encouraging a mathematical mindset through reasoning ...x(1)s(rkavmkbgrc3cfeaoxp2berur... ·...

Matt Wallace, PhD

UC Davis

School of Education

Encouraging a Mathematical

Mindset Through Reasoning &

Problem Solving

Goal for today…

… and beyond

Let’s Chat!

In the chat section please share…

• What are three words you associate with struggle?

Let’s Chat!

In the chat section please share…

• What are three words you associate with struggle?

• How many of the words that you see are positive?

What we know about struggle…

• Struggle is often perceived as bad, as evidence of a lack of success, or both (Boaler, 2019)

• Struggle conflicts with our notions of successful teaching (SanGiovanni, 2020)




• Struggle and mistakes are the best times for brain growth (Boaler, 2015, 2019)





• Mistakes and struggle are a necessary component of learning mathematics with understanding (Hiebert & Grouws, 2007)






• Struggle is an opportunity to delve more deeply into understanding mathematics instead of simply seeking correct solutions (NCTM, 2014)






• Struggle is an opportunity to delve more deeply into understanding mathematics instead of simply seeking correct solutions (NCTM, 2014)

• Teaching that embraces and uses productive struggle leads to long-term benefits, with students more able to apply their learning to new problem situations (Kapur, 2010)


• US math instruction is routinely characterized as rarely asking students to think, reason, and problem solve, i.e., struggle (Banilower et al., 2006)



• Curriculum, textbooks, and many online resources include trivial and unchallenging questions



• Curriculum, textbooks, and many online resources include trivial and unchallenging questions

• We rescue students when they’re struggling, and in doing so empty the work of challenge and opportunities for struggle (Reinhart, 2000; Stein et al, 2009)

NAEP (National Assesment of Educational Progress) Achievement Results

Percentage of US students who scored at or above proficient in mathematics

4th Grade 8th Grade 12th Grade

ALL students 41% 34% 25%

Source: The Nation’s Report Card: Trends in Academic Progress 2015 & 2019





A. Indian students 24% 15% 10%







Black students 20% 14% 7%








Hispanic students 28% 20% 12%









EL students 16% 5% 5%










Low SES students 26% 18% —











Students w/

Disabilities17% 9% 6%











Students w/ Disabilities 17% 9% 6%


What would these numbers look like if things were equitable?










Students w/ Disabilities 17% 9% 6%


What would these numbers look like if things were equitable?

What would teaching and learning look like if we were “doing better”?

How can we support struggle in learning math?

1. Create an environment that encourages struggle and makes students aware of the benefits of struggle

2. Give students opportunities and supports to engage in struggle

• Share research about struggle, perseverance, and learning

In a number of studies, people’s mindsets have changed when they have been shown the evidence of struggle and brain growth…Once people become aware of the science, growth and change starts to take place. I see it in students and hear about it from teachers all around the globe. The science also supports this.

—Jo Boaler

1. Creating an environment that encourages struggle and makes students aware of the benefits of struggle



• Establish norms for struggle




• Value struggle

• Please read the vignette

• While reading consider the following questions…

1.How are student experiences different?

2. What appears to be valued in the two classrooms?




• Value struggle

Ms. Flahive Ms. Ramirez

• Structured the work/thinking for students,

by guiding them step-by-step

• Rescued students from struggle, mistakes,

and even failure

• Deprived students of the opportunity to

make sense of the math

• Students learn that if you struggle, the

teacher will tell you what to do

• ANSWERS are what’s valued

• Supports students by focusing their

thinking

• Enables students to make sense of the

math for themselves

• Students learn that if you struggle, the

teacher will assist you but they’ll have to

figure things out for themselves

• THINKING & STRUGGLE are what’s

valued

2. Give students opportunities and supports to engage in

struggle

• Use tasks that promote reasoning & problem solving


struggle


Examples Non-Examples

Think about how you would solve each task (solve if you have time)


struggle


Examples Non-Examples

Think about how you would solve each task (solve if you have time)

A knife is used to cut off the top of a spherical orange, 4 cm from the center of the orange. The orange has a radius of 5 cm. What is the area of the circle that was created by the cut?


struggle


Other examples…


struggle


Other examples…

A square birthday cake is frosted on top and on

the four sides. How should it be cut for 7 people

if everyone is to get the same amount of cake

and the same amount of frosting?


struggle


Take a few moments to think about how you would describe the

example tasks.

Food for thought: How do these example tasks compare to the

kinds of tasks that you’ve used with your students?


struggle


Some essential features of problem solving and reasoning tasks

• The solution is not obvious

• The pathway is not suggested

• Some degree of thought or effort is required (procedures may be used,

but are not used mindlessly)

• Usually can be represented in multiple ways, such as with pictures,

figures, symbols, graphs, and manipulatives


struggle


• Support (don’t rescue students from) the struggle


struggle



Step 1: Understand the

problem

Step 2: Devise a plan

Step 3: Carry out the plan

Step 4: Look back

George Polya, 1945

Polya’s Problem

Solving Principles


struggle




problem



Step 4: Look back

George Polya, 1945

Not necessarily

linear!


struggle




problem



Step 4: Look back

Read Two Ways

Read 1: read to understand and describe the context of the problem

Read 2: re-read to identify and represent the quantities of the problem

Adapted from Kelemanik et al, 2016


struggle




problem



Step 4: Look back

Help! I don’t

know what this

problem is

asking me to

do.


struggle




problem



Step 4: Look back

Read 1: read and then give a summary of the problem in your own words without using any numbers

Read 2: read and draw a visual representation that includes the context from read 1 AND all the quantities


struggle




problem



Step 4: Look back

Affordance

1. Acknowledge student thinking & struggle

2. Paraphrase the request (instead of giving too much help)

3. Encourage the use of available resources

Adapted from Warshauer, 2015


struggle




problem


Step 4: Look back

Help! I don’t

know what to

do for this

problem.



struggle




problem


Step 4: Look back


That struggling

feeling is the feeling

of your brain

growing. Keep

going! This is a

really important

moment for brain

growth.

Don’t forget about all of

the strategies you can use

to figure this out. I know

you can do this!

So you’re trying to find how

many chicken wings are in

the bucket. How many do

you think it will be?

A Time for Telling (Schwartz & Bransford, 1998)

Reasoning tasks w/ no lecture

Lecture & practice Reasoning tasks w/ lecture


Findings…

• “Reasoning tasks w/ lecture” students significantly out performed students in other classes on all achievement

tests

• In the “reasoning tasks w/ lecture” classes, students brains were primed for learning new methods because they

were given problems first (some of which they did not know how to solve)

Conclusion…

• “The question is not should we tell or explain, but when is the best time to do it? …Clearly the best time is after

students have explored problems”


Reasoning tasks w/ no lecture

Lecture & practice Reasoning tasks w/ lecture

After the struggle…

• Take time to explain the math (not the problem) that you want students to learn—there’s always a time to “tell”!

• Wherever possible, connect your explanations to students’ work




From 9:27 to

9:36 the battery

charge looks like

it’s increasing

1% per minute




From 9:27 to

9:36 the battery

charge looks like

it’s increasing

1% per minute Let’s talk about

the line of best fit

Step 1: graph dataStep 3: use line to

predict other data

points

Step 2: draw a line

that balances an

equal # of points

above & below




From 9:27 to

9:36 the battery

charge looks like

it’s increasing

1% per minute

Step 1: graph dataStep 3: use line to

predict other data

points

Step 2: draw a line

that balances an

equal # of points

above & below

When you found

that the the

battery increased

1% per minute

from 9:27 to

9:36, you found

the slope of the

line of best fit in

that interval…

Take a moment to consider the following questions…

• How can you support struggle in learning math?

• How do these ideas reflect you stepping out of your comfort zone (if at all)?

Stepping Outside Your Comfort Zone

Questions, Comments, Insights,…

Thank You!

Please contact me for any other information.

Matt Wallace

[email protected]

mailto:[email protected]?subject=

encouraging a mathematical mindset through reasoning ...x(1)s(rkavmkbgrc3cfeaoxp2berur... ·...

Documents