We all know the story of Goldilocks and the Three Bears. When Goldilocks entered the home she kept finding things that were not quite right; they were too hot, too small, too hard, or too soft. It took a while before she would find the item that was just right for her. Sometimes I feel like math education is a bit like this. Either the problems are too sophisticated or too easy for the given child. And it takes a while before we find the task that is just right. And by “just right” I am referring to it being right for a given child at a given point in time. The “just right” task allows the child access to the goal, yet challenges them to coordinate their thoughts in new ways. Typically, however, we focus only on the problems and situations that seem right for the grade level.
At the AIMS Center for Math and Science Education we believe that children’s mathematics is a result of both their maturation and what knowledge they have constructed as a result of their interactions with the social and physical world. This is best explained by the theory of radical constructivism. This theory, and our work at AIMS, is based on the psychology of Jean Piaget, Ernst von Glasersfeld, and Leslie P Steffe, among others.
In his 2017 plenary paper to the 39th annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education, Dr. Steffe outlines seven programs of research that he believes are critical to mathematics education. The first two are:
- To construct a mathematics curricula (plan) for children who enter their first year of school as counters of discrete items.
- To construct quantitative mathematics curricula for children who enter their first year of school as counters of figural unit items -items that they can imagine even if not visible.
These two research programs are critical to mathematics education because they address half of all children in kindergarten and first grade. However, the current mathematical standards, and the curricula based on them, are written for children who enter Kindergarten having a more sophisticated understanding of number and who are no longer counters of perceptual or figural unit items. How can our mathematics curriculum be equitable if half or more of all children in kindergarten have no access to the material? If teachers only present the tasks they find in their textbooks and other standards-based materials, those are clearly not the right tasks for every student. Access does not mean that teachers talk about the material and/or put resources in the hands of students. Instead, access refers to students being able to mentally operate on the ideas presented to them. For example, if a child does not have a concept of six, then they cannot conceive of a situation in which six are joined with four more. And they certainly cannot conceive of multiple iterations of the number six.
So let us not run our mathematics curriculum like the story of Goldilocks and the Three Bears. Our teachers should not be trying to fit students into the curriculum, but rather, we should be designing the curriculum to fit our students. Student achievement should be a measure of the progress a child makes in their construction of increasingly more sophisticated schemes. This measures their learning, not simply their performance.
I hope you continue to read the blog posts on the AIMS Center webpage, as they chronicle our understanding of how children come to know number, and how teachers come to understand the learning of children, and shape their curriculum to their students.