Are you ready for more? That is the question the authors of Illustrative Mathematics and Open Up Resources ask students following engagement in problem solving routines and instructional activities. This week my students responded with a resounding, “BRING IT ON!” Not only that, they attacked the activities and problems with such zeal this educator was left with goose bumps and happy tears!
I teach 6th – 8th grade middle school students with disabilities and have the honor of looping with them throughout their junior high years. My current eighth grade students were my first sixth grade group. From day one I have incorporated Dr. Jo Boaler’s Growth Mindset research, teaching students the tenants as we have explored the Weeks of Inspirational Math. We have embraced the belief that with struggle, mistakes, problem solving and a growth mindset everyone can learn maths to high levels. In addition, my pedagogy has included constructivist ideals with heavy doses of productive discourse, collaboration, and joint construction of knowledge. I have consistently utilized the routines of Notice/Wonder, Estimation 180, Which One Doesn’t Belong, Visual Patterns, Empty Number Lines and Grayson Wheatley’s Quick Draw as tools to teach students the problem-solving process, and have been utilizing curriculum that is inquiry based and student centered. To this end my students have become familiar with self-talk and working through the problem-solving process in a way that makes sense to them. We have developed the following anchor chart as a guide if we get stuck.
We have also created an anchor chart to remind us what we expect from each other when working in groups and collaborating.
When I discovered Open Up Math Resources, a beautiful curriculum grounded in routines for reasoning, research-based practices, student centered, world connected, inquiry based instructional practices that resonate with so many of the constructivist philosophers I have come to passionately embrace, I became an instant zealot for the curriculum!
Following is my attempt to capture a moment in time that happened this week in one of my classes. For this educator this is evidence that the paradigm shift educators are being asked to make concerning their pedagogy is vital and life altering!
When students completed the exploration of the instructional activities in lesson 6.1 Tiling the Plane they were given the following prompt:
On graph paper, create a tiling pattern so that:
- The pattern has at least two different shapes.
- The same amount of the plane is covered by each type of shape.
My students have disabilities and they have learned that there are many tools available in our classroom to aid them in the removal of barriers to their access to the mathematics we are exploring. Many of my students have dysgraphia, dyslexia, fine and gross motor challenges as well as a plethora of other disabilities that require supports. For my students with gross and fine motor barriers, drawing with conventional paper and pencil as well as on a computer is too restrictive.
With that in mind one of my students went straight to the pattern blocks to tackle this problem. The student expressed the desire to create a tessellation that would satisfy the prompt.
The tiles chosen by the student were hexagons and trapezoids. After a little while working with the chosen tiles the student created the following pattern.
While I was circulating among the students to monitor understanding, strategies and misconceptions I found this student working diligently. They asked me what I thought of their creation. I find I always channel my mom in these situations and turn the tables by saying, “It doesn’t matter what I think of it, what do you think of it?” Who knew mom was a constructivist? The student said they really liked their design, but was not sure if it was correct. I reread the prompt and asked, so what do you think? Are your shapes covering the same amount of the plane as the question asks? The student counted the trapezoids and said they know it takes two trapezoids to make a hexagon so there were too many trapezoids. They then decided to try an easier problem to help them break down the design. To do so they pulled out a portion of the pattern to critique (solve an easier problem).
The student then said, “I notice that for every one hexagon there are two trapezoids. So, in this pattern there is a two to six ratio.” I asked, what relationships do you notice or wonder about that information?” The student said, “Well, I will need to think about common multiples and maybe factors.” They thought for a while and then said, “I know I can multiply 2×3 to get six, so I wonder if I start with six hexagons and 12 trapezoids, will I be able to create a hexagon pattern with them where the yellow and red cover the same amount of the plane?” With that I left the student to explore on their own for a while. When I returned, the student was experimenting with several patterns, and was starting to create a straight-line pattern like the following using six hexagons and twelve trapezoids.
At this point I was happy to see that the student was showing understanding of decomposing a shape into different shapes, and that the new decomposed shapes still cover the same area. My student on the other hand was not happy. They did not like the design and expressed the desire to create a hexagon, and a more elaborate pattern. They rotated, translated and wondered aloud about orientation and were quickly on to something! After a short period of time the student created the following beautiful piece of mathematical artistry that met the requirements of covering the plane!
The student was disappointed that they could not physically draw their design, but was thrilled when I suggested they take a picture of their work and upload it to our Google Classroom with their assignment. They were also not happy with the gap in the middle and said they were going to work on this more at home!
This one moment in time is what every educator lives for. It is a moment when all that is learned before, and what is being learned come together cohesively and flawlessly. This moment would not have been possible if this student did not feel safe to take risks, make mistakes, make conjectures, develop strategies, reason abstractly, and problem solve. This child that has been identified as having significant learning disabilities has a beautiful mathematical mindset and disposition as well as a growth mindset! This child is a critical thinker, a problem solver, a risk taker, and a world change maker! This child proves that everyone can learn maths to their highest level and if educators will make the shift to teaching students to think critically, problem solve, collaborate, communicate, take risks and have a growth mindset, the possibilities for learning are limitless!
So again, I ask, Are you ready for more? I know I am! I can’t wait to provide routines for reasoning and instructional activities for my students so they may become amazing mathematicians and thinkers! As a bonus, I will have the opportunity to stand as witness to their mind-blowing awesomeness this school year!