To help you understand the difference between concrete and formal reasoning patterns, two puzzles are described below, followed by secondary students answers and their explanations of their answers.
Materials
Paperclips, copies of the puzzles
Procedure
1. First perform the puzzles yourself, writing your answers and explanations as the students did on separate sheets of paper.
2. If you are able to have secondary students complete the puzzles, you can use their responses to respond to the following questions, as well as the students responses included here.
Minds On Strategies
When you have finished completing the puzzles, answer these questions.
1. What differences in reasoning did you find among the students' responses toa. The mealworm puzzleb. The ratio puzzle
2. What similarities did you observe in the thinking patterns of Norma, David and Dolores?
3. What similarities did you observe in the thinking patterns of Jean, John and Harold?
4. Which students are using concrete reasoning patterns? Which are using formal reasoning patterns?
5. What are some characteristics of
a. Concrete reasoning?b. Formal reasoning?
An experimenter
wanted to test the response of mealworms to light and
moisture. To do this he set up four boxes as shown in Figure
2.28. He used lamps for light sources and constantly watered
pieces of paper in the boxes for moisture. In the center of
each box he placed 110 mealworms. One day he returned to
count the number of mealworms that had crawled to the
different ends of the boxes. The diagrams show
that mealworms respond to (response means move toward or
away): A. light but not
moisture. B. moisture but not
light. D. both light and
moisture D. neither light nor
moisture Please explain your
choice.
The drawing in
Figure 2.29 is called Mr. Short. We used large round buttons
laid side-by-side to measure Mr. Short's height, starting
from the floor between his feet and going to the top of his
head. His height was four buttons. Then we took a similar
figure called Mr. Tall, and measured it in the same way with
the same buttons. Mr. Tall was six buttons high. Now please do these
things: 1. Measure the
height of Mr. Short using paper clips in a chain. The height
is_______. 2. Predict the
height of Mr. Tall if he were measured with the same paper
clips.________ 3. Explain how you
figured out your prediction. (You may use diagrams, words,
or calculations. Please explain your steps
carefully.)
Responses to the Mealworm and Mr. Short Puzzle
Students Mealworm
Puzzle Responses Mr.
Short Puzzle Responses Student 1 (Norma,
Age 12) D. "Because even
though the light was moved in different places the mealworms
didn't do the same things. Prediction for Mr.
Tall: 8-1/2 paper clips. Explanation: "Mr.
Tall is 8-1/2 paper clips tall because when using buttons as
a unit of measure he is 2 units taller. When Mr. Short is
measured with paper clips as a unit of measurement he is
6-1/2 paper clips. Therefore, Mr. Tall is 2 units taller in
comparison which totals 8-1/2." Student 2. (Jean,
Age 13) B. "I, II, and III
show that mealworms seem to like the light, but in III they
seem to be equally spaced. This leads one to believe that
mealworms like the dryness and the reason in pictures III
and IV they are by the light is because of the heat that the
light produces which gives a dryness effect. Prediction for Mr.
Tall: 9.2 paperclips. Explanation: "The
ratio using buttons of height of Mr. Short and Mr. Tall is 2
: 3. Figuring out algebraically and solving for
x: 6-1/2 2/3 = x gives you
9.2 as the height in paper
clips." Student 3 (David,
Age 14) A. "They usually
went to the end of the box with the light." Prediction for Mr.
Tall: 9 paperclips tall. Explanation: "I
figured out by figuring that Mr. Small is 2/3 as tall as Mr.
Tall." Student 4. (John,
Age 16) C. Boxes I and II
show they prefer dry and light to wet and dark. Box IV
eliminates dryness as a factor, so they do respond to light
only. Box III shows that wetness cancels the effect of the
light, so it seems they prefer dry. (It would be clearer if
one of the boxes was wet-dry with no light." Prediction for Mr.
Tall: 9 clips (pencil marks along Mr. Short). Explanation: "I
estimated the middle and then on-fourth of Mr. Short. That's
about the size of one button. I measured the button with my
clips and found 1-1/2. So then I counted out six times 1-1/2
buttons and got nine." Student 5. (Dolores,
Age 17) C. "In the
experiment 3 mealworms split 1/2 wet, 1/2 dry. So it's safe
to assume that light was not the only factor involved.
Prediction for Mr.
Tall: 8 paper clips tall. Explanation: "If Mr.
Short measures 4 buttons or 6 paper clips (2 pieces more
than buttons), then Mr. Tall should be 2 paper clips more
than buttons. Student 6. (Harold,
Age 18) A. "Because there
are 17 worms by the light and there are only 3 by the
moisture. Prediction for Mr.
Tall: 9-3/6 Explanation:
"Figured it out by seeing that Mr. Tall is half again as
tall as Mr. Short, so I took half of Mr. Short's height in
clips and added it on to his present height in clips and
came up with my prediction."
Concrete versus Formal Thinking: Implications
While reading through the responses to the two puzzles, you should have noticed that some of the answers were more complete, more consistent, and more systematic that others. Each answer represents the reasoning of the student who wrote it. Yet surprising as it may seem, inconsistent and incomplete answers were just as common as complete and consistent ones. The patterns of thinking that the students show as responses to these puzzle problems are integral aspects of Piaget's theory of cognitive development.
In the mealworm puzzle students using concrete reasoning tend to focus on one variable and exclude the others. The student cannot detect the logic of the experiment that allows one variable to be isolated and separated so that they can be dealt with as distinct causal agents. Formal reasoning patterns are shown when students realize that variables are held constant while only one is allowed to change, as in boxes II or IV, or comparing I with III. The formal thinker considers all possible causal factors to test the hypothesis that light or moisture is responsible for the distribution of mealworms.
In the Mr. Short puzzle, concrete reasoning is recognized by students who simply add the extra amount to the height of Mr. Tall. This type of thinking is a much more direct measure of qualitative difference than is the ratio, which the student makes only by making a correspondence between each individual button and paper clip. Formal thinkers understand that each button corresponds to a certain number of paper clips. Once the ratio is known, the answer is found by calculation.
Concrete thinkers need reference to familiar actions, objects and observable properties. They use reasoning patterns that include conservation, class inclusion, reversibility and serial ordering. Their thinking shows inconsistencies among various statements they make, and often contradict themselves. Most of the students you will teach are concrete thinkers, even though Piaget has suggested that the formal stage of development begins at about age 14. Studies have shown that many adolescents and adults do not develop formal thinking capacities.
Formal or abstract thinkers reasons with concepts (F=ma), indirect relationships, properties and theories. They are able to use a wide range of thinking patterns, including combinatorial reasoning, proportional reasoning, control of variables, and probabilistic reasoning. Formal thinkers are able to plan out a course of action, manipulate ideas in their minds, and actively check the validity of their own ideas.