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Using mental math, respond to the following:
1. How many thousand-dollar bills make ten million dollars?
2. What is 1/5 of the world's 6.2 billion population?
3. Estimate a 40 percent discount on a $54.90 item.
4. Solve: 3x[sup3] = -24
5. What is the hypotenuse of a right triangle with legs 10 and 24?
6. Evaluate:

(a) 81[sup3/4]  (b) 16[sup-1/2]  (c) sin (pi/3)  (d) log[sub2] 64

How would your students handle these items? How would their strategies compare with yours? Would their facility with these items make other mathematics easier for them? How can we help students do these and similar items fluently?
The first three items above are what might be called "number sense" in upper-elementary or middle school programs. (See Reys et al. [1991] for more activities at the middle school level.) Items 4-6 involve more advanced topics but, like the comparable arithmetic items, should be doable by students mentally, easily, and quickly. In my experience teaching middle school, high school, community college, and teacher-education courses, I have found that students' facility with these items is not universal. Yet for students for whom such work is easy, other mathematics is more accessible. As a result, I have been trying to develop strategies for integrating brief, frequent mental-math instruction and assessment into all my courses. In this article, I share my reasons for thinking that mental math is important beyond the middle school, sample objectives for different courses or strands, and teaching ideas to make mental math an important, ongoing goal in secondary mathematics instruction and beyond.

WHY TEACH MENTAL MATH?
One simple reason to emphasize mental math is that it is useful for workers, consumers, and citizens. Bell (1974) noted that in daily life, adults use estimation more often than exact computation. People commonly estimate or calculate mentally the time needed to travel certain distances, the cost of a shopping cart of groceries, taxes, tips, discounts, unit prices, miles per gallon, miles per hour, and others. Interestingly, many of the basic arithmetic applications of everyday life involve percents and proportions, which are part of what researchers call the multiplicative domain, or proportional reasoning (Sowder et al. 1998). Although proportional reasoning is a goal of the middle school curriculum, it is often not fully mastered at that level. Reys (1994) notes that number sense matures with experience. Further, many of the important everyday applications, such as compound interest or discounts, do not interest middle school youngsters. Nevertheless, if students are enrolled in courses that teach only algebra and geometry, they may not get focused opportunities to hone their skills and understandings in the vital multiplicative domain.
Another major benefit of mental math is that it facilitates learning many important structural topics. For example, mentally calculating a tip, say, 15 percent of $24.98, and sharing aloud the typical strategy--that of finding 10 percent and then half of that amount for 5 percent and combining--highlights the distributive property. The estimation discussion helps students recognize and extend the property to other situations, in particular, to variables. Similarly, commutativity and associativity can be highlighted through such mental-math items as 28 + 769 + 72 and 8 × 43 × 25, which employ "shortcuts." For example, in the latter problem, students learn to recognize that 8 × 25 is 200 and 200 × 43 is 8600.
As well as providing a natural springboard for studying properties, mental math makes easier the understanding of inverse operations. For example, when powers are transparent to students, for example, in 2[sup6] = 64, then initial work with logarithms is easy (i.e., log[sub2] 64 = 6.) The same holds for other inverses, for instance, trigonometric ratios and inverse trigonometric functions. In the area of modeling, we want students to recognize such special sequences as multiples, powers, squares, and triangular numbers; these topics, too, are easy when students have practiced building, extending, and identifying patterns.
Another reason that we need mental math is that many of today's students, as they readily admit, are calculator-dependent. Even though the calculator is a wonderful tool for mathematical exploration and problem solving, many occasions arise for which working mentally is faster, more appropriate, or easier. We need balanced instructional programs, including both significant, appropriate calculator-based investigations and non-calculator work (NCTM 2000, p. 32). If students have never been asked to solve problems without calculators and if they have not learned calculator-free strategies, then mental math will never be an option that they choose. We need to ensure that they have mental-math skills in their repertoire. In this vein, too, we would also serve our reform efforts by defining and helping students achieve modest but valuable mental-math objectives to demonstrate that they can do what the general public perceives as "the basics."
Finally, mental math should be included in our programs because it is rewarded. When students have regular opportunities to estimate, share orally, evaluate, compare their approaches, and transfer strategies to new settings, they feel challenged and, ultimately, empowered. They take pride in sharing instances in daily life in which they have used their new facilities and feel liberated from their calculator dependence.

WHAT MENTAL-MATH OBJECTIVES ARE APPROPRIATE IN HIGH SCHOOL AND COLLEGE?
Reys and Yang (1998) describe number sense as "a person's general understanding of number and operations. It ... includes ... [using] this understanding in flexible ways to make mathematical judgments, and to develop useful strategies for handling numbers and operations .... [T]he characteristics typically associated with number sense include using multiple representations of number, recognizing the relative and absolute magnitudes of numbers, selecting and using benchmarks, decomposing and recomposing numbers, understanding the relative effects of operations on numbers, and flexibly and appropriately performing mental computation and estimation." (pp. 225-26)
Many aspects of this definition can be extended beyond "number sense" to "operation sense," "symbol sense," "graph sense," and others. In all instances, students should be able to operate mentally, estimate, be flexible, transform, use multiple representations, and choose strategies that are based on the particular question at hand. To that end, for several courses I have developed objectives that incorporate these features.
Objectives and sample items for the areas of number sense, beginning algebra, and precalculus are shown in figures 1, 2, and 3, respectively. Having specific objectives that are clearly known to students is part of building a successful program. Pages such as these are given to students very early in the term, usually along with a partial answer key. Various subsets of the problems are then identified as goals for different units of study. The lists are used cumulatively, so that the number-sense strand, for example, is an ongoing goal, as are objectives from previous units.
Although most of the items in figures 1, 2, and 3 would be classified as "skills," many of them build on or include concepts and representations (e.g., graphs). Having a limited, well-defined set of items that students should be able to do quickly makes clear an expectation for competence in what some people call "the basics." Every curriculum strand has mental-math objectives. For example, in the area of geometry and measurement, students should be able to find basic areas, perimeters, and volumes, expressed in appropriate units; convert within the metric system; identify equal and supplementary angles in basic cases; and translate simple "if ..., then ..." statements into other forms. For statistics and probability strands, students should be able to find centers of small ordered data sets, calculate small factorials, and find probabilities of outcomes for small numbers of coin or dice tosses. For trigonometry, students should be able to identify graphs of parent functions, find functional values at standard unit-circle locations, and find missing lengths in right triangles derived from well-known primitive Pythagorean triples.

HOW: TEACHING IDEAS FOR MENTAL MATH
The first step in building success in mental math is to specify objectives clearly and to provide sample items. Adequate time should be taken early in the term to discuss those objectives seen as prerequisites to a course, and time should be allotted during the term to introduce new objectives as they are added. The objectives, as always, should be related to the core content of the course and should be appropriately developed.
The second step is to offer frequent, but brief, in-class practice. I often use mental math as a warm-up or closing activity. In five minutes at the end of a period with books packed away, for example, students can silently do three independently displayed items from the overhead projector; share answers, which they do remember, with partners; then share and compare their strategies as a class. I have also created some mental-math developmental practice sets, shown in figures 4 and 5, that introduce or reinforce basic ideas. For example, in number-sense set D (fig. 4), students are given an opportunity to compare commonly confused fractions and percents (e.g., 1/2 as opposed to 1/2%). On algebra set A in figure 5, a series of numerical problems becomes the backdrop for operations with exponents.
Following all these activities, we always debrief immediately. My students share strategies in small groups and then with the entire class. This approach usually produces a highly "teachable moment" when students are surprised by, and often impressed with, others' thinking. For example, on the item "Find 80% of 55," students have shared a variety of responses, including the following:
* I found 10 percent, which is 5.5, then multiplied by 8. To do that, I multiplied 8 by 5 to get 40, then 8 by 1/2 to get 4. The total is 44.
* I found 10 percent, or 5.5; doubled it to get 11; then subtracted that from 55 to get 44.
* I knew that 80 percent is 4/5. One-fifth of 55 is 11, so 4/5 is 4 times as much, or 44.
The last method is often an eye-opener for many students. They realize that knowing fraction-percent equivalences permits a shortcut that makes other work much easier.
Within general instruction, too, I try to highlight opportunities for students to use mental math in what I call a "mental-math moment." For example, in discussing a homework question that can be done mentally, I might say, "Here's a good chance to use our mental math." I really want students to see that they can use their "biological calculators"--their brains--more easily than their electronic calculators!
The in-class activities provide informal assessment and self-evaluation for students. For formal assessments, I use ten- to twenty-five-item quizzes presented on the overhead projector. The students are given an answer sheet with limited space for recording answers and are asked to use a clean cover sheet. I uncover and read aloud one item at a time, watch students' responses, and continue when everyone has finished doing that item. The items usually remain on the screen along with the next item or two, but then they slide off. My students are extremely attentive during these assessments. As with the practice activities, when the quiz is over, the students share their answers and strategies with their group members. We then discuss their thoughts as a class, always asking for other approaches to items for which those are likely to occur. I strongly encourage students to take notes from the discussion to capture strategies and understandings that they will want to use again in the future. If students do not suggest a particularly valuable strategy. I might say, "Here's another way that you may want to try...." I use a low-stakes grading policy on these quizzes, usually counting a given student's one best score from three mental quizzes. My thinking is that students need not only the practice and the feedback but also the chance to excel. The students are always optimistic that "the next quiz will be better," and for most of them, it is.
Another assessment strategy is to give two-part examinations, one part being "without calculators." Then students can work comparable mental-math items in a longer time frame but still without electronic assistance. This approach is particularly helpful in precalculus graph-match activities when students are studying equations and their related graph transformations or when students are associating functions with real-world situations.
A final assessment strategy that I have used in a community college developmental arithmetic class is one that I call "think twice mentally." I ask the students to find opportunities in their daily lives when they used or could have used mental math to solve a problem. They write the story, state the question, explain two mental-math approaches that they could have used, and answer their question. This activity integrates writing with mathematics, gives prominence to mental math in daily life, and forces students to find more than one approach to a problem. When I first use this activity, I share samples from my experience and make clear to students the guidelines for evaluation. At first, most students find the activity very challenging, but after one or two submissions, their work improves. They definitely become more aware of mathematics in their lives and of strategies for operating mentally. As a bonus, I learn, too, where students see mathematics in their daily lives; I can then incorporate their scenarios into applications that I use for teaching other topics.

SUMMARY
Making mental math a high priority in my classes is an asset to all my students. They become liberated from calculator dependence, build confidence in doing mathematics, become more flexible thinkers, and are more able to use multiple approaches to problem solving. Moreover, other topics requiring numerical or symbolic fluency are easier for them to learn. I hope that the rationale, teaching ideas, and sample objectives presented here will encourage others to identify the mental-math objectives in their courses and engage students in activities that build their facility with them.
ADDED MATERIAL
Rheta Rubenstein, rhetar@aol.com, teaches mathematics at University of Michigan--Dearborn, Dearborn, MI 48128. She is interested in making mathematics accessible to all students.

REFERENCES
Bell, Max. "What Does 'Everyman' Really Need from School Mathematics?" Mathematics Teacher 67 (March 1974): 196-202.
National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.
Reys, Barbara J. "Promoting Number Sense in the Middle Grades." Mathematics Teaching in the Middle School 1 (September-October 1994): 114-20.
Reys, Barbara J., Rita Barger, Maxim Bruckheimer, Barbara Dougherty, Jack Hope, Linda Lembke, Zvia Markovits, Andy Parnas, Sue Reehm, Ruth T. Sturdevant, and Marianne Webber. Developing Number Sense in the Middle Grades, Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5-8. Reston, Va.: National Council of Teachers of Mathematics, 1991.
Reys, Robert E., and Der-Ching Yang. "Relationship between Computational Performance and Number Sense among Sixth- and Eighth-Grade Students in Taiwan." Journal for Research in Mathematics Education 29 (March 1998): 225-37.
Sowder, Judith T., Randolph A. Philipp, Barbara E. Armstrong, and Bonnie P. Schappelle. Middle-Grade Teachers' Mathematical Knowledge and Its Relationship to Instruction: A Research Monograph. Albany, N.Y.: State University of New York Press, 1998.

FIG. 1 NUMBER SENSE: OBJECTIVES AND SAMPLE ITEMS
1. Use number facts with multiples of ten: 80 × 900 700 × 6000 720 ÷ 90
2. Identify and complete patterns:

Squares:      1, 4, 9, 16, 25, 36, 49, 81, 100, 121, 144, 169, 196, 225
Cubes:        1, 8, 27, 64, 125, 216
Powers of 2:  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

3. Give fraction-decimal-percent equivalences:

1/3  2/3  1/5  2/5  3/5  4/5  1/8  2  3  1/2%

4. Reduce simple fractions:

12/16  18/20  40/200  28/42  15/25  45/60

5. Identify a simple percent from a diagram. What percent is shaded? [Graphic Character Omitted]
6. Mentally calculate problems using commutative and associative properties:

42 + 87 + 58  4 X 31 X 25  5 X 17 X 20  79 + 88 + 12

7. Know and use base-ten relationships:
a) How many thousands make a million?
b) What is a tenth of a hundredth?
c) What is a hundredth of a hundredth?
d) How much is $5.06 million?
e) In 1990, about 25 million people lived in Canada and ten times as many lived in the United States. About how many people lived in the United States in 1990?
f) Notebooks cost $0.38 each. What is the cost to order 100 of them for your office supply room?
8. Find change. For example, find the change when you give a $50 bill for a charge of $38.17.
9. Calculate simple multiplication exercises using the distributive property:

99 X 3  32 X 5  96 X 4  38 ? 12 + 38 ? 88

10. Estimate tips and discounts:
a) Approximate a 15% tip on a bill of $31.96.
b) What is the sale price at a 30%-off sale on a $48.38 item?
11. Convert basic measures:
a) 1000 cm = ___ m
b) 0.25 kg = _____ g

FIG. 2 BEGINNING ALGEBRA: OBJECTIVES AND SAMPLE ITEMS
1. Use order of operations to evaluate expressions:
(a) 6 + 7[sup2] (b) 5 + 7 ? 9 (c) 6(4 + 8) (d) 18 ÷ 3 ÷ 3 (e) 18 - 3 + 3
2. Simplify expressions with exponents:
(a) a[sup3] ? a[sup2] (b) (b[sup3])[sup2] (c) c[sup7]/c[sup3] (d) a[sup6] ? a[sup2]
(e) (b[sup3])[sup4] (f) 1/a[sup6] (g) b/b[sup5] (h) (a[sup2]b[sup3])[sup5] (i) -6[sup2]
3. Use powers in multiplication estimation. For example, estimate 3.1 × 10[sup5] × 7.9 × 10[sup2].
4. Evaluate expressions involving 0 and 1:

(a) 5/0  (b) 0/4  (c) 2[sup0]  (d) 7[sup1]  (e) 1[sup78]

5. Find powers including 0's and negatives:
(a) 2[sup5] (b) 3[sup4] (c) 5[sup3] (d) 10[sup4] (e) 8[sup-1] (f) 4[sup-2]
6. Operate with integers:

(a) (-87) + 104  (b) (-87) - 104  (c) (-5)(-8)  (d) (-5)(-8)(-3)
(e) -(-3)[sup4]   (f) (-3)[sup4]    (g) -3[sup4]   (h) 8(-3 + 3)/5-6

7. Rewrite using the distributive property:
(a) 8(x - 9) (b) a(8a - 6) (c) -x(4 - x) (d) (a + 2)(a + 3) (e) (x - 4)(x + 5)
8. Translate:
a) 5 less than x
b) The reciprocal of x
c) Twice the reciprocal of x
d) The sum of a number and its square
e) Twice the sum of a number and 1
9. Solve basic equations:

(a) 3x + 1 = 13  (b) 2x - 5 = 17  (c) -3x + 4 = 19  (d) -4x - 8 = 12

10. Find special number pairs. For example, find two numbers whose--
sum is 11 and product is 24,
sum is -2 and product is -15,
sum is -8 and product is 15,
sum is -3 and product is -18.
11. Graph simple inequalities:

(a) x > -2    (b) x < 5   (c) -5 < x < 4

12. Find basic square roots, and recognize undefined expressions:

(a) 64  (b) 144  (c) -9

FIG. 3 PRECALCULUS: OBJECTIVES AND SAMPLE ITEMS
1. Find small powers:
4[sup3] = 64 2[sup4] = 16 3[sup5] = 243 10[sup6] = 1,000,000 2[sup10] = 1,024
2. Know meanings of 0 and negative exponents:

7[sup0]  9[sup-1]  5[sup-2]  10[sup-3]  (2/3)[sup-1]

3. Translate into and out of scientific notation including calculator and paper forms: [Graphic Character Omitted]

1.23 X 10[sup5]   5.009 X 10[sup-2]
680,000          .000675

4. Simplify expressions with exponents:

a[sup5] ? a[sup2]  a[sup5]/a[sup2]  (a[sup5])[sup2]  (3a)[sup4]

5. Find powers of fractions:

(1/2)[sup3]  (2/5)[sup4]   (1/3)[sup-4]  (3/4)[sup-2]

6. Find basic roots in radical or fractional power notation; recognize undefined forms:
[square root]64 [cube root]64 [Graphic Character Omitted]32 [Graphic Character Omitted]-32 64[sup1/2] 64[sup1/3] 32[sup1/5] [Graphic Character Omitted]-64
7. Evaluate expressions with rational exponents:

[cube root]64[sup2]       32[sup3/5]  (27/125)[sup-1/3]

8. Find basic logs, and recognize undefined forms:

log[sup2]0.5  log[sub7][square root]7   log[sub2]1/8  ln e  ln 1  ln 1 / e

9. Match situations and graphs:
a) You owe a friend $50 and pay back $10 per week for 5 weeks.
b) Your business's daily costs are $50 plus $10 for each part that you manufacture.
c) Your sales increased the first six months but have been dropping lately.
Graph i
Graph ii
Graph iii

FIG. 4 NUMBER SENSE: DEVELOPMENTAL PRACTICE SETS

Set A
1. 1/5 of 55
2. 2/5 of 55
3. 3/5 of 55
4. 4/5 of 55
5. 20%   of 55
6. 60%   of 55
Set B
1. 50%   of 80
2. 25%   of 80
3. 75%   of 80
4. 12 1/2%   of 80
5. 37 1/2%   of 80
6. 87 1/2%   of 80
Set C
How much does each
person get?
Share $45.70--
1. for yourself (1 person).
2. with 10 persons.
3. with 100 persons.
In $45.70 are how many--
4. ones?
5. tens?
6. hundreds?
7. tenths?
8. hundredths?
Set D                      T/F
1. 1/20 = 1/2 of 1/10.     _____
2. 1/20 = 1/2 of 10%  .    _____
3. 1/200 = 1/2 of 1/100.   _____
4. 1/200 = 1/2 of 1%  .    _____
5. 1/50 = 1/5 of 1/10.     _____
6. 1/50 = 1/5 of 10%  .    _____
Set E
Write as a percent.
1. 1/2
2. 1/5
3. 1/20
4. 1/50
5. 1/200
6. 1/500
Set F
Think of a dozen eggs. Tell the
resulting fraction of a dozen.
1. 1/2 of 1/2
2. 1/4 of 1/3
3. 1/2 of 1/3
4. 1/2 of 1/6

FIG. 5 BEGINNING ALGEBRA: DEVELOPMENTAL PRACTICE SETS

Set A
Evaluate:
1. 2[sup2]
2. 2[sup3]
3. 2[sup2] ? 2[sup3]
Write as a power of 2:
4. 2[sup2] ? 2[sup3]
5. 2[sup3]/2[sup2]
6. (2[sup3])[sup2]
Set B
Rewrite using the distributive property:
1. (100 + 2)3
2. (100 - 2)3
3. (a + 2)3
4. (a - 2)3
5. (a + b)4
6. 4(a + x)
Set C
Solve:
1. 7x = 56.
2. -7x = 56.
3. x + 2 = 13.
4. x - 2 = 13.
5. 7x + 2 = 58.
6. 7x - 2 = 58
Set D
Find two numbers that--
1. multiply to 12 and add to 7;
2. multiply to 12 and add to 8;
3. multiply to -12 and add to 1;
4. multiply to -12 and add to -1;
5. multiply to -12 and add to -11;
6. multiply to 12 and add to -13.
Set E
Evaluate:
1. 7[sup2]
2. (-7)[sup2]
Solve:
3. x[sup2] = 36.
4. x[sup2] + 2 = 38.
5. x[sup2] - 2 = 34.
6. x[sup2] = -36.
Set F
Solve, or state that the solution set is
empty:
1. [square root]x = 9.
2. [square root]x = -9.
3. [square root]x + 2 = 9.
4. [square root]x - 4 = 9.