Relational Mathematics

A Constructivist Approach To Learning

My approach to teaching children mathematics is based in part on a philosophy that says children learn from experience especially when considering instilling in children knowing, doing, and valuing. Children have a need to participate and be part of their learning environment especially in the elementary grades when they are at the height of developmental potential. Children construct their own individual interpretations of the world around them. As educators, we need to be cognoscente of this at every level, especially when educating children in the elementary years since their future building blocks are literally in our hands. As educators, we are all aware of the different theorists who have left their mark on cognitive development in terms of how it happens, what influences it, and suggestions on how to encourage it. Such famous names as Piaget , Van De Walle , Kolberg , and Erickson have all researched the development of cognition on many fronts.

While an elementary school teacher does not have to be a specialist in cognitive development, it is very helpful, pro-active, and satisfying to include the student’s developmental stage when:

  • Planning lesson structure
  • Setting student tasks
  • Directing classroom conversation
  • Assessing
  • Teaching based on our own mathematical competence
  • Communicating with people who have a stake in our student’s learning.

Most elementary school teachers are not philosophers, or psychologists, I think most would agree that we need to be aware of how impactfull our approaches are to learning on the development of child cognition during the elementary grades, where it counts most, especially when it comes to assessment. One of the challenges we face as educators is reaching all of the students in our increasingly diverse classrooms. When considering Piaget’s theory, during the age of 6 to 11 years, children are developing the concrete operational stage. In other words, this is the 4th natural stage of cognitive development following the preoperational stage. The concrete operational stage focuses on the development of the appropriate use of logic. (The very foundation of mathematics in order to make sense of or to assign meaning to the world) This stage is of the utmost importance for elementary school teachers, specifically when considering teaching methodologies around mathematics, which are the building blocks for future cognitive development, and are different for each student (different backgrounds, and different abilities).

For the purpose of sharing my philosophy with you, I have chosen to focus, or align my studies with both Piaget, and Van De Walle, 2 great influences on my approach to teaching children at the elementary level. In order to provide you with the breadth and depth of my philosophy, I have included a few key aspects of Piaget’s concrete operational stage of development, listed below, as I consider these to be the most meaningful to first understand when considering teaching children mathematics, in order to build on the diversity of the classroom. These elaborations provide insight into benchmarking teaching or facilitating techniques, and learning styles of the classroom: (The following definitions are as listed on the Wikipedia website for quick reference, and further insight)

Seriationthe ability to sort objects in an order according to size, shape, or any other characteristic. For example, if given different-shaded objects they may make a color gradient.

Classificationthe ability to name and identify sets of objects according to appearance, size or other characteristic, including the idea that one set of objects can include another. A child is no longer subject to the illogical limitations of animism (the belief that all objects are alive and therefore have feelings).

Decenteringwhere the child takes into account multiple aspects of a problem to solve it. For example, the child will no longer perceive an exceptionally wide but short cup to contain less than a normally-wide, taller cup.

Reversibilitywhere the child understands that numbers or objects can be changed, then returned to their original state. For this reason, a child will be able to rapidly determine that if 4+4 equals 8, 8−4 will equal 4, the original quantity.

Conservationunderstanding that quantity, length or number of items is unrelated to the arrangement or appearance of the object or items. For instance, when a child is presented with two equally-sized, full cups they will be able to discern that if water is transferred to a pitcher it will conserve the quantity and be equal to the other filled cup.

Elimination of Egocentrismthe ability to view things from another's perspective (even if they think incorrectly). For instance, show a child a comic in which Jane puts a doll under a box leaves the room, and then Melissa moves the doll to a drawer, and Jane comes back. A child in the concrete operations stage will say that Jane will still think it's under the box even though the child knows it is in the drawer. (See also False-belief task ).

 

Although these are some key aspects of child development which hold particular significance to elementary teachers to consider in the classroom everyday, it is also key to keep in mind that children develop at different rates, in a variety of differing ways. As educators we embrace difference in the classroom. Interdependence, and Autonomy are themes that we find every day in the classroom when considering different learning styles, or differing approaches to education, i.e. individual work, and group work. In other words, we should be teaching for process, and not so much on content, which is a transformative approach to education. We are facilitating learning and understanding, instead of directing or teaching knowledge. This pedagogical approach to teaching is considered to be a multipolar approach focusing on activism, and stresses the need for students to ask critical questions, and to suggest possible solutions to the problem, and to evaluate the probable outcomes of the solutions they suggest.

 

Teaching student-centered mathematics is a modern approach to rote learning. Van De Walle has created a professional mathematics series of books called “The Van De Walle Professional Mathematics Series: Teaching Student – Centered Mathematics”. For my purposes as an elementary school specialist, my focus has been around grades K-3, and 3-6. For the purposes of this website, I will be quoting Van De Walle, and providing to you some insightful techniques to teach mathematics successfully in the elementary grades, with many connections.

 

Van De Walle suggests that since children make sense of mathematics in their own way, problem solving is a good approach to a diverse classroom. Each student will bring the skills and ideas that they own to solve the problem. “The sophistication of the methods and approaches used will vary in accord with the range of ideas found within the class […] In contrast, in a traditional, highly directed lesson, it is assumed that all students will understand and use the same approach and the same ideas” (Van De Walle Diversity in the Classroom p 27)

 

Van De Walle suggests establishing a problem based setting in the classroom, and the following techniques to address diversity in the classroom:

Making sure that problems have multiple entry points (implement use of manipulatives, drawings, i.e. student invented methods etc)

Plan differentiated tasks (use multiple sets of numbers for computation)

Use heterogeneous groupings (avoid ability grouping)

 

An example of a differentiated task:

For problem solving involving computations, insert multiple sets of numbers, students are able to select the first, second, or third number in each bracket:

Eduardo had {12, 60, 21} marbles. He gave Erica {5, 15, 46} marbles.

How many marbles does Eduardo have now?

 

"A problem based classroom setting will assist you in addressing varying learning styles, rates, and entry points of accomplishing the curriculum outcomes set forth to achieve. When considering assessment is a thoughtful, listening based technique that is integrated within the activity with diminished focus on the testing at the end of a lesson.Relational Understanding is the measure of the quality and quantity of connections that a new idea has with existing ideas. The greater the number of connections to a network of ideas, the better the understanding." (Van De Walle & Folk 2007)

 

 

 

   
   
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