## Types of Estimation and Meaningful Estimation Activities

## Types of Estimation

Taylor-Cox (2001) suggests four distinct types of estimation.

**True approximations**are used when an estimate is acceptable, especially when dealing with very large numbers. Is it really important to know that the average distance from the Earth to the sun is 92,955,630 miles, or will 93,000,000 miles suffice? True approximations are more applicable to problems in the intermediate and upper grades. Unfortunately, the numbers that youngsters work with in the primary grades are typically smaller and less complex than the numbers that lend themselves to true approximations. Making true approximations has little advantage with simpler numbers because we can easily calculate to ensure that we know the exact amount.**Overestimating**is used when rounding up might be beneficial, such as overestimating the amount of food to buy for a child’s birthday party. The major drawback for this option is that it may be wasteful if you get way too much. But if you underestimate, some kids may not get enough food.**Underestimating**is used when rounding down is applicable. This can be helpful in certain situations. Better to underestimate the amount of profit a business will make so as to avoid overspending.**Range-based estimations**broaden the applicability and understanding of estimation. Some situations call for underestimating and some for overestimating. Range-based estimation involves thinking about quantity in terms of the upper end and the lower end that encompass an estimate: “What are the minimum and maximum quantities I need for this?” In the primary grades, teachers can design mathematical tasks that are productive and worthwhile by using range-based estimation, thereby encouraging students to become better estimators.

## Meaningful Estimation Activities

For estimation activities to be meaningful rather than futile, Taylor-Cox (2001) suggests that the activities include the following five components.

**Purpose.** Whenever you ask students to estimate a number, give them a reason for doing so. These contexts offer a purpose and give students a reason to engage in real-life mathematical problem solving. Otherwise, students may ask, “Who cares?” Making the task relevant, interesting, and significant invites students to care and, consequently, invites them to engage in mathematics. Offering a purpose does not ensure that the “Who cares?” response will disappear. But by listening to students and reflecting on their perspectives and feelings, you can manage the continuing challenge of providing meaningful mathematics.

**Referents (benchmarks)**. To help students succeed, give them a referent or benchmark they can use when making estimations. For instance, if you ask students to estimate the number of marbles in a jar, it would be helpful to provide a smaller container with a known number of marbles of the same size. This container gives students a point of reference on which to base their estimates for the larger jar.

** Pertinent information**. Clarify the actual mathematical problem to be solved so that the students can decide what type of estimation is most appropriate. For example, students do not need to estimate the number of marbles in a jar if they are going to open the jar and count the actual number of marbles. As explained earlier, doing so counteracts the purpose of, and time spent on, estimating. Rather, you should ask whether estimating or counting to find out the actual number is more appropriate. Which methods will be used to check for accuracy? What kind of information is pertinent to the given mathematical situation?

**Diverse experiences.** Students need numerous and diverse experiences with estimation in the context of other content areas, such as in time and measurement. Primary-grade students often have difficulty estimating time. Teachers are no help when they attach inaccurate time constraints to their statements. When they say, “I will be there in just a minute” or “Wait one second,” we really mean, “I will be there when I can” or “wait indefinitely.” Perhaps asking students to actually time the teacher encourages them to check estimates of time while enhancing their experiences and improving their precision with estimating time.

Young students work on measurement skills by comparing lengths, weights, and capacities. For estimating size they use comparative language, such as larger, smaller, heavier, and lighter. Figure 5.5 shows two examples of activities for different age groups that require estimating size. In these types of tasks, students engage in estimation that is related to size rather than quantity, they recognize that estimation is an important tool for dealing with real-life mathematical experiences.

**Range-based techniques.** Estimation should involve using mathematical skill to predict information within a reasonable range. If, for example, the actual number of a quantity or measurement is in the 70s, an appropriate estimation may be in a range of 10 or less. But if the actual number is in the 700s, an appropriate range may include up to 60 or 70 numbers. Although suitable ranges vary with the problem situation, the aim is to estimate within an appropriate range. However, many students still want to estimate the precise amount. To combat this need for the right answer, it may help to use terminology such as the actual answer. A range of about 10 to 20 per hundred is reasonable. This type of ranged-based estimation is particularly helpful in situations that call for approximating a quantity that may need to be overestimated.

Estimation experiences improve the students’ estimation skills, increase their confidence in their level of mathematical expertise, enhance their perception of the value of mathematics, and improve their mathematics achievement test scores (Booth & Siegler, 2006). Each estimation activity is an opportunity for teachers to connect mathematics with the everyday lives of students.

### References

Booth, J. L., & Siegler, R. S. (2006, January). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 42, 189–201.

Taylor-Cox, J. (2001, December). How many marbles in the jar? Estimation in the early grades. Teaching Children Mathematics, 8, 208–214.

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