B1.4 Estimate the number of objects in collections of up to 50, and verify their estimates by counting
Skill: Estimating the Number of Objects
An understanding of the concept of quantity is essential for developing the ability to estimate, judge magnitude, and appreciate proportional relationships. Students develop their estimating skills by becoming aware of relationships between quantities (Is this quantity closer to 10 or 20?) and by judging order of magnitude; for example, a student who estimates that there are 100 counters in a set, when there are actually only 40, has not developed a good sense of magnitude. However, someone who estimates that there are 50 counters in a set has a better sense of magnitude. Students should therefore be encouraged in the early grades to question the reasonableness of their estimates. Manipulatives, such as ten frames, help develop estimation skills because they allow students to visualize quantities.
Source: translated from Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, Numération et sens du nombre, p. 45.
Estimation is a process whereby visual and mental information is used to assess the magnitude of a quantity. Estimates play an important role in our daily communications by giving us approximate quantities - this happens through the media (nearly 40 people were waiting in line at the new local store, for example) as well as in everyday life (two litres of milk cost about $5, for example).
As with the teaching of mathematical concepts in general, estimation skills should be developed in a problem-solving context. In order to develop an understanding of estimating a quantity, it is even better to present problems in which only the estimate is sought; for example, "Approximately how many pages of the dictionary do I have to read to get through 40 words?" Sometimes it may be worthwhile to compare the estimates with the exact quantity in order to examine the effectiveness of the estimation strategies and the accuracy of the estimates. However, this should not be done routinely or students will not see the relevance of the estimates, since the answer sought seems to be the exact quantity.
Students often believe that an estimate is a guess or that only one estimate can be considered acceptable, when in fact what matters in the estimate is the magnitude, not the number. A good estimate is a number that roughly defines the quantity being estimated. It is certain, however, that efficiency in estimating results from a good sense of numbers. There are no set rules for how to do this and no set goal: the context of each problem and the size of the numbers will determine the acceptable degree of accuracy; for example, an estimate of about 10 blocks or about 15 blocks is very acceptable when the actual quantity of blocks is 12, whereas an estimate of about 30 would be the result of guesswork or an inefficient strategy. On the other hand, if the actual quantity is 35 blocks, estimates of about 28, 30, or 40 become acceptable, since the difference between the estimate and the exact quantity is negligible, given the size of the actual quantity.
In estimation situations, in order to clarify that the number represents an approximate quantity, it is important that students communicate the estimate using "approximation" language such as about, approximately, at least, near, a little more than, more or less, or between; for example, about 50 people were at the party, it is about 1 m tall, it is a little less than 5 kg, or a little more than 20. These teaching strategies help to minimize the impression that there is one "right" estimate and to recognize the value of various estimates and their degree of accuracy.
In order to estimate, a comparison must be made between the quantity to be estimated and some benchmark; for example, Rachelle and Oman are wondering how many students are playing in a specific part of the schoolyard. As they think about it, they can use a benchmark such as the number of students in their class group, which is 20 students. This known real-life situation helps them to mentally picture a certain number of people in a given space and to estimate either that there are fewer than 20 people in the field or, on the contrary, that there are 20 to 40 students (2 times more) or that there are nearly 60 people (3 times more). It is important to know that when estimating with benchmarks, students apply proportional reasoning between a known amount (20 students in a certain space) and an unknown amount (the number of students in another space).
If students do not have existing cues for a given situation, they may need to use an alternative strategy. For example, students could count a subset of objects and use that number as a benchmark to estimate the total quantity. If asked to estimate the number of marbles in a jar, students can first take a handful of marbles, count them, and use that as a benchmark to determine the total quantity of the jar. They could also count the marbles in the bottom layer of the jar (10 black marbles) and then estimate the number of rows of marbles in the jar (about 5) to be able to estimate the total number of marbles (50). Students could also divide the quantity into sections and estimate the quantity in each section.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 36-38.
Skill: Counting (Checking Estimations Using Counting Strategies)
Counting involves both reciting a series of numbers (counting) and associating them with a series of objects.
By the time most children enter Kindergarten, they have usually learned to count and may be able to count to large numbers. Students usually use their memory and recite the numbers continuously, as they do with the letters of the alphabet. However, if they are asked what number follows 5, they may start counting from 1 again without really understanding the meaning of the question. Young children may not understand that we always count the same way; for example, children may count by saying "1, 2, 3, 4, 5, 6… " one time and "1, 2, 3, 5, 4, 6, 8…" another time, without noticing this inconsistency. If asked to count objects, they may not associate each object with a number in the count (see "one-to-one correspondence" in the underlying elements) or may touch two elements with a single number (for example, fourteen), so the count may not produce the same result each time. Children also sometimes do not understand that any set of objects (even very different objects, such as cookies and apples) can be counted in the same count and that one can start counting from any object in the set and arrive at the same number.
Counting involves being able to relate a quantity to the name or numeral that represents it.
Part of the complexity of counting is that a connection must be made between the name or numeral (symbol for the number) and the quantity it represents, a connection that children do not immediately grasp. For example, the word five and the numeral 5 represent not only the 5th object counted and the 5th digit in the set of numbers they have learned to recite, but they also represent the quantity of objects counted. It is essential that children make the connection between the quantity value of a number and its role in the pattern. If asked to name the quantity of objects at the end of a count, children may begin counting again without understanding that the last number said is, in fact, the quantity of objects in the set.
Learning to count to large numbers is a valuable experience. However, students must first develop an understanding of the concept of quantity and number relationships by working with small numbers. Students may be able to count very far and still have limited knowledge of the quantity represented by the result of a count; for example, students may be able to count to 30, but be unable to show 30 objects in a larger set of objects. Therefore, they should be given multiple opportunities to make the connection between the name or symbol of a number and the quantity represented.
Source: translated from Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, Numération et sens du nombre, p. 10-11.
Key Concepts of Counting
The purpose of this section is to help teachers understand some of the basic concepts involved in the early development of counting skills. The acquisition and mastery of these concepts does not necessarily occur in a linear order; for example, students may learn parts of one concept, move on to another, and then return to the previous one.
The following list of concepts is not intended to present a process that students will proceed through in a lockstep way. Rather, it is intended to be a tool to help teachers understand the concepts embedded in the skill of counting. It is not necessary for students in the early grades to know the names of these concepts.
- Stable Order: The counting sequence is stable and constant; we always say, for example, "1, 2, 3, 4, 5, 6, 7, 8… "and not "1, 2, 3, 5, 6, 8…".
- Order Irrelevance: Counting objects in a set can begin with any object in the set without changing the total quantity.
- Abstraction: A quantity can be represented by different things; for example, 5 can be represented by five identical objects, by five different objects, by five invisible things, such as ideas, or by five points on a line. Abstraction is a complex concept, but most students grasp it fairly easily.
- Conservation Principle: Counting a set of objects remains the same whether the objects are scattered or moved closer together.
- One-to-One Correspondence: During a count, each object is associated with a single count. Initially, it is beneficial for students to touch each object and set it aside once it is counted.
- Cardinality: Cardinality is the idea that the last count of a set of objects represents the total number of objects in the set. A student who counts again when asked to count the candy in a set just counted does not make the connection between the count and the quantity of objects in the set.
- Movement is Magnitude: As one moves forward in the number sequence, the quantity increases by a constant interval of 1 and as one moves backward in the sequence, the quantity decreases by 1. By extension, skip counting in intervals other than 1 are included under this element; for example, if one skip counts by 5, the quantity increases by 5 throughout the sequence and if one skip counts backward by 2, the quantity decreases by 2 throughout the sequence. Students can use this idea to facilitate learning basic number facts.
- Unitizing or Grouping According to Base 10: Unitizing is the act of grouping in a set. Grouping in base ten involves grouping into a ten once the count exceeds 9 in a decade. Understanding this idea is foundational to the concept of place value and the symbolic writing of numbers.
Source: translated from Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, Numération et sens du nombre, p. 13-14.