Follow-Up: Common Core on the Ground
In my previous blog, I wrote about how the Common Core-based mathematical practices have improved my teaching methods in my Algebra 2 class. I said: "For the first time in my teaching career:I feel the students really understand the connections between an algebraic equation, the graph of that function, and its complex roots. The problem-solving approach makes a difference."
A commenter asked for more detail.
Earlier this year, after students had solved quadratic equations, I tried out a Mathematics Design Collaborative lesson. I hoped "Forming Quadratics" would help my students understand what a solution actually is. At this point, many of my students could follow steps to get to the correct answers. However, conversations and assessments suggested that they had not understood why they were doing what they were doingthe meaning behind their answers.
This assessment-for-learning lesson involves three of the mathematical practices: 1) make sense of problems and persevere in solving them; 2) reason abstractly and quantitatively; and 3) construct viable arguments and critique the reasoning of others.
First, students worked individually on a pre-assessment that challenged them to match graphs to equations (written in both factored form and standard form), explain their reasoning, create an equation given the y-intercept and the minimum value, and find the roots of that equation.
Students' responses verified that they didn't really understand the connection between an equation and its graph.
Next, our class discussed the key features of parabolas and how they relate to quadratic equations. Students worked in pairs to match cards: One set of cards depicted graphs, and the other shared clues such as "has no real roots" or parts of equations.
Students' conversations showed that they were really grappling with the material and were working together to make sense of it all. They asked each other questions: "Why did you place this card here?" "Could you have done this another way?" "How did you know what algebraic expression to write here?"
Before this lesson, students could successfully "find x" given a formulabut what did they really understand? If we want students to master concepts, we must ask them to analyze their own thinking and to make conclusions based on prior knowledge.
This served as a springboard for a subsequent lesson on solving polynomial equations of higher degrees. Even on the first day of the polynomial lesson, students were able to look at a cubic or quartic function's graph and state its real solutions. By the end of the lesson, they genuinely understood how to find the complex roots of polynomial equations.
I'd never taught solving polynomials (beyond quadratics) to non-advanced Algebra 2 studentsthe content standards of Common Core are often more rigorous than our previous state standards, at least in math. I wasn't sure how it would go. But with the help of the mathematical practices and new tools, my students met the challenge.
Ali Crowley teaches Algebra 2 and AP Calculus at Lafayette High School in Lexington, KY. A National Board-certified teacher with 11 years of experience, she is a member of CTQ's Implementing Common Core Standards team.