A Few Problems, Many Solutions

At this point in the year, the iPads are pretty integrated into our class's curriculum and daily routines.  We have prioritized exploring what we call "creativity apps" as opposed to games and apps with a singular focus.  Meanwhile in math, our class has begun utilizing Exemplars problems and rubrics to further student problem solving, math thinking, and math communication (see Merry Melvin's previous post for more on assessment).  Each problem is rich with multiple questions, levels, and solutions, and our students are finding them appropriately challenging.  We gave them free range to decide how to present their work.

Here are some solutions from the one about predicting the number of Sweethearts in a box:

And the one about planning the sleeping arrangements for class camping trip:

And our most recent problem about carpeting a basement:

Each solution requires a written explanation and a visual representation.  Some students decide to use the iPad to prepare both.  Some students use different apps depending on what they are being asked and some have a favorite they choose consistently.  Some students create their graphs and diagrams using physical materials such as Base Ten blocks or  tiles or puff balls and then document or enhance them by importing a photo.  Some students are comfortable emailing us their work and some students prefer working offline and turning in physical products.  We remain open to all of their preferences and meet them in their comfort zone.

While it was no surprise that many students elected to use their iPads to prepare their final solutions we were fascinated by the range of apps children chose.  Through regular explorations and many months of projects, Drawing Box, AiWriter, Educreations, Explain Everything, Keynote, and Inspiration Maps have become familiar friends. Left to their own devices, students have to decide whether the iPad and our creativity apps are the "right tool for the job" and which tool will serve them best.

Of course we do have some students who insist that the iPad is the best tool for them and it's really not .  But after a few experiences of being distracted and not completing their work on time, they learn that lesson too and change tactics.  At the beginning of the year, we set a few goals for our 4th grade 1-1 iPad program and progress is visible on many fronts.  To me, these are examples of students taking charge of their learning and our classroom becoming a place that fosters and supports creativity, independence, differentiation, and meaningful assessment.

Assessing to Strengthen Learning and to Support Teaching

While we have always had the ability to look at student work for evidence of learning, various Apps are giving us all the more reason to be learning to assess in an increasing number of formative ways. Not only can we use assessment to steer our instructional decisions, targeting instruction and/or differentiating subsequent math learning times based on evidence of student learning; we can set up routines so that the students can self-assess and support one another, simultaneously bolstering their metacognitive skills and math thinking, as well.

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 4^th 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.