Early Math Intervention
EARLY MATH INTERVENTION: CATCHING UP SOONER IS BETTER David Dockterman, Ed.D. Harvard Graduate School of Education Houghton Mifflin Harcourt
THE PLAN ➤
The math committee in our brains
➤
Nature + Nurture = variation among learners
➤
Strengthening all the subcommittees ➤
Mindset/Beliefs
➤
Executive Function/Learning Behaviors
➤
Math
Dockterman 2016
NUMBER APPROXIMATION
EXECUTIVE FUNCTION
LANGUAGE RETRIEVAL
SYMBOLIC, PROCEDURE
EMOTIONAL STATE
Dockterman 2016
8 Dockterman 2016
SENSE OF DREAD CHOKING RACING HEARTBEAT
THE ANXIOUS BRAIN
SHAKY
When are you willing to take a risk?
Dockterman 2016
DEVELOPING A MINDSET FOR LEARNING Dockterman 2016
A PICTURE OF GOOD MATH LEARNING Think, Pair, Share: 2-3 characteristics of what good math learners do
Dockterman 2016
Persevere
Surrender Dockterman 2016
I believe this is worth doing.
I believe I can learn what I need.
Knowledge, Skills, Strategies
Beliefs Behaviors
I believe I can do it.
I believe my group values my effort.
Task Dockterman 2016
LITERATURE REVIEW JUNE 2012
Teaching Adolescents To Become Learners
The Role of Noncognitive Factors in Shaping School Performance: A Critical Literature Review
SOCIO-CULTURAL CONTEXT
STUDENT BACKGROUND CHARACTERISTICS
SCHOOL AND CLASSROOM CONTEXT
ACADEMIC MINDSETS
SOCIAL SKILLS
ACADEMIC PERSEVERANCE
ACADEMIC BEHAVIORS
ACADEMIC PERFORMANCE
LEARNING STRATEGIES
PERFORMANCE Beliefs I believe it’s worth doing. I believe I can do it. I believe I can learn what I need. I believe my group supports me.
Behaviors Attendance Attention Focus Challenge-seeking Help-seeking Perseverance Strategic thinking
Knowledge, Skills Visual number sense Mathematical language Symbolic procedures Mathematical reasoning Dockterman 2016
VARIABILITY IS INNATE
panamath.org Dockterman 2016
VARIABILITY IS INNATE
panamath.org Dockterman 2016
VARIABILITY IS INNATE
panamath.org Dockterman 2016
EARLY EXPERIENCES MATTER MOST the case of absolute pitch
Dockterman 2016
EARLY MATH EXPERIENCES MATTER
Top…
Left…
Bottom… http://www.uchicago.edu/features/20120312_levine/
➤
Object manipulation (physical and mental)
➤
with language
➤
and symbols
➤
in contexts
Right…
http://www.psy.cmu.edu/~siegler/2014-early-RamSieg.pdf
Dockterman 2016
NATURE + NURTURE = LOTS OF DIFFERENCES Dockterman 2016
THE PLAN ➤
✔The math committee in our brains
➤
✔Nature + Nurture = variation among learners
➤
Strengthening all the subcommittees ➤
Mindset/Beliefs
➤
Executive Function/Learning Behaviors
➤
Math
Dockterman 2016
RISKS TO A GROWTH MINDSET FOR MATH
“I got that easily. I must be good at that.”
“It’s easy for them but not for me. Maybe I’m not good at this.”
Dockterman 2016
OUR WORDS AND ACTIONS MATTER ➤
Act like EVERYONE is a math person (including girls).
➤
Focus on growth (mastery vs performance).
➤
Beware: math anxiety can be contagious.
➤
Model how you learn from mistakes (get an error routine).
➤
Give praise like you’re giving advice.
Dockterman 2016
EFFECTIVE EFFORT
bang head here
not just effort
Dockterman 2016
nachans; creative commons
Dockterman 2016
Dockterman 2016
Thinking About Challenges
› Every day, in and out of school,
you decide whether to take on new challenges.Think about a situation when you sought out a major challenge. Describe it here:
› Why did you decide to take on this challenge?
Seeking Out Challenges in MATH 180 › There are opportunities for you to challenge yourself throughout MATH 180. Describe how you’ll challenge yourself in each of these parts of MATH 180.
› The next time I’m in the
Success Zone, I’ll challenge myself by
› The next time I’m in the
Learn Zone, I’ll challenge myself by
› The next time I’m in the
Brain Arcade, I’ll challenge myself by
› The next time I’m working on
mSpace pages, I’ll challenge
myself by
Dockterman 2016
Focusing on Concentration › Describe a situation when you were focused and concentrated
while learning something new. How did your focus and concentration make you feel about yourself?
› Describe a situation when you were
not focused or concentrated while
learning something new. How did your lack of focus and concentration make you feel about yourself?
Focus & Concentration Strategies
› Focus and concentration will help you be successful. Use these strategies to plan how to improve your focus and concentration in MATH 180. CONCENTRATION STRATEGIES
EXPLAIN HOW YOU WILL USE THESE STRATEGIES
Calm Your Mind
My mind feels a little stressed in MATH 180 during . One way I will calm my mind is to. . .
Acknowledge and Release Random Thoughts
I sometimes have random thoughts when working on . One way to release these thoughts and concentrate is to. . .
Focus on One Thing Only
The most difficult thing to focus on in MATH 180 is . One way to direct my focus and attention during class is to. . .
Identify and Eliminate Distractions
I sometimes get distracted in MATH 180 while working on . One way to eliminate distractions during class is to. . .
Dockterman 2016
STRUGGLE & CONFUSION ARE GOOD if you have a way forward
Dockterman 2016
DELIBERATE PRACTICE
It’s about doing, not just knowing Dockterman 2016
MAKE IT INTERESTING ➤
like a good story
➤
what happens next
Dockterman 2016
Unboxing is uncertain, and uncertainty motivates Dockterman 2016
Like reality TV - low stakes predictions Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
THE PLAN ➤
✔The math committee in our brains
➤
✔Nature + Nurture = variation among learners
➤
Strengthening all the subcommittees ➤
✔Mindset/Beliefs
➤
✔Executive Function/Learning Behaviors
➤
Math Right…
Left…
Dockterman 2016
REAL (PURPOSEFUL)
LINGUISTIC
“SEVEN”
REPRESENTATIONAL (MENTAL MODEL)
SYMBOLIC
7 Dockterman 2016
FROM SURFACE TO DEEP STRUCTURE 2
Explain take-away problems.
Write the following problem on the chart paper titled Take-Away Problems.
4
Solve the problem with an open number line.
Ask the three questions for drawing open number lines, and use students’ answers to draw the open number line on the board.
h Where do we start? (70)
Take-Away Problems
What jumps do we make? (back 10 and 5)
70 15
Alexander had $ . He spent $ . How much money does he have left?
Where do we end up? (55) What equation can we write for this problem? (70 2 15 5 55)
Write the equation on the board.
µ5
µ 10
55 60 70
70 µ 15 = 55
h In take-away problems, there is always a change. The change in a take-away problem with money could be spending, losing, or giving. In this problem, Alexander spent some of his money.
Dockterman 2016
3
FROM SURFACE TO DEEP STRUCTURE Comparing Problems Alexander has $___.
29 30
Nicky has $___. What is the difference between the two amounts of money?
1
Solve the problem with an open number line. h These numbers are easy. What is the difference between 29 dollars and 30 dollars? ($1)
Watch as I show this with an open number line. I know how much money each boy has, so I can write both numbers.
Write 29 and 30 on the board.
h To go from 29 to 30, I can make a jump of 1. Draw an open number line on the board.
2
Solve the problem with another open number line. h Watch as I draw another open number line for this problem. This time I’ll jump from 30 back to 29.
Draw the open number line on the board.
+1
29 30
29 + 1 = 30
µ1
29 30 +1
h The answer is the jump number, the distance
29 30
between 29 and 30 on an open number line. The answer is 1.
h The 1 tells the difference between 29 and 30.
The answer is 1. Notice that the answer is the jump number, which is the distance between 29 and 30 on an open number line.
Here’s an equation that matches the problem.
Write the equation on the board. Then underline the 1 in the equation and explain again that it is the answer to the question.
Write the equation for the new open number line and underline the 1 as you again explain that it is the answer.
+1
29 30
29 + 1 = 30
µ1
29 30
30 – 1 = 29
+1
29 30
Dockterman 2016
29 + 1 = 30
h The difference or distance is 1. It doesn’t matter
whether I jump forward or back. It is 1 either way.
4
3
FOR STUDENTS WITH MORE SUBCOMMITTEE NEEDS 1
Present a take-away problem. //Here is a take-away subtraction problem.
Erase the board, write the problem, and read it aloud.
10 children are playing basketball. 4 stop playing and go home. How many children are still playing?
1
Present a comparing problem. //Here is a comparing problem.
Erase the board, write the problem, and read it aloud.
7 girls and 3 boys are at the park. What is the difference between the number of girls and boys?
//Make a cube train with 10 cubes.
Give time for students to make the train.
//Make two cube trains—one with 7 cubes for the girls and one with 3 cubes for the boys.
//Now take off 4 cubes from your train. These 4
Give time for students to make the trains. Choose a student to hold up the two trains and point to the 4 cubes that show the difference.
cubes show the number of children who go home. How many cubes are left? (6) What equation can we write? (10 – 4 = 6)
Choose a student to demonstrate breaking off 4 cubes from the 10-cube train.
//Put the trains next to each other. What is the difference? (4) What equation can we write to show the difference? (7 – 3 = 4)
Write 10 – 4 = 6 on the board.
Write 7 – 3 = 4 on the board.
Dockterman 2016
CLEAR STEPS FOR GROWTH + WAYS TO SHORE UP ALL PARTS OF THE COMMITTEE
Dockterman 2016
BETWEEN MINDSET AND PERFORMANCE ➤
MINDSET: Students (and their teachers) must believe they can learn math.
➤
BEHAVIORS: They need clear steps (coaching/teaching) for how to grow their skills.
➤
NORMS: Growth takes focused, deliberate effort (with different paces for different kids and lots of mistakes along the way).
➤
MOTIVATION: Seeing yourself grow can help sustain motivation.
➤
PERFORMANCE will follow.
Dockterman 2016
WHAT ARE YOUR NEXT STEPS?
[email protected] [email protected] Twitter: @dockterman
THE PLAN ➤
The math committee in our brains
➤
Nature + Nurture = variation among learners
➤
Strengthening all the subcommittees ➤
Mindset/Beliefs
➤
Executive Function/Learning Behaviors
➤
Math
Dockterman 2016
NUMBER APPROXIMATION
EXECUTIVE FUNCTION
LANGUAGE RETRIEVAL
SYMBOLIC, PROCEDURE
EMOTIONAL STATE
Dockterman 2016
8 Dockterman 2016
SENSE OF DREAD CHOKING RACING HEARTBEAT
THE ANXIOUS BRAIN
SHAKY
When are you willing to take a risk?
Dockterman 2016
DEVELOPING A MINDSET FOR LEARNING Dockterman 2016
A PICTURE OF GOOD MATH LEARNING Think, Pair, Share: 2-3 characteristics of what good math learners do
Dockterman 2016
Persevere
Surrender Dockterman 2016
I believe this is worth doing.
I believe I can learn what I need.
Knowledge, Skills, Strategies
Beliefs Behaviors
I believe I can do it.
I believe my group values my effort.
Task Dockterman 2016
LITERATURE REVIEW JUNE 2012
Teaching Adolescents To Become Learners
The Role of Noncognitive Factors in Shaping School Performance: A Critical Literature Review
SOCIO-CULTURAL CONTEXT
STUDENT BACKGROUND CHARACTERISTICS
SCHOOL AND CLASSROOM CONTEXT
ACADEMIC MINDSETS
SOCIAL SKILLS
ACADEMIC PERSEVERANCE
ACADEMIC BEHAVIORS
ACADEMIC PERFORMANCE
LEARNING STRATEGIES
PERFORMANCE Beliefs I believe it’s worth doing. I believe I can do it. I believe I can learn what I need. I believe my group supports me.
Behaviors Attendance Attention Focus Challenge-seeking Help-seeking Perseverance Strategic thinking
Knowledge, Skills Visual number sense Mathematical language Symbolic procedures Mathematical reasoning Dockterman 2016
VARIABILITY IS INNATE
panamath.org Dockterman 2016
VARIABILITY IS INNATE
panamath.org Dockterman 2016
VARIABILITY IS INNATE
panamath.org Dockterman 2016
EARLY EXPERIENCES MATTER MOST the case of absolute pitch
Dockterman 2016
EARLY MATH EXPERIENCES MATTER
Top…
Left…
Bottom… http://www.uchicago.edu/features/20120312_levine/
➤
Object manipulation (physical and mental)
➤
with language
➤
and symbols
➤
in contexts
Right…
http://www.psy.cmu.edu/~siegler/2014-early-RamSieg.pdf
Dockterman 2016
NATURE + NURTURE = LOTS OF DIFFERENCES Dockterman 2016
THE PLAN ➤
✔The math committee in our brains
➤
✔Nature + Nurture = variation among learners
➤
Strengthening all the subcommittees ➤
Mindset/Beliefs
➤
Executive Function/Learning Behaviors
➤
Math
Dockterman 2016
RISKS TO A GROWTH MINDSET FOR MATH
“I got that easily. I must be good at that.”
“It’s easy for them but not for me. Maybe I’m not good at this.”
Dockterman 2016
OUR WORDS AND ACTIONS MATTER ➤
Act like EVERYONE is a math person (including girls).
➤
Focus on growth (mastery vs performance).
➤
Beware: math anxiety can be contagious.
➤
Model how you learn from mistakes (get an error routine).
➤
Give praise like you’re giving advice.
Dockterman 2016
EFFECTIVE EFFORT
bang head here
not just effort
Dockterman 2016
nachans; creative commons
Dockterman 2016
Dockterman 2016
Thinking About Challenges
› Every day, in and out of school,
you decide whether to take on new challenges.Think about a situation when you sought out a major challenge. Describe it here:
› Why did you decide to take on this challenge?
Seeking Out Challenges in MATH 180 › There are opportunities for you to challenge yourself throughout MATH 180. Describe how you’ll challenge yourself in each of these parts of MATH 180.
› The next time I’m in the
Success Zone, I’ll challenge myself by
› The next time I’m in the
Learn Zone, I’ll challenge myself by
› The next time I’m in the
Brain Arcade, I’ll challenge myself by
› The next time I’m working on
mSpace pages, I’ll challenge
myself by
Dockterman 2016
Focusing on Concentration › Describe a situation when you were focused and concentrated
while learning something new. How did your focus and concentration make you feel about yourself?
› Describe a situation when you were
not focused or concentrated while
learning something new. How did your lack of focus and concentration make you feel about yourself?
Focus & Concentration Strategies
› Focus and concentration will help you be successful. Use these strategies to plan how to improve your focus and concentration in MATH 180. CONCENTRATION STRATEGIES
EXPLAIN HOW YOU WILL USE THESE STRATEGIES
Calm Your Mind
My mind feels a little stressed in MATH 180 during . One way I will calm my mind is to. . .
Acknowledge and Release Random Thoughts
I sometimes have random thoughts when working on . One way to release these thoughts and concentrate is to. . .
Focus on One Thing Only
The most difficult thing to focus on in MATH 180 is . One way to direct my focus and attention during class is to. . .
Identify and Eliminate Distractions
I sometimes get distracted in MATH 180 while working on . One way to eliminate distractions during class is to. . .
Dockterman 2016
STRUGGLE & CONFUSION ARE GOOD if you have a way forward
Dockterman 2016
DELIBERATE PRACTICE
It’s about doing, not just knowing Dockterman 2016
MAKE IT INTERESTING ➤
like a good story
➤
what happens next
Dockterman 2016
Unboxing is uncertain, and uncertainty motivates Dockterman 2016
Like reality TV - low stakes predictions Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
Dockterman 2016
THE PLAN ➤
✔The math committee in our brains
➤
✔Nature + Nurture = variation among learners
➤
Strengthening all the subcommittees ➤
✔Mindset/Beliefs
➤
✔Executive Function/Learning Behaviors
➤
Math Right…
Left…
Dockterman 2016
REAL (PURPOSEFUL)
LINGUISTIC
“SEVEN”
REPRESENTATIONAL (MENTAL MODEL)
SYMBOLIC
7 Dockterman 2016
FROM SURFACE TO DEEP STRUCTURE 2
Explain take-away problems.
Write the following problem on the chart paper titled Take-Away Problems.
4
Solve the problem with an open number line.
Ask the three questions for drawing open number lines, and use students’ answers to draw the open number line on the board.
h Where do we start? (70)
Take-Away Problems
What jumps do we make? (back 10 and 5)
70 15
Alexander had $ . He spent $ . How much money does he have left?
Where do we end up? (55) What equation can we write for this problem? (70 2 15 5 55)
Write the equation on the board.
µ5
µ 10
55 60 70
70 µ 15 = 55
h In take-away problems, there is always a change. The change in a take-away problem with money could be spending, losing, or giving. In this problem, Alexander spent some of his money.
Dockterman 2016
3
FROM SURFACE TO DEEP STRUCTURE Comparing Problems Alexander has $___.
29 30
Nicky has $___. What is the difference between the two amounts of money?
1
Solve the problem with an open number line. h These numbers are easy. What is the difference between 29 dollars and 30 dollars? ($1)
Watch as I show this with an open number line. I know how much money each boy has, so I can write both numbers.
Write 29 and 30 on the board.
h To go from 29 to 30, I can make a jump of 1. Draw an open number line on the board.
2
Solve the problem with another open number line. h Watch as I draw another open number line for this problem. This time I’ll jump from 30 back to 29.
Draw the open number line on the board.
+1
29 30
29 + 1 = 30
µ1
29 30 +1
h The answer is the jump number, the distance
29 30
between 29 and 30 on an open number line. The answer is 1.
h The 1 tells the difference between 29 and 30.
The answer is 1. Notice that the answer is the jump number, which is the distance between 29 and 30 on an open number line.
Here’s an equation that matches the problem.
Write the equation on the board. Then underline the 1 in the equation and explain again that it is the answer to the question.
Write the equation for the new open number line and underline the 1 as you again explain that it is the answer.
+1
29 30
29 + 1 = 30
µ1
29 30
30 – 1 = 29
+1
29 30
Dockterman 2016
29 + 1 = 30
h The difference or distance is 1. It doesn’t matter
whether I jump forward or back. It is 1 either way.
4
3
FOR STUDENTS WITH MORE SUBCOMMITTEE NEEDS 1
Present a take-away problem. //Here is a take-away subtraction problem.
Erase the board, write the problem, and read it aloud.
10 children are playing basketball. 4 stop playing and go home. How many children are still playing?
1
Present a comparing problem. //Here is a comparing problem.
Erase the board, write the problem, and read it aloud.
7 girls and 3 boys are at the park. What is the difference between the number of girls and boys?
//Make a cube train with 10 cubes.
Give time for students to make the train.
//Make two cube trains—one with 7 cubes for the girls and one with 3 cubes for the boys.
//Now take off 4 cubes from your train. These 4
Give time for students to make the trains. Choose a student to hold up the two trains and point to the 4 cubes that show the difference.
cubes show the number of children who go home. How many cubes are left? (6) What equation can we write? (10 – 4 = 6)
Choose a student to demonstrate breaking off 4 cubes from the 10-cube train.
//Put the trains next to each other. What is the difference? (4) What equation can we write to show the difference? (7 – 3 = 4)
Write 10 – 4 = 6 on the board.
Write 7 – 3 = 4 on the board.
Dockterman 2016
CLEAR STEPS FOR GROWTH + WAYS TO SHORE UP ALL PARTS OF THE COMMITTEE
Dockterman 2016
BETWEEN MINDSET AND PERFORMANCE ➤
MINDSET: Students (and their teachers) must believe they can learn math.
➤
BEHAVIORS: They need clear steps (coaching/teaching) for how to grow their skills.
➤
NORMS: Growth takes focused, deliberate effort (with different paces for different kids and lots of mistakes along the way).
➤
MOTIVATION: Seeing yourself grow can help sustain motivation.
➤
PERFORMANCE will follow.
Dockterman 2016
WHAT ARE YOUR NEXT STEPS?
[email protected] [email protected] Twitter: @dockterman
