The first sentence from the cover nicely captures the basic premiss of Cognitively Guided Instruction (henceforth, CGI): “By the time they begin school, most children have already developed a sophisticated, informal understanding of basic mathematical concepts and problem-solving strategies.” Indeed, kids seem quite capable of using concrete modeling using fingers, tokens, and such to solve problems that, judging purely by the mathematical knowhow they posses, should be beyond them.
As an example, this morning, Suhana solved this problem: “Suhana had 15 cookies that she distributed equally to Anna, Betty, and Calvin. How many cookies did each get?” Suhana is unfamiliar with division. Quite possibly, she has never heard the word used in the mathematical sense. But using tokens (poker chips, in case you are curious), she distributed fifteen tokens into three rows (she put three in each row and then distributed the rest), and could easily tell that each will get 5 cookies. Just because grown ups reach for division for this problem it does not follow that division is needed for this problem. Carpenter et al. are ascribing just such inventiveness to children, and have over two decades of classroom observation to back them up. Far from unusual, what Suhana did is par for the course.
The authors lay the groundwork in the first chapter, “Children’s Mathematical Thinking”. It begins with three problems that, to adults, are all merely instances of “eight minus three” but to kids are quite distinct. The distinct strategies children routinely use to solve these “identical” problems lays bare their differences.
- Eliz had 8 cookies. She ate 3 of them. How many cookies does she have left?
- Eliz has 3 dollars to buy cookies. How many dollars does she need to earn to have 8 dollars?
- Eliz has 3 dollars. Tom has 8 dollars. How many more dollars does Tom have than Eliz?
- She layed out 8 tokens, removed 3, and counted what was left.
- She layed out 3 tokens, added tokens until 8 tokens were laid out, and counted the new tokens.
- She laid out two rows of tokens side by side, 3 and 8, and counted how many more the longer row had.
The point to emphasize here is that “[children] do not need to be taught specific derived facts. In an environment that encourages children to use procedures that are meaningful to them, they will construct these strategies for themselves.”. Different strategies are suggested by different problems, and exposing children to a range of problems is beneficial. Chapters 2 and 4 lay out one way to classify addition/subtraction and multiplication/division problems, respectively. Here, I will merely list one example of each class of addition/subtraction:
- (Result unknown) Connie had 5 marbles. Juan gave her 8 more. How many marbles does Connie have altogether?
- (Change unknown) Connie had 5 marbles. How many more does she need to have 13 marbles altogether?
- (Start unknown) Connie had some marbles. Juan gave her 5 more. Now she has 13. How many did she have initially?
- (Result unknown) Connie had 13 marbles. She gave 5 to Juan. How many marbles did she have left?
- (Change unknown) Connie had 13 marbles. She gave some to Juan. Now she has 5 left. How many did she give Juan?
- (Start unknown) Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many did she have to start with?
- (Difference unknown) Connie had 13 marbles. Juan had 5 marbles. How many more marbles did Connie have than Juan?
- (Compare quantity unknown) Juan had 5 marbles. Connie had 8 more than Juan. How many marbles did Connie have?
- (Referent unknown) Connie had 13 marbles. She has 5 more than Juan. How many marbles does Juan have?
- (Whole unknown) Connie has 5 red marbles and 8 blue marbles. How many marbles does she have?
- (Part unknown) Connie had 13 marbles. 5 are red and the rest are blue. How many blue marbles does Connie have?
Chapter 5 is a defense of such modeling strategies used by children, and gives examples of seemingly far more complex problems that very young children can solve using modeling: “19 children are taking the minibus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit 3 to a seat?”. The authors deplore that “It may seem obvious that children would attempt to perform the action in a problem if they have no other way to solve it. Many older children, however, appear to approach word problems by looking for superficial clues to decide whether to add, subtract, multiply, or divide. In other words, many students seem to abandon a fundamentally sound and powerful general problem-solving approach for the mechanical application of arithmetic skills.”
Subsequent chapters deal with multidigit concepts and furnish suggestions for starting to use CGI in a classroom. The appendix provides concrete numbers demonstrating the effectiveness of CGI in classrooms.
This, then, was a brief taste of what CGI is. Standard disclaimers such as “I am not a practitioner, perhaps I am even an outsider to education” apply. I liked the book — and if you are trying to teach math to young children, I whole-heartedly recommend this book.