This is a copy of an article written by Dr. Bohan and Susan Bohan, Dr. Bohan's daughter in law, a mathematics consultant for Region VI ESC in Huntsville, Texas.
Extending the Curriculum Through
Creative Problem Solving
Standard 2 of the NCTM's Professional Standards for Teaching Mathematics (1991) challenges elementary school teachers of mathematics to encourage classroom discourse by --
1. posing questions and tasks that elicit, engage, and challenge each
student's thinking;
2. listening carefully to students' ideas;
3. asking students to clarify and justify their ideas orally and in writing.(p.35)
There is also increased attention in school mathematics in having teachers provide students an environment in which they can use prior knowledge and experiences to construct new knowledge for themselves. As Wolfe and Reising (1988) suggest, in order to really own a piece of knowledge, the learner must reconstruct that knowledge and make it new.
It can be a real challenge for an elementary teacher to encourage mathematical discourse and provide opportunities for student construction of knowledge in a regular classroom. However, help in meeting that challenge can be found for teachers through extensions of the regular elementary mathematics curriculum. The role of the teacher is to promote creative problem solving activities which provide opportunities for students to utilize their prior mathematical knowledge in solving problems. To encourage discussion, it is best if the problems are open ended and non-routine.
The example that follows is one in which creative problem solving has been incorporated into the teaching of our numeration system. It was initially taught in a fourth grade class but can be modified for use at almost any grade level.
The Regular Lesson
Ancient numeration systems are a very useful tool in elementary school mathematics. Because of common elements between our numeration system and that of the ancient Egyptians, this system, and others like it, provide a great vehicle for enhancing a child's understanding of base and place value. The Ancient Egyptians used the symbols in Figure 1 to represent whole numbers through 9,999.
FIGURE 1: Egyptian symbols used to write numbers through 9,999
Normally, such a lesson begins by introducing students to the Egyptian System and providing practice converting from the Egyptian system to ours and vice-versa. Then students are asked to compare and contrast the two system. What are the commonalities? What are the differences? Students make conjectures as to things in our system that might indicate it evolved from a system such as this. In addition to making a nice connection between mathematics and history, students should be able to establish the following:
* The values of the Egyptian symbols are the values of our places.
* The Egyptians represented the number 8,000 by writing the symbol for
1,000 eight times. We represent this number by writing the symbol for eight
in the thousands' place.
* They never use a given symbol more than nine times. Nine is the value
of
our greatest digit.
The Extension
To extend the lesson through creative problem solving, start by having students represent several rather long numbers in the Egyptian system. Record one of these numbers on the chalkboard as indicated in Figure 2. To make the point of how cumbersome this system is, several other questions might be asked. "What is a number that would take up even more space than the ones we have written?" If prompting is necessary you may ask, "Why does the space for lotus flowers take up more room than the space for heel bones?" (There are more of them.) "What number would require more heel bones than 6,042?" (9,692) "What four-digit number would take up the most space?" (9,999) If prompting is needed, ask, "What is the greatest digit we have in our numeration system?" (9) "Can you write the number that would take up the most space now?" Ample time should be provided for student interaction.
FIGURE 2: Large numbers are cumbersome.
You are now ready to present the class with the following challenge. "It sure takes up a lot of space to write some numbers such in the Egyptian system, doesn't it ? Suppose you were living in Egypt around 3000 BC and had to use this system. Your job is to make changes in the system that would make it easier to use. When you have completed your task, you will get to share your system with the entire class."
At this point, students should be allowed to work in cooperative groups or individually to solve the problem. Any modification that makes the system easier to use must be accepted. Activities like this seem to differentiate themselves. More capable students tend to make more complex modifications. If you use cooperative groups, you may wish to structure the groups heterogeneously. The following are three "notation creation" products that came from a fourth grade class.
Jim's System
Jim decided to work alone. He thought the main problem was that it was that the symbols were too hard for him to draw. Jim simply changed the symbols using a mark for 1, an X for 10, an H for hundred and a T for thousand. He finished quickly. Jim was encouraged to find ways of shortening the writing of the numbers even more. He eventually decided to work in a group with Mark and Bob.
Mary's King Tut Notation
Mary worked alone to develop Mary's King Tut Notation. What a nice name! Mary's creation used the same basic Egyptian symbols with modifications. Ones were represented with tally marks. As is often done, Mary used a diagonal tally to represent the numbers five or greater. She explained that this practice makes it possible to recognize the number without having to count the tally marks each time. Several ones are represented in Figure 3(a).
A heel bone was still used to represent the number ten. However, only a single heel bone was used. Mary drew ladybugs on that single heel-bone to show the number of tens. (The use of ladybug may be because students in this class usually called heel bones croquet hoops rather than heel bones.) Several tens are represented in Figure 3(b). Mary also used a single rope to represent hundreds. Knots were tied in the rope to indicate the number of hundreds. Figure 3(c) shows Mary's representations of several hundreds. The lotus flower also remained the symbol for 1,000. Again, only a single flower was used for a given number of thousands. Petals were placed on the flowers...the number of petals indicating the number of thousands. Figure 3(d) shows several thousands represented in Mary's notation.
FIGURE 3: A student's modification of the numeration system
3 (a) Slanted tally marks help read the number of ones.
3 (b) "Ladybugs" indicate the number of tens.
3 (c) Knots on the rope indicate the number of hundreds.
3 (d) Petals on the flower indicate the number of thousands.
Figure 4 shows several numbers written in the Original Egyptian System and in Mary's King Tut Notation. A comparison of the two systems makes one wonder if such a change might have been adopted had Mary lived in those ancient times!!
FIGURE 4: A comparison of the Egyptian System with Mary's System
| Number | The Ancient Egyptian System |
Mary's King Tut |
The Markjimbob System
Mark, Jim, and Bob worked together to develop the Markjimbob system.
In their creation, additional symbols were invented to be used in conjunction
with the Egyptian system. Figure 5 contains their new symbols and their
values.
FIGURE 5: Examples of the new symbols in the Markjimbob system
The new symbols were used to tell how many of a given one, ten, hundred,
or thousand the number contained. For example, 
represented 3
hundreds or 300, 
4 thousands or 4,000,
and 
7 tens or 70. Figure 6 (a) shows
several numbers written in the Markjimbob System. These numbers should have
a familiar ring to them.
FIGURE 6: Numbers written in the Markjimbob system.
6 (a) The original Markjimbob system
6 (b) The first modification
When the Markjimbob system was presented to the class, someone asked why they didn't have a symbol for the number two. They said they had considered that but it wasn't needed. "Either way you would need to write two symbols so why make up a new one? To write a number like 200 you might as well just use two ropes as make up a new symbol." No need to ask about needing symbols zero and one!
During the class discussion of the Markjimbob system, it was suggested
that to avoid confusing the new symbols with the original symbols, we might
write the newly created symbols over the Egyptian symbols
rather than beside them. Figure 6 (b) shows several numbers written in this
form. Looking at the representation of the number 2,025 in Figure 6 (b),
we can see that recording two's is a little messy. The symbol " 
" was invented to take care of this difficulty.
Now 2,025 would be written like this.
It was then suggested that after working with the system for a while,
maybe the Egyptian symbols could be left off. People would know that the
first symbol on the right was always the ones, the second place the tens,
then hundreds, and thousands. If this were done, a symbol for 0 and 1 would
also be needed. The following symbols were chosen.
Several numbers had been written in this system (below) when a student
said, "Wait! That's just like our
FIGURE 7: The second modification of the Markjimbob system.
system!" The class realized that (except for using different symbols), Mark, Jim, and Bob had re-constructed our numeration system! This provided their teacher with an opportunity to lead s discussion as to how our numeration system evolved and to point out that at different times in that evolution, systems just like those we had built had actually been in use. This experience provided the whole class with a better understanding and a feeling of ownership of our base ten system. As Wolfe and Reising(1988) suggest, in order to really own a piece of knowledge, the learner must reconstruct that knowledge and make it new.
Summary
In his book To Think, Frank Smith (1990) suggests that all of us think all of the time, freely and effortlessly. The main block to thinking and creating in a specific subject stems, not from an inability to think, but rather from a lack of knowledge about that subject. If we want students to be creative, you need to provide them something to be creative with ... knowledge. If we want students to construct knowledge, they can only do so from prior knowledge and experiences. Once knowledge on a topic has been attained, creative problem solving lessons applying this knowledge can be helpful in many ways. Among other things, such lessons can:
* help promote the kind of classroom discourse that makes the
mathematics classroom
an exciting, vibrant place to work;
* provide meaningful activities for successful learners extending
the content of the day
while the teacher re-teaches material to unsuccessful learners;
* be effective used individually, in cooperative groups, or as a
whole class activity;
* empower students to construct in the mathematics arena and give
them a feeling
of ownership of a piece of mathematics;
* provide products be used in evaluation of higher level thinking.
Such creations
are good exhibits to place in a portfolios.
There are many opportunities for creative problem solving in the elementary mathematics curriculum. For example:
* While teaching one-to-one correspondence, have students pretend
to be shepherds
in a civilization that only has numbers for one, two, and many. Have them
find
a way of being sure, at the end of a day in the pasture, that no sheep have
been lost.
* While teaching the concept of fractions use models to develop
the meaning
of numbers like one-half, one-fourth, three-fourths. Use only verbal names.
Pause
and give students an opportunity to construct their own symbols for these
new
numbers. Have them defend their choices.
* Do the same for negative numbers. (A seventh grade class called
negative five
a five-destroyer because it could destroy fives.)
Try a lesson like the ones described above. Don't be discouraged if the initial responses aren't up to the level of Mary, Mark, Jim, and Bob. You will find that the depth of student responses will increase with experience. The only way to improve student's thinking is to immerse them in an environment where thinking is taking place and is valued at least as much as getting the right answer.
Also See:
For extensive help in using the Ancient Egyptian system for enrich the understanding
of our base ten system see articles by Payne (1986) and Krusen (1991)
listed below.
References
Krusen, Kim. "A Historical Reconstruction of Our Numeration System."
Arithmetic Teacher 38 (March 1991): 46-48.
National Council of Teachers of Mathematics. Constructivist Views on the Teaching and Learning of Mathematics. Reston, Va.: The Council, 1990.
National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989.
National Council of Teachers of Mathematics. Professional Standards for Teaching Mathematics. Reston, Va.: The Council, 1991.
Payne, Joseph N. "Ideas." Arithmetic Teacher 34 (September 1986): 26-32.
Smith, Frank. To Think. New York: Teachers College Press, 1990.
Wolfe, Denny and Reising, Robert. Writing for Learning in the Content Areas. Portland, Maine: J. Weston Walch, 1986.