, are the following questions: What instructional approach can best foster the adaptive expertise of school mathematics among the majority of children with mental retardation? For example, will a purposeful, meaningful, inquiry-based approach that integrates mathematics with children's literature, science, and other subjects be more effective than a traditional skills approach in helping most of these children learn basic skills such as mastering the basic number facts, constructing key concepts such as place value and functions, inventing procedures such as counting-on or carrying, and applying problem-solving heuristics? What form of the investigative approach might best be suited for fostering adaptive expertise with different topics? For example, might guided discovery learning be helpful or not in inventing informal addition strategies, devising reasoning strategies to determine simple sums and differences, understanding key place-value concepts, or constructing multidigit arithmetic, none of the above, or all of the above?

There are significant, but not insurmountable, barriers to implementing and evaluating an investigative approach with children with mental retardation. One is that inadequate attention has been given to developing effective instructional methods for implementing the NCTM reforms with children having learning difficulties (Giordano, 1993; Hofmeister, 1993; Mallory, 1994; Rivera, 1993). Clearly, much more thought and effort must go into developing mathematics programs that are engaging, understandable, and thought provoking for all children, but especially for children with mental retardation. Another barrier is a widespread pessimism about children with mental retardation, a disposition that has been fueled, in part, by their lack of progress when taught in a developmentally inappropriate manner. For example, because many special education teachers do not believe that children with learning difficulties are capable of adaptive expertise, they incorporate the NCTM's (1989, 1991) recommendations into their traditional instructional approach in such a way as to distort or defeat the intent of the reforms (Grobecker, 1999). Clearly, educational programs are needed to help special education teachers, administrators, parents, curriculum developers, and others overcome the stereotypes associated with mental retardation—the stereotypes that result all too often in a self-fulfilling prophecy.

Although there is no guarantee that the investigative approach will be effective in promoting even a modest level of adaptive expertise among children with mental retardation, how will we know if we do not give it a genuine try? Until it is empirically established to be invalid, the mathematical instruction of these children, whatever its form, should be evaluated by the criterion proposed by Trafton and Claus (1994): "Are we teaching mathematics so that all students will be empowered to use it flexibly, insightfully, and productively?" (p. 19)

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The Development of Early Childhood Mathematics Education

Arthur J. Baroody , in Advances in Child Development and Behavior, 2017

Abstract

Addressed are four key issues regarding concrete instruction: What is concrete? What is a worthwhile concrete experience? How can concrete experiences be used effectively in early childhood mathematics instruction? Is there evidence such experiences work? I argue that concrete experiences are those that build on what is familiar to a child and can involve objects, verbal analogies, or virtual images. The use of manipulatives or computer games, for instance, does not in itself guarantee an educational experience. Such experiences are worthwhile if they target and further learning (e.g., help children extend their informal knowledge or use their informal knowledge to understand and learn formal knowledge). A crucial guideline for the effective use of concrete experience is Dewey's principle of interaction—external factors (e.g., instructional activities) need to mesh with internal factors (readiness, interest). Cognitive views of concrete materials, such as the cognitive alignment perspective and dual-representation hypothesis, provide useful guidance about external factors but do not adequately take into account internal factors and their interaction with external factors. Research on the effectiveness of concrete experience is inconclusive because it frequently overlooks internal factors.

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Evidence for Cognitive Science Principles that Impact Learning in Mathematics

Julie L. Booth , ... Jodi L. Davenport , in Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts, 2017

Recommendations for Further Research

As noted by a number of researchers, there is a lack of studies exploring the effects of spacing of instruction or practice on mathematics learning. Spaced practice is not widely evident in mathematics instruction ( Bahrick & Hall, 2005; Dempster, 1988; Mayfield & Chase, 2002; Rohrer, 2009; Willingham, 2002), and mathematics textbooks often promote blocked practice rather than spaced practice (Rohrer, 2009). Further study is needed to evaluate the effect of distributed practice on a variety of mathematics skills, especially those beyond rote fact memorization.

Another clear gap in the research is whether the effects of distributed practice vary with individual differences in learner characteristics. Several potential characteristics of interest have been suggested, including prior knowledge, and motivation (Dunlosky et al., 2013), and cognitive capacities (Delaney, Verkoeijen, & Spirgel, 2010). However, there have been no direct tests of whether the benefits of spacing and lag effects in mathematics learning are moderated by these individual differences variables.

Delaney et al. (2010) suggest that researchers must examine whether other classroom activities may be effectively spaced to achieve the same goals. For example, classroom discussions may similarly serve to remind students of the content, and distributing discussion of those topics over time may be even more effective than distributing practice on those ideas over time (Delaney et al., 2010).

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Insights from Cognitive Science on Mathematical Learning*

David C. Geary , ... Kathleen Mann Koepke , in Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts, 2017

Challenges for instruction

As described in the Booth et al. chapter (this volume), several key principles that have emerged from cognitive science research over the last few decades could have important implications for the design and delivery of mathematics instruction. Booth and her colleagues review evidence from laboratory-based studies in addition to evidence from classroom-based studies for a number of powerful learning principles including scaffolding, worked examples, interleaved problem types, and analogical comparison. While the classroom-based body of evidence for these principles has been growing in recent years, there remains little evidence that such principles are being systematically and widely incorporated into mathematics instruction in the classroom via curriculum materials or textbooks. This is somewhat surprising given that (1) the empirical evidence in support is often fairly strong and convincing and (2) many of the principles would be quite simple to incorporate into instruction.

As an example, a recent national research and development effort was funded by the Institute of Education Sciences (IES) to revise and then evaluate in a large randomized trial the efficacy of Connected Mathematics Project (CMP) after incorporating principles of cognitive science. CMP was revised to explicitly include (1) graphics with verbal descriptions to promote the integration of concepts, (2) interleaved worked examples with new problem types, (3) spaced learning of critical content over time, and (4) focused feedback on quizzes and homework to promote learning. The revised CMP materials are being experimentally contrasted against the original CMP materials. The findings from this center are still being disseminated but preliminary results seem promising with the revised materials demonstrating improved learning in three of the four CMP units (Davenport, Kao, & Schneider, 2013).

There are recent examples where principles of cognitive science have been explicitly incorporated into mathematics interventions for students with learning disabilities. Fuchs, Malone, Schumacher, Namkung, and Wang (in press) developed one such intervention for a fourth grade fractions curriculum. The program builds on Siegler et al.'s (2011) integrated theory of number development to explicitly emphasize the magnitude representation of fractions and regularly provides students the opportunity to experience and compare rational numbers situated on a number line, just as whole numbers are commonly represented. This approach to fractions instruction has proven to be quite effective, with students consistently outperforming students in business-as-usual conditions. The authors conclude that when intervention is delivered in small groups and relies on systematic, explicit instructional principles and state-of-the-art thinking about the content area (i.e., fractions magnitude understanding), struggling students' performance improves on complex curricular content beyond what is achieved for matched control group students who receive school-based classroom and more traditional interventions.

This research by Fuchs and colleagues highlights the possibility for more explicit cognitive strategy use for students who struggle and may have, for example, poor language comprehension and/or limited working memory. For many of these students, typical classroom instruction is not sufficiently meeting their needs particularly in mathematics where the content sequentially builds and becomes increasingly layered (Fuchs et al., 2015). If students with limited working memory capacity are struggling to master arithmetic operations, such as single-digit addition and subtraction, when fractions or more complicated concepts are introduced, then mastery of such new content becomes that much more difficult. Thus, as mentioned earlier, a better understanding of students' cognitive profiles (e.g., working memory or attention) could help provide clues for more or less effective intervention approaches (see also Sweller, 1988). For example, Fuchs et al. (2014) demonstrated that students with low working memory capacity performed better on fraction number lines in a conceptual practice condition (e.g., self-explanation similar to "worked-examples") as compared to a fluency condition (e.g., speeded practice). Conversely, the fluency condition seemed to benefit students with higher working memory capacity. Focusing such research and development efforts on interventions for students with learning disabilities who often exhibit memory or executive function deficits may prove particularly fruitful given that many of the approaches described in this volume help to minimize the working memory or cognitive load demands of problem solving. These cognitive principles, perhaps when coupled with more explicit forms of instruction, could be quite powerful for improving the education outcomes of struggling learners. More research is needed with these populations to better understand when and to what extent students may benefit from more explicit use of these potentially powerful cognitive science principles.

It is worth considering the role that such cognitive research could play in influencing teacher preparation and practice. For example, a principle, such as interleaved problem types demonstrates fairly robust evidence in laboratory and classroom based studies showing that mathematics learning improves as students develop stronger associations between a given problem type and its corresponding strategy (Rohrer, Dedrick, & Stershic, 2015; Rohrer, Dedrick, & Burgess, 2014). Interleaved practice is a relatively simple strategy that could be quite easily implemented with little training or support in nearly every mathematics classroom in the country. However, it is not clear if and how such a principle is being included as part of preservice teacher preparation or in-service teacher professional development. As principles of cognitive science increasingly inform education research, more research is necessary to better understand how such new knowledge is being utilized both by teacher preparation programs and by school districts providing learning opportunities for in-service teachers.

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An Overview of Technology and Learning

B. Means , J. Roschelle , in International Encyclopedia of Education (Third Edition), 2010

Mathematics

Graphing calculators have been the leading technology in mathematics classrooms; most secondary students in the United States and several other countries have access to a graphing calculator. While some critics have warned that the introduction of calculators into mathematics instruction will lead to an inappropriate de-emphasis on calculation skills, mathematics educators recommend that the calculators be used not as a substitute for skill learning but to support students when they are engaged in learning mathematical concepts. Calculators can handle calculations and graph construction, freeing students' cognitive resources to concentrate on solution strategies and understanding concepts. Other software tools supporting students' mathematics problem solving include the Geometer's Sketchpad and Cabri Géomètre, which can be used to support students' exploration and development of geometry proofs, and Tinkerplots and Fathom, which provide tools for organizing, analyzing, and representing data in ways that support understanding of statistics. SimCalc MathWorlds, a research-based tool, similarly provides representations that support students' learning of concepts in Algebra and the transition to Calculus. Advanced calculators incorporate Computer Algebra Systems, which can support students in reasoning about algebra strategically, while allowing the calculator to handle tedious rewriting of symbols. Libraries of virtual manipulatives provide similar capabilities to explore mathematical situations and concepts, but within a narrower focus. Overall, these tools provide support for both calculation and representation of mathematical concepts.

Cognitive tutors represent another advance in the use of ICT in mathematics learning. A cognitive tutor observes the step-by-step process of a students' problem solving and intervenes when the student deviates from an expert solution process. Carnegie Tutors are available for pre-algebra, algebra I and II, geometry, and integrated mathematics. Half a million students in roughly 2600 US schools have used one or more of these tutors.

The commercial market also contains many examples that reflect earlier computer-assisted instruction approaches. These approaches typically present students with tutorials and then offer feedback and hints as students solve problems.

Recent advances in classroom networking are leading to mathematical tools that allow groups of students to participate in constructing mathematical objects. For example, the commercial TI-Navigator system allows students to simultaneously submit graphs or equations to the teacher and offers the teacher controls to compare and contrast students' work. Research projects have been exploring the potential of collaboration with such specifically mathematical tools.

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Fostering Metacognitive Development

Linda Baker , in Advances in Child Development and Behavior, 1994

1 Observations of Teachers' Classroom Behaviors

One source of evidence regarding the teacher's role in fostering metacognitive development comes from naturalistic observations of teachers' spontaneous behaviors in the classroom. Moely et al. (1992) observed 69 teachers in grades K through 6 during language arts and mathematics instruction. A detailed coding scheme was devised to characterize the teachers' verbalizations. A factor analysis of the categories in the coding scheme yielded four factors, one of which is of particular interest in the present context because it included teachers' suggestions to children about how to study. Teachers gave strategy suggestions during only 2.3% of the lesson intervals in which they were observed, and they gave a rationale for why a strategy should be used in less than 1% of the intervals. Although Moely et al. found little evidence of differential strategy instruction across grade levels, teaching activities coded as strategic peaked in grades 2 and 3. Moely et al. regarded this trend as consistent with existing research; although children in this age range do not generate effective strategies on their own, they are capable of learning to use strategies they are taught. Whether or not teachers are aware of this capability is open to question, but in their 1986 summary of this study, Moely et al. suggested that teachers are sensitive to the developmental competencies of their students. However, Moely et al. also acknowledged that the instructional shift may have occurred because the curriculum in grades 2 and 3 puts a greater demand on memory skills than in kindergarten or first grade.

The teachers' strategy suggestions were further described by Moely et al. (1992) with respect to specific content. Twelve categories of suggestions were identified. The majority of these dealt with cognitive strategies; they included recommendations to use elaboration, simple repetition, prior knowledge, and general aids such as dictionaries. Cognitive strategies accounted for 86% of the strategy suggestions made. Two categories were metacognitive in nature: self-checking (i.e., the teacher suggests that children check their work or use self-testing); and metamemory (i.e., the teacher tells children that some procedures will be more helpful than others and why, and otherwise provides children with information about their memory processes). These categories accounted for 8.2% and 5.8% of the strategy suggestions, respectively, with no grade-related differences in emphasis. Teachers were more likely to provide rationales for strategies they suggested to children in grades 4, 5 and 6 than to the younger children. This trend was interpreted as evidence of teachers' sensitivity to the developing metacognitive ability of their students, but again, whether or not teachers are aware of this change requires empirical documentation. In addition, teachers rarely provided instruction in strategy generalization. In general, Moely et al. expressed disappointment in the infrequent provision of metacognitively oriented instruction. Nevertheless, they acknowledged that the teachers' suggestions for cognitive processing, though limited, were appropriate and potentially helpful aids to children's learning.

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