# Creating a Math-Rich Environment

### From Foundations For Numeracy

A math-rich classroom is not one that requires fancy and expensive tools, but is instead one that is led by a teacher who is knowledgeable in the area of age-appropriate math concepts and who can incorporate them into daily activities (Sarama & Clements, 2005)
^{[1]}.
Increasingly, the role of the teacher is viewed as a mentor who helps children connect their informal numeracy knowledge with their increasingly explicit knowledge of mathematics (Sarama & Clements, 2005)^{[1]}.
Teachers can also help children connect various math topics to each other, such as geometry and numbers (e.g., count the sides of a desk and then measure each side; Clements & Sarama, 2006)
^{[2]}.

Recent thinking in math education has shifted from an emphasis on the teaching and practicing of algorithms (e.g., memorizing arithmetic facts or completing pages of math problems) to focusing on reasoning and problem solving and their application to real world problems. There is also a focus on developing positive attitudes and feelings of self-efficacy toward mathematics (De Corte & Verschaffel, 2006)
^{[3]}. For example,
De Corte and Verschaffel (2006) ^{[3]}developed five guidelines which should be included in an effective mathematics learning environment:

- Teach and support active, productive learning processes in all students maintaining a balance between individual exploration and systematic instruction/direction.
- Promote the development of self-regulation so that students learn to be in charge of their own learning.
- Create activities based on real-life situations applicable to future knowledge and skills and that enable the opportunity to collaborate with others.
- Create occasions to gain widely applicable thinking and learning skills within the math content.
- Foster a classroom atmosphere and culture that promotes students’ open reflection and discussion on learning and problem-solving strategies.

## Contents |

## Number Sense

It is important for children to master the connection between quantities and numbers in the first two or three years of school (Nunes, 2008)
^{[4]}.
Quantities do not need to be a numerical value (e.g., you can compare two people’s heights without knowing their exact numerical value or you can determine which pile of candies has more visually). Working with quantities helps children develop early mathematic abilities. Initially, this process can begin as children learn the relation between a set of objects and the numeric symbol (word or digit) that represents it (e.g., ** is 2). It is important that children understand that a numeric concept like “2” can apply to any set of two objects (Gelman & Gallistel, 1986)
^{[5]}.
As a result, a math-rich environment at this early stage will provide children with multiple opportunities to make judgments about quantity (e.g., *Which of these boxes has more blocks? Which tower is taller?*) and to associate quantities with meaningful symbols (e.g., *Let’s count the blocks and see which box has more. Let’s measure which tower is taller using this ruler*). In this way, teachers promote the understanding that quantity is an important feature of sets, that number symbols represent specific quantities, and that quantity is a characteristic that can be applied to any set.

As children develop basic number sense, a math-rich classroom will present children with more complex understandings of quantity. For example, if we take one away from a group of blocks but add a different block we still have the same number as before. This concept can easily be demonstrated with any number of objects in the classroom (e.g., pinecones during fall theme or balls during a play session). By using everyday objects and demonstrating how quantity is affected by the addition and/or subtraction of additional elements, children learn that quantity is impacted in consistent and predictable ways by the addition or subtraction of objects. Teachers can also use these practical, real-world experiences to make links between changes in quantity and written Arabic statements of the change (e.g., 10–1+1=10). The relating of real-world terms and operations to Arabic symbols will be an important skill for solving word problems (Cummins, 1991)
^{[6]}.

## Number Line

As discussed in the research summary, a child’s development of a mental number line is essential to the ability to add and subtract as well as to estimate (Siegler & Booth, 2004)
^{[7]}.
Forming a mental number line requires the ability to visualize abstract numbers, to order numbers by quantity, locate a given number along a line, and generate any portion of the number line that may be required for problem solving (Gervasoni, 2005)
^{[8]}.
For example, understanding that 50 is a number beyond 40, but is half the way between 0 and 100, reflects that a child has the ability to understand the relation between numbers.

The classroom environment can be used to develop this skill. For example, count the first 100 days of school and create a number line banner around the room. Each day add another number to the line and create activities that use the number line. Also, refer to the calendar number line in a way that reflects the relations between milestones (i.e., *What day was half-way between the start of the year and a holiday?*). Have the children collect items (e.g., paperclips); find where their number of paperclips fits on the line, is it more or less than another child’s? Refer to the number line to help children with their addition or subtraction.

Use the number line to present number patterns, such as counting by 2s, 5s, or 10s. The development of a child’s mental number line is also important for estimation. Children will be better able to understand the relation between numbers if they are asked to approximate the relative position of various numbers on a number line (e.g., label a number line with only 1 and 100, then ask a child where on a number line they think 84 will fall, then have them check your visual number line to determine the actual position (Sigler & Opfer, 2003)
^{[9]}.
The accuracy of a child’s number line continues to develop throughout grade school (Siegler & Opfer, 2003)^{[9]}.

Work with a number line can be adjusted for children with more advanced skills. For example, the visual number line can be changed to include decimals or can become a positive and negative integer number line. This will allow children to answer questions and visualize the distance from zero, whether it be positive or negative (e.g., *Which one is further from zero, 54 or negative 54?;* de Hevia & Spelke, 2009)
^{[10]}.

## Base-10

The base-10 system is foundational to later math skills; many children require instruction that focuses on this repeating decade structure (Geary, 2006)
^{[11]}.
From Kindergarten to Grade 2, children need to understand that the word “ten” can represent one unit (1 ten) or ten individual units (10 ones) and that these representations are exchangeable (National Council of Teachers of Mathematics, 2004)
^{[12]}.
Any number of concrete materials can be used to help children learn the concepts of place value. For example, you can have blocks or connecting cubes representing ones, tens, hundreds, or you could have children group popsicle sticks into groups of tens using elastic bands and then use the groups of 10 as well as individual sticks to count or represent various numbers depending on the activity. Children can then use these blocks to represent the base-10 composition of numbers (e.g., 36 is represented as 3 ten-units and 6 one-units and not just 36 separate units; Geary, 2006)^{[11]}. Using popsicle groupings that are banded together can also be used to effectively represent the process of borrowing and carrying with demonstrations of multi-digit arithmetic (e.g., When subtracting 9 from 32, there are not enough “sticks” to take 9 from 2. So, we take one of the ten-units apart to create “12” in the ones category. Then take away 9, leaving 3 one-unit popsicle sticks and 2 ten-unit sets).

## Addition/Subtraction/Fractions/Division/Multiplication

Simple arithmetic is likely one of the easiest techniques that teachers can incorporate into a math-rich classroom. Even the relatively complex concept of division can be easily demonstrated with sharing (e.g., *If there are 10 cookies and 5 children, how many cookies will each child get if we share them fairly?*). In preschool and early elementary school, sharing gives children a basic understanding of simple fractional relationships. For example, read the book “The Doorbell Rang” by P. Hutchins (1986)
^{[13]},
then ask the students questions about how the cookies can be shared as more and more children arrive at the house.

Children also benefit from learning that there are multiple strategies that can be used to solve a problem. A simple way to demonstrate different problem solving methods is to have children share how they arrived at an answer with the class; if several students share their strategy, then likely the class will be exposed to several solution methods (Caliandro, 2000)
^{[14]}.
Another way to facilitate this type of learning is by having the children solve math problems in small groups and then have the groups present their work to the class (De Corte & Verschaffel, 2006)^{[3]}.

Learning about money is a skill commonly acquired in the primary grades; teachers can have a jar full of change ready for children to use to solve real-world problems. Skills targeted with money include: coin recognition, values of coins, counting sets, equivalent collections, choosing coins to make a specific amount, and making change (Van de Walle, Karp, & Bay-Williams, 2010)
^{[15]}.

## Geometry

A classroom can be rich in the variety of shapes available for students to manipulate. Desks may be square, the table a rectangle or circle, and a funnel cone shaped. The block set may contain a variety of triangles and other shapes to be manipulated, allowing the formation of new shapes (e.g., a parallelogram). Having these shapes easily visible allows students to begin making shape connections, measuring shapes, and estimating their size or volume depending on their grade level. In a math-rich classroom, it is important for teachers to demonstrate and model the use of the correct terms for geometric shapes (e.g., cube, sphere, rhombus, cone). Geometry is especially easy to incorporate with art, such as by asking the children to create pictures from a group of shapes or by labeling the shapes that children spontaneously create. Geometry can also be emphasized in explaining the effects of shading on three-dimensional objects.

## Technology

The use of simple technology such as calculators and computers is a valuable component of elementary mathematics education. However, it is important to consider technology as an addition to learning “mental” or “by-hand” strategies, not as a replacement for them. Calculators may be particularly valuable, for example, for checking answers (Langrall, Mooney, Nisbet, & Jones, 2008)
^{[16]}.
Students could be asked to use a calculator to check either their own or another student’s work. It is also important to show children that errors can occur very easily, even with technology.

Computer programs that are designed to teach children math concepts using games have become very popular. Understandably, it is very compelling for both busy teachers and parents to use a self-directed math program that can provide children with a potentially engaging, fun, and consistent method to present math concepts while providing them with consistent practice (with instant feedback on accuracy). However, computer programs’ efficacy for teaching purposes can be challenging. For example, Wilson, Maisterek, and Simmons (1996)
^{[17]}
noted that many computer games have amusing repercussions if children make errors. As a result, in their study, it seemed that children found it entertaining to deliberately make errors, which undermined the efficacy of the practice. Teachers must be aware that children may not be using a computer program effectively and that monitoring to ensure that a child stays “on task” is still required. As well, teachers and parents should remember that interaction with others cannot be replaced by even a well-designed computer program.

If children have reliable access to computers, teachers can use simple programs to teach data management and presentation methods to children. For example, even young children can learn to create a database for simple information and then use this information to create graphs. Unfortunately, the consistency of access to technology can pose a challenge to many classroom instructors. Taking stock of classroom resources and skills with computers can help to maximize the efficacy of technology’s incorporation into the classroom.

## References

- ↑
^{1.0}^{1.1}Sarama, J., & Clements, D. (2005). How children “think math”.*Early Childhood Today, 20*(2), 11. - ↑ Clements, D. H., & Sarama, J. (2006, June).
*Scaling up the implementation of a pre-Kindergarten mathematics curriculum: The Building Blocks curriculum.*Paper presented at the Institute of Education Sciences Research Conference, Washington, D.C. - ↑
^{3.0}^{3.1}^{3.2}De Corte, E., & Verschaffel, L. (2006). Mathematical thinking and learning. In K. A. Renninger & I. E. Sigel (Vol. Eds.), W. Damon, R. M. Lerner (Eds-in-Chief),*Handbook of child psychology: Volume 4. Child psychology in practice*(6th ed., pp. 103-152). Hoboken, NJ: John Wiley & Sons Inc. - ↑ Nunes, T. (2008). Mathematics instruction in primary school: The first three years.
*Encyclopedia of Language and Literacy Development*(pp. 1-9). London, ON: Canadian Language and Literacy Research Network. Retrieved January 22, 2010, from http://www.literacyencyclopedia.ca/pdfs/topic.php?topId=250. - ↑
Gelman, R., & Gallisten, C. R. (1986).
*The child’s understanding of number*. Cambridge, Massachusetts: Harvard University Press. - ↑
Cummins, D. D. (1991). Children’s interpretations of arithmetic word problems.
*Cognition and Instruction,*8(3), 261-289. - ↑
Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children.
*Child Development, 75*(2), 428-444. - ↑
Gervasoni, A. (2005).
*Opening doors to successful number learning for those who are vulnerable.*Mathematical Association of Victoria Annual Conference, 2005. - ↑
^{9.0}^{9.1}Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity.*Psychological Science, 14,*237-243. - ↑
de Hevia, M.-D., & Spelke, E. (2009). Spontaneous mapping of number and space in adults and young children.
*Cognition, 110*(2), 198-207. - ↑
^{11.0}^{11.1}Geary, D. C. (2006). Development of mathematical understanding. In D. Kuhl & R. S. Siegler (Vol. Eds.), W. Damon , R. M. Lerner (Eds-in-Chief),*Handbook of child psychology: Volume 2. Cognition, perception and language*(6th ed., pp. 777-810). Hoboken, NJ: John Wiley & Sons Inc. - ↑ National Council of Teachers of Mathematics. (2004). Overview: Standards from Prekindergarten through Grade 2. Retrieved January 22, 2010, from http://standards.nctm.org/document/chapter4/index.htm.
- ↑
Hutchins, P. (1986).
*The doorbell rang*. New York: Greenwillow Books. - ↑
Caliandro, C. K. (2000). Children’s inventions for multidigit multiplications and division.
*Teaching Children Mathematics, 6*, 420-424. - ↑
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010).
*Elementary and middle school mathematics. Teaching developmentally*(7th ed.). New York: Pearson Education Inc. - ↑
Langrall, C., Mooney, E., Nisbet, S., & Jones, G. (2008). Elementary students’ access to powerful mathematical ideas. In L. English (Ed.),
*Handbook of international research in mathematics education*(2nd ed.). NY: Routledge. - ↑
Wilson, R., Majsterek, D., & Simmons, D. (1996). The effects of computer-assisted versus teacher-directed instruction on the multiplication performance of elementary students with learning disabilities.
*Journal of Learning Disabilities, 29*(4), 382-90.