From understanding is like a light bulb, OFF or ON…
To understanding is a complex process that ebbs and flows
As we learn we repeatedly move into and out of understanding. Embracing the complexity of what it means to truly understand an idea can push us to employ new teaching methods and create a different kind of classroom. Engaging in the practices requires students to explore and revisit ideas in different ways allowing new insights to emerge over time.
If a teacher views understanding like a light bulb: with each new concept, students begin with the bulb turned OFF and the teacher’s job is to find a way to switch it ON. When a student presents an answer clearly, this indicates the light bulb is now ON. The teacher can mark this concept as “completed” and move on to the next concept, at least for that student.
When some students indicate that they understand the teacher’s explanation, the teacher might assume that other students also understand—the teacher might assume he/she has found a way to switch the light bulb ON for the class. This mindset views understanding as a single event: crossing the line from “I don’t get it” to “Now I get it.”
Understanding is a complex process, not a single event. Most concepts have multiple layers and facets. Students might understand one layer but not deeper layers. They may understand a concept from one perspective but not from others. Why do students seem to understand an idea one day but then seem to forget it the next day? Why do students appear to understand an example that is delivered by the teacher but then get stuck trying to answer the next example on their own? We have found that the more we work with a concept, the more we see these kinds of things playing out in our classrooms. Rather than become discouraged, we’ve changed our view of what it means to teach for understanding.
Understanding ebbs and flows. People often need to hear, see, and experience an idea in a number of ways, a number of times, and in a number of contexts in order to integrate and understand it more deeply. This is where the practices of math and science come into play in powerful ways. By asking questions, seeking multiple solutions, investigating phenomena and developing models, students have many interactions with concepts and may learn them in multiple contexts. When a student can recognize and use concepts in varied settings, we see this as a mark of real understanding.
Allow students to use multiple methods and representations. When students are encouraged to plan and use their own strategies and representations they rely on what they know and understand (correctly or not). As a result, each student may help reveal a different piece of the puzzle or layer of understanding, which will help fill out the idea for others. At the same time each student can reveal to the teacher gaps in his/her understanding to help guide subsequent instruction.
Pose open-ended tasks and tasks with “low floor, high ceiling” attributes that allow students to enter at a variety of levels, and provide space for student thinking. Avoid giving instructions that are too specific or directive. Don’t tell students what tools to use or start them down a certain path. Take care when implementing tasks so as not to close down student thinking or innovation with overly specific directions that remove opportunities for variation. Add language like:
1. “Solve in a way that makes sense to you.”
2. “Show your work in a way that others would understand.”
3. “Draw a diagram to show your thinking."
Encourage peer discussion as it takes advantage of divergent thinking and allows students to clarify and revise their thinking. Sentence frames and questions can help students develop productive academic discussions.
Do not leave an idea at first mention and move on. It is important to engage all the students in thought about the big ideas when a single student brings them up. Direct questions to the whole group rather than just the student who shared the correct thinking. Inquire with questions or statements such as:
1. “What do you think he/she means by that?”
2. “Can someone show us what that might look like?”
3. “Discuss in your groups why that makes sense”
4. “Who can add on to/respond to that idea?”
5. “What would that look like in a … (table), etc.?”
Ask questions/pose problems that challenge traditional examples offered in many textbooks. For example: if students see visual examples in the simplest or most limited form, their understanding may become narrow or one-dimensional. Jo Boaler in Mathematical Mindsets says:
"It is important to revisit mathematical ideas, but the ‘practice’ of methods over and over again is unhelpful. When you learn a new idea in mathematics, it is helpful to reinforce that idea, and the best way to do this is by using it in different ways.When looking at examples and visuals in textbooks, consider other ways to represent the idea."
For example, textbooks often illustrate partitioning thirds into the same size and shape fractional pieces. Instead, provide students with a figure in which same size partitions are not possible and ask them what thirds would look like. This thinking holds for science as well – applying ideas in different contexts deepens understanding. Have students use an idea to explain a novel phenomenon.