SECOND GRADER’S DISCOVERIES OF ALGEBRAIC GENERALIZATIONS
Steven T. Smith
Northwestern University
ssmithla@earthlink.net
This paper will, necessarily
briefly, indicate one pathway by which children can begin to build algebraic
generalizations across additive and multiplicative domains. The approach is Vygotskian, and, more
particularly, influenced by the work of Davydov and his colleges (Davydov
1975). A central issue for a Vygotskian
approach is to indicate precisely how the teacher scaffolds children’s entry
into what Davydov terms ‘theoretical generalization’ (Davydov 1990). I will focus here on how the teacher’s
questions do so.
I will first discuss the earliest
algebraic teaching experiment I attempted, with second graders in a low-income
inner-city public school. After children had some experience modeling and
solving a range of word problems, I just asked them, “What can we find out
about any situation in which we put quantities together or take them
apart? I wanted to find out if
discussing a general kind of situation made sense to them at all. There was a very strong response. Children immediately invented a wide range
of ways to model general groupings of quantities and name them. Figure 1 shows some of these ways. It also
shows the model the class chose for collective use and discussion. Notice its similarity to a Euler diagram,
though it was invented from the ground up.
Children were able to use their models to make discoveries about what
statements are true or false of any Put-Together or Take-Apart situation, as
you can see in Figure 2. Notice that
this process of discovery builds up a set of statements rather like rudimentary
theorems. The models function like rudimentary proofs: Children used them to demonstrate why
statements were true or false. They
were also able to orally explain why the statements they generated were true or
false. Figure 3 gives some examples of
their explanations.
Let’s consider the role of the
question in this process. First,
general natural language understandings (e.g., children’s understanding of the
‘any situation in which we put quantities together or take them apart’ question,
and their namings of them: ‘everything’, ‘piece’, ‘piece’ etc.) together with
prior experience in constructing groupings are important constituents of the
process. We are not seeing
generalization built up de novo here.
Natural language and cultural discourse are suffused with
generalization. The question does not so much impart generalization as signal
that generalization is being asked for.
We are exploring how to continue, focus, redirect and aim children’s
existing language and experienced based generalizing orientations towards
relationships between quantities in general.
In particular, what the question focuses children upon is powerful
grouping experience they already have (‘putting together’, ‘taking apart’), and
apparently can intentionally explore,
discuss, and model in general terms (in response to the ‘any situation in which
we put quantities together or take them apart’ question) if asked to do
so. Once they do so
they are able to construct models supporting the discovery of true or false
statements about any put-together or take-apart situation, and demonstrate why
those statements are true or false of any additive situation, via the model and
verbally as well (Figures 2 and 3). But while an existing cultural base of
generalization is being drawn upon, mathematical generalization (indeed, any
theoretical generalization) requires selection and special constructions of
language and experience to direct discovery along lines that are likely to be
productive. The role of the question is to help select from and direct towards.
Which brings us to the next
point. If the question is only aiming,
rather than imparting, a generalizing orientation, what is it aiming children
at? Essentially, ‘What can we find out
about any situation in which we put quantities together or take them
apart?’ explores aiming children towards the kind of research questions
mathematicians undertake, the kind that constructs the field of mathematics
itself. Notice that it is not
focused at a problem solving level.
Problem solving, even of the sort focused on discovering alternative
solution methods, by definition focuses on the solution of a particular
problem. The longer term aims of a
sequence of problem solving episodes may remain tacit, unarticulated. Researchers, curricula and teachers of
course have such aims, but they are not necessarily discussed with children.
Yet mathematics, indeed, any intellectual field, is primarily about the
construction of questions that overtly address such long term aims. We see above that children can in fact
respond to such questions and begin to intentionally take on such aims. A
trajectory of generalization across kinds of quantities and varying kinds of
meaningful operations on them is not only under construction, but discussed
with children and intentionally aimed for as such. Notice that in Figures 4 through 7 length quantities and
comparing operations are inquired about in the roughly same way (discovering
what statements are true of lengths, where A = B (figure 4) and where B > S
(figures 5-7). Figure 7 shows a collective
representation of a range of true statements discovered, selected, posted, and
discussed by groups of children, not unlike the way an intellectual field might
collect, publicize and discuss knowledge.
In sum, the role of the question is
not merely to extend or ‘transfer’ tacitly
general grouping dispositions from situation to situation, but to overtly
direct attention towards those very wide ranges of possibility, and the means
of capturing them, at which intellectual fields aim, in this case, any
put-together or take-apart situation.
Prior, general, grouping orientations are overtly focused by the
question on the discovery of models and statements about them, publically
displayed, and used to demonstrate true or false statements about ‘any’. The existence, means, and value of
intellectual aims are thereby signaled.
And a wide range of potential use (any put-together or take-apart
situation) is pointed out.
To the extent that the results
touched upon above continue to hold, this ‘from cultural generalization to
mathematical generalization’ perspective suggests that something more like a
flow than a barrier, more like rapid and universal, rather than difficult and
exceptional appropriation of mathematical generality, should be expected,
unsurprising. It may be, in some
domains at least, that the ‘transfer dilemma’ of learning psychology, and the
‘discontinuities’ of developmental psychology are artifacts of misdirecting
children’s attention. It is not just exclusive foci on ‘skills’ and ‘the
answer’ that miscommunicates mathematical subject matter, but exclusive foci on
problem solving and domain specific contexts as well. We need to communicate the aims and means of intellectual
construction if we expect children to have some clue of the value possibilities
of mathematics, so that they can orient their attention, aims, and constructive
actions accordingly. The argument above is that when they do, in the situation
above at least, mathematical generalization is not only enabled, but rapid.
Let’s consider an expansion of that
argument: mathematical generalization
not only in the sense of some range that statements and models about ‘any’ may
cover, but the effective leverage such statements and models promote in
situated use. Efficiency is often
claimed by basic skills approaches, effectiveness by problem solving
approaches. I will discuss 2 further
teaching experiments, because there the algebraic development occurs before
certain ‘skills’ have been developed and problems encountered (2nd graders
already have considerable prior additive experience). Hence they bear on the question of efficient skill and problem
solving development.
First, one cluster of teaching
experiments aims kindergartners and older children towards the solution of
broad ranges of word problems by not focusing them on solving the
problem initially. Instead, I ask ‘how
can you show me which’ questions. That
is, given any additive word problem context, simple ones at first, such as
Marta and Lucille sharing cookies, ask “How can you show me which are: All the cookies? Marta’s cookies?
Lucille’s cookies?” This elicits
a rich variety of grouping models and discourse, very like the algebraic models
depicted in Figures 1 and 2 above, except that they do not refer to ‘any’ group
or subgroup. By this means not only the
simplest word problems (combine: total unknown, change: total unknown) but
problems of intermediate difficulty (combine: part unknown, change: change
unknown) become immediately accessible to kindergartners and first
graders. At a computational level, once
kindergartners can model groupings in a situation, they can use the simplest,
most widely known computational method, counting-all, to solve more difficult
problems (by counting-all within groups or subgroups they have
constructed). The ‘how can you show me
which’ question draws from prior general grouping understandings (‘which’), and
via grouping models kindergartners invent, directs them towards an expanded
range of problem solving and computational skill, quite rapidly.
Now let’s go to the other end of the
spectrum, 4th graders discovery of algebraic relationships in the
multiplicative domain, and briefly touch on how this builds proportional
problem solving and computational skill.
Table 1 displays a progression of mathematical questions posed by the
teacher (top row), and summarizes the algebraic discoveries children made in
response to them. Again, children
compare quantities and discover a broad range of relational statements about
them,
construct models and verbal explanations to demonstrate why
these statements are true, and build inferential connections between
alternative statements of the same relationship (If L = 3 S, then S = 1/3 of L,
and L/3 = S, and L/S = 3, etc.), so that inferential shifts between
multiplicative, divisive, fractional, and ratio perspectives on the same relationship
can be made. But let’s focus narrowly
on ‘skill’ and problem solving objectives only. On the 3rd day of this teaching experiment, shown at the bottom
of the 5th column of Table 1, children begin to solve proportion problems, an
important skill (and problem solving) objective. They are tested on proportional problem solving for unknowns in
all 4 possible positions in the proportion the next day. 17/21 get every problem right (6 problems
total), 3/21 make 1 mistake, 1/21 makes 2 mistakes. Even if we ignore all else
children learned in those 3 days, 3 days for nearly all of the class to learn
to solve all 4 types of proportion problems at that level of accuracy
constitutes very rapid and efficient learning (and strictly speaking, they
learned proportional problem solving in 1 day). A look at another aspect of the test suggests why. Children are asked to also describe each
proportional relationship (as well as find the unknown) for all the problems
(e.g., 9/3 = ?/5, ?/2 = 16/8, 15/? = 25/5, etc.). 17/21 give at least 2 descriptions for each problem, 11/21 give 3
(e.g., 15/? = 25/5 is described as “5/1 ratio and L/5 = S and S x 5 = L). The ability to describe the relationship
from both multiplicative and divisive perspectives, and shift from one to the
other, enables solving for any unknown in a proportion problem. The algebraic focus on discovering multiple
relational statements and inferential connections between them enables flexible
proportional problem solving and speeds up the development of computational
skill.
To sum up: We’ve considered the role questions may have in scaffolding
children’s entry into mathematical generalization. Questions can select from
common (cultural) generalizing orientations and experience (about grouping,
comparing, any), and direct towards
models, statements, and rudimentary proof-like demonstrations about ‘any’
grouping or comparing relationships.
This jointly communicates the existence of an expanding range of
mathematical value possibilities and means to capture them. It may also scaffold effective,
situation-specific problem solving and a rapid and efficient development of
computational skill.
References
Davydov, V. V. (1975). The psychological characteristics of
the “prenumerical” period of mathematics
instruction. In L.P. Steffe (Ed.), Soviet studies in the psychology of
learning and teaching
mathematics: vol. 7, (pp. 109 -
206). Chicago: University of Chicago
Davydov, V. V.
(1990). Types of generalization in
instruction: Logical and psychological problems in the structuring of school
curricula. Reston, VA: NCTM.
The research reported in
this paper was supported in part by the National Science Foundation under grant
# REC- 9806020. The opinions in the
paper are those of the author or authors and do not necessarily reflect the
views of the National Science Foundation