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## Teaching Number Sense at All Grade Levels

Gurganus (2004) agrees that number sense is analogous to phonemic awareness. However, she takes the broader view and notes that, unlike phonemic awareness, number sense develops throughout a student’s mathematics education and applies to a wide range of concepts. Here are her suggestions to teachers for promoting number sense across the grade levels.

**Pair numbers with meaningful objects.**To help young students view numbers as values rather than labels, associate numbers with concrete objects. For example, there are two wheels on a bicycle, three wheels on a tricycle, and four wheels on a car.**Use language to gradually match numbers with objects and symbols.**Model using talk to create sentences about number activities so that students can use self-talk to describe these relationships. For instance, “Two blocks and three more blocks give us five blocks.”**Incorporate counting activities.**Ask younger students to count to 10 and back. Challenge older students to count by 2s, 5s, 10s, and even 3s, 4s, or 7s. Counting up and back builds understanding of number relationships and magnitudes. Have students challenge each other to guess a counting pattern. For example: “500, 525, 550, 575—What is my pattern?”**Provide experiences with number lines.**Create a large number line across the classroom floor using colored tape. Have students move from number to number to show counting, operations, or even rounding. Draw number lines using whole numbers, integers, or decimals.**Plan meaningful estimation experiences.**Students need to recognize that many things cannot and need not be measured precisely. Provide lots of practice with estimation. Stress that estimation is not guessing but that there should be a reasonable range for the estimation based on experience. For example, “How many students do you think ate in the cafeteria today?”**Measure and then make measurement estimates.**Have students use measurement tools to measure length, area, volume, mass, temperature, and other attributes of meaningful things in their environment. Young students can start with measuring the teacher’s desk or distances on the classroom floor. After some practice, ask students to estimate before they measure. This builds a stronger sense of measurement units and what they represent.**Use number charts.**Charts in different arrangements (e.g., 1 to 100) offer many opportunities for students to explore number patterns. Cover up specific numbers on the charts and challenge students to discover the underlying relationships of difficult concepts such as factors and primes.**Introduce materials that involve numbers or number representations.**Ask students to examine items such as dice, dominoes, playing cards, coins, clocks, and rulers. Ask them to search for ways they can adapt these items for counting, pattern making, number operations, and number comparisons.**Read literature that involves numbers**. Books such as The Mud Flat Olympics, by James Stevenson (1994), or Anno’s Counting Book, by Anno Mitsumasa (1977), provide a different way to take a mathematical journey.**Create magic number squares.**Show students how to determine the missing numbers and have them create new squares to challenge their classmates.**Manipulate different representations of the same quantity.**Model moving back and forth between decimals, fractions, and percentages (e.g., 0.25 = 1/4 = 25%). Point out the same length in millimeters, centimeters, and meters (e.g., 35 mm = 3.5 cm = 0.035 m).**Explore very large numbers and their representations.**Students love the sound of large numbers, like billion and trillion, but often have difficulty conceptualizing them.**Use calculators to investigate the effects of squaring and other exponents.**Where appropriate, express large numbers with scientific notation (e.g., 500,000 can be written as 5 × 105).**Collect and chart data.**At every grade level, students can collect meaningful data. Ask the students to use concrete objects whenever possible, such as counting each type of bean in a mixture or the number of marbles of each color in a collection. Also ask the students to examine the data using graphs, formulas, and other comparisons.**Compare number representations in other cultures.**Students can gain insights into number relationships by exploring how other cultures count, use symbols for numbers, and solve algorithms. Students often find these activities fascinating. They can read about gesture counting and the various symbols and systems that various cultures have used to represent numbers (see Zaslavsky, 2001).**Set up spreadsheets.**Commercial spreadsheets are a great way for teaching students how to encode formulas for cells that will compute and compare values within other cells. Ask “What if. . .?” questions and manipulate values within the spreadsheet.**Solve problems and consider the reasonableness of the solution.**Remind students that the last step in problem solving should be to ask, “Does this answer make sense?” Have them practice selecting solutions by estimation without actually working out the problems.**Find everyday, functional uses of numbers.**Explore every opportunity for students to see the practical applications of mathematics. For example, they could follow their favorite sports team’s averages, track a company on the stock market, look for sales at the department store, or determine distances on a road map for the school field trip. Whenever possible, ask the students to graph, compare, predict, and discuss their data and measurements.**Explore unusual numbers.**Older students might find adventure in special numbers with intriguing patterns. Examples are Fibonacci and the golden ratio; abundant, perfect, and weird numbers; and number patterns that form palindromes.**Model the enjoyment of numbers and number patterns.**Research studies show repeatedly that the teacher is the most critical factor in establishing a climate for curiosity and enjoyment of mathematics. Keep learning and searching for new ways to have fun with numbers. The Internet is a valuable resource for number games. See follow up posts for a list of resources.

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