Wonder #4: What do We Learn About in a Mathematics Community?

With the voices of community members in mind, a big question I have had is what the learning should look like in a mathematics community. Professional learning outcomes should include mathematical content knowledge, reflection on teaching practices, and understanding assessment for learning. Shulman (1986) identified three different categories of content knowledge necessary for success in teaching mathematics.  These include subject, pedagogical, and curricular knowledge. Increased knowledge of teaching, reflection, and creating a robust assessment plan are all part of the work of a mathematics community.

Content Knowledge for Teaching Mathematics

Shulman (1986) emphasized that teacher understanding must go beyond ‘the math’ or ‘the teaching pedagogy’ or even ‘the curriculum’ individually and teachers must know all of these things in a balanced, thorough way.

Subject-Specific Content

Subject knowledge must go beyond facts or concepts of a domain to create an understanding of the structures of a subject, including

  •          ways that basic ideas are organized;
  •          how we determine whether something is right or wrong; and
  •          knowing specific academic mathematics vocabulary.

Teachers must connect specific concepts to understanding the larger subject. This can be approached through identifying instructional sequences and seeing connections between mathematical ideas.

Pedagogical Content

Pedagogical content is the knowledge of a subject for teaching where teachers understand

  •          what makes topic learning easy or difficult;
  •          ways to differentiate learning experiences by process and product;
  •          use of rich tasks and varying instructional strategies; and
  •          specific preconceptions and misconceptions that may occur.

Pedagogical content knowledge includes the most useful forms of representing, analogies, examples, and demonstrations to teach a specific concept. Error analysis allows teachers to understand concepts deeply and understand how their students construct understanding.

Curricular Knowledge

Curricular knowledge is teachers understanding

  •          teaching tools such as manipulatives or technology that may present or exemplify particular content;
  •          alternative resource materials that might be useful;
  •          what student learning has already occurred in earlier grades; and
  •          where student learning leads to in other grades.

Foundational documents to develop teacher curricular knowledge are Curricular Through Lines. These are a summary of the continuum of each concept over time within curriculum. These documents allow teachers to identify pre-skills to develop assessments and instructional interventions, as well as know how concepts develop in later grades to ensure appropriate emphasis during instruction. Curricular Through Lines are useful tools for teachers to be able to see the concepts that were taught prior to their grade, as well as where concepts develop in future grades. The example documents are unique to the Saskatchewan Curriculum, but their structure can inform professional developers in order to create documents based on your own curriculum.

Grades K – 3 Through Lines Grades 9 – 10 Including Modified
Grades 1 – 4 Through Lines Grades 9 – 12 Workplace and Apprenticeship
Grades 3 – 6 Through Lines Grades 9 – 12 Foundations
Grades 5 – 9 Through Lines Grades 9 -12 PreCalculus
Grades 7 – 10 Through Lines

Another purpose for Curricular Through Lines is to show the differences between mathematics in different pathways. For instance, communicating with teachers, students, and parents regarding the differences between Saskatchewan’s Modified, WorkPlace & Apprenticeship, and Foundations & PreCalculus courses at the grade 10 level.

Teaching as Reflection-In-Action

Schon (1987) believes that educational reform is possible through professional reflection which leads to action. Reflection-in-action is a teacher’s ability to respond to surprise. The art and talent of teaching is reflection-in-action, where teachers have a surprise and respond to it.

Reflection in Action

Reflection-in-action causes teachers to think about how they understand and react to new phenomena. Teachers must be given the opportunity to experience the art of teaching by reflecting on their responses to unpredictable events and how they responded. This is possible by setting up conjectures and experimenting with scenarios.  The goal of this cycle is to improve a learning situation.  Learning and teaching is unique to an individual and we need to recognize that there must be less emphasis on the technical application of knowledge.  Time to reflect and discuss is essential, as this change is not easy. Enhancing the ability for teachers to make professional decisions in the face of unique student circumstance improves student learning.

Assessment for Learning Practices

Wiliam (2011) states that professional development should focus on formative assessment, as a regular assessment-teacher-action cycle produces substantial increases in student learning. Teacher learning should include

  •          understanding base knowledge of assessment practices.
  •          planning for implementation of strategies to respond to the assessment; and
  •          discussing instructional changes made and results on student learning.

Mathematics concepts build by concept over time. While the focus for many provinces, school divisions, and schools is to increase student achievement in mathematics, that requires increased opportunities for students to be able to engage in grade level mathematics. This can only occur through opportunities to fill gaps in skills and understanding, which begins with identification of those gaps.

Gap Filling

Diagnostic Assessments

The first step in providing the opportunity for students to engage in grade-level mathematics is to identify which essential skills students are proficient at and which skills are barriers to engagement. A grade-level Pre-Assessment built on Essential Learning Outcomes is a tool that can help inform students, teachers, and parents.  A Pre-Assessment can be administered in its entirety at the beginning of the school year, or broken apart into concepts needed as pre-skills for each unit of study in the new year.

The structure of a continuum of Pre-Assessment Diagnostics is

Diagnostic Design

The questions in a Grade 3 Pre-Assessment are identical to those questions in the Grade 3 Post-Assessment. In addition to those core questions, concepts from Grade 3 are added. A suggestion is that the Post-Assessment would be administered in early May to allow for reteaching and redirection in order to best prepare students for the next grade level.

Not all concepts are included in these diagnostic assessments. Only those concepts that are skill based are included. For instance, the concept of Area is not included, as a student can understand the concept of area as an application of multiplication. Multiplication appears in the PreAssessment, but knowing the area of a rectangle does not.

These assessments are meant to be formative only. They are not meant to be a part of a reporting document, as they do not fully test conceptual understanding in the depth that curriculum requires. These are only a tool to know which preskills students are struggling with, and which preskills students are proficient with.

The DRAFT diagnostics below were created by a working group from our Mathematics Community, including: Dulcie Puobi, Victoria MacMillan, Jennifer Brokofsky, Michelle Naidu, Lisa Bryden, Sharon Harvey, Terry Johanson.

Grade 3 Pre-Assessment Grade 3 Post-Assessment
Grade 4 Pre-Assessment Grade 4 Post-Assessment
Grade 5 Pre-Assessment Grade 5 Post-Assessment
Grade 6 Pre-Assessment Grade 6 Post-Assessment
Grade 7 Pre-Assessment Grade 7 Post-Assessment
Grade 8 Pre-Assessment Grade 8 Post-Assessment
Grade 9 Pre-Assessment Grade 9 Post-Assessment
Grade 10 Pre-Assessment

To see Post-K to Grade 3 Performance-Based assessments, visit the Saskatoon Public Schools Curriculum and Instruction WordPress.


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