Learning and Mathematics

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Learning to Count. Computing with Written Numbers. Mistakes - Ginsburg (1977)

Ginsburg draws heavily on the idea of assimilation -- the incorporation of new ideas into an existing body of knowledge -- to explain how children acquire or misacquire arithmetical skills and concepts. He looks at both the informal, concrete understanding of basic concepts that children acquire before entering school and the abstract, formal concepts and computations they are expected to learn in the classroom. The difference between such formal and informal knowledge often results in a gap between the ability to do paper-and-pencil calculations and intuitive understanding; sometimes students actually have strong informal abilities not indicated by their performance on school tasks, and sometimes they master formal algorithms without understanding the concepts behind them. Ginsburg focuses on students just entering school, but his ideas generalize to older students, for example calculus students who can take derivatives but can't explain the problems or their answers.
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Chapter:

Ginsburg, H. (1977). Learning to Count. Computing with Written Numbers. Mistakes. In Ginsburg, H., Children's Arithmetic: How They Learn It and How You Teach It (pp. 1-29, 79-129).

Overview:

Ginsburg emphasizes that students do not encounter arithmetic solely in the classroom; in addition, they often possess a substantial body of informal knowledge. For example, younger students are exposed to basic numbers and ideas of counting before entering school. Older students also frequently have some knowledge of topics such as algebra before learning the subject formally: without knowing formal algebraic principles, a child can often determine how many gum balls can be bought for a dollar if each costs 25.

Assimilation of formal concepts into their body of informal knowledge often poses a problem for children because of the highly symbolic nature of formal arithmetic. To a child, the representation ".25x=1.00" may not mean the same thing or seem to require the same type of computation as "How many gum balls can be bought for a dollar if each costs 25?"

Because formal arithmetic is abstract and the relation between the two problems stated above may be unclear, children who can easily figure out how many gum balls they can buy when standing in the store with a dollar cannot necessarily do the same problem when it is written symbolically on paper. Thus students who fail at the paper-and-pencil, symbolic calculations so prevalent in the classroom do not necessarily lack intelligence or ability; they may simply have assimilated formal concepts differently into their informal body of knowledge.

Errors in arithmetic, Ginsburg claims, generally stem from misassimilation or misinterpretation of the symbols. For example, given a column of numbers of different lengths to add, students may get the wrong answer because they don't understand the place value system and so misalign the columns. Similar errors can be seen in the setting up of algebra problems.

Students go through three main phases in understanding formal computation:

  1. The child can state the rules but not the ideas behind them. For example, in our gum ball algebra problem students may divide both sides by .25 because they have been taught to do so, but may know no other reason why the method is correct.

  2. The student still cannot explain his or her rationale, but can give examples of methods that would lead to an incorrect answer, such as the idea that if .25 were added to each side it would yield 1.25 -- which tells how much, not how many.

  3. The student can explain the rationale behind the method, in this case that division reverses multiplication, so dividing each side by .25 tells us by how much .25 was originally multiplied. Crucial to this last stage is a student's ability to explain the method in his or her own words, not merely to repeat what the teacher has said. This stage, Ginsburg states, is not reached easily or by many children.

Direct Quotes (and some comments):

"Children learn a great deal about numbers [and arithmetic] outside of school, without instruction or special help; indeed their parents are often unaware that they are trying to learn.... Also, they manage to learn even though their experience is often chaotic and confusing, not to say unplanned." (p. 10)

"Children's understanding of written symbolism generally lags behind their informal arithmetic." (p. 90) (This corresponds to the gum ball problem I mention above. The child could solve the problem if he or she was buying gum balls, but can not relate that to the algebra problem ".25x=1.00" either to read or to write it.)

"There is usually a reason for [children's] mistakes, and the reason is a systematic rule. Most often it does not help to think of errors in terms of 'low IQ' or 'learning disabilities'. These labels are too general.... The use of such vague labels conveys the false impression of having explained something while they merely serve to distract attention from the real problem -- children's faulty rules and the gaps between their formal and informal knowledge." (p. 129) (For example, a student who adds a column of numbers by aligning the leftmost rather than the rightmost column may do so because he or she is following the rules for reading and writing which say to proceed from left to right.)

"While they make many errors in written arithmetic, children may in fact possess relatively powerful informal knowledge. This can be used as the basis for effective instruction." (p. 129)

-- summarized by Andrea Hall

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