The complexity of mathematical thought and the quality of learning: portfolio assessment
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This document presents a study on portfolio assessment in a pre-service teacher training course, focusing on students' problem-solving approaches in geometry. The findings reveal different pathways used by two students when tackling the same geometry problem, highlighting the effectiveness of this assessment method in fostering mathematical thought. The analysis is supported by a model integrating SOLO taxonomy and activity theory to evaluate the quality of learning outcomes.
![The complexity of mathematical thought and
the quality of learning:
Portfolio assessment
Fernando Luís Santos
Instituto Piaget, Almada
fernando.santos@almada.ipiaget.pt
António Domingos
Universidade Nova de Lisboa
amdd@fct.unl.pt
INTRODUCTION
We 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 DIVIDE
process
procedure
concept
PRO+CEPT
ACTIVITY THEORY
Based on the 3th generation of Activity Theory (Engeström, 2001) to rise
the context and to give a more structured analysis.
PORTFOLIO ASSESSMENT
Portfolio 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: 𝐼𝑓 𝐴𝐵 = 𝑥, 𝐵𝐶 =
8
9
, 𝐴𝐶 = 10 𝑡ℎ𝑒𝑛
𝑥9
=
𝑥
2
9
+ 109
𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦𝑖𝑛𝑔 𝑥9
=
𝑥9
29
+ 100 ⇔
4𝑥9
= 𝑥9
+ 400 ⇔ 4𝑥9
= 𝑥9
+ 400 ⇔ 3𝑥9
= 400 ⇔
𝑥 ≅ 11,55 ⇔ 𝐴𝐶 =
8
9
≅ 5,77 𝑐𝑚
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+
then
180+
= 90+
+ 30+
+ ∡ABC so ∡ABC = 60+
If AB = 2a, BC = a, AC = a 3 ∧ AC = 10 therefore
a 3 = 10 ∧ a =
10
3
so BC = a so BC ≅ 5,77 cm
• 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.
References
Biggs, 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](https://image.slidesharecdn.com/cerme10poster-170205113920/85/The-complexity-of-mathematical-thought-and-the-quality-of-learning-portfolio-assessment-1-320.jpg)
The complexity of mathematical thought and the quality of learning: portfolio assessment
- 1. The complexity of mathematical thought and the quality of learning: Portfolio assessment Fernando Luís Santos Instituto Piaget, Almada fernando.santos@almada.ipiaget.pt António Domingos Universidade Nova de Lisboa amdd@fct.unl.pt INTRODUCTION We 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 DIVIDE process procedure concept PRO+CEPT ACTIVITY THEORY Based on the 3th generation of Activity Theory (Engeström, 2001) to rise the context and to give a more structured analysis. PORTFOLIO ASSESSMENT Portfolio 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: 𝐼𝑓 𝐴𝐵 = 𝑥, 𝐵𝐶 = 8 9 , 𝐴𝐶 = 10 𝑡ℎ𝑒𝑛 𝑥9 = 𝑥 2 9 + 109 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦𝑖𝑛𝑔 𝑥9 = 𝑥9 29 + 100 ⇔ 4𝑥9 = 𝑥9 + 400 ⇔ 4𝑥9 = 𝑥9 + 400 ⇔ 3𝑥9 = 400 ⇔ 𝑥 ≅ 11,55 ⇔ 𝐴𝐶 = 8 9 ≅ 5,77 𝑐𝑚 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+ then 180+ = 90+ + 30+ + ∡ABC so ∡ABC = 60+ If AB = 2a, BC = a, AC = a 3 ∧ AC = 10 therefore a 3 = 10 ∧ a = 10 3 so BC = a so BC ≅ 5,77 cm • 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. References Biggs, 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