Teacher-Designed Mathematical Portfolio Assessments: Motivations, Potential Benefits, and Lessons Learned

Abstract

In order to identify the potential benefits and challenges of implementing student portfolios as quality mathematics assessment, a pilot study was conducted with teachers in various secondary school settings. The multi-case study consisted of five teacher participants from geographically and demographically differing contexts, four in the USA and one in Canada. They were interviewed and surveyed, exploring their motivations for utilizing a portfolio system of mathematics assessment for the 2020–2021 school year, the design of their portfolio system, and resultant impacts on student dispositions around mathematics. Each participating teacher implemented different structures of portfolio assessment, including the types of items included and how the items were assessed. Results showed that compared to traditional multiple-choice tests, teachers felt portfolios were a better reflection of their students’ “extra-mathematical” skills, such as problem-solving and persistence. Teacher surveys and interviews also demonstrated confidence that the use of portfolios as assessment was superior to more traditional measures in terms of adhering to good pedagogical practices. Through questionnaire data and interviews, teachers reported portfolios were particularly beneficial for students for whom English is not their primary language, as well as students with special needs. The paper lays the groundwork for additional research studies in the area of student mathematics portfolios based on the provided framework.

Résumé

Afin d’identifier les avantages et les défis potentiels associés à la mise en œuvre de portfolios d’élèves pour évaluer la qualité des mathématiques, on a mené une étude pilote auprès d’enseignants de différents établissements du secondaire. L’étude de plusieurs cas comprenait cinq participants, des enseignants issus de contextes géographiques et démographiques différents, quatre aux États-Unis et un au Canada. On a interviewé et sondé les enseignants pour explorer leurs motivations derrière l’utilisation d’un système de portfolio d’évaluation des mathématiques pour l’année scolaire 2020–2021, ainsi que celles derrière la conception de leur système de portfolio, et enfin, pour en évaluer l’effet sur les dispositions des élèves à l’égard des mathématiques. Chaque enseignant participant a mis en œuvre différentes structures d’évaluation du portfolio, y compris les types d’éléments inclus et la façon dont ces éléments ont été évalués. Les résultats ont montré qu’en comparaison avec les évaluations traditionnelles avec choix multiples, les portfolios, selon les enseignants reflétaient mieux les compétences « extra-mathématiques» de leurs élèves, telles que la résolution de problèmes et la persévérance. Les sondages et les interviews menés auprès des enseignants ont également révélé une confiance à savoir qu’en tant que mode d’évaluation, l’utilisation des portfolios était supérieure aux mesures plus traditionnelles en ce qui concerne l’adhésion aux bonnes pratiques pédagogiques. Selon les données obtenues des questionnaires et des interviews, les enseignants ont indiqué que les portfolios étaient particulièrement bénéfiques aux élèves dont l’anglais n’est pas la langue maternelle, ainsi que ceux qui ont des besoins particuliers. L’article établit un contexte pour d’autres études de recherche dans le domaine des portfolios de mathématiques des élèves fondées sur le cadre proposé.

Appendices

Appendix 1

A sample portfolio prompt from Aaron’s Calculus class, along with a few select examples of potential tasks for students to include.

Calculus Assessment 1

Below attached are many optimization problems. Demonstrate your understanding of the material that we have covered in this course so far by solving a chosen problem and providing an analysis of the graph. Some topics to include are:

• Determining a derivative through first principles

• Limits

• Continuity

• Relate the graph of a function to the graph of its derivative and it’s second derivative

• Instantaneous rates of change (first derivative)

• Instantaneous rates of change of instantaneous rates of change (second derivative)

• Average rates of change

• Curve sketching

Here are the possible optimization problems to choose from (pick one only).

Quadratic Optimization Problems.

We are going to fence in a rectangular field. If we look at the field from above the cost of the vertical sides are $10/ft, the cost of the bottom is $2/ft and the cost of the top is $7/ft. If we have $700 determine the dimensions of the field that will maximize the enclosed area
We are going to fence in a rectangular field. Starting at the bottom of the field and moving around the field in a counter clockwise manner the cost of material for each side is $6/ft, $9/ft, $12/ft and $14/ft respectively. If we have $1000 to buy fencing material determine the dimensions of the field that will maximize the enclosed area
The sum of two positive numbers is 48. What is the smallest possible value for the sum of their squares?
A rectangular field is to be enclosed by fence and then divided into two by another fence parallel to one of the sides. If there are 1800 m of fencing available, find the dimensions of the field of maximum area. What is the area?
Find two positive numbers x and y that add to 30, and for which 2x2 + 5y2 is a minimum

Appendix 2

A sample portfolio problem regarding matrices from Jess class.

Trail Mix Problem

There are three varieties of trail mix offered by a catering company, each variety is made in six- pound batches. The varieties are:

• Healthy Style (HS)

• Perfectly-Even Style (PES)

• Sweet-Tooth Style (STS)

One batch of each trail mix requires:

• HS: A pounds of fruit, B pounds of nuts, and C pound of M&Ms ○

• PES: D pounds of fruit, E pounds of nuts, and F pounds of M&Ms

• STS: G pounds of fruit, H pounds of nuts, and J pounds of M&Ms

The catering company gets an order for a summer camp for their last week of camp:

• Monday: K batches of HS, L batches of PE, and M batch of STS

• Tuesday: N batches of each style

• Wednesday: P batch of HS, Q batches of PE, and R STS

• Thursday: S batch of each style

• Friday: T batches of HS, U batches of PE, and V batches of STS.

The goal of this problem is to determine the total amount of each ingredient needed for the entire week's order using ONE sequence of matrix multiplications.

Your journal entry will need to explain your rational for setting up the matrices as you did and what the appropriate labeling needs are so that your final answer is a column or row matrix that shows the total amount of each ingredient the catering company will need to fulfill this week’s orders.

Remember to explain how you are certain that your method of solving this problem will result in the desired solution matrix.