Assessāre is a Late Latin frequentative verb derived from assess, the inflectional stem of the past participle assessus, from the Latin verb assidēre “to sit next to or by ”.1
Assessment Of, As, and For Learning
Assessment and evaluation require thoughtful planning and implementation to support the learning process and to inform teaching. All assessments and evaluation of student achievement must be based on the outcomes in the provincial curriculum and allow for flexibility determined by the needs of the student. “Mathematics is as much an aspect of culture as it is a collection of algorithms,”2 Assessment practices must be culturally valid as well as algorithmically valid.
There are three interrelated purposes of assessment. Each type of assessment, systematically implemented, contributes to an overall picture of an individual student’s achievement.
Assessment for learning (formative assessment) involves the use of information about student progress to support and improve student learning, inform instructional practices, and:
- is teacher-driven for student, teacher, and parent use;
- occurs throughout the teaching and learning process, using a variety of tools; and
- engages teachers in providing differentiated instruction, feedback to students to enhance their learning, and information to parents in support of learning.
Assessment as learning (formative assessment) actively involves student reflection on learning, monitoring of his/her own progress, and:
- supports students in critically analyzing learning related to curricular outcomes;
- is student-driven with teacher guidance; and
- occurs throughout the learning process.
Assessment of learning (summative assessment) involves teachers’ use of evidence of student learning to make judgements about student achievement and:
The Western mathematics’ ideology of quantification (i.e., one makes the best assessment decisions about learners when based on quantitative data) dominates the domain of student assessment, according to Boaler and Confer (2017).3 However, “The misalignment between test content and valuable content makes the testing frenzy in our schools illogical” (p. 1). They explain:
When we place an excessive value on test scores or grades, we communicate fixed mindset messages to our students. Parents and teachers see children as the words that label them; “high,” “average” or “low.” Students quickly take ownership over their status. For example, students may take on the label of a “C” student and resign themselves to an incorrect belief that this reflects some innate mathematical capability and that working harder will not change that. (p. 2)
Their conclusion is that quantitative data have no place in formative assessment. It should only be used in summative assessment.
Formative assessment must be diagnostic feedback that clearly offers learners a “growth mindset pathway focused upon giving students the information they need in order to learn” the content (p. 2). This feedback serves as a constructive analytical critique of what the learner accomplished, written on a note in a personal supportive genre. Simply put, formative assessment is all about “show what you know” (p. 4). The realistic goal for all students becomes, “strive hard to do the very best you can” (p. 5).
A growth mindset is incompatible with teaching for a specific standardized test.
Assessment using Triangulation of Data
Triangulation of data involves collecting evidence from multiple sources over time. The term triangulation is often accompanied by a drawing of a triangle labeled: Conversations, Observations, and Products.
Assessment should be balanced and include evidence of all three of conversations, observations, and products, or in student terms, “say, do, and write evidence.”
Assessment Tools for Observation and Data Collection
Many mathematics assessment tools exist, and educators must make decisions on which tools are best suited not only for the subject material, but also for the students. The list and examples below are certainly not exhaustive, but are intended to be a starting point for teachers. In addition, tools can be used in different ways and the same tool, depending on how it is used, can often be formative (for or as learning) or summative (of learning). The following table groups some assessment ideas according to how they might be used most frequently, but certainly not exclusively. Each link connects to a page that provides a brief explanation and example(s).
|Assessment For |
|Assessment As |
|Assessment Of |
|Pre-Assessments||Vertical non-permanent surfaces||Rubrics|
|Exit Slips||Math Talk (Number Talk)||Project Based Learning|
|Diagnostic Assessments||Co-constructed Assessments||Tests and Prototypes|
How Much Assessment Evidence is Enough?
Since Saskatchewan math curricula contain learning outcomes by grade, the simple answer is: When the student has demonstrated clear understanding of the outcome. Of course, it is never that easy, and there are many factors for educators to consider, However, once there is enough evidence for a teacher to defend the decision that the student has met the learning outcome, there is little value in assigning more assignments, homework or assessment tasks for that outcome. This does not mean that there is no place for “spaced practice”. Teachers may assign more tasks that are not necessarily evaluated. There is no timeline for when a student must demonstrate an understanding. Evidence-based practice would assume that the student is able to demonstrate their understanding at any point in time and retention of learning will need to be revisited.
The Alberta Assessment Consortium maintains that “having a myriad of marks in the marks book is not necessarily a desirable objective.”
Quality and a balanced assessment is the goal, keeping in mind the importance of holistic assessment (gathering data from multiple perspectives), triangulation of data, and the importance of cultural relevance.
When students are pressured to earn a decent grade in a “foreign” abstract mathematics course, Simeonov (2016, pp. 442-443) pointed out that most students memorize without meaningful understanding. Students learn to hate mathematics, and then as parents, they infuse their attitude into their children for elementary teachers to confront (Duchscherer et al., 2019, p. 49).4
Education Sector Strategic Plan (ESSP) Math Data Collection
When gathering data to inform the placement of students on the holistic math rubric, the intent is to look at multiple samples of work done throughout the year. Any of the assessments included in this section could be used to inform the ESSP Math data collection using the holistic rubrics.
See Culturally Valid Assessment Section5
Rubrics, checklists, and portfolios have been helpful techniques in the culturally valid assessment. Moreover, teacher interviews with students can draw on the recurrent learning strength of the oral tradition for communicating. Interviewing is an excellent assessment technique for culturally responsive mathematics teaching, as long as students believe that their teacher wants to find out what they do know.
No matter what is being assessed, the most promising practical strategy in a teacher’s repertoire of culturally valid assessment is the portfolio.6 Using portfolios nurtures students’ responsible autonomy in the classroom, strengthens students’ collaborative relationships with their teacher, and encourages students to develop a capacity for self-assessment. Portfolios draw upon students’ cultural assets. In short, portfolios naturally emphasize important cultural resources of Indigenous students, while greatly benefiting non-Indigenous students at the same time.
This SaskMATH resource can help you move towards developing a portfolio assessment for your students. However, it can not offer a particular strategy or a particular set of assessment tools that will work for you, because these involve a host of decisions unique to each teacher or school.
1Definition of assess | Dictionary.com. (2020). Retrieved 6 July 2020, from https://www.dictionary.com/browse/assess
2Carl Boyer (1906-1976) American historian of mathematics. Quoted from an unknown 1949 calculus textbook.
3Boaler, J. (2020). Retrieved 5 July 2020, from https://www.youcubed.org/wp-content/uploads/2017/03/1439422682-AssessmentPaper.pdf
4Culture-Based School Mathematics for Reconciliation and Professional Development | McDowell Foundation. (2020). Retrieved 5 July 2020, from http://mcdowellfoundation.ca/research/culture-based-school-mathematics-for-reconciliation-and-professional-development/