From Abacus to Algebra: Growing Young Mathematicians
This post is from Michelle Navarre, Head of School, at Polaris Charter Academy in Chicago, and Linda McEvoy Grein, Former Math Instructional Guide at Polaris Charter Academy and Executive Director, EduQuate LLC.
Kahari sits on the corner of the alphabet rug in his 2nd grade classroom. He watches his teacher model how to add 26 and 17.
"First you stack the numbers like this," she says, drawing the 26 on top of the 17 on her chart paper. Then she draws the plus sign next to the stacked numbers. "Then we add the numbers on the right. Jay, why do we start there?"
Kahari watches Jay look up at the anchor chart. "Cuz the right is the ones," he says.
"That's right, because it is the ones place!" Ms. M says enthusiastically.
When will she let me give the answer? Kahari wonders. It's 43. I knew it the second she put it up. He stops listening and starts to think about what he's going to do when he gets home.
Yazmine sits across from Kahari on the alphabet rug. She too watches Ms. M walk through the problem on the board. She tries hard to memorize the steps. First stack. Then add the right. Then add the left. Then circle your answer.
"Now it's your turn to try." says Ms. M. "You will all complete 10 problems on your own. Don't forget to start with the ones place and regroup when you need to!" Ms M exclaims as she passes out papers, the signal to go off and solve on your own.
Kahari is done in 5 minutes and wiggles looking for something else to do. Yazmine confidently solves 27 + 16 as 313. Later that evening as Ms. M is grading her students' independent work she sees a wide range of answers and wonders, "Where did I go wrong?"
The story of Ms. M is not unique. In classrooms all over the country teachers are teaching math just as they learned it growing up:
- Model the steps of solving a problem
- Support kids to memorize the rules
- Ask them to spit the rules back
- Then give them problems to work independently
Photo Credit: Linda Grein
Back in 2013, our school was just another school teaching another generation of Americans how to do math by following the rules rather than by understanding how math works. Unfortunately, this method doesn't teach students to understand concepts of math deeply enough to solve real world problems--or even the problems on high quality math assessments. Just read Elizabeth Green's piece in The New York Times, Why do Americans Stink at Math?. It wasn't long before we realized we needed to change how we taught math.
Polaris Charter Academy is a K-8 school in the EL Education network, serving 450 students in the West Humboldt Park neighborhood of Chicago. In our early years we relied on the curriculum map, terminology, and resources of a well respected, mass published elementary math curriculum. Initially, we saw solid gains among our new students, but within four years our scores stagnated and held there.
To regain our momentum, we decided to pilot new mathematical teaching strategies based on Cognitively Guided Instruction or CGI (based on the book Children's Mathematics: Cognitively Guided Instruction by Thomas Carpenter). Teachers learned that young children have natural problem solving strategies, which they build on over time to develop counting skills and deep conceptual number sense. In the elementary grades, students continue to move along this trajectory so that by middle school they are ready for complex math tasks involving rational numbers and algebra.
The challenge for primary and elementary school teachers, then, is to identify where students are on the math trajectory and to nudge them along with pre-planned strategic questions and tasks that take them to the next point in their math growth. Our primary teachers learned to use JARS pre-assessments developed by Stephanie Z. Smith & Marvin Smith to know where students are in their number sense development. This weekly number sense routine tracks students' development from pre-counting through flexible skip counting to factor fluency. Once students have developed strong number sense, teachers use the algebraic anticipatory framework (developed from Children's Mathematics) to identify students' chosen strategy for problem solving.
When teachers know where students are on the trajectory, they can design the daily grapple, the entre of the math lesson. It begins with a strategic, problem-based question that engages students in solving an interesting problem. Then teachers listen. Students' discussion of how they are working the problem both reveals their current understanding and builds new insights. The dialogue between young mathematicians as they try to prove mathematical concepts is the sound of young mathematicians growing! Take a peek into Mona Iehl's third-grade classroom to hear it for yourself.
Readers might wonder how teachers ensure that all students in a classroom are able to engage in the math dialogue and solve challenging problems aligned to grade level standards. What about the student who is still on the lower levels of the math trajectory? To support all students, we've created a "balanced math" diet that supplements the main course with missing math vitamins. This includes:
- Math Workshop: Unit-based work pulled from Engage New York that defines key vocabulary and engages students in application games, geometry, measurement and data, etc.
- Fluency: Daily exercises to develop fact fluency.
- Priority Standard Reteach: A weekly strategic intervention/reteach time focused on supporting students' mastery of the highest prioritized standards (as identified by John Van de Walle et al. in Teaching Student-Centered Mathematics, 2006)
Implementing this balanced math approach has required a culture shift in math education at Polaris as teachers themselves built their background knowledge in math and changed their practice. We've invested time and resources in professional development where teachers grappled with the Common Core standards on, above, and below their grade level. They color coded the trajectory of student learning in each strand of math and developed a deep understanding of what the standards are really asking students to know and do. They learned to identify where students are on the learning trajectory and to anticipate students' problem-solving strategies. They practiced facilitating student conversation about math and engaging all students in showing their thinking to their peers. Margaret Schwan and Mary Kay Stein's 5 Practices for Orchestrating a Productive Math Discourse has been an invaluable resource, along with the National Council of Teachers of Mathematics', Principles to Actions: Ensuring Mathematical Success For All Students.
Now, three years after shifting from traditional math to CGI, teachers are seeing first hand what they had previously thought was impossible. Students speak loudly and proudly of their math understanding. Their voices, not teachers' voices, are the most prevalent in the classroom. In the 2014-2015 school year students scored on average at the 20th percentile of math proficiency. In the 2015-2016 school year, the average stands at the 40th percentile. This means Polaris mathematicians are making far more than a year's growth in a year's time. We're hopeful that our end-of-grade math achievement will reflect the growth we see. Meanwhile, teachers and students at Polaris are newly energized about grappling with math. They feel more confident in their own math knowledge and skills. And, what's most exciting is, they see a clear pathway from abacus to algebra.