Assessing Inquiry-Based Mathematics Education with Both a Summative and Formative Purpose

Abstract

In this chapter I present some of the results of a study aiming to learn if and how using an assessment tool such as a grid of criteria, can be useful for summative assessment and encourage formative assessment processes in the case of inquiry-based mathematics education. I focus my research on a course centred on problem solving in the canton of Geneva (Switzerland). In the first part, I provide a synthesis of teachers’ points of view about this course relative to problems they submitted to students, research narrative chosen as a means to assess students and the assessment of students’ problem solving competencies. I then describe how it leads to a collaborative work that aims to elaborate a grid of criteria of research narratives to assess students’ problem solving competencies with both a summative and formative purpose. Finally, I describe an exploratory study; an analysis of two lessons taught by a teacher who uses the grid of criteria, in order to understand if and how she refers to these criteria to develop informal formative assessment.

Appendices

Annexe 1: A Student’s Research Narrative (in French)

Annexe 2: Programme of Mathematics Development Course (MDC)

• I. Organization

  […] This weekly period is intended to support a teaching that contributes to the strengthening and development of problem solving strategies and mathematical situations activities.

• II. Programme

  • The suggested activities are linked with three main topics:

    • Numbers and Operation

    • Space and measure

    • Function and algebra

  • The problem solving strategies are:

    • Analogy

    • Trial and error—Example/counter-example

    • Inductive and deductive reasoning

    • Organized study of all cases and exhaustion of solutions

    • Introduction to proofs

  • These strategies contribute to the development of:

    • Scientific procedures

    • The rules of scientific debate

• III. Mathematics development course: Introduction

  […] The allocation of an additional period in the curriculum for grade 8 students with scientific profile, aims to enable these students to learn and become familiar with this important part of mathematical activity. The aim is not simply to solve problems “one by one”, but also to discover and systematize problem-solving methods. In particular, the aim is to place the student in a learning situation where she/he will have to implement a “scientific approach”, that leads her/him to the following scheme:

  • Try–Conjecture–Test–Prove

  This part of mathematical activity is required when students are confronted with the so-called open-ended problems. This places the pupil in the most typical situation of mathematical activity, that of confronting a problem which enables her/him to work as a mathematician who is confronted with a problem to which she/he does not know the solution.

• IV. Open Ended Problem

  • a. According to a definition proposed by a group of researchers at the IREM of Lyon, an “open-ended problem” has the following characteristics:

    • the wording is short

    • the wording does not introduce the method or the solution, the solution must not be reduced to the use or an immediate application of recent coursework

    • the problem must be situated in a conceptual field that students are familiar enough with, so that they can easily “take possession” of the situation and engage in trials, conjectures, draft resolutions, or counter-examples.

  • b. Solving a problem consists of a series of steps outlined in the official textbook:

    • 1. Appropriation of the wording: “understanding the problem to identify its purpose”

      At this stage, the teacher must ensure that all students are involved in the problem.

      That is to say that they are able to construct a correct representation of the data, understand the constraints and the goal to achieve. If necessary, the teacher answers questions, rephrases or makes the student rephrase the problem.

    • 2. Data processing: “design a plan”, then “put the plan into action” and “get back to the solution”.

      This stage corresponds to the research and resolution of the problem itself. A relatively short time slot can be allocated for individual research, followed by a second group work time.

      During the individual research phase, the teacher can verify that each student has actually read the problem, has at least partially assimilated it, and that, during the group work, she/he will not only follow the ideas of the one who speaks first. Group work helps to avoid the discouragement of certain pupils, to stimulate the exchange of ideas among students, to learn how to collaborate, to listen to each other, to defend their point of view, to respect each other.

    • 3. Communication of research procedures and results: “Write the results in a form that anyone can understand and follow the work done”

      At this stage, the student must account for all the resolution of the problem, individual phase and the group work included. Such a written report gives the teacher the first insight into the student’s research work and provides an occasion for evaluation. The writing of this report is a basis for the evaluation and is therefore an important competence for the student. This is why the practice of “research narrative” has been chosen as a thread for this course. According to the textbook, a research narrative is “a comprehensive account of research, including trials and errors that didn’t lead to a satisfactory result, or wrong conjectures, and the reasons which lead them to abandon them.”

• V. Research narrative

  • 1. Presentation of research narrative

    […] The objectives of this pedagogical practice can evolve throughout the year. They may initially be:

    • to develop students’ curiosity and critical thinking

    • to provide a communication tool that facilitates students’ writing

    • to put in place the rules of mathematical debate, in particular the following ones: a counterexample is sufficient to invalidate a statement, examples that verify a statement are not sufficient to show its validity, an observation on a drawing is not sufficient to prove that a statement is true

    • to allow the teacher to get a much better knowledge of the procedures of his pupils.

  • 2. Correction and assessment

    • Criteria for a good research narrative

    The action of narration is not an easy activity, but one can retain some elements that are to be emphasized and encouraged by the corrector of the copies.

    • Writing style

    • The accuracy of the narrative: all ideas, all trials are described thoroughly

    • The sincerity of the narrative

    • Criteria for Good Research

    To help students to better understand what is expected, it could be useful to refer to intermediate assessment means.

    • An assessment of the analysis of a problem by the formulation and explanation of conjectures

    • An assessment of the research phase: identifying and comparing strategies

    • Another assessment of the “research” phase: using hints

    • An assessment of the overall attitude

    • An assessment of an oral presentation.