Judy Roitman wrote: I'd be interested in hearing from the folks who do teach this, from upper grades in elementary on -- how do you develop this "algebra sense"?
If one teaches arithmetic as a separate topic in mathematics without an underlying understanding of algebra, there is danger that students will come away with memorized algorithms which they will automatically use, regardless of the numbers involved or the real situation to which the numbers and the operations refer. Having an understanding of the arithmetic/algebra connection enables the teacher to teach basic computational skills (whether by written algorithm or calculator) AND the underlying algebraic relationships. If one has really learned arithmetic in a mathematically sound manner, then it is hypothesized that there will be no shock when the need arises to introduce an unknown factor in the problem. There are ways to do this from the primary grades through the intermediate elementary school grades. There is some interesting work coming from Ken Milton of the University of Tasmania. Two articles of his are: Getting the Teaching of Algebra Right, and Fostering Algebraic Thinking in Children. Both of these articles are from the australian mathematics teacher (lack of capitals is their idea!) He is very specific about how to do this with actual lessons and dialogues as examples. Also, Robert Davis has been doing some wonderful work with algebra that is accessible to intermediate level students. Davis, R. (1986) Algebra in the Elementary School Proceedings of the Fifth International Congress on Mathematics Education Birkhauser, Boston. pp.66-67 He is at Rutgers in New Jersey and has continued to investigate this topic. If you are on-line Dr. Davis, can you give us some other sources for your work?
Second grade students can develop algebraic concepts as they name five as 4+1 or 3+2 or 1+1+1+1+1..and as they demonstrate this understanding with real materials. A popular activity for primary teaching is renaming the date as the children do their calendar work.
Fourth graders can explore multiplication by breaking up the factors into *friendly* parts. 27*34 can be shown as an array and the array can be split into parts that are easily computed. (20+7)*(30+4) Investigations with arithmetic can lead to the understanding of multiplication of binomials. But children can also see that 3*3*(3*34) gives the same result as 27*34
Mental multiplication techniques can be used to gain understanding of the distributive property of multiplication over addition or subtraction. 6*(24) is 6*20 plus 6*4 or 6*12 doubled or ...
They can also explore that dividing by nine is like dividing by 3 and again dividing the result by 3. 63/9 is the same as 63/3 and 21/3 is 7. b/9 is (b/3)/3. They can generate and test these findings as they relate to arithmetic or work on the calculator.
Finding patterns is another algebraic activity that young children enjoy. Expressing the patterns in words and eventually with invented symbols bring the children closer to algebra. Function machines also intrigue them. Games such as Krypto in which children use 5 randomly selected numbers from a given range of 0-20 and any operations they wish in order to express another randomly chosen number in that range.
The pan balance in the primary classroom gets in touch with equations. How many chestnuts will balance 10 peanuts? and so on.
Any experience that deals with ratio is helpful to understanding algebra. If 10 garbanzos fill a tablespoon and there are 20 tablespoons in a small cup how many garbanzos will fill the cup. That's just multiplication. But ask how many kidney beans will fill that cup? - or whatever you happen to be measuring in your class that day. Then you have a door open to ratios.
If a teacher has an understanding of algebraic concepts and accepts that elementary students can develop those concepts, there will be many opportunities to invent algebraic experiences that are relevant to what your children are doing in all their school experiences. Even in the gym!
These are just a few examples of activities that can lay groundwork for the understanding of algebra. It's fun to select an algebraic concept and sit down and design an elementary level activity that develops that concept concretely or invites the child to invent a solution to a real problem that employs that concept. One can do algebra without ever seeing an X. Algebra is arithmetic. Deriving the principles of algebra while inventing arithmetic is fruitful activity. However, transmitting algebra as in an 8th or 9th grade textbook to children in the 4th grade is an inappropriate approach, IMHO.
Lucille L. Peterson Math Leadership Program Bank Street Graduate School of Education Tel: 212-875-4665 610 West 112th Street Fax: 212-875-4753 New York, NY 10025 E-mail: firstname.lastname@example.org