Last year, I had the opportunity to coach math teachers in my school. Our discussion seemed to inevitably turn to how so many students view mathematics as a collection of disparate topics, each with certain types of problems, words, and typical student errors. Students failed to make connections between lessons and only saw a grouping of problems that led to that unit test. Math is made up isolated topics and “applications” are usually introduced at the end of the study of the topic. We have all seen teachers that follow textbooks in demonstrating techniques using worked examples, and then inviting learners to follow their worksheets. Learners consolidate their learning by completing tests, which test them on their techniques they learned to use to solve routine questions.
In my classroom, I want to provoke all learners into taking initiative, into engaging fully with the mathematical ideas and to show evidence of mathematical thinking. This type of learning is accomplished when pedagogic tasks call upon learners to make use of their powers of making sense. Human sense making was described with examples by Polya (Polya, 1962) and Dewey (Dewey, 1938). This sense making involved imagining and expressing what is imagined, generalizing, conjecturing and convincing yourself and others, organizing and characterizing and focusing.
What I think activates human sense making is a disturbance, which can be experienced as surprise, puzzlement or perplexity. This element of surprise can be described as a cognitive dissonance. In my classroom, this surprise is what drives my teaching and my students’ learning.
Noddings in Philosophy of Education describes an example of caring as a way of demonstrating a phenomenological approach (Noddings, 2012). For my understanding, I needed to work through what a phenomenological approach to teaching would look like. This approach begins with the premise that all empirical knowledge must start with sensory impressions and concepts are formed based upon sense impressions or a combination of senses and other concepts. Observations are made and are focused on after mentally removing the details from the whole of the environment in which we are observing. We then find relationships or order within these observations. The activity of looking for relationships between the perception is not linear and not necessarily arrived through a logical process. This could be described as the “aha” experience of the learner. In that moment, a new relationship is seen and it is after that moment when the learner will use logic to determine fi the relationship will hold true in the context of the other relationship that are known. The process of looking for a relationship among the phenomena observed is the activity of thinking. Thinking is different from recalling facts.
Mathematical thinking can be initiated through experiencing some phenomena which triggers questions. Instead of giving the student the laws, definitions, relationships up front, and guided through the reasons why they are true, thinking becomes an activity and students are interested in looking at a problem and “making sense” of the new situation. Polya in his classic math problem solving book How to Solve It (Polya, 1957) pointed out that the most vital for actual learning and not just participating is the phase of looking back, of making sense of what has been done and to strengthen the links between the state of being stuck and strategies for getting unstuck.
In order to create the conditions in which learners can experience this energy, the teacher needs to experience and recreate the surprise in and for themselves. I pose questions to my students in order to intrigue them sufficiently to answer the question or explain what they have observed. The type of question is open-ended allowing all students to participate and take part in the learning conversation. A question is open when it is framed in such a way that a variety of responses or approaches are possible. This will undoubtedly result in richer conversations and almost any student will be able to find something appropriate to contribute to this conversation. This approach is different from asking a question that students cannot answer and then asking a simpler question to for them to respond to. Students need to gain confidence by answering the teacher’s questions right from the start. In addition to engaging students, this type of questioning is in response to both Rousseau and Dewey’s comments on the differences in talents and interests as described in Noddings’ book Philosophy of Education (Noddings, 2012). The issue is not necessarily about whether or not you change curriculum content for differences in student interests or learning styles, but more about changing how the curriculum is taught. It is about the questions and tasks that are being posed to students in math class. Many more examples of these types of questions designed to engage all students can be found in More Good Questions to Differentiate Mathematics (Small and Lin, 2010).
Example of phenomenal approach: The Ice Cream Puddle Problem
Students viewed a video problem called the Ice Cream Puddle Problem. The video shows an ice cream cone melting and the question that students determined to answer was “What is the diameter of the puddle?”.
Students asked questions and determined what information they needed to solve the problem.
Students created models and drawings. Through their collaborative work they discover that the puddle is not two-dimensional but rather a three-dimensional cylinder with a very tiny height. Eventually, they figured out that they could calculate the diameter of the puddle when they knew the volume of the ice cream used.
What is constructivism?
In Noddings’ Philosophy of Education, constructivism can be described as “a philosophy, an epistemology, a cognitive position, or a pedagogical orientation.” (Noddings, 2012, p. 126) Constructivism is in sharp contrast to the more traditional mathematics instruction based on the transmission view, where students passively absorb mathematical structures invented by others and consists of teachers telling them about facts, skills, and concepts.
In constructivism, knowledge is actively created or invented by the child, and passively received from the environment. The idea goes back to the Piagetian position that mathematical ideas are made by children, not found like a pebble or accepted from others like a gift. (Noddings, 2012). Children invent new ways of thinking about the mathematical ideas.
Children create new mathematical knowledge by reflecting on their physical and mental actions. Ideas are constructed or made meaningful when children integrate them into their existing structures of knowledge. The interpretations that children make are shaped by their experience and social interactions. Learning mathematics for me is about the process of adapting to and organizing one’s quantitative world, not discovering preexisting ideas that have been imposed by others.
In constructivism, students are learning through a social process. Mathematical ideas and truths are established by creating a cooperative culture for students to learn in. The constructivist classroom is seen as a culture in which students are involved in discovery and invention but also in learning conversations that involve explanations, debates, questioning, sharing and evaluations.
Teachers who use traditional instruction for teaching math to children puts values only on established mathematical procedures and concepts. Even the use of concrete materials is used for introducing ideas with the goal to get to the abstract and symbolic established mathematics as referred to in the earlier blog post. In constructivist instruction, students are encouraged to use their own methods for solving problems. All methods and ideas are valued and all students become active learners. Through interaction with authentic and relevant mathematical tasks and with other students, the student’s own intuitive mathematical thinking becomes more powerful.
Example of constructivism in mathematics:
Example: Constructivism in Mathematics Class
From the book Eyes on Math, (Small & Lin, 2012)
BEING FAMILIAR with the distributive principle for multiplication will, in the short term, help students to learn new facts by using known facts. Actually, knowing the facts for 5 and the facts for 2, along with the distributive principle, would allow a student to efficiently figure out any unknown fact. The picture provided here is designed to show that a strategy for multiplying two numbers is to break one of them into parts, multiply each of the parts by the other factor and add the results.
QUESTIONS to supplement the question with the picture and to include in a conversation about the picture include:
• What multiplications does the picture show?
• How can you rearrange the rows or columns to show other ways to figure out 7 x 6?
• What does the picture tell you about how you can figure out multiplication facts you don’t already know?
Dewey, J. (1938). Experience and education. New York: Macmillan.
Noddings, N. (2012). Philosophy of education (3rd ed.). Boulder, Colo.: Westview Press.
Polya, G. (1957). How to solve it; a new aspect of mathematical method. (2d ed.). Garden City, N.Y.: Doubleday.
Polya, G. (1962). Mathematical Discovery: on understanding, learning, and teaching problem solving. (Combined ed.). N.Y.: Wiley.
Small, M., & Lin, A. (2010). More good questions: great ways to differentiate secondary mathematics instruction. New York: Teachers College Press.