 Two trains of thought  Behaviorism (Skinner)  Constructivism (Piaget) Behaviorism – One can affect learning by changing behavior Constructivism – One affects “understanding” by constructing his/her own knowledge

 The prevailing theory in math education is constructivism  Constructivism is seen to entail the following six items by most experts.

“The particular problem of mathematics lies in its great abstractness and generality, achieved by successive generations of particularly intelligent individuals each whom has been abstracting from, or generalizing, concepts of earlier generations.” R. Skemp

“The present day learner has to process not raw data but data-processing systems of existing mathematics.” R. Skemp

Mathematics cannot be learnt directly from the environment, but only indirectly from other mathematicians, in conjunction with ones own reflective intelligence. It makes students dependent on teachers (including textbook authors).

It can expose one to acquiring a life- long dislike of mathematics.

 First two principles to learning mathematics  Concepts of a higher order than those which a person already have cannot be communicated to him/her by a definition, but only by arranging for him/her to encounter a suitable collection of examples.  Since in mathematics these examples are almost invariably other concepts, it must first be ensured that these are already formed in the mind of the learner.

Rule 1 is broken by most textbooks and many teachers. Good teachers give ample number of examples for each definition.

 Pythagorean Theorem  Given a right triangle with right angle C then c 2 =a 2 +b 2 where c is the side opposite angle C.

Concept is widely used but is hard to define. Mathematical concepts are among the most abstract.

Concept development starts with classifying. Pre-verbal examples.  Baby finishes drinking his bottle and places it with empty bottles that he sees set in the corner.

 Boy sees a baby for the first time and it is crawling. He pets it on the head like he would a dog.

 Low Level Start with one object experienced from multiple perspectives until it is classified. c 1 c 2 c c 3 c 4

Similar objects are then grouped or classified with the original object. cc’c’’c’’’c’’’’

From this group we abstract invariant properties by which we recognize object ch as a member of the class {c, c’, c’’, c’’’, c’’’’}. Classifying is collecting together experiences on the basis of similarities. Abstracting is an activity that makes us aware of similarities.

Naming an object helps to classify it. However, naming a concept can be limiting. A name and the concept can often be confused. 4vs. 3

 Two Types of Concepts  Primary – Those derived from sensory and motor experiences  Secondary – Those abstracted from other concepts Defining a concept can short circuit the intended process of defining it.

 Two uses of definitions  Specifies the limits, boundaries of a larger class of objects  Provides means of describing new concepts of lower order from known concepts Example: Magenta can be described as between red and blue

Beginning examples should have little “noise” More advanced examples should become noisier Collections of examples require inventiveness and a clear knowledge of the concepts/definitions they are being used for

If a particular level of abstraction is imperfectly understood, all subsequent levels are in peril. Contributory concepts necessary for each new stage of abstraction must be available when needed.

Pairing concepts together with a connecting idea is known as a relation. A transformation is applied to a relation to change it or combine it with another relation. The entire process of transforming relations creates complex structures.

The study of structures is an important part of mathematics, and the study of the ways in which they are built up and function is at the very core of the psychology of learning mathematics.

Conceptual structures are known as schema. One of the most basic mathematical schema is the set of Natural numbers with the operations of addition and multiplication.

One of the most misunderstood mathematical schema is the set of positive fractions with the operations of addition and multiplication.

Understanding – to assimilate into an appropriate schema not an all-or-nothing state

 Three failures to understand  Wrong schema is used  Gap between new idea and the (appropriate) existing schema  The existing schema is not capable of assimilating the new idea without undergoing expansion or restructuring

 Five Dimensions of Understanding  Nature of task  Role of the teacher  Social culture of classroom  Mathematical tools support learning  Equity and accessibility

 Nature of the Task  Students develop mathematical understanding as they invent and examine methods of solving problems  Paper-and-pencil worksheets – faster execution of skills  Watching a teacher at the board – imitation  Doing and reflecting – UNDERSTANDING  Role of the teacher

 Problematic – a task is problematic if it promotes new understanding. The task is an exercise if it involves only previously learned material and skills.  Problematic tasks promote reflection and communication  Problematic tasks require the use of tools  Allow students time to explore new tools  Use tools only when needed  Make sure the tool is suitable to the task

 Problematic tasks should leave behind a residue  Outcomes of a task: (1) insights into structure, and (2) strategies for solving problems   Task should connect with previously learned knowledge (in and out of math class)  Task should be significant

 Role of Teacher  Facilitate conceptual understanding  Make sure task fits goal and is a genuine mathematical problem  Don’t be a professor of truth and correctness  Select appropriate problems and sequencing to promote learning  Don’t intervene too much or too deeply – stems initiative and creativity  Provide essential information when it is needed  Control the social culture of the classroom

 Social Culture of the Classroom  INTERACTING IS ESSENTIAL  Reflection and communication depends on the social culture of the classroom  Ideas are currency  Autonomy of students with respect to methods  Appreciate mistakes  Authority for reasonability and correctness lies in the logic and structure of the subject  Students choose and share methods  Mistakes are seen as opportunities  Correctness based on arguments

 Mathematical tools support learning  Tools are not necessarily calculators and computers  Meaning does not reside in the tool. It is formed by using the tool  Tools are used only to accomplish a task  Tools should be versatile  Tools should help to communicate effectively  Tools should aid thinking