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What is Mathematics?

December 4, 2012 4 comments

To most people, mathematics is about calculating numbers. Some may even expand the definition to include the study of quantity (arithmetic), space (plane and solid geometry), and change (calculus). But even this definition does not encompass the many areas where mathematics and mathematicians are found. A broader definition of mathematics comes from W. W. Sawyer. In the 1950s, he
described mathematics as the “classification and study of all possible patterns.” He explained that pattern was meant “to cover almost any kind of regularity that can be recognized by the mind” (Sawyer, 1982).

Other mathematicians who share Sawyer’s view have shortened the definition even further: Mathematics is the science of patterns. Devlin (2000) not only agrees with this definition but has used it as the title of one of his books. He explains that patterns include order, structure, and logical relationships and go beyond the visual patterns found in tiles and wallpaper to patterns that occur everywhere in nature. For example, patterns can be found in the orbits of the planets, the symmetry of flowers, how people vote, the spots on a leopard’s skin, the outcomes of games of chance, the relationship between the words that make up a sentence, and the sequence of sounds we recognize as music. Some patterns are numerical and can be described with numbers, such as voting patterns of a nation or the odds of winning the lottery. But other patterns, such as the leopard’s spots, are visual designs that are not connected to numbers at all.

Devlin (2000) further points out that mathematics can help make the invisible visible. Two thousand years ago, the Greek mathematician Eratosthenes was able to calculate the diameter of Earth with considerable accuracy and without ever stepping off the planet. The equations developed by the eighteenth-century mathematician Daniel Bernoulli explain how a jet plane flying overhead stays aloft. Thanks to Isaac Newton, we can calculate the effects of the unseen force of gravity. More recently, linguist Noam Chomsky has used mathematics to explain the invisible and abstract patterns of words that we recognize as a grammatical sentence.

If mathematics is the science of patterns and if visible and invisible patterns exist all around us, then mathematics is not just about numbers but about the world we live in. If that is the case, then why are so many students turned off by mathematics before they leave high school? What happens in those classrooms that gives students the impression that mathematics is a sterile subject filled with meaningless abstract symbols? Clearly, educators have to work harder at planning a mathematics curriculum that is exciting and relevant and at designing lessons that carry this excitement into every day’s instruction.

I will leave the discussion of what content to include in a Pre K to 12 mathematics curriculum to experts in that area. My purpose here is to suggest how the research in cognitive neuroscience that we have discussed in the previous chapters can be used to plan lessons in mathematics that are more likely to result in learning and retention.

Resources:

Devlin, K. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. New York: Basic Books.

Sawyer, W. W. (1982). Prelude to mathematics. New York: Dover Publications.

Types of Mathematical Disorders

November 20, 2012 3 comments

The complexity of mathematics makes the study of mathematical disorders particularly challenging for researchers. Learning deficits can include difficulties in mastering basic number concepts, counting skills, and processing arithmetic operations as well as procedural, retrieval, and visual-spatial deficits (Geary, 2004). As with any learning disability, each of these deficits can range from mild to severe.

Number concept difficulties. An understanding of small numbers and quantity appears to be present at birth. The understanding of larger numbers and place value, however, develops during the preschool and early elementary years. A poor understanding of the concepts involved in a mathematical procedure will delay the adoption of more sophisticated procedures and limit the child’s ability to detect procedural errors. Studies show that most children with mathematical disorders nevertheless have their basic number competencies intact. However, they often are unable to use their number concept skills to solve arithmetic problems (Geary, 2004).

Counting skill deficits. Studies of children with mathematical disorders show that they have deficits in counting knowledge and counting accuracy. Some may also have problems keeping numerical information in working memory while counting, resulting in counting errors.

Difficulties with arithmetic skills. Children with mathematical disorders have difficulties solving simple and complex arithmetic problems, and they rely heavily on finger counting. Their difficulties stem mainly from deficits in both numerical procedures (solving 6 + 5 or 4 × 4) and working memory. They tend to use developmentally immature procedures, such as counting all rather than counting on.

At the same time, they do not show the shift from procedure-based problem solving to memory-based problem solving that is found in typically achieving children, most likely because of difficulties in storing arithmetic facts or retrieving them from long-term memory. Moreover, deficits in visual-spatial skills can lead to problems with arithmetic because of misalignment of numerals in multi-column addition. Although procedural, memory, and visual-spatial deficits can occur separately, they are often interconnected.

Procedural disorders. Students displaying this disorder:

  • Use arithmetic procedures (algorithms) that are developmentally immature
  • Have problems sequencing multistep procedures, such as 52 × 13 or 317 + 298
  • Have difficulty understanding the concepts associated with procedures
  • Make frequent mistakes when using procedures

The exact cause of this disorder is unknown, but research studies have yielded some intriguing findings. Children with developmental or acquired dyscalculia can still count arrays of objects, say the correct sequence of number words while counting, and understand basic counting concepts, such as cardinality. However, they have difficulties in solving complex arithmetic problems. Researchers suspect one possible cause may be a dysfunction in the brain’s left hemisphere, which specializes in procedural tasks.

Memory disorders. Students displaying this disorder:

  • Have difficulty retrieving arithmetic facts
  • Have a high error rate when they do retrieve arithmetic facts
  • Retrieve incorrect facts that are associated with the correct facts
  • Rely on finger counting because it reduces the demands on working memory

This disorder likely involves the manipulation of information in the language system. Here again, a dysfunction of the left hemisphere is suspected, mainly because these individuals frequently have reading disorders as well (D’Amico & Guarnera, 2005). This association further suggests that memory deficits may be inheritable.

Memory disorders can be caused by two separate problems. One involves disruptions in the ability to retrieve basic facts from long-term memory, resulting in many more errors than typically achieving children. Research findings indicate that this form of memory disorder is closely linked to the language-processing system and may indicate developmental or acquired deficits in the left hemisphere.

The second possibility involves disruption in the retrieval process caused by difficulties in inhibiting the retrieval of irrelevant associations. Thus the student seems impulsive. For example, when asked what is 7 + 3, a student might quickly blurt out 8 or 4 because those numbers come next in counting (Passolunghi & Siegel, 2004). Solving arithmetic problems becomes much easier when irrelevant information is prevented from entering working memory. When irrelevant information is retrieved, it lowers working memory’s capacity and competes with correct information for the individual’s attention. This type of retrieval deficit may be caused by deficits in the brain’s executive areas of the prefrontal cortex responsible for inhibiting working-memory operations.

Visual-spatial deficits. Students with this disorder:

  • Have difficulties in the spatial arrangement of their work, such as aligning the columns in multicolumn addition
  • Often misread numerical signs, rotate and transpose numbers, or both
  • Misinterpret spatial placement of numerals, resulting in place value errors
  • Have difficulty with problems involving space in areas, as required in algebra and geometry

Studies indicate that this disorder is closely associated with deficits in the right parietal area, which specializes in visual-spatial tasks. Individuals with injuries to this area often show a deficit in spatial orientation tasks and in the ability to generate and use a mental number line (Zorzi, Priftis, & Umiltá, 2002). Some studies suggest that the left parietal lobe also may be implicated.

Many students eventually overcome procedural disorders as they mature and learn to rely on sequence diagrams and other tools to remember the steps of mathematical procedures. Those with visual-spatial disorders also improve when they discover the benefits of graph paper and learn to solve certain algebra and geometry problems with logic rather than through spatial analysis alone. However, memory deficits do not seem to improve with maturity. Studies indicate that individuals with this problem will continue to have difficulties retrieving basic arithmetic facts throughout life. This finding may suggest that the memory problem exists not just for mathematical operations, but may signal a more general deficit in retrieving information from memory.

Resources:

D’Amico, A., & Guarnera, M. (2005). Exploring working memory in children with low arithmetic achievement. Learning and Individual Differences, 15, 189–202.

Geary, D. C. (2004, January-February). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4–15.

Passolunghi, M. C., & Siegel, L. S. (2004). Working memory and access to numerical information in children with disability in mathematics. Journal of Experimental Child Psychology, 88, 348–367.

Zorzi, M., Priftis, K., & Umiltá, C. (2002). Neglect disrupts the mental number line. Nature, 417, 138.

Alleviating Math Anxiety in the Classroom

November 5, 2012 Leave a comment

Shields (2005) suggests that five areas contribute in one way or another to math anxiety: teachers’ attitudes, curriculum, instructional strategies, the classroom culture, and assessment. Let’s take a look at what research studies say about each of these five areas as well as what can be done to lessen anxiety and improve student achievement in mathematics.

Teacher attitudes. Research studies confirm that teacher attitudes greatly influence math anxiety and represent the most dominating factor in molding student attitudes about mathematics (Harper & Daane, 1998; Ruffell, Mason, & Allen, 1998). Here are some things you can do to maintain a positive attitude in yourself as well as in your students:

  • Present an agreeable disposition that shows mathematics to be a great human invention.
  • Show the value of mathematics by how it contributes to other disciplines as well as society.
  • Promote student confidence and curiosity by assigning appropriate, interesting, and relevant tasks.
  • Reduce the weight given to tests in determining grades, ranking students, or measuring isolated skills.
  • Assess students on how they think about mathematics.
  • Include multiple methods of assessment such as oral, written, or demonstration formats.
  • Provide feedback that focuses on a lack of effort rather than a lack of ability so that students remain confident in their ability to improve (Altermatt & Kim, 2004).
  • Use the six NCTM Assessment Standards for School Mathematics (1995) as a guide for their testing practices. In brief, these standards state that assessment should (1) include real life activities, (2) enhance mathematics learning, (3) promote equity, (4) be an open process, (5) promote valid inferences about mathematics learning, and (6) be a coherent process.

Research studies clearly indicate that student performance in mathematics improves when anxiety is alleviated (Ashcraft, 2002). Teachers alleviate that anxiety when they demonstrate excitement and confidence in the subject, develop a relevant mathematics curriculum, use effective instructional strategies, create classrooms centered on discovery and inquiry, and assess students in a meaningful and fair manner (Shields, 2005).

Resources:

Altermatt, E. R., & Kim, M. E. (2004). Can anxiety explain sex differences in college entrance exam scores? Journal of College Admission, 183, 6–11.

Ashcraft, M. H. (2002). Math anxiety: Personal, educational, and cognitive consequences. Current Directions in Psychological Science, 11, 181–185.

Harper, N. W., & Daane, C. J. (1998). Causes and reduction of math anxiety in preservice elementary teachers. Action in Teacher Education, 19, 29–38.

Ruffell, M., Mason, J., & Allen, B. (1998). Studying attitude to mathematics. Educational Studies in Mathematics, 35, 1–18.

Shields, D. J. (2005, Fall). Teachers have the power to alleviate math anxiety. Academic Exchange, 9, 326–330.

Students With Both mathematics and Reading Difficulties

October 29, 2012 Leave a comment

Students who have both reading and mathematics difficulties are obviously at a double disadvantage. However, even though the reading and mathematical processing areas of the brain are separate from each other, these two cerebral regions interact whenever the learner must translate word problems into symbolic representations. Here are some strategies that are effective with these students.

Cue words in word problems. Help these students decode language into mathematical operations by alerting them to common phrases or cue words found in word problems that identify which operation to use.

Word problem maps. Give students with reading problems a story map to highlight certain important aspects of the story such as introduction, plot line, characters, time line, and story climax. Gagnon and Maccini (2001) have developed a similar learning aid, called a word problem map, to help students with mathematics difficulties organize their thoughts as they tackle word problems. The map can be completed by an individual student or by students working in groups of two or three.

The RIDD strategy. The RIDD strategy was developed by Jackson (2002) in 1997 for students with learning disabilities. In practice, it has shown to be particularly helpful to students who have difficulties in both reading and mathematics. RIDD stands for Read, Imagine, Decide, and Do. The following is a description of these four steps.

Step 1: Read the problem. Read the passage from beginning to end. This helps students focus on the entire task rather than just one line at a time. Good readers often skip words within a text, or they substitute another word and continue reading. In this step, students decide ahead of time what they will call a word that they do not recognize. In mathematics word problems, substitutions can be made for long numbers rather than saying the entire number on the first reading. Teachers should model this substitution when they read the problem aloud to the class.

Step 2: Imagine the problem. In this step, the students create a mental picture of what they have read. Using imagery when learning new material activates more brain regions and transforms the learning into meaningful visual, auditory, or kinesthetic images of information. This makes it easier for the new information to be stored in the students’ own knowledge base. Imagery helps students focus on the concept being presented, and provides a way of monitoring their performance.

Step 3: Decide what to do. In order to generate a mental picture of the situation, this step encourages students to read the entire mathematics problem without stopping. They then decide what to do and in what order to solve the problem. For example, in a word problem requiring addition and then subtraction, students would read the problem, create a mental picture, and then decide whether to add or subtract first. For young students, teachers can guide them through this step with appropriate questioning so the students can decide what procedures to use. Note how this step combines reading, visualization, and problem solving.

Step 4: Do the work. During this step the students actually complete the task. Often, students start reading a mathematics problem, stop part way through it, and begin writing numerical expressions. This process can produce errors because the students do not have all the information. By making this a separate step, students realize that there are things to do between reading the problem and writing it down. Jackson (2002) observed that when students used RIDD to solve mathematics problems, they liked this strategy because they perceived the last step as the only time they did work. Apparently, the students did not realize that what they did in the first three steps was all part of the process for solving problems.

Computer assistance. Computer programs are now available for elementary level students that address both reading and mathematics weaknesses. For example, Knowledge Adventure has several software titles that focus on teaching basic mathematics and reading skills while adhering to national and state standards. Each program provides instruction at a student’s own pace and includes automatic progress tracking for each student so teachers can provide additional instruction to those who need it.

References:

Gagnon, J., & Maccini, P. (2001). Preparing students with disabilities for algebra. Teaching Exceptional Children, 34, 8–15.

Jackson, F. B., (2002, May). Crossing content: A strategy for students with learning disabilities. Intervention in School and Clinic, 37, 279–282.

Students With Nonverbal Learning Disability

October 22, 2012 3 comments

Students with nonverbal learning disability (NLD) have good verbal processing skills but will have problems comprehending the visual and spatial components of mathematics skills and concepts,especially when dealing with geometric shapes and designs. Although it may be difficult for students with NLD to understand mathematics concepts and solve problems, they may have no trouble applying a mathematical formula that has been explicitly taught. They generally learn verbal information quickly. But when they look at a diagram for the first time, they look at a detailed piece. When they look a second time they see a different piece and then another piece when they look for the third time. Because there is no visual overview, the diagram may not make sense. Additionally, due to their poor spatial organization ability, they may have difficulty aligning problems on a page to solve them correctly.

Teachers of arithmetic and mathematics who work with students with NLD should consider the following strategies (Foss, 2001; Serlier-van den Bergh, 2006):

  • Rely heavily on the student’s verbal and analytic strengths. These students begin to work when speech is used, so use speech as the starting point. For example, have students read the mathematics problem aloud before attempting to solve it.
  •  Gain a commitment from the student to collaborate to improve visual and spatial weaknesses. Drawing diagrams and graphic organizers that are related to mathematics concepts and problems may help considerably.
  • Use words to describe visual and spatial information. Ask the student to do the same while pointing to the corresponding places on the diagram or concrete model.
  • Provide sequential verbal instructions for nonverbal tasks.
  • Young students with NLD may feel awkward handling manipulatives because their tactile sense is not developed. However, manipulatives can help students develop mental images of geometric shapes and visualize spatial relationships as well as improve their visual memory skills. Ask them to touch objects first with their dominant hand, then with the non-dominant hand, and finally with both hands at once.
  • Encourage the student to slowly integrate sensory information: Read it, say it, hear it, see it, write it, do it.

Resources:

Foss, J. M. (2001). Nonverbal learning disability: How to recognize it and minimize its effects. (ERIC Digest E-619). Arlington, VA: ERIC Clearinghouse on Disabilities and Gifted Education.

Serlier-van den Bergh, A. (2006). NLD primary materials: Basic theory, approach, and hands-on strategies. Paper presented at the Symposium of the Nonverbal Learning Disorders Association, March 10–11, 2006, San Francisco, CA.

Research Findings on Addressing Mathematics Difficulties

October 15, 2012 Leave a comment

Numerous research studies have looked at the effectiveness of instructional strategies on improving achievement by students with mathematics difficulties. As expected, some strategies work better than others, and a particular strategy’s effectiveness can depend on the nature of the learning difficulties found in the individuals being studied. Three recent projects looked at more than fifty studies to determine which instructional strategies worked best for students with difficulties in mathematics (Baker, Gersten, & Lee, 2002; Gersten et al., 2006; Kroesbergen & van Luitt, 2003). Their meta-analyses focused on six aspects of instruction and their effectiveness with low achieving students in mathematics and special education students. Effectiveness of the strategies was determined by effect sizes (0.2 small effect, 0.4 moderate effect, and 0.6 or above large effect). Table 7.4 lists the six strategies along with the effect size for teaching low-achieving students with learning difficulties in mathematics and special education students.

Table 7.4

 

 

 

In these studies, systematic and explicit instruction provided consistently strong effects for both groups. The effect size of 1.19 for the special education students would indicate that more than 80 percent of the participants improved their test scores after the strategy was used. This strategy involved teachers demonstrating a specific plan for solving a problem and the students using that plan to find their way to a solution. These plans provided highly explicit models of steps and procedures or of the questions that students should ask when solving problems. Student think-alouds showed a strong effect size for special education students. This strategy encourages students to express their thinking by talking, writing, or drawing the steps they used to solve a problem. This process may be effective in part because it reduces the impulsive approach that many of these students use to solve problems. Strategies that involved the visual and graphic depictions of problems resulted in a moderate effect size (0.50) for special education students. Of particular interest was the finding that the specificity of the visual representation determined the effectiveness of the intervention. Effect sizes were much larger when teachers (1) presented graphic depictions with multiple examples, (2) helped students select which visuals to use and why, and (3) had students practice with their own graphic organizers. Although using peer-assisted learning and formative data improved the performance of both groups, the effect sizes were larger with low-achieving students than with the special education groups.

Resources:

Baker, S., Gersten, R., & Lee, D-S. (2002). Asynthesis of empirical research on teachingmathematics to low-achieving students. Elementary School Journal,103,51–73.

Gersten, R., Chard, D., Jayanthi, M., & Baker, S. (2006). Experimental and quasi-experimentalresearch on instructional approaches for teaching mathematics to students with learningdisabilities:A research synthesis. Signal Hill, CA: Center on Instruction/RG Research Group.

Kroesbergen, E. H., & van Luitt, J. E. H. (2003). Mathematics interventions for children with specialeducation needs: Ameta-analysis. Remedial and Special Education,24,97–114.

Detecting Mathematics Difficulties

October 8, 2012 5 comments

As with any learning difficulty, the earlier it is detected, the better. Studies have shown that using intense tutoring with first graders who display problems with calculations significantly improved their end-of-year achievement in mathematics (Fuchs et al., 2005). The key, of course, is early detection so that interventions can begin as soon as practicable.

Determining the Nature of the Problem

The first task facing educators who deal with students with mathematics difficulties is to determine the nature of the problem. Obviously, environmental causes require different interventions than developmental causes. Low performance on a mathematics test may indicate that a problem exists, but tests do not provide information on the exact source of the poor performance. Standardized tests, such as the Brigance Comprehensive Inventory of Basic Skills—Revised, are available and provide more precise information on whether the problems stem from deficits in counting, number facts, or procedures.

Educators should examine the degree to which students with mathematics difficulties possess the prerequisite skills for learning mathematical operations. What skills are weak, and what can we do about that? They also should look at the mathematics curriculum to determine how much mathematics is being taught and the types of instructional strategies that teachers are using. Are we trying to cover too much? Are we using enough visual and manipulative aids? Are we developing student strengths and not just focusing on their weaknesses?

Prerequisite Skills

Examining the nature of mathematics curriculum and instruction may reveal clues about how the school system approaches teaching these topics. A good frame of reference is the recognition that students need to have mastered a certain number of skills before they can understand and apply the principles of more complex mathematical operations. Mathematics educators have suggested that the following seven skills are prerequisites to successfully learning mathematics (Sharma, 2006). They are the ability to

  1. Follow sequential directions
  2. Recognize patterns
  3. Estimate by forming a reasonable guess about quantity, size, magnitude, and amount
  4. Visualize pictures in one’s mind and manipulate them
  5. Have a good sense of spatial orientation and space organization, including telling left from right, compass directions, horizontal and vertical directions
  6. Do deductive reasoning, that is, reason from a general principle to a particular instance, or from a stated premise to a logical conclusion
  7. Do inductive reasoning, that is, come to a natural understanding that is not the result of conscious attention or reasoning, easily detecting the patterns in different situations and the interrelationships between procedures and concepts

Students who are unable to follow sequential directions, for example, will have great difficulty understanding the concept of long division, which requires retention of several different processes performed in a particular sequence. First one estimates, then multiplies, then compares, then subtracts, then brings down a number; and the cycle repeats. Those with directional difficulties will be unsure which number goes inside the division sign or on top of the fraction. Moving through the division problem also presents other directional difficulties: One reads to the right, then records a number up, then multiplies the numbers diagonally, then records the product down below while watching for place value, then brings a number down, and so on.

Resources

Fuchs, L. S., & Fuchs, D. (2002). Mathematical problem-solving profiles of students with mathematical disabilities with and without comorbid reading disabilities. Journal of Learning Disabilities, 35, 563–573.

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