## 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.

## From Memorization to Understanding

It is imperative to teach children the meaning of what they are doing when they manipulate numbers during arithmetic computations. Meaning not only increases the chances that information will be stored in long-term memory, but also gives the learner the opportunity to change procedures as the nature of the problem changes. Without meaning, students memorize procedures without understanding how and why they work. As a result, they end up confused about when to use which procedure. Teachers who use primarily a declarative approach emphasize not only arithmetic facts but how they are related to each other and connected to other concepts the students have already learned. They use elaborative rehearsal and provide for cognitive closure.

### Are We Teaching Elementary-Grade Arithmetic for Understanding?

In some schools, we teach too much arithmetic through procedural approaches and very little with declarative methods. Could it be because that is how most teachers learned arithmetic themselves? Could this explain why arithmetic instruction in the primary grades has not changed very much over the years? We teach students a procedure for solving computation problems, which they then repeatedly practice (procedural memory). But the practice does not result in computational fluency because we rarely talk about how and why the procedure works. Consequently, when we give the students a problem to solve, they reflexively draw on their knowledge of the practiced procedure and apply that procedure quickly and efficiently, but with little understanding of the mathematical concepts involved.

Of course, students need to learn some basic procedural activities, such as memorizing a short version of the multiplication tables along with a few number facts. But the emphasis should be on showing students (at the earliest possible age) why they are performing certain arithmetic operations. The more arithmetic we can teach through declarative processes involving understanding and meaning, the more likely students will succeed and actually enjoy mathematics.

### Example of a Declarative-Based Approach

A declarative-based approach focuses on capitalizing on the students’ innate number sense, intuitive notions of counting by finger manipulation, and an understanding of a base-10 model for expressing quantities. It includes allowing students to create their own procedures for arithmetic computations so that they truly understand the algorithms involved. Researchers have long recognized that students in the primary grades are capable of constructing their own methods of computation (Carpenter et al., 1998; Fuson et al., 1997). In doing so, the primary-grades students pass through three predictable developmental levels.

- At the first level, students deal with all the quantities in a problem. To add a group of objects, they count out separate groups of objects, combine the groups, and then recount the total. To subtract, students count out and separate a group, and then recount what is left.
- At the next level, students consider all parts of the problem before solving it. They demonstrate this ability by counting on from, or back to, a quantity to determine an answer.
- At the most advanced level, students use abstract knowledge and consider quantities in flexible ways. They make use of what they already know to solve new problems. For example, students might use their prior knowledge to realize that 6 + 7 is equal to 6 + 6 + 1, or that 7 + 9 is equal to 6 + 10, by decomposing and recombining tens and ones.

Understanding the development of mathematical thinking in young students allows teachers to anticipate procedures that students are apt to invent and find ways to support students as they progress through the different levels. When teachers encourage students to invent alternative problem-solving strategies, the learning objectives are different from those that result from instruction using standard memorization procedures. The emphasis is on making sense and finding meaning in the methods that students create and successfully use (Scharton, 2004).

Mathematics educator, Susan Scharton, has been a strong advocate for giving primary-grade students opportunities to solve computational problems, to create their own procedures for solving them, and to explain their methods to others. She found that this approach improved the students’ accuracy as well as their understanding of the methods they had created. When she asked students

to explain their methods, their understanding of their own procedures deepened as a result of this elaborative rehearsal. Listening to the methods that others had used prompted some students to experiment with other students’ methods of computing.

#### Resources

Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998, January). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. *Journal for Research in Mathematics Education*, 29, 3–20.

Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., Carpenter, T. P., & Fennema, E. (1997, March). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. *Journal for Research in Mathematics Education* 29, 130–162.

Scharton, S. (2004, January). I did it my way: Providing opportunities for students to create, explain, and analyze computation procedures. *Teaching Children Mathematics*, 10, 278–283.

## Estimation and Methods of Estimatation

A close correlate to number sense is estimation. NCTM’s Curriculum Focal Points (NCTM, 2006) state that students in Grade 3 should be able to “develop their understanding of numbers by building their facility with mental computation . . . by using computational estimation, and by performing paper-and-pencil computations.” In Grade 4, students should “extend their understanding of place value and ways of representing numbers to 100,000 in various contexts. They use estimation in determining the relative sizes of amounts or distances.”

Estimation is an extension of the brain’s innate ability to subitize. Estimating how many animals to hunt or how many crops to plant to feed the village was a survival skill. Our ancestors were good at it. Are we? Mathematics educators often comment on the poor estimation skills of students. A frustrated teacher once told me that a middle school student felt very pleased with himself after calculating the size of a molecule to be just over one meter in length. The unreasonableness of this measurement never occurred to him. Yet, ironically, youngsters often successfully use estimation skills outside of school. For example, they can quickly make the computations needed to cross a street with traffic, decide if a sibling is sharing equally, or accurately throw, catch, or hit a ball in sports. Poor estimation skills, it seems, are more likely to appear inside school when dealing with arithmetic estimation, and they can result from at least three factors.

- First, students at an early age are programmed to get the exact answer in a problem, so they have few experiences with estimation. Furthermore, activities that ask students for both an estimated and exact answer undermine the value of estimation. Why should students estimate if they are going to find the exact answer, too?
- Second, when students use a calculator in their work, they assume the calculator’s answer must be right, with no thought that they could have inadvertently entered an incorrect number or a misplaced decimal. Consequently, they rarely reflect on the reasonableness of their answers.
- Third, because students want to get the answer quickly, estimation is avoided because it often takes more time.

Activities involving estimation should begin as early as possible in the primary grades. However, they should not be isolated as a single unit of instruction, but rather should be taught in the context of other mathematics skills throughout all grade levels. If we want to emphasize the value of estimation, then students should be given assignments that require them only to estimate.

## Methods of Estimation

The common methods of estimation include (1) rounding, which involves finding a number to the nearest ten, hundred, thousand, or the nearest one, tenth, hundredth, thousandth; (2) front-end estimation, which entails computing the higher place values or leftmost digits, then adjusting the rounded sum using the lower place values or digits to the right; and (3) clustering, which involves grouping numbers, and is useful whenever a group of numbers cluster around a common value. These methods of estimation are most helpful when students are doing computational tasks. They can check whether their answers come close to the estimated answer and to determine if their answer makes sense.

Students need to be aware that methods of estimation may not work in the real world. If you want to buy a shirt for $17.45, rounding down to the nearest dollar will not give you enough money to buy it. This is also true for estimations related to measurement. If you need exactly three and one-quarter yards of fabric to make a dress, you will not succeed if you round down to just three yards. Thus, rounding down of estimations of quantity in real-life situations will not give you enough. So what other types of estimation are available?

### Resources

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

## Developing Multidigit Number Sense

Students in primary grades have developed a notion of counting but have a difficult time studying subject matter that contains large numbers, such as the population of a country, distances to the planets and stars, and the cost of running a space mission. Although they are fascinated by large quantities, they have a limited understanding of them and often express exaggerated amounts in their conversation as in, “There were thousands of people at my birthday party.” When students lack an understanding of large numbers, they cannot reason effectively with the information they are given. In this situation, teachers need to develop the students’ ability to process large numbers, that is, develop their multidigit number sense.

The concept of multidigit number sense refers to the students’ understanding of, and flexibility in, using numbers of more than one

digit. It includes intuitive feelings for large numbers and their uses as well as the ability to make judgments about the reasonableness of multidigit numbers in different problem situations (Jones, Thornton, & Putt, 1994). Because of the complexity of this topic, teachers should select meaningful activities that help students make sense of how large numbers are used in context.

Diezmann and English (2001) have found success working with students in the primary grades by selecting activities that help the students read large numbers, develop meaningful examples for large numbers, and understand large numbers that represent quantity, distance, and money.

**Reading large numbers**. In this activity, students are introduced to the pattern in reading large numbers. Numbers of increasing magnitude are displayed for the students, starting with the ones column, progressing to the thousands column, and finally, the millions column. The name of each column is added to facilitate students’ reading.

**Developing physical examples of large numbers.** Concrete examples help students understand the nature of ever-increasing numbers. One activity to show visually the quantities 1, 10, 100 and 1,000 is to use colored sprinkles (confectionery decoration) on buttered bread that is cut into four pieces. The students add 1 sprinkle on the first piece of bread, 10 sprinkles on the second piece, approximately 100 sprinkles on the third piece (by estimating groups of 10), and approximately 1,000 sprinkles on the final piece (by estimating groups of 100).

The sprinkles activity provides a meaningful example for the students’ understanding of the relative magnitude of numbers to a thousand. Some students may extrapolate beyond the physical examples, and observe that you probably cannot fit one million sprinkles on one piece of bread.

**Appreciating large numbers in money**. What sized container would be needed to carry a million dollars? Before solving this problem, the students should complete two tasks. The first involves making posters that are labeled with the amounts $1, $10, $100, $1,000, $10,000, $100,000 and $1,000,000. The students identify items in magazine and newspaper advertisements that approximately cost each of these amounts, and glue the pictures of items under the corresponding amounts. This activity raises students’ awareness of the monetary value of expensive items. In the second task, the students calculate how much money is in a Monopoly game.

After completing these tasks, the students tackle the main problem of determining the container size needed to hold a million dollars. The students should use the Monopoly money to help them solve this problem. No containers are provided as the students are encouraged to model different container sizes with their hands. Through discussion, the students should realize that there is more than one answer to the problem. For instance, the size of the container is dependent on the denomination of the notes that are used to make one million dollars. Some students may observe that a larger-sized container would be required if notes of low value are used and vice versa.

**Appreciating large numbers in distance**. How far away are the brightest stars? The purpose of this activity is to develop the students’ understanding of large distances within the context of space travel. One approach is to have the students make 10 paper stars and label them with the names of the 10 brightest stars in the sky, their brightness, and their distance from Earth. The stars can be fastened onto upturned paper cups for ease of mobility. The students initially arrange the stars by order of brightness, beginning with the brightest star.

Next, consideration is given to the stars’ distances from Earth. After the students discuss the notion of measuring stellar distances in light years, they rearrange the stars in order from the closest to Earth to the most distant. Extend the activity by asking the students to discuss whether there is a relationship between the brightness of a star and its distance from Earth.

To represent the stars’ relative distances from Earth in light years, draw a time line and mark it in 100s from 0 to 1,000. Ask the students to position each star at the correct number of light years from Earth. Then they can discuss the fact that when we see a star today, the light from that star was actually emitted many years ago. Older students may be able to connect the year when light was emitted from particular stars to significant historic events on Earth. In this way, students make links between their mathematical understanding and their scientific knowledge.

### References

Diezmann, C. M., & English, L. D. (2001, Fall). Developing young children’s multidigit number sense. Roeper Review, 24, 11–13.

Jones, G., Thornton, C., & Putt, I. (1994). A model for nurturing and assessing multidigit number sense among first grade children. Educational Studies in Mathematics, 27, 117–143.