What is Mathematics?

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

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

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

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

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.

Detecting Mathematics Difficulties

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.

Making Mathematics Meaningful to Teenagers

It is important for students to find meaning in what they are learning because meaning is one of the criteria the brain uses to identify information for long-term storage. One way to help learners find meaning is to connect what they are learning to their daily life. Yet too often students in secondary school mathematics classes have difficulty seeing the practical and concrete applications of mathematics to  everyday living. Here are just a few suggestions as to how mathematical concepts can be meaningfully related to common experiences.

Probability

Determining odds. Millions of people visit casinos, buy lottery tickets, play the stock market, join in the office football pool, and meet with friends for a game of poker. They invest their money in chance, believing they can beat the odds. The mathematical principle of probability can tell us how often we are likely to win, helping us decide whether to risk the odds and our money.

How do we determine probability? Let’s say there are 12 apples in a fruit basket. Five are red and seven are green. If you close your eyes, reach into the basket, and grab one apple, what is the probability that it would be a red apple? Five of the 12 apples are red, so your chances of picking a red apple are 5 out of 12, or as a fraction, 5/12, which is about 42 percent. Or, if you are choosing between two colleges, one in Texas and one in Connecticut. You decide to flip a coin. The chances are one out of two, or 1/2, of getting heads or tails. The odds are 50 percent for each.

What could the odds be for winning the state lottery if you buy only one ticket?

Does gambling pay off? Odds in roulette. Is roulette a good bet at a casino? Actually, the casino will win more often than the player. Here’s why. The roulette wheel is divided into 38 numbered slots. Two of these slots are green, 18 are red, and 18 are black. To begin the round, the wheel is spun, and a ball is dropped onto its outside edge. When the wheel stops, the ball drops into 1 of the 38 slots. Players bet on which slot they believe the ball will land in. If you bet your money that the ball will land in any of the 18 red slots, your chances of winning are 18 out of 38, or about 47 percent. If you bet your money on a certain number, such as the red slot numbered with a 10, your chances of winning fall to 1 in 38, or 2.6 percent.

The mathematics of probability guarantee that the roulette wheel will make money even if the casino doesn’t win every time. Remember there are 18 each of the red and black slots. There are also 2 green slots. Whenever the ball lands in one of those green slots, the house  wins everything that was bet on that round. So again, let’s say you bet that the ball will land in a red or black slot. This is the safest possible bet in roulette, since the odds are 18 out of 38 (47 percent) that you will win. But there are 20 out of 38 chances (53 percent) that you will lose.

Calculating Interest on Buying a Car

How much are you actually paying when you finance a car purchase? Understanding interest can help you manage your money and  help you determine how much it will cost you to borrow money to pay for your car purchase. Interest is expressed as a rate, such as three percent or 18 percent. The dollar amount of the interest you pay on a loan is figured by multiplying the money you borrow (called the principal) by the rate of interest.

Suppose you want to buy a used car for $10,000. The car salesman says that the dealership will finance your car at a rate of 8.4 percent, and estimates your monthly payments at about $200 over a period of five years. How much money are you actually paying back to the dealer over the term of the loan? Is this a good deal, or should you shop around? What if a bank offered to loan you the $10,000 at a rate of 9.0 percent for four years? Which offer is better?

Exponential Changes/Progressions

Population growth. The number of people living on Earth has grown dramatically in the last few centuries. There are now 10 times more people on our planet than there were 300 years ago. How can population grow so fast? Think of a family tree. At the top are two parents, and beneath them the children they had. Listed beneath those children are the children they had, and so on down through many generations. As long as the family continues to reproduce, the tree increases in size, getting larger with each passing generation.
This same idea applies to the world’s population.

New members of the population eventually produce other new members so that the population increases exponentially as time passes. Population increases cannot continue forever. Living creatures are constrained by the availability of food, water, land, and other vital resources. Once those resources are depleted, a population growth will plateau, or even decline, as a result of disease or malnutrition.

How fast will population grow? Arriving at a reasonable estimate of how the world’s population will grow in the next 50 years requires a look at the rates at which people are being born and dying in any given period. If birth and death rates stayed the same across the years in all parts of the world, population growth could be determined with a fairly simple  formula. But birth and death rates are not constant across countries and through time because disease or disaster can cause death rates to increase for a certain period. A booming economy might mean higher birth rates for a given period.

The rate of Earth’s population growth is slowing down. Throughout the 1960s, the world’s population was growing at a rate of about two percent per year. By 1990, that rate was down to one and a half percent, and is estimated to drop to one percent by the year 2015. Family planning initiatives, an aging population, and the effects of diseases such as AIDS are some of the factors behind this rate decrease. Even at these very low rates of population growth, the numbers are staggering. Can you estimate how many people will be living on Earth in 2015? By 2050? Can the planet support this population? When will we reach the limit of our resources? How could this affect the lifestyle of your children or grandchildren?

Is this job offer a good deal? Looking to make a million dollars? Let us examine a plan for earning a million dollars based on a contract between an employee and an employer. First let us agree upon a contract.

Contract for Employment

Employee ____________ (enter your name)
Employer ____________ (a company agreeing with these terms)

Points of Agreement

1. The employee will work a five-day work week.
2. The employee will be paid for the week’s wages each Friday.
3. The employee will be hired for a minimum of 30 work days.
4. The salary schedule is as follows:

• The base pay for Day 1 is one penny.
•  Each subsequent day, the salary is double that of the previous day.

Signed ____________________________ (Employee)
Signed ____________________________ (Employer)
Date: _______________________

Is this a good deal? Take a guess how much money this employee will have earned in the 30 working days: My guess: $__________________. Calculate the amount one would earn working six weeks (40 hours a week) at minimum wage? Minimum wage salary (before taxes and other deductions) $___________. Now let’s calculate the earnings for this contract and see whether the employer or the employee has made the better deal. In week one, the wages would be: Monday, 1 cent; Tuesday, 2 cents; Wednesday, 4 cents; Thursday, 8 cents; and Friday, 16 cents, for a total weekly earnings of 31 cents. Doesn’t seem like much does it? Now continue calculating the daily wages for the next five weeks.

There is a formula that allows one to calculate a particular day’s wages without having to calculate every step. This is an example of a geometric progression, a sequence of numbers in which the ratio of any number to the number before it is a constant amount, called the common ratio. For example, the sequence of numbers 1, 2, 4, 8, 16, … has a common ratio of 2. A geometric progression may be described by calling the first term in the progression X (in this example X is one cent), the common ratio as R (in this example, R = 2), and in a finite progression, the number of terms as n. Then the nth term of a geometric progression is given by the expression: Xn = X1Rn–1

Questions about this job:

1. How does the total amount of money earned compare with your original guess?
2. Suppose you wanted to buy a car. On which day could you purchase your car and pay in cash?
3. Can you develop a formula for the daily salary? (Answer: Daily Salary = 2n–1 X where n = the number of days you’ve been working, and X = your base pay on Day 1.

This counting principle can be applied also to social causes. Efforts to address social issues are often started by just a scant few individuals who are committed to a cause. Suppose you tell one person a day about your issue. A one-on-one plea will be much more effective in convincing the listener. On the second day, there will be two of you who can approach two more people. On day three, there are four of you to approach four more people. On day five, the eight of you convince eight more people, and so on. By day 12, there are over 2,000 people who know about your cause, and by day 30, over one billion people are talking about the issue that is so close to your heart! Yet you personally talked to only 30 people. By the way, now you know why unfounded rumors spread
so quickly.

Three Steps to Layering the Curriculum

Layering the curriculum is a simple way to differentiate instruction, encourage higher-level thinking, prepare students for adult-world decision making and hold them accountable for learning. Any lesson plan can be converted into a layered unit with three easy steps (Nunley, 2004, 2006).

Step One: Add some choice. Choice transforms a classroom instantly. Choice suddenly turns unmotivated students into motivated ones, ensures student attention, and gives students the perception of control. Choice is the centerpiece to student-centered, differentiated classrooms. Traditionally, mathematics teachers have seen their subject as one that is so regimented and sequential that it has little room for student choice. But within even the tightest structured curriculum, some student choice is possible.

• Take your teaching objectives and offer two or three assignment choices as to how students can learn those objectives. Not all objectives need to be taught through choices, but offer as many as you can. These can include teacher lecture, small-group peer instruction, hands-on tactile projects, or independent study.
• For example, if your objective is that students can determine the area of a triangle, you may offer a quick chalkboard lesson on that topic. Then allow the students to do some practice problems themselves, work in small groups, play a computer game that practices that concept, or complete a task using manipulatives.
• One suggestion worth considering is to make your lectures optional and award points for them. Tell the students that they can either listen to your lecture (direct instruction) or work on another assignment from the unit instead. What you will discover is that all students will probably listen to the lecture. But the fact that it is now their decision, rather than the teacher’s mandate, changes the whole perception of the task and increases attention.

Step Two: Hold students accountable for learning. One of the unfortunate developments in our traditional grading system is the wide variation in how grading points are awarded in our classrooms. Some teachers award points simply for practicing a skill, some for just doing assignments, and of course some points are eventually awarded for demonstrating mastery in the test. Because grading schemes are nearly as numerous and varied as the number of teachers, a heavy weight is frequently put on the points awarded for doing assignments. This means that students can earn enough points to pass a course without actually learning much at all. In fact, so many points have been awarded for doing class work and homework that many students never understand that the purpose of doing an assignment is to actually learn something from it. They say, “I did it; doesn’t that count?”

• A key to layering the curriculum is to award grade points for the actual learning of the objective rather than the assignment that was chosen for the learning.For example, if our objective is that students learn how to determine the area of a triangle, then points are awarded for the assignment based on whether or not the student can do that. Whether they chose to do the book work, a manipulative exercise, or a computer game is immaterial. What is important is that they learned the objective. This can be done through oral defense, small-group discussions, or unannounced quizzes. Have sample problems on index cards that you or their classmates can pull at random. Two or three sample problems can easily check for the skill. Award points for acquiring the skill rather than for the journey chosen to get there.
  

Step Three: Encourage higher-level thinking. One of the main components of braincompatible learning is helping students make complex connections with new information. Finding relationships, hooking new learning to previous knowledge, and cross-connecting between memory networks. These are the keys to real learning. Layering the curriculum encourages more complex learning by dividing the instructional unit into three layers: (1) basic rote information, (2) application and manipulation of that information, and (3) critical analysis of a real-world issue. Rather than just calling them layer 1, layer 2 and layer 3, the complexity of the learning is tied into the actual grade a student will earn, so the layers are called C layer, B layer and A layer.

• The C layer consists of all the objectives that have to do with the lower levels of Bloom’s taxonomy. This layer consists of rote learning and concrete facts. All students begin in this layer. Even the highest-ability students can add to their current bank of knowledge, so the entire class starts here.
• After students complete this C layer, they move to the B layer, which asks them to connect the new information gained in the C layer to prior knowledge. This layer includes assignments that require problem solving, application, demonstration of mastery, or unique creations. The purpose of this layer is to attach new knowledge to prior knowledge to make a more complex picture or network in the student’s brain. Interdisciplinary assignments work beautifully in this layer. A student who satisfactorily completes the C and B layer would then earn the grade of a B on this unit.
• Finally, the A layer asks students to mix the facts and basic information they have learned with more sophisticated brain concepts such as values, morality, and personal reflection, in order to form an opinion on an adult-world issue or current event. This layer asks for critical thinking and prepares students for their role as voters and decision makers in the real world. Many educators may refer to this area as the essential question. A student who successfully completes this layer will earn the grade of an A on this unit.

All students are expected to complete the three layers. Many students may not be able to show sufficient mastery of a skill or handle an Alayer issue with the sophistication needed to gain enough points for a letter grade of A or B. Nonetheless, they all must still tackle the three layers. We are preparing these students for an adult world that will ask them to gather and manipulate information and to make community decisions based on that information. Thus all students need to practice these types of thinking. At the outset, teachers help students walk through all the layers so they experience success and understand the process. As the year progresses, units may be left more open in their structure so that students are free to move among the layers as they are ready.

References:

Nunley, K. (2004). Layered curriculum: The practical solution for teachers with more than one student in their classroom (2nd ed.). Amherst, NH: Brains.org.

Using Practice Effectively With Young Students

Practice allows the learner to use the newly learned skill in a new situation with sufficient accuracy so that it will be correctly remembered. Before students begin practice, the teacher should model the thinking process involved and guide the class through each
step of the new learning’s application.

Since practice makes permanent, the teacher should monitor the students’ early practice to ensure that it is accurate and to provide timely feedback and correction if it is not. This guided practice helps eliminate initial errors and alerts students to the critical steps in applying new skills. Here are some suggestions by Hunter (2004) for guiding initial practice, especially as it applies to young students:

Limit the amount of material to practice. Practice should be limited to the smallest amount of material or skill that has the most relevancy for the students. This allows for sense and meaning to be consolidated as the learner uses the new learning. Remember that most preadolescents can deal with only about five items in working memory at one time.

Limit the amount of time to practice. Practice should take place in short, intense periods of time when the student’s working memory is running on prime time. When the practice period is short, students are more likely to be intent on learning what they are practicing. Keep in mind the 5- to 10-minute time limits of working memory for preadolescents.

Determine the frequency of practice. New learning should be practiced frequently at first so that it is quickly organized (massed practice). Vary the contexts in which the practice is carried out to maintain interest. Young students tire easily of repetitive work that lacks interest. To retain the information in long-term memory and to remember how to use it accurately, students should continue the practice over increasingly longer time intervals (distributed practice), which is the key to accurate retention and application of information and mastery of skills over time.

Assess the accuracy of practice.As students perform guided practice, give prompt and specific feedback on whether the practice is correct or incorrect, and why. Ask the students to summarize your feedback comments in their own words. This process gives you valuable information about the degree of student understanding and whether it makes sense to move on or reteach portions that may be difficult for some students.

Testing as a Form of Practice

Most people think the purpose of a written test is to evaluate a student’s achievement in the area being tested. That is a very limited view. Written tests can tell us so much more. For example, written tests can

• Allow students to practice what they have learned
• Give teachers information about what each student has learned
• Help teachers analyze how successful they were at teaching their lesson objectives

With younger students, teachers should consider using written tests mainly for practice and recording the score of only every third or fourth paper. Oral tests are a good substitute because they are less stressful, and some younger students are better at telling you what they know than writing it.

Resources:

Hunter, M. (2004). Mastery teaching. Thousand Oaks, CA: Corwin Press.

Multiplication with Understanding

An elementary school principal recently told me of conversations she had with parents about the third-grade mathematics curriculum. The parents felt there should be heavy emphasis on memorizing multiplication facts. To them, third-grade mathematics should include memorizing facts through drill and practice, worksheets, flash cards, and other memorization aids. But this school principal was promoting an approach that encouraged problem solving and understanding. She explained to the parents that this approach would help children remember the processes of multiplication for a much longer time. She recounted from her own experiences that students who had mastered their multiplication tables during third grade were barely able to remember them the following year. Apparently, memorizing multiplication facts during third grade had accomplished little because it did not build understanding of multiplication concepts. Despite having experienced a “back to basics” curriculum, they still did not know what multiplication is.

The Principles and Standards for School Mathematics (NCTM, 2000) states that “learning mathematics with understanding is essential” and that research shows “the alliance of factual knowledge, procedural proficiency, and conceptual understanding makes all three components usable in powerful ways.” NCTM’s Curriculum Focal Points (NCTM, 2006) emphasizes that “Students understand the meanings of multiplication and division of whole numbers through the use of representations.”

Students typically develop the ability to add quite naturally, but multiplication is much more complex than addition and requires guidance to understand the actions that are important elements of the process. By memorizing facts before developing an understanding of multiplication, students get the mistaken impression about the need to understand what it means to multiply and the
situations in which multiplying is the appropriate thing to do.

So what does it mean to understand multiplication? The mathematics education literature suggests that a basic understanding of multiplication requires four interconnected concepts:

(a) quantity, (b) problem situations requiring multiplication, (c) equal groups, and (d) units relevant to multiplication. Most of these understandings can develop from experiences using counting and grouping strategies to solve meaningful problems in the early grades (Smith & Smith, 2006).

• Understanding quantity. The meaning of quantity often gets overlooked in addition, but it provides an important foundation for understanding multiplication. A quantity is a characteristic of objects that can be counted or measured, and it consists of a number and a unit. Seven dollars is an example of a quantity because it includes both the number 7 and the unit, dollars. Number words (e.g., seven) are often used to describe the number portion of a quantity, but other representations, such as pictures (e.g., 7 bills representing 7 dollars), can be used. In addition to the number, a unit must be specified to know the complete quantity. A count is a particular type of number that is part of the quantity characteristic of collections of objects. It answers the question, “How many.” Counting begins with counting by ones and progresses to skip counting using larger, equal-sized units. Students need sufficient experience in counting collections of objects to clearly understand these two aspects of quantities and the various ways of representing them. A measure (e.g., length) is a particular type of quantity that is a continuous characteristic of individual objects. Measuring includes selecting an appropriate unit of measure (e.g., an inch) and determining the number of these units in the continuous characteristic of the object. Thus, to fully understand quantity, students need to understand the differences between discrete and continuous quantities, recognizing they use both different units and different processes (counting versus measuring) to determine the number portion of the quantity.
• Understanding problem situations requiring multiplication. Students need experience interpreting word problems that require multiplication and distinguishing them from other situations requiring addition, subtraction, or division. Students also need to understand the relationships between multiplication and division and be able to find each of the three possible unknown quantities, providing any two of these three pieces of information are given (e.g., 3 × 7 = ? or 3 × ? = 21).
• Understanding equal groups.Students need experience arranging objects into groups to understand the role of equal groups in multiplication and to understand the efficiency of multiplying equal groups instead of counting all of the objects in the problem. Number sense includes the ability to compose and decompose numbers. Reasoning in multiplication includes using factors and multiples as equal groups when composing and decomposing numbers instead of using adding. For example, eight objects can be arranged into groups  representing multiplication (one group of eight, two groups of four, four groups of two, or eight groups of one) rather than groups representing addition (one and seven, two and six, four and four, and eight and zero). Visual images are particularly helpful in understanding grouping (e.g., the difference between a disorganized collection of 60 items and the same 60 items organized into 5 groups of 12 items or an array of 6 rows and 10 columns).
• Understanding units relevant to multiplication. Students need experience with counting and arranging objects into groups to understand the differences between various kinds of units that are relevant to multiplication. Addition most often involves the joining of unequal quantities of the same unit (e.g., adding 35 cents and 24 cents). However, the two factors in multiplication most often refer to different units (e.g., multiplying 12 dogs by four legs for each dog). Students also need to understand how units are sometimes transformed in multiplication. For example, adding 7 oranges to 7 oranges makes 14 oranges, but multiplying the same units, such a 7 inches times 3 inches equals 21 square inches.

One way to increase the students’ deeper understanding of the process of multiplication is to show different ways that multiplication can be carried out by hand. Figure 5.8 shows how to multiply a three-digit number by a two-digit number using the traditional method (a) as well as another way known as lattice multiplication (b). Multiplication requires three steps: multiply, carry, and add. In the traditional method, the multiplying and carrying steps are done together, so it is easy to get confused. In lattice multiplication, each step is clear. It was introduced to Europe by the famous mathematician, Fibonacci, in 1202 in his treatise, Liber Abacii (Book of the Abacus).

The process is simple. If we wish to multiply 427 by 36, we write 427 across the top of the lattice and 36 down the right-hand side of a 3 × 2 rectangle (because we have three- and two-digit numbers). We fill in the lattice by multiplying the digits at the head of the columns by the digits to the right of the row. If the partial product is two digits, the tens digit goes above the diagonal and the units digit is written below the diagonal. If the partial product is only one digit, a zero is placed above the diagonal and the unit digit below.

When all the combinations have been multiplied, we add the numbers along the diagonal, beginning in the upper right and placing the sum on the diagonal to the left outside the grid. If the sum is two digits, the tens digit is placed in the top row of the diagonal to the left and added to that diagonal’s sum. Reading the digits outside the grid from upper left down across the bottom gives the final product of 15,372. This approach is not a cure-all, but it does provide novelty, and it may be just what some students need to better understand the process of multiplication.

References

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (2006). Curriculum focal points for mathematics in prekindergarten through grade 8. Reston, VA: Author.

Smith, S. Z., & Smith, M. E. (2006, March). Assessing elementary understanding of multiplication concepts. School Science and mathematics, 106, 140–149.