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TRANSCRIPT
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)
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)
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)
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)
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)
• 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)
What we know about struggle…
• US math instruction is routinely characterized as rarely asking students to think, reason, and problem solve, i.e., struggle (Banilower et al., 2006)
What we know about struggle…
• 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
What we know about struggle…
• 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
• 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
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%
A. Indian students 24% 15% 10%
Source: The Nation’s Report Card: Trends in Academic Progress 2015 & 2019
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%
A. Indian students 24% 15% 10%
Black students 20% 14% 7%
Source: The Nation’s Report Card: Trends in Academic Progress 2015 & 2019
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%
A. Indian students 24% 15% 10%
Black students 20% 14% 7%
Hispanic students 28% 20% 12%
Source: The Nation’s Report Card: Trends in Academic Progress 2015 & 2019
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%
A. Indian students 24% 15% 10%
Black students 20% 14% 7%
Hispanic students 28% 20% 12%
EL students 16% 5% 5%
Source: The Nation’s Report Card: Trends in Academic Progress 2015 & 2019
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%
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% —
Source: The Nation’s Report Card: Trends in Academic Progress 2015 & 2019
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%
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%
Source: The Nation’s Report Card: Trends in Academic Progress 2015 & 2019
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%
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/ Disabilities 17% 9% 6%
Source: The Nation’s Report Card: Trends in Academic Progress 2015 & 2019
What would these numbers look like if things were equitable?
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%
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/ Disabilities 17% 9% 6%
Source: The Nation’s Report Card: Trends in Academic Progress 2015 & 2019
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
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
• 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
• 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
1. Creating an environment that encourages struggle and makes students aware of the benefits of struggle
• Share research about struggle, perseverance, and learning
• Establish norms for struggle
1. Creating an environment that encourages struggle and makes students aware of the benefits of struggle
• Share research about struggle, perseverance, and learning
• 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?
1. Creating an environment that encourages struggle and makes students aware of the benefits of struggle
• Share research about struggle, perseverance, and learning
• Establish norms for struggle
• 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
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
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
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
Examples Non-Examples
Think about how you would solve each task (solve if you have time)
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
Examples Non-Examples
Think about how you would solve each task (solve if you have time)
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
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?
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
Other examples…
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
Other examples…
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
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?
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
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?
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
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
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the struggle
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the 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
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the 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
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the struggle
Step 1: Understand the
problem
Step 2: Devise a plan
Step 3: Carry out the plan
Step 4: Look back
George Polya, 1945
Not necessarily
linear!
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the struggle
Step 1: Understand the
problem
Step 2: Devise a plan
Step 3: Carry out the plan
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
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the struggle
Step 1: Understand the
problem
Step 2: Devise a plan
Step 3: Carry out the plan
Step 4: Look back
Help! I don’t
know what this
problem is
asking me to
do.
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the struggle
Step 1: Understand the
problem
Step 2: Devise a plan
Step 3: Carry out the plan
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
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the struggle
Step 1: Understand the
problem
Step 2: Devise a plan
Step 3: Carry out the plan
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
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the struggle
Step 1: Understand the
problem
Step 2: Devise a plan
Step 4: Look back
Help! I don’t
know what to
do for this
problem.
Step 3: Carry out the plan
2. Give students opportunities and supports to engage in
struggle
• Use tasks that promote reasoning & problem solving
• Support (don’t rescue students from) the struggle
Step 1: Understand the
problem
Step 2: Devise a plan
Step 4: Look back
Step 3: Carry out the plan
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?
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
A Time for Telling (Schwartz & Bransford, 1998)
Reasoning tasks w/ no lecture
Lecture & practice Reasoning tasks w/ lecture
A Time for Telling (Schwartz & Bransford, 1998)
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”
A Time for Telling (Schwartz & Bransford, 1998)
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
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
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 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
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
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