What is the Mismatch between school arithmetic and real-life arithmetic? (or Why must the teaching of arithmetic change in school?)
Back in 1992 Marilyn Burns wrote this essay about the mismatch in her book, About Teaching Math.
What amazes me is that this essay was published in 1992. It embodies the ideas of the Standards in terms of Number and Operation. Luckily, the classrooms of Region 10 do not have this mismatch. The activities our students engage in often match the type of work people do in real life.
The Mismatch Between School Arithmetic and Real-Life Arithmetic
By Marilyn Burns
Published in About Teaching Math, Math Solutions Publications, 1992
ISBN (0-201-48039-5).
Because arithmetic is a much-used life skill, it makes sense to look at real-life applications when making decisions about what children really need to learn. One way to begin is to identify the daily uses people have for arithmetic. (It would be useful to make a list of actual occasions during the past month when you used arithmetic. Include as many situations as you can in which you had to add, subtract, multiply, or divide to arrive at an answer you needed.)
Adults who make such a list generally include the following types of situations: when I figure my checkbook balance; when I’m in supermarket figuring how much I’m spending or doing price comparisons; when I want to know how much wallpaper or floor covering I need; when I need to determine the tip in a restaurant; when I want to know what mileage I’m getting on my car; when I need to figure how long a roast or turkey needs to cook; when I have to decide what time to leave to arrive at the movies on time.
A list of everyday applications of arithmetic is testimony to the usefulness of arithmetic. For further insights, two questions can be asked about any specific situation:
*What method is used most often to do the arithmetic calculation - a calculator, paper and pencil, or figuring mentally?
*Is it necessary to be accurate or is an estimate sufficient (or even preferable)?
When adults analyze the arithmetic methods they most often use, they find that they have usually relied on two methods – using a calculator and figuring mentally. (A study has reported that 93 percent of adults’ uses of arithmetic do not involve paper and pencil.) When the lists are evaluated to determine how often accuracy is necessary and how often an estimate will suffice, there is generally a 50-50 split.
In classroom practice, however, teachers admit to spending more than 75 percent of their math time, and closer to 90 percent for many, on paper and pencil arithmetic skills, with students practicing the skills in isolation from any problem situations. Arithmetic exercises are usually provided on worksheets or textbook pages, neatly arranged and ready for children to apply the computation methods they’ve been taught. Estimating is often encouraged, but arriving at accurate answers is always required.
The mismatch between the work students are asked to do in school and the arithmetic needs they will have in real life is revealed even more clearly when we examine the typical uses of arithmetic. For example: finding yourself in a supermarket without your checkbook and with only $20.00. You need to keep track of the cost of the groceries you put in your cart to avoid arriving at the checkout counter with a grand total more than you can pay. A running total is required, and you most likely will keep it mentally. (There are some supermarkets, however, that now provide shopping carts with built-in calculators!) Each time you select an item you decide how much to add to your running total. If a chicken costs $2.37, you’ll most likely add $2.50; if a sack of chips costs $.89, you’ll probably add a $1.00. Although you’ll try to be fairly accurate with your calculations, it’s sensible to estimate each item to avoid the unnecessary hassle of laborious computation. Not only is it more sensible to use estimates than accurate amounts, it’s also advisable to overestimate each time, or at least enough of the time, to be sure not to be caught short. Your arithmetic calculations may help you to decide, in order to keep within your limit, not to buy items that aren’t absolute necessities.
Another example: tipping in a restaurant. The check for dinner shows a total of $33.92. You examine the check for accuracy, adding the amounts listed or, more likely, estimating to sure the total is reasonable. Of the total, the amount for the meal was $32.00 and tax was $1.92. Suppose the food was good and the service was prompt and cordial. What tip do you leave? How do you figure it? Standard protocol calls for at least 15 percent of the total without tax. You most likely do the calculation mentally and most likely by a method different from the standard multiplication algorithm you learned in school. Some people figure that 10 percent of $32.00 is $3.20, half of that is $1.60, and the amount together is $4.80.
Still, how much do you leave? $4.80? $5.00 More? Less?
One more example: figuring gas mileage. Some people regularly check their gas mileage when filling the tank to check their car’s performance. Suppose it takes 11.7 gallons of gas to fill the tank. The odometer indicates that you’ve traveled 253 miles since the fill-up. How do you figure the miles per gallon? You know you have to divide. If you have a calculator handy, you may use it. If, however, you’re figuring mentally, chances are you’ll adjust the numbers to make them friendlier. Instead of 11.7, 12 will do; instead of 253, you use 250. Calculating mentally you get a little more than 20 miles per gallon. That’s close enough to know that your car doesn’t need immediate attention. But suppose you had arrived at 8 miles per gallon. What would you do? Most likely you’d first recheck your calculations, since you know what a reasonable ought to be. Then you would decide if emergency car care was needed.
Contrast these real-life situations with the work children generally face in school. Most of children’s arithmetic practice is separated from problem situations. They do not have to choose the operations nor decide on reasonable numbers to use. They are not asked to decide how to do a calculation but are expected to perform algorithms using pencil-and-paper procedures they were taught. Although encouraged to recheck their computations, most students are not motivated to do so. Since there is no context for the calculation, there is no way to readily notice when an answer is unreasonable, or in some cases, ridiculous.
Estimates for answers are generally not acceptable on children’s arithmetic work. (A cartoon in a national magazine showed two students at the chalkboard, one struggling with 6 + 7. The struggling student commented to the other, “The problem is, they require such pinpoint accuracy.”) Textbook directions often encourage students to estimate first for a ballpark figure against which to judge their answers. But students rarely do so. Why estimate when an accurate answer is required? Why do the work twice? Since there is no context for the calculations, students are not given opportunities to decide when accuracy is demanded and when estimates are sufficient. Their interest is very different from the interest an individual has when in the supermarket, the restaurant, or at the gas pump.
What are the basics in Arithmetic?
The real-life situations people face that call for using arithmetic generally require that they do the following:
1. Choose the operation (or operations) needed.
2. Choose the numbers to use.
3. Perform the calculation, either using a calculator, paper and pencil, or figuring mentally.
4. Evaluate the reasonableness of the answer and decide what to do as a result.
The situations students face in school, however, most often only require the third step listed and allow only one option – using paper and pencil to perform the calculation as it was taught. An emphasis on this narrow aspect of arithmetic in no way provides even minimal arithmetic proficiency.
Why has pencil-and-paper arithmetic computation become the mainstay of the elementary curriculum? It seems clear enough to understand. The ability to do calculations has long been seen as essential for the successful use of arithmetic. This practice was in place and well entrenched long before calculators and computers removed the drudgery from computation. There was no way to arrive at arithmetic answers other than doing the calculations by hand. Because of the present availability of calculators, having students spend more than six years of their schooling mastering paper-and-pencil arithmetic computation is as absurd as teaching them to ride and care for a horse in case the family car breaks down.
With or without calculators, however, learning to do paper-and-pencil arithmetic and practicing on isolated examples has never ensured that children learned to use these skills when needed. This deficiency is obvious to teachers when word problems are assigned and children repeatedly ask, “Do I need to add or subtract?” Arithmetic practice in isolation does not lead children to notice when they make a division error, such as omitting a zero in the quotient, and produce an answer that is 10 times too small.
This does not mean children do not need to learn any arithmetic. Arithmetic skills are necessary life tools. Doing arithmetic mentally demands mastery of basic facts along with the ability to estimate. Using a calculator successfully and evaluating the answer requires an understanding of the necessary arithmetic processes and the ability to identify a reasonable solution. Being able to use paper and pencil to solve an arithmetic problem has its place. But there is no place in our schools for focusing on computation apart from problem situations and then claiming that children are being taught arithmetic. The very definition of teaching arithmetic needs to be revised. Arithmetic competence cannot be measured solely by evaluating students’ mastery of computational algorithms. Mastery of arithmetic must include, as basic and integral, knowing which operations are appropriate to particular situations, which numbers are most reasonable to use, and what decisions can be made once the needed calculations have been done. It makes no sense to say that a child can do arithmetic but cannot apply it situations. The basics of doing arithmetic must include being able to apply arithmetic operations to situations, as well as being able to calculate answers.
You would never say that a person who can only play scales knows how to play the piano. You would not assume that a person who can catch a football knows how to play the game. You would not make the judgment that a person who can saw a board knows how to build a bookcase. The rudiments are necessary, but they would be senseless to learn as ends in themselves.
The motivation for learning skills in life is their eventual use. Children who practice scales have heard music. Children who practice catching and throwing a football have seen football games. In school, however, children are expected to practice their arithmetic skills without any sort of mathematical bookcase to build. No wonder so many are not motivated to develop their proficiency. What sense does it make? Instead of seeing arithmetic skills as useful tools that save work, children see them as making work, their only purpose the completion of pages of exercises.
The reality is that many children cannot apply arithmetic skill even to the simplest of work problems. Many students are not able to tell if it is cheaper to buy things two for a nickel or three for a dime. They are lost when asked if one-third of a cup and one-half more will give more or less than a full cup. However, when these same children can demonstrate proficiency with paper-and-pencil computation, it is common for teachers to say that, because the children can perform addition, subtraction, multiplication, or division calculations, they have mastered arithmetic.
The simplistic and useless definition of arithmetic competence that has resulted in the achievement of computational proficiency as the major mathematics goal of the elementary school must change. The definition of skill basic to arithmetic must broaden. Children must develop understanding of arithmetic concepts in problem-solving contexts. They must learn to calculate mentally, to use numbers comfortably to come to reasonable estimates, to develop understanding of relationships among numbers and operations, to be confident and competent in their number understand, and to develop an appreciation and fascination for numbers.