Measurement depends on several general concepts. Children need to learn: 

• the concepts of more, less, and same. Children need to know that this glass has more milk in it than another; that this field is larger in area than another; and that this object is heavier than that. At a very young age, even during infancy, children have intuitive concepts of more, less, and same, even though they may not know these words and of course cannot talk about them. Even three- or four-year-olds cannot say, “This glass has more because it is filled to the top.” At the outset, children’s intuitive concepts are based not on precise measurement, but on perception. That dog looks bigger than my dog. My sister’s bowl looks like it has more ice cream than mine. This block feels heavier than that one. Judgments like these are often correct, but, in the absence of precise measurement, can be wrong.
• that one object can be measured in different ways, depending on the attribute being considered. For example, you can measure a box’s height, width, depth, capacity or weight. Yet young children tend to focus on only one attribute.

• the language of measurement. The terms you use vary depending on the attribute or dimension being considered. For example, you describe a brick with a greater weight as heavier, not larger. You could compare a bookcase to a file cabinet using a variety of words, such as taller (height), longer (depth), or wider (width).
• that comparisons require attending to relevant dimensions. For example, suppose two brownies are each the same length and width, but one brownie is one inch thick and the other is one and a half inches thick. You need to consider the thickness to determine which has more heavenly brownie flavor. 
• the order and relative sizes of quantities along some attribute. 

  

  Like Goldilocks, you must know that Baby Bear is smaller than Mama Bear and that Mama Bear is smaller than Papa Bear. Ordering objects requires understanding that size is relative. The same object can be both smaller than one object and at the same time larger than another. For example, Mama Bear is shorter than Papa Bear but taller than Baby Bear.
• inferences can be made from information about order. For example, if a tree is taller than a sunflower and a daisy is shorter than a sunflower, then the tree must be taller than the daisy. If you know that a flea is lighter than a book, and that an elephant is heavier than a book, you can conclude that an elephant is heavier than a flea. Both of these are examples of transitivity: if a < b and c > b, then c > a.

General, often intuitive, concepts are not enough, however. It’s fine to know that this is more than that, but for many purposes you must be able to determine how much longer or heavier this is than that, how much louder this sound is than that, and how much more cake is on this plate than that. Children need to overcome their perceptual approach—relying on how things look—and adopt a more explicit, formal and rigorous method.

Sometimes you need to measure objects exactly. Precise, quantitative measurement involves several features that children need to learn over time, from early preschool through Kindergarten and beyond. Children need to learn to:

• use constant, standard, and identical units of measurement. If they do not, their measurements will be meaningless to others and cannot be shared. For that reason, children need to learn to use units like inches or centimeters for measuring length.
• understand how precise measurement is different from counting objects. When you count to determine the number of objects, the collection can be a mixture of anything—one big dog, one small dog, and one pencil; or one unicorn and two blocks. The sum is three in both cases. By contrast, when you measure, you must count only identical units such as inches or centimeters. Also the final value represents a measurement, such as a distance or a weight, not a quantity. 
• be able to use and understand the system of whole numbers. Unless children know that five comes after four and nine before 10, they will not understand that five pounds is heavier than four pounds, and nine inches is shorter than 10 inches. They also need to understand that the starting point of any measurement is zero, not one.
• understand written symbolism to read a measuring tool, such as a ruler, scale or measuring cup. Sometimes, children find this difficult because rulers are divided into fractional units, such as halves, quarters, and more, and measuring cups may also show fractions.  
• understand, in later years, how fractional numbers are used to make precise measurements, such as determining that this object is 4.6 meters long and this other object weighs 15 ½ pounds.  And after that, they must understand negative numbers in order to measure temperatures or debts.
• recognize, in later years, that scientific measurement makes more mathematical sense than customary measurement. The metric system is based on tens, and uses prefixes like kilo-, centi- and milli-, which make explicit the relationships among metric units. For example, there are 1,000 meters in one kilometer and 1,000 grams in one kilogram. By contrast, equivalent customary units must be memorized. There are 12 inches in one foot, but 16 ounces in one pound.

Clearly the use of precise measurement is the educational goal. One pedagogical approach might be to start with standard units (customary or scientific) from the beginning of schooling, and to avoid non-standard units altogether. Another, and likely more effective approach, is to begin with non-standard units, such as children using their feet to step off a distance, with the goal of showing that this kind of measurement is unreliable. (For example, Harry needs three steps to get from point A to point B, but Sally needs only two.) Having established this, the teacher might have children measure objects using an agreed-upon reference object, like same-sized paper clips, to produce reliable measurements. After that, the teacher might introduce children to the use of customary or scientific units.

Sometimes, precise measurements are critical. For example, if you are making bread, you probably should measure the amount of yeast you put in the dough. However, there are occasions in which precise measurement is not needed (such as putting a pinch of salt in a stew). In that case, you can approximate, in the sense that your estimate need only be “close enough” to the correct answer. But how close is close enough? It depends on context. If you are estimating an elephant’s weight, an error of a pound or two makes no difference. But if you are estimating the weight of a letter to be mailed, a pound or two can cost you a tidy sum.

To arrive at an estimate, you can’t just guess the number. Instead you need to follow a strategy based on experience and on mathematical concepts. Suppose you are estimating the height of an elephant. You know from previous experience that elephants are tall. You also know from experience that Professor Ginsboo is about four feet tall. You then may use a mental image of those four feet—a kind of standard in your mind—to determine that the elephant’s height is equal to about three of the imagined four-foot Professor Ginsboo lengths. That means that the elephant is about 12 feet tall, which in fact is not a bad estimate.

So the process of estimation depends on experience (you have seen elephants at the zoo) and on your ability to use mathematical concepts (you understand the foot measure so you can add the three lengths of four feet, or you may even know that three times four is 12). Estimation depends also on the idea of approximation and on your willingness to deal with uncertainty. Estimation may exploit perception (you see that the elephant is very tall). But children need to learn that perception is not enough and must be guided by mathematical concepts to achieve a reasonable approximation. Children often crave regularity and tend to think that an answer is either right or wrong. They need to learn that the idea of “close enough” can be useful when a precise answer is not possible or not needed.

Measurement is more complex than often imagined. Sometimes you can rely on perception to describe an attribute (“This elephant looks very tall.”); sometimes you can estimate (“The elephant is about 12 feet tall.”); and other times you must measure precisely (“The elephant is 12 feet and three inches tall.”). Further, to produce accurate measurements and communicate the results to others, you must understand several mathematical concepts, particularly the fundamental idea of a standard unit. Happily, children don’t need to learn all of this at once. But it is helpful for teachers to have the whole picture in mind as the help them gain their understandings of measurement.