A couple of years ago, I began hearing about a “must-read” book in math education circles. The book, Building Thinking Classrooms in Mathematics (BTC), by Dr. Peter Liljedahl outlines 14 practices to move school math into a thinking practice. Since that original read, I have done professional development with teachers, taught classes with students using the ideas introduced in the book, and even interviewed Peter.
As someone who has spent the last several years researching math anxiety and looking at how we teach math, the idea of tasks that allow students to use all their problem-solving skills, not just computation, resonated with me. When I introduced the easiest and most quickly adopted the idea of having students stand in groups and work on whiteboards or other VNPS (vertical non-permanent surfaces in BTC talk) teachers often immediately started to incorporate this practice into their classrooms. However, some of the elementary teachers I support began wondering how they would use their manipulatives and VNPS. Did they need to invest in magnetic manipulatives? Did manipulatives take away the “thinking”? This led to a number of really critical conversations.
Sometimes in education, we run after the new shiny thing. As educators, we are always looking for ways to help our students become more engaged in the process of learning. We want them to own their thinking and educational development. However, there are many practices, like using manipulatives that are proven to help students develop conceptual understanding. We don’t stop using these practices in a BTC-style classroom. We also don’t need to have manipulatives on the whiteboards.
Karen Karp (2022) writes “using multiple representations creates a mental residue” for students. Representations (concrete-pictorial-abstract) are not islands of learning for our students. They should see these representations concurrently in order to build connections. Manipulatives can be used in groups on the floor, a table, or cart and then students can transfer their thinking either pictorially or in a numerical form onto VNPS. This helps students make connections between representations.
I have created some manipulative-based tasks that I have been using in classrooms. The first are two examples of single “thinking tasks” with push cards that allow the teacher to give students an extension without changing the task. When students are given the task/shape card and a collection of pattern blocks (purposefully a mix of shapes so they have to use a combination of shapes versus all one type) I ask them to answer the question “How Much?” I don’t give them more guidance than that. Often students will try to ask additional questions and I respond by saying “oh that is such an interesting question” and walk away. As students “finish” their ideas I use push cards to change the value of a piece. This re-engages them in the task that has adjusted but not changed completely.
I originally designed this task as a potential fraction task (grades 3-5 depending on your push card). However, younger students can also explore this task with whole numbers. Several middle and high school teachers have also mentioned that the task can be used to teach equations and substitution. If you try these in your classroom, I would love to hear how it went!
Challenge: I used the same number and type of shapes to make both shape cards. What shapes and how many did I use?
The second area I have been exploring is using manipulatives for thin-sliced tasks. So look for a part two shortly!
