The complexity of mathematical thought and the quality of learning:Portfolio assessment

Fernando Luís SantosInstituto Piaget, Almada

fernando.santos@almada.ipiaget.pt

António DomingosUniversidade Nova de Lisboa

amdd@fct.unl.pt

INTRODUCTIONWe present the highlight of an experience of portfolio assessment by two students on a geometry problem in a pre-service teacher training course analysed through a model designed to access the mathematical thought and the quality of student leaning outcomes. Results allowed us to see the different pathways used by the students on the same problem.

The analytical model, supported by SOLO taxonomy uses Activity Theory as a contextual framework that allows to integrate the different relations regarding to mathematical thought, namely advanced mathematical thinking concepts like procept and proceptual divide.

SOLO TAXONOMY

PROCEPT & PROCEPTUAL DIVIDEprocessprocedure

conceptPRO+CEPT

ACTIVITY THEORYBased on the 3th generation of Activity Theory (Engeström, 2001) to rise the context and to give a more structured analysis.

PORTFOLIO ASSESSMENTPortfolio assessment brings an open evaluation method into the mathematical classroom and allows to foster the mathematical thought of a student, in this experiment, students chose three of 15 problems and had close to a month to solve them and to explain in detail their resolution process. This solution process involves brainstorming sessions centred on the best solution, and the detailed explanation necessarily involved self-regulated learning processes. This teaching methodology aims to extend mathematical knowledge of future teachers, involving them in activities more open and less structured than the traditional ones.

Find the length of BC (to the second decimal place) knowing that AC = 10cm and ∡BAC = 30+ (do not use any trigonometry).

The outcomes of the two students were chosen because they show a similar path (12th grade mathematics) and took different approaches to the same problem.

Raquel

I sketched another triangle making an isometry using [AC] as a symmetry axis, creating with B′ an equilateral triangle ABB′ because if ∡BAC = 30+ then the resulting isometry makes ∡CAB′ = 30+ therefore ∡BAB′ = 60+ and a equiangular triangle is also an equilateral triangle. By using the Pythagorean theorem, I calculated: 𝐼𝑓𝐴𝐵 = 𝑥, 𝐵𝐶 = 8

9, 𝐴𝐶 = 10𝑡ℎ𝑒𝑛

𝑥9 =𝑥2

9+ 109𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦𝑖𝑛𝑔𝑥9 =

𝑥9

29 + 100 ⇔4𝑥9 = 𝑥9 + 400 ⇔ 4𝑥9 = 𝑥9 + 400 ⇔ 3𝑥9 = 400 ⇔

𝑥 ≅ 11,55 ⇔ 𝐴𝐶 = 89≅ 5,77𝑐𝑚

Mariana

This taxonomy brought by Biggs and Collis (1982) identifies five levels of growing complexity: (i) prestructural; (ii) unistructural; (iii) multistructural; (iv) relational and (v) extended abstract.

If∡BAC = 30+, ∡BCA = 90+then180+ = 90+ + 30+ + ∡ABCso∡ABC = 60+

IfAB = 2a, BC = a, AC = a 3� ∧ AC = 10therefore

a 3� = 10 ∧ a =103�soBC = asoBC ≅ 5,77cm

• Surpasses the proceptual divide;• Outcome classified as extended abstract;• Makes connections to other concepts;• Explains her pathway;• Justifies the outcome;• Gives evidences to support the resolution;• Different mediating artifacts.

• Broke the rules of the problem;• Written outcome classified as multistructural;• Latter classified as unistructural after oral presentation;• Makes simple connections;• Descriptive pathway without explaining;• Does not give evidences to support the resolution;• Contradictions in the rules and in the mediating artifacts.

These students are well acquainted with paper examinations, but, portfolio assessment was a different approach and they were in unknown territory.Results showed different pathways used by the students on the same problem, and the portfolio assessment allowed to foster the different mathematical thought of both students.

ReferencesBiggs, J., & Collis, K. (1982). Evaluating the quality of learning. London: Academic Press. Engeström, Y. (2001). Expansive learning at work: toward an activity theoretical

reconceptualization. Journal of education and work, 1(14), 133-156. Gray, E., & Tall, D. (1994). Duality, ambiguity and flexibility: a proceptual view of simple

arithmetics. The journal for research in mathematics education, 2(26), 115-141. Tall, D. (Ed.). (2002). Advanced mathematical thinking. New York: Kluwer Academic Press.

More info here:

This work is supported by national funds through FCT - Foundation for Science and Technology in the context of the project UID/CED/02861/2016