As the world quickly moves from the industrial age into the age of STEM (science, technology, engineering and mathematics), it is becoming imperative that today’s schools prepare students to become critical thinkers who can reason, problem solve, and collaborate. Weckbacher and Okamoto assert that spatial skills, the foundation of geometrical thinking, is paramount to success in STEM careers (Weckbacher & Okamoto, 2015). According to Smith and Stein (Smith & Stein, 2011), children in America are not required to think critically, problem solve or wrestle with concepts or ideas. Instead, they are taught rote procedures that require memorization of ideas and facts that do not prepare them for college and careers where problem solving and critical thinking will be expected. The National Council of Teachers of Mathematics (NCTM) (National Council of Teachers of Mathematics, 2014) puts forth that effective teaching practices must include:
- The establishment of mathematics goals to focus learning
- Implementing tasks that promote reasoning and problem solving
- Using and connecting mathematical representations
- Facilitating meaningful mathematical discourse
- Posing purposeful questions
- Building procedural fluency from conceptual understanding
- Supporting productive struggle in learning mathematics
- Eliciting and using evidence of student thinking.
This review will examine instructional routines as a teaching strategy as well as the Quick Draw and Block Building instructional routines/tasks that aid in the development of critical thinking, spatial reasoning and geometric thinking and utilize the effective teaching practices called for by NCTM.
Mathematical Instructional routines are learning activities designed with a predictable flow and structure of the learning experience. As outlined in Routines for Reasoning (Kelemanik, Luenta, & Creighton, 2016), the predictability of these routines supports students by answering these questions, “What is it that I am supposed to be doing? What question will I be asked next? or, How will things work today in the lesson?” When students internalize the components of the routine they are free to attend to problem solving, critical thinking and the strategies of their classmates. Likewise, educators are able to attend more effectively to the strategizing and sense making by students as the structure of the lesson is predictable. Furthermore, the authors assert practicing predictable instructional routines allow for the strategies, questions, methods, and skills used to become intrinsic. Thus, creating a mathematical disposition that students use in other problem-solving situations. Effective instructional routines should be focused on the development of critical thinking and should support all learners by including the following components:
- Articulation of math practice goal- math action process identified. This is the why.
- Think time – provides adequate processing opportunities, especially for those with disabilities and those who are English Language Learners, promotes independence, builds on strengths. Think time should be provided throughout the routine.
- Partner work – aids in developing collaboration skills, critiquing the thinking of others, constructing viable arguments and allows for more processing time
- Whole group discussion – is not just explaining the process or strategy, but what students noticed or wondered during strategizing and how those observations helped in the strategizing. Also aids in language development by exposure to vocabulary, restating and comparing and contrasting strategies.
- Math practice reflection – utilization of sentence starters to support language and focus the reflection.
- Access through multiple modalities and representations – integrating universal design for learning by utilizing concrete and abstract, graphic organizers, gestures, tables, graphs, and drawing.
- Liberal use of math practice-focused prompts – emphasis on thinking and not answer getting.
When utilized correctly, Kelemanik, Luenta, and Creighton, indicate the use of quality instructional routines remove barriers for learning mathematics by providing “authentic contexts, multimodal techniques, rich opportunities for language use embedded in mathematical learning experiences and, an instruction that scaffolds students’ development of increasingly abstract thinking,” When instructional routines are facilitated as outlined, students will participate in cognitively demanding tasks that will develop conceptual understanding and math pracitices.
Quick Draw is a geometry instructional routine developed by Grayon Wheately. In this routine the teacher projects a geometric line drawing for three seconds and then instructs students to draw what they saw. Student drawings are completed on lineless plain paper. Once students have been given time to draw, the image is revealed for a second quick look, and students are again instructed to draw what you saw. When students are comfortable with their drawings the teacher facilitates a whole class discussion about what students saw and how they drew what they saw. According to Richardson and Stein (Richardson & Stein, 2008) the use of the Quick Draw routine promotes the development of spatial sense and communication skills in students. The predictable questions students can expect in this routine include, “How did you see it? and How did you draw it?” The conversations generated by these questions aid in the development of a common language that becomes an intrinsic part of students. Expanding on the benefits of this routine, Wheatley and Reynolds (Wheatley & Reynolds, 1999) espouse Quick Draw is a low floor, high ceiling routine that promotes growth mindset, flexibility, and each of the mathematical action processes. In addition, they support Weckbacher and Okamoto and believe that the industrial age ideals of rote memorization and mass production no longer serve the best interest of students in our modern STEM driven world. The use of Quick Draw promotes the development of spatial sense, mathematical language and vocabulary, offers think time/processing time, encourages attention to precision, and inspires the joint construction of knowledge. Therefore, Quick Draw meets the five guiding principles set forth by Kelemanik, Luenta, and Creighton by being a high cognitive demand task that builds on students’ strengths, uses multimodal access in an environment that is rich in language and promotes growth mindset. Utilizing Quick Draw then promotes the critically needed spatial sense needed in our modern STEM world.
Problem Solving with Cubes is another potential geometry instructional routine. This activity is outlined by Browning and Channell (Browning & Channell, 1992) as follows: A picture of a 2-D cube model is given. In addition, a pictoral representation of the front, right, top, and base view is shown on dot grid paper. Students are provided with fifteen one-inch cubes and a piece of paper. Students are instructed to work in their groups to construct the cube model. This activity was presented as a lesson/activity and progresses from the perceptual level through the representational level of geometry by scaffolding the images and eventually removing the blocks and encouraging students to visualize rather than build. Throughout the activity, students are encouraged to communicate, explain, and reflect. The authors also suggest that this activity be followed by abstract activities available on the internet. Browning and Channell indicate this activity supports the development of reasoning skills, the ability to visually manipulate images for understanding, and forming generalizations. These are vital skills when students are exposed to images and visualizations in subjects such as mathematics, science and history as well as in their perceptions in the world. The Problem Solving With Cubes activity has the potential to be a great geometry routine with adaptations such as creating a predictable routine and questions to support students and teachers and by delivering the images one at a time over several class periods rather than a multi-page worksheet.
Instructional routines are a powerful tool for promoting the mathematical action processes. They support students as well as teachers with their predictable nature and provide a framework for students to take into other mathematics explorations. In my class the math questions that have become intrinsic are, “What do I notice and wonder? Have I done a problem like this before? Is there an easier problem I can solve that will help? Will working backwards help? What tools, tables, graphs, pictures or models will help me? and What patterns and relationships do I see, and what do they tell me?” The use of instructional routines is a non-threatening and often fun way for students to practice problem solving and critical thinking strategies and practices and they lead to the development of growth mindset. I was disappointed that I could not find more peer reviewed instructional routines that are specific to geometry. I wonder I there is more research and exploration needed in this area. I was able to locate several resources that are not peer reviewed, but meet all of the requirements as established by NCTM and Kelemanik, Luenta, and Creighton. These resources and others may be accessed here:
Browning, C. A., & Channell, D. E. (1992). Problem Solving with Cubes. The Mathematics Teacher, 447-450: 458-460.
Kelemanik, G., Luenta, A., & Creighton, S. J. (2016). Routines for Reasoning; fostering the mathematical practices in all students. Portsmouth: Heinemann.
National Council of Teachers of Mathematics. (2014). Principles to action: ensuring mathematical success for all. Reston: National Council of Teachers of Mathematics.
Richardson, K., & Stein, C. (2008). Developing spatial sense and communication skills. Mathematics teaching in the middle school, 101-107.
Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston: The National Council of Mathematics.
Weckbacher, L. M., & Okamoto, Y. (2015). Discovering space in the elementary school. Journal of education and learning, 35-39.
Wheatley, G., & Reynolds, A. M. (1999). Image maker: developing spatial sense. Teaching Children Mathematics, 374-378.