Strategies for Successful Learning, Volume 7, Number 1, September 2013
Brought to you by Learning Disabilities Worldwide (LDW®) through the generosity of Saint Joseph's University.
Joan Gujarati, Ed.D.
In his book, 1+1=5 and Other Unlikely Additions, La Rochelle (2010) poses mathematics problems such as 1+1=11? In the aforementioned example, with colorful illustrations, he portrays pictures of a sneaker, a basketball, a skate, and a puck and the reader has to figure out that 1 basketball team + 1 hockey team = 11 players. In another example, 1+1=14?, he depicts a nature scene with grass, flowers, an ant, and a spider with a large web and the reader has to deduce that 1 ant + 1 spider = 14 legs. The book is filled with 1+1 = [some number] problems in which the reader uses the context clues (illustrations) to reach a solution.
Students in the primary grades are generally taught that 1+1=2. So how are all these other problems, which La Rochelle showcases in his book for 1+1=?, a possibility? It takes a strong understanding of number sense to be able to think flexibly to go beyond procedural understanding to really attach meaning to numbers to move toward development of conceptual understanding. After all, numbers do not exist in isolation; they should mean something and have some type of units attached to them. Students are more apt to pick up and be more interested and invested in mathematics concepts which hold personal meaning and are made relatable to lived experiences. As Gersten and Chard (1999) posit, developing number sense is a prerequisite for succeeding in mathematics.
What is Number Sense?
Howden (1989) defines number sense as “good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways which are not limited by traditional algorithms” (p. 11 cited in Van de Walle, Karp, & Bay-Williams, 2013). Number sense involves flexibility in thinking about numbers. Someone with strong number sense has a general understanding of numbers and operations and the ability to use this knowledge flexibly to develop useful strategies for solving complex problems. In the primary grades, students move from basic counting to understanding number size and relationships, place value and operations. In general, we have number sense because numbers have meaning for us (Sousa, 2008). Number sense develops when students connect numbers to their own real-life experiences. It constitutes a way of thinking that should permeate all aspects of mathematics teaching and learning (Berch, 2005).
Strategies to Promote Real-World Number Sense
Since no two students learn exactly the same way, the ideas presented below can tap diverse learners’ needs because different learning modalities are presented. When children use a variety of senses to learn about numbers, they can learn more efficiently (Zaslavsky, 2001). Concrete examples that teachers can utilize in their classrooms to promote number sense, especially early number sense, are showcased. Central to these ideas are connections to the real world and students’ lived experiences. I have used these strategies with elementary school students and with preservice elementary teachers in my mathematics methods courses. These conceptual experiences are the ones that let children develop number sense and flexibility. Although early number sense is generally a concept stressed in the lower elementary grades, the suggested activities can span different grade levels because of the open-ended nature of number sense acquisition skills, and the developmental levels of students.
Most students need many opportunities to deeply understand a topic. Particularly for those students who may exhibit difficulties retaining and applying information, promoting more conceptual experiences is even more critical (Rathmell & Gabriele, 2011). It is important to provide students opportunities over time to make sense of mathematics ideas to conceptually understand them and not just to merely practice skills. Distributed instruction presents mathematics topics repeatedly throughout the school year rather than one chapter or unit at a time. The constant exposure to concepts is more likely to stay with children (Rathmell & Gabriele, 2011), particularly children with Learning Disabilities. The activities presented below can be done over time and revisited in different forms throughout the school year(s).
- Number card selection. Using cards with a number from 1 through 10 written on them (these can be made using index cards or can be store bought), have students randomly pick a number and think of as many connections they can make with it to the real world. For example, a child chooses “5.” How does five relate to their lived experiences? Some possibilities could include 5 fingers on a hand, 5 toes on a foot, 5 pennies equal a nickel, a pentagon has 5 sides, or 5 players on a basketball team. Depending on the classroom needs, students could work individually or in pairs, show their answers through oral, written, or artistic means. This activity could also be done with varied grade levels as the responses would likely change as levels of mathematical thinking increase.
- Favorite number collage. Students choose their favorite number and make a collage about how that number relates to the real world. Students can use pictures from newspapers or magazines, download images from a computer, or draw them and arrange them in a meaningful way to highlight their number. Collages can then adorn the classroom and can also be used for students to get to know one another better, mathematically speaking.
- Internet number research. Depending of the level of the child, bridge the use of technology with mathematics by having students do some internet research on their favorite number or even a number randomly assigned to them. This task can be done individually or in pairs. Students can then share out what they found using different modalities: oral, written, pictorially, and even dramatically.
- The hundreds chart. Typically, the hundreds chart is just thought of as a tool for counting and can often be seen hung up around a primary grade classroom. However, there are so many more possibilities to use a hundreds chart to develop number sense. Have students take the time to explore patterns found on the hundreds chart such as row, column, and diagonal patterns. For example, students may discover that diagonal patterns could yield ±9 or ±11 depending on the direction of the diagonal. This tool can be used to understand number relationships and magnitudes, and those numbers on the chart can be related to real-world items.
- Class graphs. Classroom communities are rich data sources. Many possibilities exist to collect and chart data. For example, have the class make a graph about their favorite mode of city transportation and then ask questions about quantities to compare the results (e.g., how many more students selected car than subway?). The “favorite” could even be linked to another curricular unit currently being studied.
- Estimation experiences. Provide students with meaningful estimation experiences. For example, the class can have an “estimation jar” in which each student gets to take it home and fill it up with something for the class to estimate weekly; it could be snack foods or other items, depending on a teacher’s preference. Do a “guess and check” where students have to predict the amount in the jar and then the class figures out a way to check the answer. It is important for a teacher to stress a reasonable range for the estimation based on experience.
- Home-school connections. Students can choose a number and count/locate objects that relate to that number in their homes. Through this experience, numbers are paired with meaningful, concrete objects.
- Link numbers to other cultures. Given the diverse student populations in classrooms today, have students talk about how numbers are represented in other cultures; how different cultures may express and communicate numerical quantities.
Number Sense as a Prerequisite for Later Mathematics Development
The classroom activities showcased in this article are only a sampling of the possibilities to promote real-world number sense, notably early number sense. For students to understand a key concept, they must learn multiple ways to represent the idea, learn multiple strategies to come up with solutions, and become fluent using varied tools so that their thinking is flexible and efficient (Rathmell & Gabriele, 2011). The importance of highlighting the classroom activities above is to showcase inquiry-based learning which can be done over short and long periods of time. With greater engagement with real-world number concepts, students can learn to think flexibility about numbers and build a strong foundation on which more advanced concepts at the upper elementary, middle, and high school levels are based.
Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333-339.
Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. Journal of Special Education, 33(1), 18-28.
Howden, H. (1989). Teaching number sense. Arithmetic Teacher, 36(6), 6-11.
LaRochelle, D. (2010). 1+1=5 and other unlikely additions. New York, NY: Sterling Publishing Co., Inc.
Rathmell, E. C., & Gabriele, A. J. (2011). Number and operations: Organizing your curriculum to develop computational fluency. In F. Fennell (Ed.), Achieving fluency: Special education and mathematics (pp. 105-139). Reston, VA: National Council of Teachers of Mathematics.
Sousa, D. A. (2008). How the brain learns mathematics. Thousand Oaks, CA: Corwin Press.
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Upper Saddle River, NJ: Pearson Education, Inc.
Zaslavsky, C. (2001). Developing number sense: What other cultures can tell us. Teaching Children Mathematics, 7(6), 312-319.
Joan Gujarati, Ed.D., is an Assistant Professor in the Department of Curriculum and Instruction at Manhattanville College in Purchase, New York where she teaches the childhood mathematics methods courses. She is a former elementary school teacher and Math Teacher Leader. Dr. Gujarati has presented at numerous professional conferences and has published in the field of childhood mathematics education. Dr. Gujarati’s research interests include early childhood and elementary mathematics education, teacher beliefs and identity, teacher quality, effectiveness, and retention, and curriculum development. Dr. Gujarati can be reached at