Academic journal article The Mathematics Enthusiast

Problem Solving in the Primary School (K-2)

Academic journal article The Mathematics Enthusiast

Problem Solving in the Primary School (K-2)

Article excerpt

Abstract: This article focuses on problem solving activities in a first grade classroom in a typical small community and school in Indiana. But, the teacher and the activities in this class were not at all typical of what goes on in most comparable classrooms; and, the issues that will be addressed are relevant and important for students from kindergarten through college. Can children really solve problems that involve concepts (or skills) that they have not yet been taught? Can children really create important mathematical concepts on their own - without a lot of guidance from teachers? What is the relationship between problem solving abilities and the mastery of skills that are widely regarded as being "prerequisites" to such tasks? Can primary school children (whose toolkits of skills are limited) engage productively in authentic simulations of "real life" problem solving situations? Can three-person teams of primary school children really work together collaboratively, and remain intensely engaged, on problem solving activities that require more than an hour to complete? Are the kinds of learning and problem solving experiences that are recommended (for example) in the USA's Common Core State Curriculum Standards really representative of the kind that even young children encounter beyond school in the 21st century? ... This article offers an existence proof showing why our answers to these questions are: Yes. Yes. Yes. Yes. Yes. Yes. And: No. ... Even though the evidence we present is only intended to demonstrate what's possible, not what's likely to occur under any circumstances, there is no reason to expect that the things that our children accomplished could not be accomplished by average ability children in other schools and classrooms.

Keywords: Common core standards; elementary mathematics education; problem solving in elementary school;

Lesh: "Do you really think your children can do this? "

Riggs: "So far, nobody has taught them yet about what they can't do.

Can Children Solve Problems involving Concepts they have not been Taught?

Most people's ordinary experiences are sufficient to convince them about the truth of two important assumptions about learning and problem solving.

* First, the kinds of things that students can learn, and the kinds of problems that they can solve, tend to be strongly influenced by the things they already know and are able to do. So, the accompanying "common sense assumption" is that these prerequisites must be mastered before students are expected to learn relevant new ideas, or solve relevant new types of problems. And consequently, learning is viewed as a long step-by-step process in which prerequisites are checked off one at a time.

* Second, concepts and abilities do not go from unknown to mastered in a single step. They develop! And, so do associated abilities. In fact, especially for the most important "big ideas" in the K-12 curriculum, development typically occurs over time periods of several years, and along a variety of dimensions - such as concrete-abstract, intuition-formalization, situated-decontextualized, specific-general, or increasing representational fluency, or increasing connectedness to other important concepts or abilities. So, in situations which are meaningful and familiar to students, rapid developments often occur for clusters of related concepts and abilities. And, in these contexts, students' ways of thinking often integrate ideas and abilities associated with a variety of textbook topic areas - so that the resulting knowledge and abilities are organized around experiences as much as around abstractions.

For readers who are familiar with Vygotsky's zones of proximal development, the title of this section poses a question that is clearly naïve. Learning does not occur in this all-or-nothing manner. For example, in a series of projects known collectively as The Rational Number Project (RNP, 2011), it is well known that the "difficulty level" of a given task can be changed by years - simply by changing the context or the representational media in which problems are posed (e. …

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