I haven’t taken much time to keep up this website. I write today to reflect and share a new pbl that is being experimented with in a couple classrooms at my school. Recently, a coworker and I planned and started a math pbl. We were searching for more knowledge and experience with pbl in math in order to coach other teachers at school. We were very hesitant to preach math pbl when we hadn’t done one ourselves. So, we set out to learn!
This pbl began with brainstorming math solutions in a leadership meeting. What are the biggest needs in math? What areas need to be concentrated on? We kept coming back to problem solving. We found that many times our students wanted to show their answer, but not their work. Rush, rush, rush! We thought thoroughly upon how we were teaching students to “show their work and thinking,” but also what does that even mean. What does it mean to “show your work” in math? How do we show our thinking? Why do we keep asking them to show their work if we haven’t properly explained and explored those terms? AHA! That’s our driving question, How can we, as mathematicians, show our math thinking effectively? We soon listed what we as teachers needed to know and what we already knew.
What does it look like? What does it sound like? What tools and parts of speech can we use? Showing your thinking involves showing the steps and the plan you made to solve a problem. Metacognition thinking stems need to be used. It means slowing down… thinking critically, sharing your ideas, and gaining feedback.
We use Everyday Math at our school. We needed to see how this could be thread through a curriculum that spirals through different concepts each lesson. We had to ask ourselves: Can this fit into each concept? Can it show up everyday as we explore something new or old? YES! Whether you’re extending a pattern, creating a shape, using an algorithm… we’re always thinking something. The question is how can we verbalize and show that thinking?
Our entry event theme: Math is Art! Have you ever read “Ish” by Peter Reynolds? This book is about art, but is greatly founded in the idea of perfection, creativity, and thinking “ishly.” How many of us come to math with the anxiety of perfection or that just right answer? Did you know that high anxiety causes a lower function of the math sense in the brain? We read “Ish” to students and emphasized the connections they might make between math, art, and the young character in the book. How many of you think you’re a mathematician? Do you think your math could be framed? What might math look like in a frame? What frustrates you in math and what motivates you? What does it mean to think “ishly”? What does it mean to make something “look right”? Students began thinking about our driving question. How can we, as mathematicians, show our math-ish thinking effectively? In the beginning, students thought very broadly for their Knows and Need to Knows. “It looks like our ideas. It looks like math.” Few questions were raised. We planned ahead for such thinking…
Then, we unleashed the exploration! Students needed a better idea of what this could look like. We had an “art museum” set up for them. Black table cloths were on student tables. Frames and “paintings” were created with math thinking. Some frames had math thinking stems. Some frames showed several solutions or algorithms to a problem. Some frames showcased problem solving steps of circling, boxing, and highlighting among word problems. A student favorite said, “Math is art. Live in the numbers.” They walked around this museum in awe. They noticed the key words and deep quotes. They were thoughtful about the numbers and words being shown. Students were able to come back to our Knows and Needs to Know with a fresh perspective. These frames strategically had math thinking that we as teachers always dreamed to see.
We then dove into the lesson in our curriculum… solve and add 3 to 4 two digit numbers. We came at these problems abstractly. We created problems for students that allowed them to think for themselves and in the order they wanted. Many times, as math teachers, we set some numbers down in a set column or row. This strategy often creates an environment where students think they have to use the order we wrote it in. Instead we placed 3 numbers in different areas of a circle with the instructions: Add these numbers, however you like, but show your math-ish thinking. Students really took off! They were drawing arrows and lines and showing algorithms. Some used words. Some added all numbers at once and others added two numbers first and then the third number. They practiced on a few problems. Some got to stand in front of the class and share.
Finally, we sent them off with their very own artistic circle of numbers (as an assessment). Add these numbers, however you choose, just show your math-ish thinking. Do it all by yourself. Students posted these circles on our own mural of thinking. Upon reflection, all of the students showed their thinking. Some students got the answer wrong, but we were able to see why they came to their answer because they had shown their thinking. Why do we show our thinking? We gain a better understanding of the steps we take and we gain feedback on how to improve. If students had just shown a right or wrong answer with no thinking, how would we know what to do to help them?
The students have been so affected by the fact that math is art. They write this phrase sometimes around their new found math-ish thinking. They shout out in lessons, “Can I show you my math-ish thinking?” As readers, I challenge you to think about two questions. How can you model your math-ish thinking for students? How can we, as math teachers, think more abstractly in order to allow for more creativity in math? To be continued…