ED6639 Technology and the teaching and learning of mathematics.

Winter 2008 with Margo Kondratieva.

Meeting 6 on February 26.

 

Note on Reading: Chapter 28

Flux in school algebra: curricular change, graphing technology and research on student learning and teaching knowledge.

 

 

 

The Questions and Ideas:

 

  1. What and how should be studied in school algebra?

·       Content (syllabus) is well defined: experiences related to solving linear and quadratic equations, factoring and simplifying algebraic expressions etc.

·       A variety of different teaching approaches (curriculum) are known, particularly, the “equational” approach and “functional” approach.

 

  1. Is there essential core or set of understandings that teachers and students should develop?

 

  1. What should be role of technology in supporting teaching of algebra?

 

  1. What are cognitive issues of students learning algebra?

·       What does a letter represent in school algebra content?  (unknown, variable, parameter.)

·       What does equal sign mean in different circumstances? (equivalence relation, constraint, or assignment.)

·       What is solving equation?

      (Find x and y such that the statement  x+y=22 is TRUE.)

·       What is function? 

       (Let x be a number and calculate y=22-x . )

·       What is a parametric family of functions?

      (Linear functions y=mx+b, where m and b are fixed numbers (parameters), x is independent variable, y is dependent variable.

 

  1. How we can analyze from the cognitive point of view each of the following?

·   A textbook in algebra;

·   Use of technology in teaching algebra;

·   Teachers’ methods in algebra;

·   Students learning transition and ways they generalize.

 

 

 

 

 

 

 

  1. How does a curriculum approach define the choice of technological tool?

 

·       For development of linked understanding between formulas, graphs and expressions Graphing calculator was the first choice.

·       For modeling (e.g. exponential growth), experimenting with data, and problem solving spreadsheets were found a better tool.

 

  1. How could it work in concrete mathematical examples?

Problem: Find the dimensions of a rectangle whose length is double its width and perimeter is 102 meters. Students experimented with Ecxel to find the solution by programming the given relations and varying the unknown until get required perimeter. This is a good preliminary exercise before the introduction of formal approach to this type of problems leading the algebraic equation 6*x=102.

 

  1. What does it add to improvement of pedagogy?

Use of technology and reflection on in-class experiences helps teachers to realize potential tension and conflicts between different aspects of their teaching. In the article we have an example when a teacher understood that she was trying to introduce a technique that the approach she was taking did not support well.

(Using graphing techniques for solving algebraic equations requires a clarification and shift from equational to functional point of view with their different roles of letters, equal sign etc..)

 

  1. How does it support students in their transitions from simpler to more complex general situations?

In the example presented in the article students were trying to develop a method of solving linear equations with two variables from a graphing approach for solving equations in one variable. Their paradigm was that solution is related to the intersection points of two curves each of which represent one side of given equation.

For example to solve x*x=2*x-1 one graphs y=x*x and y=2*x-1 and finds x=1 as an abscise of the intersection point. Thus in case of two variables (solving x+y=2*x-y) the students generalized this approach, which led them to the surfaces intersection (planes, in their case) problem. Although a valid technique, it is too complicated to be applied in this particular problem. A good thing is that the students developed a valid strategy, which in principle leads to correct solution. A bad thing is that while doing so the students overlooked a faster approach of combining like terms and simplifying the equation first to get x=2*y.

 

That brings us back to the question: what is the goal of teaching algebra?