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* *Reflection: Student Ownership
Develop fluency with positive and negative numbers - Section 2: Play of the Day

Another early-year idea to think about... ratio. Specifically, the ratio of how much thinking the student is doing vs. the teacher.

Remember, your job is not actually to “teach,” whatever you think that means. Your job is to ensure that students learn. Some people might think that doesn’t make a difference, but I think the lens you choose does alter how you approach any task.

A popular way that some people have framed how to structure a lesson is “I do/we do/you do,” where the teacher demonstrates a skill…say, the pattern for multiplying special products. For example:

Teacher says, “Here is the pattern for squaring binomials.”

(a + b)^2 = a^2 +2ab + b^2

Students write it down. Students practice it with the teacher on a few examples. Students practice it on their own. Students (supposedly) master the skill. Better teachers explain why this works. Some teachers don’t bother.

This happens with a high level of regularity in math classrooms in this country. And some students do learn what the pattern is… but it’s not clear to me that this is the best way.

The issue I have with this approach is that it takes the learning away from the student. The student has not really been pushed to think, and it frames mathematics as a series of seemingly irrelevant algorithms to apply rather than a beautiful, elegant, sexy (yes, I said it) framework of logical properties that builds upon itself and that holds the keys to our universe.

How much more powerful would it be if students were able to engage in a series of tasks that pushed them to do the thinking and see the beauty themselves? Now, this does NOT mean that the pendulum should swing too far. We shouldn’t say, “today, we are going to learn how to square binomials. Try all these out. Figure out what’s going on.” And then leave the kids for 30 minutes, come back, and everyone’s a genius. Instead, I am advocating an approach that engages kids in a carefully constructed series of tasks and questions that lead students to see/uncover/realize/reveal/understand the same pattern as in the example above, in a way where they know why it works and how it was derived.

For example, continuing with the example of squaring binomials, imagine, instead, that you ask the kids to tackle a few examples with you, each of which squares a binomial. Then, you ask kids – “what pattern do you notice here? Why does this pattern exist?” You might use a turn-and-talk, cold call, or any number of techniques, but the idea is to push students to do the thinking. Eventually, you would formally introduce the algorithm, but only at the point where students can see why it works. Doing this effectively requires a deep understanding of the mathematical concepts, a nuanced knowledge of the curriculum, and predictions of student misconceptions. It requires a *lot* of planning and thinking on the front end… but, like anything else in teaching, it becomes easier and easier with experience and deliberate practice.

Another example of this is when students ask you questions. I try to often push the question to the other students. “What do you all think of so-and-so’s question?” It has the simultaneous benefits of both getting kids to do more of the thinking and pushing kids to listen to each other. There are so many more examples, but the guiding principle is to get the students to do as much of the cognitive lifting as possible.

I posit that this way of teaching and learning, when applied to an entire curriculum, will lead to much better results.

*Ratio - Get your kids to do more of the thinking and doing*

*Student Ownership: Ratio - Get your kids to do more of the thinking and doing*

# Develop fluency with positive and negative numbers

Lesson 4 of 9

## Objective: SWBAT develop fluency with positive and negative numbers

#### Warm-up

*10 min*

Each day, students complete a warm-up that usually consists of spiraling the previous day's material, in addition to older material. Warm-up problems also sometimes extend lessons that students have encountered before to more unfamiliar contexts.

*expand content*

#### Homework

*15 min*

The homework includes questions related to this lesson, as well as spiraled review. I also provide answers to the problems on page 2.

**Allen Iverson on Practice for the Walk out the Door**

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I love your videos although the last one for Allen Iverson is not showing, it is saying the video does not exist. I love your lesson though, thanks for sharing!

| 3 years ago | Reply##### Similar Lessons

###### Adding and Subtracting Integers on a Number Line

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- UNIT 1: FLUENCIES AND THE LANGUAGE OF ALGEBRA
- UNIT 2: SOLVING LINEAR EQUATIONS
- UNIT 3: INTRODUCTION TO FUNCTIONS
- UNIT 4: INTERPRETING AND COMPARING LINEAR FUNCTIONS
- UNIT 5: SYSTEMS OF LINEAR EQUATIONS
- UNIT 6: EXPONENTS AND SCIENTIFIC NOTATION
- UNIT 7: PARALLEL LINES, TRANSVERSALS, AND TRIANGLES
- UNIT 8: CONGRUENCE AND SIMILARITY THROUGH TRANSFORMATIONS
- UNIT 9: PATTERNS OF ASSOCIATION IN BIVARIATE DATA
- UNIT 10: SLOPE REVISITED
- UNIT 11: VOLUME OF CYLINDERS, CONES, AND SPHERES
- UNIT 12: POLYNOMIALS AND FACTORING
- UNIT 13: QUADRATIC FUNCTIONS

- LESSON 1: Day One
- LESSON 2: Add and subtract positive and negative numbers
- LESSON 3: Multiply and divide positive and negative numbers
- LESSON 4: Develop fluency with positive and negative numbers
- LESSON 5: Add, subtract, and multiply decimals
- LESSON 6: Divide decimals
- LESSON 7: Evaluate expressions using substitution (Day 1 of 2)
- LESSON 8: Evaluate expressions using substitution (Day 2 of 2)
- LESSON 9: ASSESSMENT #1