

Concordia University Department of Mathematics and Statistics 1400 de Maisonneuve blvd. West Montreal, Quebec H3G 1M8 Canada 


Concordia University Department of Mathematics and Statistics 1400 de Maisonneuve blvd. West Montreal, Quebec H3G 1M8 Canada 
The Measurement Approach to teaching fractions to prospective elementary teachers was inspired by an approach under the same name but addressed to children developed by the psychologist V.V. Davydov and described in the paper by Davydov and Tsvetkovich (1991). Our approach was developed, by myself and Georgeana BobosKristof, in a design experiment – in the sense of (Cobb, diSessa, Lehrer, & Schauble, 2003) – conducted in the years 201315 with groups of students enrolled in each of these years in semesterlong "Teaching Mathematics II" courses. In year 2014 the course was taught with a textbook (in two parts) which we called "Course Notes", that we have written specially for this course: One text was on fractions of quantities and the other  on rational numbers, ratio and percent. As a result of our observations in 2014, we revised the text and used the revised version in 2015.
EDUC 387 Teaching Mathematics II. Course Notes. Part I  Fractions of quantities.
EDUC 387 Teaching Mathematics II. Course Notes. Part II  Rational numbers. Ratio. Percent.
The “Measurement Approach” directs students’ attention to multiplicative relationships between quantities defined in terms of concrete units (such as kilos, inches or cups) and not in terms of indeterminate objects such as pizzas or cakes. A systemic study of these relationships and operations on them – as a theoretical system, using definitions derived from measurement situations, mathematical reasoning based on these definitions and generalizations of observed patterns – is intended to support the gradual process of abstraction of the notion of fraction as an abstract number representing a measure of a relationship between two quantities. Some results of our experiment have been presented in Georgeana BobosKristof's PhD thesis. A link to this thesis is available under PhD Theses on this page.
The study, conducted by Helena P. Osana (Department of Education, Concordia University) and myself (Anna Sierpinska), sought to describe the content of Elementary Mathematics Methods courses (EMM) offered to prospective elementary school teachers in six universities in three Canadian provinces: Québec, Nova Scotia and New Brunswick. Three universities were Francophone and three  Anglophone. The universities were labeled FU1, FU2, FU3 (FU="Francophone University"), and AU1, AU2 and AU3 ("AU"=Anglophone University). Our research focused on the tasks that instructors offered EMM students to carry out in classroom activities, graded assignments, test and examinations. On this page, we will post the sets of tasks collected in our research. The tasks will be organized into 'Catalogues' for each course and 'indexed' using a system of categories of ACTIONS in which the tasks engage EMM students and a system of categories of ANALYTIC TOOLS offered the students to guide them in performing the actions. The 'index' is not a fixed set of categories. It expands as we analyze more courses. So far (November 2011), it contains only categories of actions and analytic tools identified in EMM courses at two universities: AU1 and AU3. This is why we call it 'a seed index'. The seed index is posted on this page, and it will be updated to include actions and analytic tools used in the remaining four universities.
A seed index of categories for analyzing and planning 'Teaching Mathematics' courses for prospective elementary school teachers
Catalogues of tasks assigned to prospective elementary school teachers in EMM courses
Tasks in the first of two mandatory EMM courses at university AU1
Tasks in the first of two mandatory EMM courses at university AU3
The research was conducted by a team of three: myself (Anna Sierpinska), Georgeana Bobos, and Andreea Pruncut. Three approaches to teaching absolute value inequalities were tried. We labeled them: Procedural Approach (PA); Theoretical Approach (TA), and Visual Approach (VA).Short (2030 minutes) lectures were designed for each approach.
The PowerPoint slides for the lectures can be viewed here.
Each approach was tried with 6 students. After the lecture, students were asked to solve some exercises. The exercises were the same in each approach.
After solving the exercises, students were interviewed about their solutions.
Some preliminary results have been mentioned in Sierpinska, 2007 (PME), see List of Publications
and in a ppt presentation prepared for the pedagogical day at Vanier College, Montreal, November 11, 2008:
"Algebra: the Calculus student's Achilles' heel": PowerPoint version ; PDF version
NOTE: The above presentation was done in collaboration with Dr. Hong Yue, and the ppt refers to her
part, which is available from here.
Published papers: see List of Publications : Sierpinska, 2006; 2007; Sierpinska, Bobos & Knipping, 2007; 2008
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Last Modified: 01/26/16 11:57