By: Jeff Heyck-Williams
At Two Rivers we have made a conscious effort to change the way that everyone thinks about, talks about, and engages in math. We have shifted the expectations to place the emphasis away from just memorizing math facts to also understanding the math. We have shifted from pushing students to memorize lots of algorithms so that they can solve specific kinds of problems on tests to applying the math concepts they understand to solve any complex math problem. We have shifted from a school where it was OK to say, “I’m not really a math person” to a place where everyone celebrates working hard to solve a math problem.
The foundations of this shift can be described in six shared expectations for math: 1. Math is fun. 2. All students can learn math. 3. We have a common approach to problem solving. 4. We value multiple representations. 5. We value concise and precise communication. 6. We value reasoning and proof. These six expectations, shared publicly with staff, students, and families, codify our beliefs about mathematics and have made the shift in the culture of numeracy at our school real.
1. Math is fun: Time is built into lessons to play with math.
I don’t recommend to anyone that they ever lie. In fact, I encourage people to tell the truth. Math is hard… or at least it can be. However, being difficult is a long way from being impossible, and there never is a reason to say, “I can’t do math.” First this statement is a gross overgeneralization. Everyone can do math. Second, it only reinforces negative stereotypes about math and math ability that can become a self-fulfilling prophecy for our children.
Instead, I recommend that we start from a place of positivity. Math is fun. Spending time working on something that is challenging leads to huge rewards as you discover the underlying patterns and, yes, beauty of the mathematics. Simply shifting our orientation in the way that we talk about math has huge benefits.
2. All students can learn math: Every student has the capacity to learn deeper conceptual knowledge in mathematics.
The emphasis here is in two places. First I mean ALL students when I say all students. I mean special education students with IEPs. I mean English Language Learners. I mean students who qualify for free or reduced price lunches because their household income is below the poverty line. I mean girls as well as boys. I mean all students.
Second is that learning math is not just learning a set of algorithms. Instead we mean, that students learn to not just compute fluently, but also to understand the mathematics that they are working with and to apply it to appropriate situations.
3. A common approach to problem solving: We use the K-W-I to teach steps to problem solving in and outside of math.
To aid students in developing the strategic competence to solve any problem that they face, we utilize a common approach to problem solving across the school, both in and out of math classes. This approach emphasizes three key points. First we work for students to first understand the problem by describing what they know. Then we have students describe what they need to find out. Finally, before problem solving they identify ideas for how they might approach the problem. By slowing students down and using the same process with every problem we face, students develop a habit of mind towards all problems that emphasizes first understanding a problem before searching for answers.
4. Representations: We value the multiple ways that ideas can be modeled or demonstrated and encourage making connections between various representations.
As students work and produce final products in mathematics, we don’t value one way of doing things over another. Rather the emphasis is first on understanding. Students utilize representations of mathematics whether manipulatives, drawings, or numbers to understand and solve the problems. No one representation is valued over another.
Once a student feels confident in their understanding of their work, we then shift the emphasis to communicating. Students explore whether or not their representations clearly express their understanding of mathematics for their classmates, teachers, and often an outside audience. If not, they revise their representations to better express their understanding to others.
Only after we have ensured that students understand the problem and have communicated their solution effectively do we shift our attention to whether their solution method and representation was efficient.
5. Concise and precise communication: We reinforce precision and brevity during class discussion.
With the question of efficiency, we push students to utilize accurate vocabulary and language to express their ideas. As students explore ideas as a class, we have opportunities to reinforce concept development and vocabulary learning as we refine the way that we talk about mathematics.
6. Reasoning and proof: Mathematical arguments are weighed on the merits of their logic NOT on the status of the speaker or beauty of the language.
Last but not least, we emphasize that it is logic that is weighed most heavily in our classrooms. As many of us know, many arguments are won by the popularity of the person speaking or the complexity of the words that he or she uses. However, we want students to develop a critical ear for arguments that places value on what makes sense.
It is with these six expectations for how students, staff, and families engage in mathematics that we have created a foundation for changing the culture of numeracy at Two Rivers.
1. Math is fun: Time is built into lessons to play with math.
I don’t recommend to anyone that they ever lie. In fact, I encourage people to tell the truth. Math is hard… or at least it can be. However, being difficult is a long way from being impossible, and there never is a reason to say, “I can’t do math.” First this statement is a gross overgeneralization. Everyone can do math. Second, it only reinforces negative stereotypes about math and math ability that can become a self-fulfilling prophecy for our children.
Instead, I recommend that we start from a place of positivity. Math is fun. Spending time working on something that is challenging leads to huge rewards as you discover the underlying patterns and, yes, beauty of the mathematics. Simply shifting our orientation in the way that we talk about math has huge benefits.
2. All students can learn math: Every student has the capacity to learn deeper conceptual knowledge in mathematics.
The emphasis here is in two places. First I mean ALL students when I say all students. I mean special education students with IEPs. I mean English Language Learners. I mean students who qualify for free or reduced price lunches because their household income is below the poverty line. I mean girls as well as boys. I mean all students.
Second is that learning math is not just learning a set of algorithms. Instead we mean, that students learn to not just compute fluently, but also to understand the mathematics that they are working with and to apply it to appropriate situations.
3. A common approach to problem solving: We use the K-W-I to teach steps to problem solving in and outside of math.
To aid students in developing the strategic competence to solve any problem that they face, we utilize a common approach to problem solving across the school, both in and out of math classes. This approach emphasizes three key points. First we work for students to first understand the problem by describing what they know. Then we have students describe what they need to find out. Finally, before problem solving they identify ideas for how they might approach the problem. By slowing students down and using the same process with every problem we face, students develop a habit of mind towards all problems that emphasizes first understanding a problem before searching for answers.
4. Representations: We value the multiple ways that ideas can be modeled or demonstrated and encourage making connections between various representations.
As students work and produce final products in mathematics, we don’t value one way of doing things over another. Rather the emphasis is first on understanding. Students utilize representations of mathematics whether manipulatives, drawings, or numbers to understand and solve the problems. No one representation is valued over another.
Once a student feels confident in their understanding of their work, we then shift the emphasis to communicating. Students explore whether or not their representations clearly express their understanding of mathematics for their classmates, teachers, and often an outside audience. If not, they revise their representations to better express their understanding to others.
Only after we have ensured that students understand the problem and have communicated their solution effectively do we shift our attention to whether their solution method and representation was efficient.
5. Concise and precise communication: We reinforce precision and brevity during class discussion.
With the question of efficiency, we push students to utilize accurate vocabulary and language to express their ideas. As students explore ideas as a class, we have opportunities to reinforce concept development and vocabulary learning as we refine the way that we talk about mathematics.
6. Reasoning and proof: Mathematical arguments are weighed on the merits of their logic NOT on the status of the speaker or beauty of the language.
Last but not least, we emphasize that it is logic that is weighed most heavily in our classrooms. As many of us know, many arguments are won by the popularity of the person speaking or the complexity of the words that he or she uses. However, we want students to develop a critical ear for arguments that places value on what makes sense.
It is with these six expectations for how students, staff, and families engage in mathematics that we have created a foundation for changing the culture of numeracy at Two Rivers.
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