According to the IMPACT, faculty can foster student ownership through three tasks: 1) by “providing students with open-ended questions or utilizing inquiry-based learning” allowing students to discover concepts (page 32), 2) by supporting students to develop meaning through productive struggle (page 33), and 3) by providing the “opportunity for transfer of ownership, practice, and application to [the] student” (page 32). In his book Building Thinking Classrooms in Mathematics Peter Liljedahl listed 14 practices teachers can use to increase mathematical thinking by students and thus enhance learning:
- Types of tasks to use
- Forming collaborative groups
- Where students work
- Arranging the furniture
- Answering questions
- When, where, and how tasks are given
- What homework looks like
- Fostering student autonomy
- Using hints and extensions
- Consolidating a lesson
- Student note-taking
- What is evaluated
- Use of formative assessment
To illustrate how most of these practices can be used to foster student ownership, I chose an activity I wrote over 30 years ago. Both documents recommend using open-ended, rich problems. The group project Polygon Numbers, which can be found in the All Access Library here on myAMATYC, meets Liljedahl’s definition of a rich problem, a problem that “require[s] students to draw on a rich diversity of mathematical knowledge and to put this knowledge together in different ways to solve the problem.”
Let’s consider this problem. Originally, the only direction given to students is to extend the patterns for triangle, square, pentagon, etc. numbers given as a series of dots that form the corresponding polygon. As suggested by Liljedahl in the chapter “When, where, and how tasks are given”, additional information is given after students start the problem and ask for assistance. For example, the instructor does not mention that they need to extend the patterns both horizontally and vertically until a group asks about this. It is open-ended because there are so many ways to approach the problem such as a geometric approach, a numeric approach, and a graphic approach. Students can enter the problem with very little prior knowledge: recognize and extend patterns, addition of counting numbers. Students with an algebra background may choose to organize the data into tables and convert the tables to graphs or quadratic-equation models. The problem solution is enhanced by using small groups, since different students will suggest different approaches to solving the problem. Learning is increased by using small groups. (Liljedahl’s research suggests 3 students per group is best and that the groups be changed every time a new topic is introduced.) Groups control how they approach the problem. They can assess their own progress in solving the problem by noting if they are forming the desired polygons and if their conjectures are confirmed or not. Liljedahl’s research shows that groups will stay on task and persist in productive struggle, if they record their thinking on what he calls vertical, non-permanent surfaces such as whiteboards. Students will not assume ownership of their learning if the instructor solves the problem or converts the problem to a routine problem when answering student questions. Instructors must also be careful to remain neutral, not stopping the student’s thinking process by stating that students are correct (or incorrect) in their thinking. Liljedahl’s research shows that instructors should only answer what he calls “keep-thinking questions.” However, if a group has become so frustrated that they are close to “shutting down”, a hint may be appropriate. A group that has used one approach successfully should be given an extension; in this case that would be to suggest using a different approach to solve the problem.
If we want students to own their learning, we need to change how we grade. In the 1990’s the work of Black and Wiliam showed that learning ceases as soon as a grade is assigned. They suggested limiting the amount of grading as ranking of students. Liljedahl goes further, suggesting that more learning occurs if we eliminate the use of points to determine a student’s grade (on a concept, unit or course) and instead use rubrics and the associated data to indicate the extent to which a student has met a learning outcome. Tests are one of many ways a student can demonstrate their attainment of a learning outcome. With points students are penalized for requiring more time to master a concept. When using data a student’s “grade” is based on what they now know and are able to do. This requires us as instructors to be clear about what we want students to know and be able to do and to communicate this to our students. The learning outcomes for the Polygon Numbers project will vary depending on the course in which it is used. However, in every case one learning outcome will involve inductive reasoning, the ability to recognize patterns, extend them, fill in missing elements in a pattern, and creating their own patterns. Assessing this will require additional assessments. Instead of traditional homework, Liljedahl’s research shows that it is better to use Check Your Understanding questions that are not checked. However, answers are given with the questions and step-by-step solutions are given a day or so later. A Check Your Understanding question after doing the Polygon Numbers project might be: Fill in the missing numbers: 1, 3, 6, ___, 15, ___, 28.
To Check Your Understanding of this blog:
- Choose one of your favorite activities or projects
- Decide how you are going to introduce the project and what hints and extensions should be available before implementing the project in a class
- Anticipate student questions and decide what additional instructions you will give after the initial introduction of the activity
- List the student learning outcomes to be attained
- Create a rubric for each learning outcome
- Decide how you will randomly choose groups for the activity
- Decide how students will receive feedback
American Mathematical Association of Two-Year Colleges. (2018). IMPACT: Improving Mathematical Prowess And College Teaching. Memphis, TN: Author.
Black, Paul J. and Dylan Wiliam. (September, 2010). Inside the Black Box Raising Standards Through Classroom Assessment. Phi Delta Kappan.
Liljedahl, Peter. (2021). Building Thinking Classrooms in Mathematics, grades K-12: 14 teaching practices for enhancing learning. Corwin Press, Inc.