Enhancing Understanding

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Using Picture Books To Improve Learning

One of the most effective tools that can be used to teach is picture books. Allowing children to see the idea illustrated or to hear an analogy through story helps them make sense of difficult concepts or those ideas foreign to their experiences. Most of us are visual learners, so actually seeing theory or thought helps us grasp its meaning.

Using Picture Books

Building Number Sense

An illustrated counting book can help little ones see that adding another item to the original amount increases the number of items in that set; and a different word is used to identify each set that has a different number of items. Ten Black Dots by Donald Crews is an excellent example to demonstrate this idea. Reading the book alone is an engaging activity for kindergarteners and first graders. Multiple exposures can result in excited group participation. Children will begin to anticipate the next page and will count the dots on each picture, patterning the correct number sequence. Following several readings, the teacher may ask the children to make either classroom or individual books using original child-created pictures. To address multiple concepts, the children could make books entitled Ten Black Triangles or Ten Black Squares. HINT: Using dot or square-shaped stickers is fun and easy for young children. The work will probably go a little faster and be a bit less frustrating for everyone. Perhaps Ten Green Dots would be fun or Ten Blue Squares. After ten is mastered, the number could be expanded. What about Twenty Purple Rectangles? That might be an interesting challenge. The possiblities are numerous and could be very entertaining without sacrificing educational benefit.

Operations

The Frog and Toad Together by Arnold Lobel is a delightful book about two very different amphibians who just happen to be good friends. This type of book lends itself nicely to making up math problems as well as the obvious science discussions about the real differences between frogs and toads. Children have fun making up their own problems after watching the teacher do it. I personally like the philosophy and approach provided in the book Children's Mathematics: Cognitively Guided Instruction by Thomas P. Carpenter, Elizabeth Fennema, Megan Loef Franke, and Linda Levi. I feel that the underlying principles in CGI are beneficial to building and solidifying student understanding.

There are basically four different types of problems. They are broadly defined as joining, separating, part to part to whole, and comparing. Within each of these classifications are additional specifications. If a problem deals with joining, the most common situation is to have the sum (result) be unknown. However, it is also possible to have the number being added (the change) unknown or the number that the problem begins with (start) unknown. The latter two are traditionally classified in textbook jargon as missing addends. If the problem calls for separation, it is possible to have the same variations. When working with part-part-whole problems, the unknowns may be the whole or a part. In comparing problems the difference, the compare quantity, or the referent set may be unknown. Higher level problems may include multiplication, measurement division, and partitive division. There is a sample downloadable worksheet in FILES that is based upon The Frog and Toad Together including six problems that offer a variety of CGI categories. One problem is identified as joining result unknown (JRU) and is also labeled multiplication (M). Either way works for me. Click here to download The Frog and Toad Worksheet

For children finding the connection is better than being told the rule or the relationship. In the early years making sense of the mathematical processes is the intent. Letting young children use whatever they need or want to solve the problems is essential in the success of doing math. If drawing helps the child, then drawing is what he or she should be allowed to explore when solving mathematical problems. If using counters helps the child make sense of the problem, then counters should be supplied. Abstracting doesn't just automatically happen. There needs to be a mental image or analogy connected to the process to embed understanding. That is what primary level teachers should focus on. One of the reasons that intermediate elementary or middle school children don't "get it" is that they have little or no background information, no conceptual understanding on which to build. Memorizing facts, although extremely beneficial to save computation time, does not create comprehension. As with reading, children develop math ability at varying rates. Taking away the crutches too soon may cause the child to fail and thereby incur permanent "damage". Let the child use whatever they feel they need until they realize that the manipulatives or drawings are in the way and are just slowing them down.

The Doorbell Rang by Pat Hutchins is one of my all time favorite books to use with children to help with a foundational understanding of division. It is a pattern book that encourages participation. The anticipation and prediction of yet another set of visitors is fun. Questions like "how many" and "who" can be discussed, especially at the end of the book when there are twelve children and twelve cookies and the doorbell rings yet again. With upper elementary students this might be a good time to discuss fractions. When I introduced my class to a math concept using children's literature, I simply read the story first, sharing the illustrations and allowing general conversation at least once so the children were somewhat familiar with the content. Prior to a subsequent read of The Doorbell Rang, I passed out a paper mat that looked like a plate on a checkered table cloth along with a sandwich bag of cookies. (Any teacher knows that food is a great motivator. Actually, the cookies were cereal shaped like little chocolate chip cookies. Cheap, but this served my purpose.) The children followed along as we read the book again, dividing their cookie cereal on the mats provided between or among the children in the story as that number increased with every ring of the doorbell. Allow ample time for most everyone to complete the thought process. Children could be paired, but I found that unnecessary. They were allowed to talk to neighbors if they wanted to help each other. We then checked ourselves before moving on to the next page.

Seeing the pattern is key. Keeping track of the information on a table also helps most children. (Graphic organizers are wonderful and a table or chart aides in clarifying understanding of content or concepts for many students.) So an abbreviated reading, more like a question-answer session with reference to the text as needed, was done with students using manipulatives and graphic organizers. As the children divided the dozen cookies once again, they filled in their division charts. Click here to download The Doorbell Rang Worksheets

This activity worked very well in my classroom. The story is interesting, well illustrated, and lends itself beautifully to multiple reads. The concept embedded is obvious and easily understood. You can probably see how this can then be expanded, more cookies to begin with, more children visiting, etc.

There are many resources available that detail ways to use children's literature. For example, Marilyn Burns, one of my heroes, has provided several books with applications for classroom use of children's literature. They offer actual classroom experiences and dialogue from real teachers and students pertaining to applications. Some of Marilyn Burns' titles include:

Math and Literature, Grades 4-6;

Math and Literature, Grades 2-3; and

Lessons for Introducing Division, Grades 3-4. These books are wonderful and are highly recommended by DragonLadyBooks.com.

Another excellent resource is a book entitled:

Math Through Children's Literature: Making the
NCTM Standards Come Alive

by Kathryn L. Braddon, Nancy J. Hall and Dale Taylor.

Geometry

Tangrams are fascinating to most students and can be used in a number of ways to enhance understanding of multiple concepts. There are several literature books that can be used to introduce children to this ancient Chinese puzzle. Grandfather Tang's Story by Ann Tompert and The Warlord's Puzzle by Virginia Walton Pilegard and Nicolas Debon are two of my favorites. The stories are very different from one another and are suitable for students from second grade up through fifth or sixth grade.

One application for tangrams is obviously geometry, the development of visual and spatial sense. Working with the puzzle pieces and sample patterns, students can replicate the images. When using Grandfather Tang's Story with third graders, I read the story through once sharing the illustrations. We predicted what form each fox fairy would take on the next page in the story. For example, when one fox becomes a rabbit, I would ask, "What do you think the other one will turn himself into?" (An interesting writing prompt is to ask children to write about what they would become as a fox fairy and why.) After the initial read each child was given a plastic tangram puzzle. (Children can make their own tangrams. See the downloadable FILES for instructions.) As we began reading the story again, the children were given time to create the various animals that the fox fairies assumed throughout the story. I prepared for the activity by creating the various tangram shapes illustrated in the book by cutting them from black paper and pasting them on white paper. I also made overheads which I used while I read the story. I placed the paper copies on the chalk tray so children could continue working on their puzzles with a reference in front of them. (It is also important to note that my third graders had tangrams available to them prior to this activity. Giving students opportunities to interact with manipulatives before actually using them for this kind of learning activity is a good idea. Otherwise children experiment with the new manipulative and are preoccupied with alternative explorations rather than the teacher defined learning task.)

When working with older children and reading The Warlord's Puzzle, the challenge was a bit different. I simply gave my fifth graders the puzzle (the seven tangram pieces) and let them try to figure out how these could be put together to create a square. Sounds easy, but it can be a bit of a trial. Once one or two students figure it out, they all get it. Amazing how that happens.

Which brings up another issue: what kind of help is really help and not just telling the answer? This is difficult in and of itself. Even teachers have trouble with this sometimes. Asking leading questions instead of just telling the child the answer can be tricky. It is better to help children through the frustration by encouraging them to experiment and try different possibilities, and by allowing them to struggle some. Reassure students that making mistakes is how we find out what doesn't work. Risk-taking and trial & error should be accepted and encouraged in classrooms. There is inherent reward involved in finding the solution with little or no assistance. This creates innate ownership and a deeper understanding.

Tangrams can be used to begin building an understanding of fractions. For example how many small triangles does it take to make a large triangle? The answer is four; so that means that one small triangle is 1/4 the size of one large triangle. The relationships between the shapes can be established and written on a data collection sheet. Once the chart has been completed, fractions can be identified and assigned. To extend this approach even further, begin to put these relationships in equations using abbreviations for the size and shape of the pieces. For example, one BT (big triangle) equals four ST (small triangles). Move students from this to creating their own number sentences by combining shapes and building equivalencies. One BT plus one P (parallelogram) equals how many small triangles? To download How to Make a Tangram Puzzle, Working with Tangram Fractions and Equivalencies, and Tangram Number Sentences click here. Download Tangrams pdf

Three Pigs, One Wolf, and Seven Magic Shapes written by Grace Maccarone and illustrated by David Neuhaus is an adaptation of a traditional story with a nifty tangram twist. The book includes math activities by Marilyn Burns which is a real plus!

There are so many other resources available that provide tangram patterns on cards or in book form. One of these is The Ancient Mystery of Tangrams which has 500 variations of the puzzle. It is a wonderful addition to your classroom library, especially for center time and indoor recess. Kids love it!

Links For More Tangrams Resources

Center for Innovations in Mathematics Teaching

Tangrams

Enchanted Learning

The Math Forum

About Teaching Mathematics

One of my most favorite resources is another book by Marilyn Burns, About Teaching Mathematics: A K-8 Resource. While working as a curriculum director for a local district, I made sure all the K-8 math teachers had a copy of this book to reference. It is easy to use and full of excellent ideas that encourage students to explore mathematics to better understand various concepts. The book is divided into four parts: Raising the Issues, Problem-Solving Activities in the Strands, Teaching Arithmetic, and Mathematical Discussions. The strands are reflective of NCTM and include: Measurement; Probability and Statistics; Geometry; Logical Reasoning; Patterns, Functions, and Algebra; and Number. Marilyn Burns has included activities for all these areas as well as for specific concepts like place value, division, and fractions. If you can only purchase one resource book to help you teach math, pick this one!

Other Great Resources

Another set of books that provides a wealth of ideas for using children's literature in every content area is Story Stretchers by Shirley Raines and Robert J. Canady/Karen Miller. The book shown here is for the primary level literature. However there are several others Story Stretchers for different elementary groups. Check them all out. They are wonderful idea generators.

How To Purchase The Books

JWMercantile.com has most of the books referenced on this Website (and many others as well). A large selection of used children's books in great condition is available. Many First Edition children's books are offered. Click here to search for a book at JWMercantile.com.

To purchase a new book described here, simply click the book title(s) or picture(s) and a window will open to the appropriate page at Amazon.com.

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