Students' Ideas at the Center Part 1: Knowledge is Connected
I’m on a lifelong journey to put students’ ideas at the center of math class.
Children’s Mathematics (2nd Edition) helps me understand the power of students’ connections:
This helps me rethink knowledge in math class. 7+7 makes 14, great! Now what? What is connected to it? Are the connections strong? Flexible?
If I’m given an abstract equation such as 7+7=14 I can reverse-engineer a meaningful context that draws upon actions and relationships I understand, like having 7 blueberries and getting 7 more. Yum – now my five senses, imagination, and emotions are getting involved!
If I forget the total, I can access my knowledge about 7+7 through many avenues. For example, I can compose 7+7 in many different but equivalent ways.
As I learn about what numbers can represent, 7+7=14 adapts and connects to other units, like tens. How does 7+7 help me think about 14 tens (also known as 140 ones)?
I can also adapt it to fractional units like fourths. 7 fourths plus 7 fourths is 14 fourths (also known as 14/4 tomato or 3 and 2/4 tomato).
When I need to figure out 6+7, I might notice a connection that draws upon my ability to compose (and therefore decompose) numbers flexibly.
It isn’t just 6+7. If I go looking for connections I’ll realize that 7+7 helps me learn most of the “hardest” addition facts. Any generalizations I make will exercise my algebraic reasoning, proving useful throughout my math journey. In other words, connections allow me to learn this year’s standards in a way that supports every year’s standards.
What is the role of the teacher, then?
Let’s say we’re learning single digit addition and I want students to be able to “Make 10.” My 1st graders used to fall into two categories: those that could already “Make 10” before they got to me, and those that dutifully tried to mimic my steps but would eventually revert to less efficient strategies like counting on. I began to create visuals like this, thinking it would help (press play):
Believe it or not, this wasn’t any more effective than marking up an abstract equation. What was wrong?
The animated visuals I showed my students were lucid – they showed exactly what I was thinking. The problem wasn’t the visual in and of itself; visuals can show powerful connections. The problem was ownership. The visuals showed exactly what Berkeley Everett was thinking, not Aaliyah, not José, not Sara.
As I see it now, my role is to put students into a position to make their own connections. I use problems and questions to help students make sense of the math and leave the construction of knowledge up to them.
Okay, maybe you agree, but you’re wondering how students will “discover” a concept without being shown?
A Problem String (also called a Number String) is one of my favorite ways to help students collaboratively construct knowledge. This 5-15 minute number sense routine features a series of related problems that allow students to notice and discuss important patterns. Students will not “master” an idea after one Problem String, instead, they make sense of it over many weeks or months. Here’s what one Problem String might look like (press play):
“Make 10” is the content goal I have, but larger messages are at play: you can solve the same problem in many ways, you can strategically compose/decompose numbers to make problems easier, conversing with your peers helps you better understand your own thinking, kids can solve problems without waiting for adults to show them how, etc.
It is hard to refrain from showing students connections that are right in front of them. However, with the right kind of problems and structures for collaboration, the light bulbs will go off and the connections will be made. Connections they own. Think of the self-affirming power of an idea a child creates with their peers.
This is the classroom I strive for, with students’ ideas at the center.
Continue to Part 2: Liz Romero’s Math Class