ScreenChomp, Explain Everything, Showbie, and Educreations are examples of some of the Apps my colleagues are using in math class to have students explain their math thinking. When a student records his or her explanation of how he or she solved a problem using one of these Apps (or on paper), that student reveals all sorts of evidence of their learning. The beauty of these Apps is that they include the student’s voice overlapping with what they have written and/ or drawn. From our experience, it seems the students are more likely to go over and revise their recordings and written work, which was more like “pulling teeth” when just paper, was involved. For many of them, each time they re-record, they clarify their thinking and hone their ability to articulate mathematical ideas.
Can we scaffold this process to ensure the students can become even more facile in communicating their math thinking? I think the answer to that is yes. We can give the students check lists and rubrics to guide their work. Not only can we provide rubrics, so they know the components to focus on, we can set up partnerships wherein they can peer conference with one another and collaboratively strengthen their understanding and ability to communicate that understanding.
There are many sources for rubrics online and we can create our own. The best thing about choosing and developing these assessment tools is that it forces us to think deeply about our teaching. The standards-based Exemplars rubric prompts us to look at the students’ work in five ways: Problem-Solving and/or basic Understanding, Reasoning and Proof, Communication, Representation, and Connections. To help students assess their understanding of a problem, as teachers we can ask them (or their peers can ask them): What do you know and what are you trying to find out? What strategies are you going to try and why? What tools do you need? Could there be more than one solution? Could you solve this problem two different ways? To help students explain their reasoning and proof, we can ask them (or their peers can ask them): Does that make sense? Why is that true? Is that true for all cases? Could you prove it? Can you think of a counter example? To help students communicate clearly, we often remind them to use their mathematical vocabulary and to describe their thinking with words, numbers (equations), and pictures, diagrams, models or graphs. We often have to prompt them to create a diagram, make a table, use a number line, put things in order, or act a problem out. In order to encourage students to make connections, we need to give them time to be reflective. We can ask (and they can ask one another): Does this remind you of other problems we have solved? How does it relate to ………..? Why does your answer seem reasonable? Do you see a pattern? Can you explain it?
In making check lists and rubrics to scaffold our students’ efforts to explain their mathematical thinking, we ourselves can think about how much time we spend developing mathematical thinking with effective questioning in our classrooms. When we are asking good questions, we are modeling thinking strategies. As we are making our thinking visible, we are highlighting the ways our students think and explain their thinking. We also can use models, diagrams, drawings, graphs, and equations to show how we are thinking and how we imagine our students are thinking. We need to listen carefully to honor our students’ thinking authentically. Elementary school age students need daily exposure to representations of mathematical ideas before they can create these independently.
One of the best results of the iPad program in 4th grade at our school has been its inherent student-centeredness. The students now have the ability to communicate their mathematical thinking both orally and by drawing and writing as well as the ability to easily revise and refine their work. They feel empowered when they use these Apps to explain their mathematical thinking.
Let it remind us again as teachers to continually reflect on our teaching when we look and listen to that student’s work for evidence of learning. Teaching ≠ learning: in other words, just because we “taught it” does not mean the students “learned” it. Assessments can reflect many nuances in student understanding and inspire many forms and variations of differentiated, intentional instruction.