Big Ideas for Small Mathematicians: Kids Discovering the Beauty of Math with 22 Ready-to-Go Activities
By Ann Kajander
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- Rating: 5 out of 5 stars5/5this book really help me to know how to teach math
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Big Ideas for Small Mathematicians - Ann Kajander
Introduction
Many children and adults believe mathematics to be a rather boring collection of difficult facts. Nothing is further from the truth! Learning mathematical facts in isolation is a bit like learning to spell but never reading or writing a story, or learning all about the technical skills of visual art but never creating a picture or examining the great works of other artists. Of course it would be boring to learn a subject this way.
Learning mathematical facts in isolation is a bit like learning to spell but never reading or writing a story.
It is just as possible to play with mathematical ideas before mastering the technical skills as it is to fingerpaint before learning about perspective and vanishing points. And creation is fun, even in math.
Practicing mathematicians find math to be exciting and fascinating. They love to play around and explore. In this book I have tried to collect some of the important ideas in mathematics that can be explored in a hands-on way. Some topics are a usual part of elementary school mathematics programs, such as activity 8 on tessellations or activity 10 on the number system, but I approach them in a new way. Some are topics not usually discussed with children, such as topology in activity 14. Nearly all of the topics are effective with older students as well. I have used many of the topics successfully with secondary-level students, and I often have to chase them out of the classroom when the bell rings!
These activities will work in a variety of situations:
elementary classroom small-group problem-solving and exploration sessions
math centers in the classroom
math fairs
special parent-and-child math days or evening sessions
parent-and-child math exploration at home
examples for mathematics teacher education
I know some elementary teachers who devote a period a week to mathematics exploration, and many parents are in search of engaging activities they can do with their children to stimulate their creativity and love of mathematics. The adult doesn’t need a particular math background—all required background is supplied in the Mathematical Idea section of each activity.
Although the activity sheets can be used as is with children whose reading ability is sufficient, I prefer to deliver the instructions orally to children working in small groups. In this way, I can better control the pacing. One would hope that the next part of an activity would flow naturally from the students’ questions. I find it is the interaction of the students with each other and with the adult that makes these activities richer. Often if children are led orally into answering questions and discussing the ideas while, say, they are making a tessellation, they don’t even realize they are talking math.
Several of the activities can also be done in art class (such as activities 1, 6, 8, 9, 12, 21, and 22) or science class (activities 13 and 17), freeing up more time for investigation in math class.
The activities are roughly ordered from activities that all children can do to activities slightly better suited for children with a little more arithmetical skill, such as understanding the idea of multiplication. Within this ordering, activities are generally grouped by topic. The charts on pages ix–xi (Content Areas in Each Activity, Process Skills Used in Each Activity, and Prerequisite Knowledge and Skills for Particular Activities) detail the basic skills and topics of each activity as well as giving you an idea of what the children should know before embarking on a particular activity. That is not to say children cannot tackle a harder idea. I don’t think it’s always necessary to finish each activity or answer everything. Rather, the activities are meant to convey the idea that mathematics can be interesting, open-ended, uncertain, surprising, and highly creative.
Parents and teachers often think these ideas are only for bright kids. Given that the gifted are often underachieving, this may be a good use for the activities, but I have had surprising success using these activities with children branded as less skilled in mathematics. I remember a mother who came to me at wits’ end with her fifth-grade daughter who hated mathematics and had poor technical skills. When I spoke with her teacher, she told me the girl was failing mathematics
and wondered why the mother would send her to an enrichment program
(referring to the Kindermath program). This young lady managed to surprise everyone in the new environment. She discovered the trick to winning 3-D Tic Tac Toe (activity 20) in no time, and came up with a solution to the Streamers Problem (activity 19) on her own, which she shared with her class and her teacher. This experience changed her attitude toward mathematics and herself, and she gave learning the missing technical skills in school a much stronger effort. Three years later, I am told, she is now an A student in mathematics!
These activities are meant for exploration, enjoyment, and to stimulate curiosity. They can involve whole families in the wonder of mathematics. Enjoy the journey!
Many of the activities lend themselves to classrooms set up in groups.
How to Use This Book
Each activity in this book is divided into informative and instructional sections to make it easy to understand the ideas behind the activity as well as guide the students through the activity successfully.
The Big Idea: This first section of each activity summarizes the underlying mathematical concept, so you can get the big picture before diving into the details.
Content Areas in This Activity: This section lists the main mathematical content areas the activity involves. This information is also summed up for all activities in the chart on pages ix–x.
Process Skills Used in This Activity: This section lists the main thinking and learning skills the students will use throughout the activity. This information is summed up for all activities in the chart on page xi.
Prerequisite Knowledge and Skills: This section lists whether the children will need particular content knowledge before embarking on the activity. Most of the activities will be enjoyable for children on some level regardless of their previous knowledge and skills, but having these technical skills will make it easier for them to understand the math behind the activity. The chart on page xi lists this information for the activities as a whole.
Age Appropriateness: Most of the activities in this book are suitable for all ages, with varying degrees of adult assistance. In this section, you will find suggestions for adapting various aspects of the activity to suit different age groups. For example, younger children can often enjoy an activity without having to understand all of the mathematics behind it, but each activity is also set up to allow for further exploration of concepts as well as the introduction of terminology for older and more advanced students.
The Mathematical Idea: This section describes the mathematical idea at the heart of the activity, to provide background for you and to assist you in following and facilitating the children’s thinking. Often children have a correct but incomplete intuitive sense of a problem, and it is helpful for the adult to have a sense of the possible approaches to use.
Helpful Terms: This box includes a list of terms relevant to the activity and their definitions. The purpose of this box is to enable you to review the basic terminology and concepts as well as to give you wording to use when explaining the ideas to the children. You will also find a complete glossary at the end of the book (page 138).
Making It Work: This is the nuts-and-bolts section of the activity. Here you will find objectives, a list of materials you will need, preparation instructions, the activity procedure, suggestions for making the most of the activity and for helping the children through trouble spots, assessment ideas for determining if the activity was successful, and an extension activity or two for those kids who want to explore more.
Activity Sheet: The activity sheet lists each step of the activity, as in the procedures section of Making It Work, but this sheet is directed to the children, with illustrations and hints to help them work through the activity successfully. For children able to read well and follow instructions, a photocopy of this activity sheet will assist you in taking them through each step, although this sheet is not meant to replace your active involvement and guidance. You may decide not to use the activity sheet for younger kids who won’t be able to read it well, or if you prefer to take the class through the activity orally, without the help of written instructions.
The activities are meant to convey the idea that mathematics can be interesting, open-ended, uncertain, surprising, and highly creative.
Activity 1
PATTERN SHAPES
The BIG Idea
Geometric patterns are the very essence of discovering new mathematical relationships.
Content Areas in This Activity
Geometric patterning
Pattern rules
Process Skills Used in This Activity
Communication (optional)
Creativity
Aesthetics of mathematics
Prerequisite Knowledge and Skills
None
Age Appropriateness
This simple activity is appropriate for all ages.
The Mathematical Idea
Patterns are a central idea in mathematics. Almost anything that has a pattern contains some mathematics, and most mathematical ideas contain some rule or pattern. Recognizing increasingly subtle patterns is an important mathematical skill. This activity will introduce children to the concept of patterns, with an emphasis on enjoying their visual appeal. Showing examples of aesthetically pleasing patterns will enhance the activity and encourage children to be creative. Many quilts, for example, show remarkable patterns.
The central notion of a pattern is that it is predictable, once we see what is repeated. The pattern may change as we progress, but in a predictable way. Children should be able to identify what is repeated in their patterns, and what would come next. Patterns can be linear, nonlinear, or rotational. (See the box on page 2 for definitions of these terms.)
Wherever there is a pattern, there will be some math. Even if they can’t yet name the patterns mathematically, children can still enjoy inventing them. Have fun creating!
HELPFUL TERMS
Patterns: Mathematical objects, such as numbers or shapes, that are continued in a predictable way are called patterns.
Geometric patterns: Patterns created using shapes are called geometric patterns.
Linear patterns: Linear patterns change by the same amount each time: for example, 2, 4, 6, 8, . . . (changing by 2) or red, blue, red, blue.
Nonlinear patterns: Nonlinear patterns change by a different amount each time: for example, 2, 4, 7, 11, 16, . . . (changing by 2, then 3, then 4, then 5, and so on) or red, blue, red, blue, blue, red, blue, blue, blue, . . .
Rotational patterns: Rotational patterns are patterns created by rotating a shape or image. For example, a minute hand traces a rotational pattern around a clock face.
An important aspect of a pattern is that we can predict what would come next.
Pattern Shapes: Making It Work
Objectives
Children will create geometric patterns.
Children will connect the skills of pattern recognition, pattern creation, and spatial reasoning.
The activity will encourage creativity and an aesthetic sense of mathematics.
Materials
a few 8 ½x 11
pieces of light cardboard or construction paper in several colors for each child
scissors for each child
glue stick for each child
11 x 17
piece of background paper for each child
photographs of quilts and other geometric designs
colored pens or pastels (unless using multicolored cardboard or paper)
photocopy of the Pattern Shapes Activity Sheet (on page 5) for each child
Try cutting the shapes out of heavy wrapping paper or textured paper as well as cardboard. Alternatively, you could purchase pattern blocks, as shown in the photo on page 4.
Preparation
You could speed up the process by creating shapes ahead of time. Copy and cut out the shapes on the activity sheet (on page 5). You can enlarge the shapes as desired. Trace them onto light cardboard or construction paper, cut them out, and color them (or use different colors of construction paper).
Precut shapes in a variety of colors may help inspire children to be creative. You could even put together a few patterns as examples for the children.
Most children enjoy creating their own patterns from precut shapes.
If you don’t cut out the shapes ahead of time, make enough copies of the activity sheet for each child to have one.
Enhance child creativity and comprehension by gathering photographs of quilts and other geometric designs.
The book Mathematical Quilts: No Sewing Required, by Diana Venters and Elaine Krajenke Ellison (Emeryville, Calif: Curriculum Press, 1999), is a source of many wonderful ideas. The mathematics in each design allows for application at the secondary and even post-secondary levels. The Long Island Children’s Museum website (www.licm.com) also has a hands-on quiltmaking feature. After entering the site, click on their hands-on activities and scroll down to the QuiltMaker
link.
Procedure
Show children pictures of geometric patterns, such as quilts, to stimulate discussion and understanding of what a pattern is. Ask children what they think a pattern is and what it is not. Guide the discussion to include the idea that a pattern involves repetition.
Hand out the activity sheet (page 5), the background paper, glue sticks, cardboard or construction paper (and heavy wrapping paper or textured paper, if desired), and colored pens or pastels (unless using multicolored paper).
If you precut the shapes, just hand out the shapes themselves. Give each child several of each shape so that everyone has plenty of shapes to choose from and can repeat shapes to form a pattern as desired.
Tell the children that they are to create any pattern they wish with the shapes. Encourage them to make a pattern, not just a pretty picture or set of random shapes. They do not need to use all of the shapes. For example, a young child could simply make a checkerboard pattern with two colors of squares.
The pattern on the left contains both linear and rotational elements.
Choose questions to prompt children appropriately, depending on the situation. For example:
How about a star?
In what other ways could you arrange the pieces?
Can you make this pattern repeat?
What would it look like if you made another ring of shapes?
Can you make your own quilt pattern?
After they design patterns they like, children glue them onto their big piece of background paper.
Suggestions
Doing this activity in a group will encourage the children to share ideas.
Encourage