Structure of Agile Mind

Units 1-3: Functional thinkingx and y are variables linked by cause and effect, and this relationship can be represented in various ways

Units 4-9: Slope, intercept, and linear modelsObtain fluency with moving between representations: tables, graphs, and equations

Units 10-14: Solve for unknownsAlgebra and systems

Units 15-16: Sequences and patterns (we’re skipping)

Units 17-18: ExponentsLaws of exponents

Units 19-23: QuadraticsFactoring and finding roots

Units 1-3: Functional thinking

x and y are variables linked by cause and effect, and this relationship can be represented in various ways

Article: ‘Functional Thinking as a Route to Algebra in the Elementary Grades’

An Example: The Trapezoid Problem• Completed in Teacher Study Group• Discussed in a class reading (Blanton & Kaput, 2003)

• How many people can sit if there are…– Two trapezoids,– 14 trapezoids,– 127 trapezoids,– n trapezoids?

• How do you know?• Can you come up with multiple solutions?

Sixth Annual Noyce Conference7/07/2011

Where we’re at this year

Semester 1:

Functional thinking & graphing

Semester 2:

Algebra, exponents, and quadratics

Where we need to be

Semester 1:

Finish unit 10 by December (currently not till February).

Semester 2:

Systems, exponents, and quadratics

Bottom line: we fell a month behind in Nov/Dec

Structure of a Unit

M T W R F

Block 1 X Block 3Guided

Assessment X

Block 2 X Test X

Classroom models for Agilemindthat work

• Four students per group

• Only one computer out

• Students have class sets of activity sheets

Independent Practice

Direct Instruction

• NO computers out• Either notebooks or

activity sheets• For each slide, students

write / draw (especially predictions), then check it.

• Students share out answers

Guided Practice• Assignment is on

some instant-feedback site

• Students write their answers on whiteboards, so the teacher can monitor

• Teacher controls pace of the questions, so that any student who doesn’t have answer on whiteboard has to finish it after class/after school

Flipping

• Pre-recorded tutorial on a problem is played on the board (some teachers are recording their own videos or podcasts)

• Students solve either that example, or work a similar example at their desk

• Teacher roams the room

Simulation

• Student demonstrates simulation on board, then class does so independently

• Students ‘act out’ simulation at end

• Article: ‘What levels of guidance promote engaged exploration with interactive simulations?

What doesn’t work

• Giving students an assignment from the agile mind textbook and have them work at their seats– This is the first thing you’d try– It doesn’t work because the students don’t produce

anything– It doesn’t work because you wind up being the

facebook police, not teaching– Students can change their answers three times, so

they just wind up guessing– You’ll get assignment results between 80 and 100%,

even though the students haven’t learned a thing.

Some examples of what I mean by ‘backwards’

From Unit 3 Test

Students must:•Find the rate of change (before we’ve defined rate or slope or practiced any such calculations)•Find the intercept (before we’ve even defined it)•Write an equation in slope-intercept form, even though they’ve never even seen such an equation

Students must find equivalent expressions, before they have seen a linear equation, been told what a like term is, or even worked on order of operations.

The a-coordinate? Whoever heard of the a-coordinate?

This problem ran throughout. Whenever agilemind used ‘m’ to mean ‘miles’, for example, students with a tenuous grasp of slope got totally lost.

• Students have roughly two weeks to gain this level of mastery, without the curriculum mentioning algebra beforehand.

But don’t worry…

• Even though only the top 25% might grasp these problems through intuition, agilemind comes back and does the same problem again once students have developed a formal understanding

From Unit 3 Test

From Unit 7 Test

From Unit 9 Test

From Unit 1 Test

From Unit 11 Test

From Unit 10 Test

From Unit 2 Test

• From a math teacher’s normal perspective, you don’t need to teach the same thing again. The skill is the same.

• But I found, again and again, that what I thought I was saying, and what the students understood me to be saying, were totally different.

It’s NOT because these skills were supposed to be prerequisite

It’s the intention of the curriculum: students gain a sense of what a function is (in a real-world sense), and how variables are linked

by cause and effect.

Only after students have been given the context several times, and in fact solved the

relevant problem, do they learn the formalism

Q & A