# Three Vertices

### CHECKING FOR UNDERSTANDING WITH PROCESS AND PRODUCT…

**Why Three Vertices? **

The main purpose behind the Three Vertices strategy is for students to feel comfortable being wrong, stuck, or just unsure as to where to start. Three Vertices focuses on the strategy, approach, and process, rather than strictly on the answer itself and is an opportunity for the teacher to manage student learning rather than merely focus on teaching. Allwright (2005) stresses that the unit of analysis on teaching and learning is too focused solely on the teaching and needs to be redirected to the learning, specifically to learning opportunities provided to students. This activity can serve both as a focus on student learning and as a formative assessment to guide teaching.

**The Set Up**

There are three vertices or sections of the classroom. Each vertex represents a different result:

(1) For students who are confident about their answer and process: This is the answer I got and how I got it.

(2) For students who are confident about their answer, but not process: This is the answer I got and I don’t know what I did

(3) For students who are not confident in answer or process: I did something but didn’t get an answer.

Depending on your classroom setup, “vertices” might include three walls; three specific desks or tables; or front of the room, middle of room, and back of the room.

**Getting Started**

Provide your students with a task, question, activity, prompt, or some other engaging item for them to produce a response. Allow some time for the students to work - about the time you would reasonably expect for students to complete the task you have given them. As they are working, be sure to circulate the room to get a feel for which vertex you believe each student should choose. This also gives you the opportunity to change vertices if needed before you turn students loose. If you notice, for instance, that you have many students clustered in vertices 1 and 2, but none in 3, you might split vertex 1 into those who could teach it to another student and those who aren’t yet at that point.

**The Chosen Vertex**

Once you feel the students have had enough time to think through the question, let them know you are going to engage them in the Three Vertices strategy. Give your students about 15 seconds to decide on which of the above options makes sense to them. Then give them an additional 30 seconds to safely move to that section of the room.

**The Discussion**

Depending on the number of students at each vertex, create subgroups of 3-4 students to discuss. You can do this randomly or based on your notes during circulation. Within their subgroups, students discuss their approaches, definitions, theorems, struggles, etc. with those in the group. Set the expectation for this discussion. Students should be sharing their thoughts on the process, approach, or rationale for their work, rather than the answer itself. Also, as with any discussion, it is useful to set the expectation that every student talks about what they bring to the discussion and every student listens without interrupting.

Regardless of the vertex chosen or the position each student is in, there is value in all approaches, struggles, and uncertainties. Understanding the thought process of students as they work through their approaches, rather than simply labeling answers right or wrong, is key to ** expert learning**. Ertmer and Newby (1996) define expert learning as a process that involves accessing one’s knowledge, selecting an approach that matches ones resources and the desired task, developing a plan, and monitoring one’s progress through this plan. In this activity, students are asked to access what they know and select an approach that matches what they know and can do. Students use discussion with others to help them develop a plan and make progress toward the desired outcome.

Give students about 3-4 minutes to discuss in their subgroups. Then have students form new subgroups of 3-4 students made up of students from different vertices. While the numbers will rarely work out to be able to have one student in each subgroup from each vertex, try to make these groups as heterogeneous as possible. Students at all vertices will get more out of the process by discussing with those who weren’t in their original vertex.

After some time to share and discuss, guide students to approach the problem again using a different strategy, keeping in mind a theorem that was mentioned, or other such suggestions based on their discussions. Ask students to offer “helper credit” for things they have learned or helped another student to learn as part of the original problem.

**Conclusion**

Regardless of the level of the learner, Three Vertices helps students on their way to becoming expert learners. The Three Vertices strategy is not about the answer necessarily, but about the why, how, or what that led to an answer and provides a good opportunity for both learning and formative assessment after a concept has been introduced, but not yet mastered. It allows students to struggle, to question themselves, and to not complete the problem or come up with a response. In fact, three vertices praises the student who perseveres and allows themselves to open up to the possibility of learning from their misunderstandings, misconceptions, or mistakes. Ultimately, this leads to deeper learning that students are more likely to retain and be able to leverage in the future.