brain compatible strategies boomerang brain compatible strategies boomerang teachers spend on...
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Brain Compatible Strategies
Boomerang
Brain Compatible Strategies
Boomerang
Teachers spend on average a total of 26 school days of a year
trying to quiet students down.
Brain Compatible Strategies
Boomerang
WhistleMusic
SingingMusic Instrument
Hear My VoiceClapping
Processing Questions
• What
• So What
• Now What
ACTIVITYSPORTS
Football
_____________
Baseball
_______________
Hockey
_____________
Basketball
____________
ACTIVITYSPORTS
Football
Jets
Giants
_____________
Baseball
Yankees
Mets
_______________
Hockey
Rangers
Islanders
_____________
Basketball
Knicks
Nets
____________
Make an Make an AppointmentAppointment
Draw a clock face. Write 3, 6, 9, and 12.
Draw lines outward from each number. Make an appointment with four different, new people in the room.
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But we want different results.
Change is difficult.
We want to bring the joy and passion back into the classroom.
need
10
Discussion QuestionThink-Pair-Share
• Find one person you do not know and be ready to report the following to the large group in 5 minutes:– Other person’s name– Other person’s teaching experience– Other person’s neuroscience strategies already in
use
Brain Compatible Strategies
Building a Group Ladder
Start any activity or discussion by having everyone write down a response, alone. Next
have them share with one other person, pairs. After that they share in a small group and finally in the large group you will have
created a level of comfort for most students in the group. By using this ladder you will never face a room of blank stares when you hope
for a response.
Brain Compatible Strategies
Mathgician
Mathgician
Mathgician
I promised answers.Understand math and have fun?
Accessible for all students without sacrificing rigor?
How?Instruction informed by brain-based learning research
will develop students’ visual understanding into symbolic math competency.
Students’ develop a rich and rigorous conceptual understanding… for continued math
achievement.
Magic Square
Magic Square
Have the students create a Magic Square by placing the numerals 1, 2 , 3, 4, 5, 6, 7, 8, and 9 in three columns of three.
Have them arrange the numbers so that when you add them vertically, horizontally, and diagonally, the answer will continue to be 15.
Magic SquarePlace the numerals 1, 2 , 3, 4, 5, 6, 7, 8, and 9 in three columns of three so that when you add them vertically, horizontally, and diagonally, the answer will continue to be 15.
Magic Square
Place the numerals 1, 2 , 3, 4, 5, 6, 7, 8, and 9 in three columns of three so that when you add them vertically, horizontally, and diagonally, the answer will continue to be 15.
8 3 4
1 5 9
6 7 2
Magic SquarePlace the numerals 1, 2 , 3, 4, 5, 6, 7, 8, and 9 in three columns of three so that when you add them vertically, horizontally, and diagonally, the answer will continue to be 15.
6 1 8
7 5 3
2 9 4
•Increases student achievement by 22%
•Cues, questions and advance organizers help students use what they already know about a topic to enhance further learning.
•These tools should be analytical, focus on what is important, are most effective before a learning experience.
Cues, Questions, and Advance Organizers
CONNECTING TO PAST LEARNINGS
InterviewsShort StoryGraphic OrganizersMural or Collage
CONNECTING TO PAST LEARNINGS
Music Activity
Models
Student Ideas
Brain Compatible Strategies
Quotes
Are you running out of wall space to hang posters? I bought a 10 ft. section of 1/2 inch PVC pipe and four ninety
degree corners at my hardware store. I then cut the pipe into equal lengths
and put it together using the four corners. I hung it form my ceiling with
a swivel and string. Then I attached four posters I had made on my
computer relating to operations of integers. Now my kids can look at the
rules for adding, subtracting, multiplying, and dividing integers at a
glance.
Research QualityGrade Level of
EvaluationExamples
A 1 High quality experimental study, blind comparison, reference standard, appropriate population
B 2
or
3
Low quality experiment
Or
High quality observational study, population is specified and appropriate, referenced standard, not blind
C 4 Low quality observational study, evaluations were completed without sensitivity
D 5 Expert opinion
Khan, Riet, Kleinjen, 2001, Undertaking Systematic Reviews of Research on Effectiveness
OUTCOMES% of Participants who Demonstrate Knowledge, Demonstrate
New Skills in a Training Setting, and Use New Skills in the Classroom
TRAINING
COMPONENTS
Knowledge SkillDemonstration
Use in theClassroom
Theory and Discussion
10% 5% 0%
..+Demonstration in Training
30% 20% 0%
…+ Practice & Feedback in
Training
60% 60% 5%
…+ Coaching in Classroom
95% 95% 95%
Joyce and Showers, 2002
Coaching Impact
TRUST
RESPECT
ATTITUDE
BEHAVIORS/MANNERISM
KNOWLEDGE/SKILLS
SMART MODEL
Physician’s Creed:
First, Do No Harm
Educator’s Creed:
Above all, do nothing to
diminish hope
Wisdom from Dr. Bill…
• If your students like you, there is nothing they won’t do for you.
• If your students don’t like you, there is nothing they won’t do to you.
TRUST ACTIVITIES
Grapho
Remote Control Math
Dicated Math
• Student populations do not look or act like they did 50 years ago or even 20 years ago.
• Educators and parents need to let go of what worked 10 years ago.
Diverse Subpopulations of LearnersCity White African
AmericanLatino/ Hispanic
Asian Other
Los Angeles
8.9 11.2 73.3 3.8 2.8
Atlanta 8.37 85.98 4.1 0.59 0.96
St. Louis 14.1 81.8 2.3 1.7 0.3
Philadelphia
13.3 64.4 15.8 5.6 0.3
Detroit 3.0 90.73 5.1 0.8 0.3
Chicago 8.1 48.6 37.6 3.2 2.5
Oakland 6.3 40.0 33.2 16.3 4.2
Cleveland 16.7 70.3 10.4 0.3 2.0
New York City
14.3 33.1 39.0 13.5 0.0
Dallas 5.1 29.7 64.1 0.9 0.2
2005-2006 Enrollment Demographic Information gathered from district websites.
The Achievement Gaps Continues…Comparison of CST Math Achievement Over Time By Demographic Groups
0%
10%
20%
30%
40%
50%
60%
Year and Grade
Perc
en
t P
rofi
cie
nt
or
Ad
van
ced
All Students
Econ Disadv (55%)
Parent no college(35%)
EL (20%)
African Am. (7%)
Latino(46%)
Students from traditionally underserved demographic population continue to under-achieve over the past 7 years.
Data from star.cde.ca.gov
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2007 TIMSS DATA2007 TIMSS DATA
• The Trends in International Mathematics and Science Study (TIMSS) measures mathematical content domains including number-sense, geometrical shapes, measurement, algebraic skills, and data displays in both 4th and 8th grades. There were 36 countries participating in grade 4, and 47 countries participating in grade 8. Key findings include:
In 2007, U.S. 4th graders (529) and 8th graders (508) were higher than the TIMSS average of 500.
Compared to 1995, the average math score for U.S. 4th graders (511) and 8th graders (492) was higher.
In 2007, 10% of 4th graders and 6% of 8th graders scored at or above the advanced international benchmarks for mathematics.
At grade 8, there were no measureable differences between U.S. males and females, though at grade 4, there was a 6 point male advantage.
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2007 TIMSS DATA 42007 TIMSS DATA 4thth Grade GradeCountry Average Score International Average 500Hong Kong 607Singapore 599Chinese Taipei 576Japan 568Kazakhstan 549Russian Federation 544England 541Latvia 537Netherlands 535Lithuania 530UNITED STATES 529Germany 525Denmark 523Australia 516Hungary 510Italy 507Austria 505Sweden 503
Slovenia 502
Armenia 500
Slovak Republic 496
Scotland 494
New Zealand 492
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2007 TIMSS DATA 82007 TIMSS DATA 8thth Grade GradeCountry Average Score
International Average 466Chinese Tapai 598Korea 597
Singapore 593Hong Kong 572Japan 570Hungary 517England 513Russian Federation 512*UNITED STATES 508 Lithuania 506Czech Republic 504Slovenia 501Armenia 499Australia 496Sweden 491Malta 488Scotland 487 Serbia 486Italy 480Malaysia 474Norway 469Cyprus 465Bulgaria 464Israel 463Ukraine 462
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PISA DATA: 15 yr. oldsPISA DATA: 15 yr. olds(Program for International Student (Program for International Student
Assessment)Assessment)
A test of mathematical literacy for 15 year old students which focuses upon the direct application of mathematical principles. The test is administered every three years, with 57 countries participating in 2006. The test was not designed to measure curricular outcomes, but rather to assess mathematics within a real world context.
In 2006, the average U.S. score in mathematics literacy was 474, lower than the international average score of 498.
Among the 57 countries in the sample, the U.S. was outperformed by 31 countries in math and 22 countries in science (m = 489).
There was no measureable change in either the U.S. mathematics literacy score from 2003 to 2006, or the U.S. position compared to the international average.
40
PISA DATA: 15 yr. oldsPISA DATA: 15 yr. oldsCountry Average ScoreInternational Average 498
Chinese-Taipai 549Finland 548Hong Kong-China 547Korea 547Netherlands 531Switzerland 530Canada 527Macao-China 525Liechtenstein 525Japan 523New Zealand 522Belgium 520Australia 520Estonia 515Denmark 513Czech Republic 510Iceland 506Austria 505Slovenia 504
41
PISA DATA: 15 yr. oldsPISA DATA: 15 yr. olds
Country Average ScoreInternational Average 498
Germany 504Sweden 502Ireland 501France 496United Kingdom 495Poland 495Slovak Republic 492Hungary 491
Luxembourg 490
Norway 490
Lithuania 486
Latvia 486
Spain 480
Russian Federation 476
UNITED STATES 474
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PISA TragedyPISA TragedyNational Science Board (2004)
Enrollment in science and engineering programs is expanding ten times faster in China than in the United States.
Approximately two-thirds of Chinese undergraduate students earn math, science, or engineering degrees compared to just one-third of United States undergraduate students.
Science and engineering graduate programs have seen a 10 percent decline in the enrollment of U.S. students over the past decade, while enrollment of foreign graduate students has increased by 35 percent.
Approximately 5.7 percent of 24 year-olds in the United States have a science or engineering degree, which is half that of Taiwan (11.1 percent), South Korea (10.9 percent), and the United Kingdom (10.7 percent). (Sivy, et al., 2004)
Given the global demand for high tech workers, there is a greater exportation of jobs overseas due to a combination of cheaper wages, as well as a better educated workforce in mathematics and science.
No Child Left Behind does not emphasize Science in the curriculum. Therefore, the No Child Left Behind does not emphasize Science in the curriculum. Therefore, the applicational possibilities of mathematics often overlooked!applicational possibilities of mathematics often overlooked!
43
4 Reasons for U.S Decline4 Reasons for U.S Decline
1. The language of math matters! Building number connections centered around a base-10 principle is crucial in the development of mathematical efficiency when problem solving.
2. Dry and boring material. Mathematical skill building needs to be FUN, and therefore needs to be presented in the format of games and activities.
3. Too much focus on the answers. In order to become facilitators of mathematical knowledge, students should practice multiple methods of problem solving from both a visual-spatial and verbal approach.
4. Time on task. Most elementary math instruction occurs in the afternoon, just 45 minutes per day.
Percent of First Year College Students in 2-year and 4-year Institutions Requiring Remediation
Those who require remediation in college often fail to earn a degree
Many High School Graduates Report Gaps in Their Preparation
Impact of When Algebra is Taken
Problem SolvingProblem Solving
What’s the problem with problem solving?
Basic computation: 80%-90% accuracy
vs.
Comprehension:Comprehension:33% accuracy33% accuracy
Optimizing the Optimizing the Problem Solving ProcessProblem Solving Process
• ProcessingProcessing by encouraging students to restate the problem and state their solution plan.
• Vocabulary developmentVocabulary development for consistent and clear verbal and written communication.
• Effective questioningEffective questioning to foster independent, creative thinking and curiosity.
• Equitable practicesEquitable practices to ensure student engagement and to foster resourcefulness, independence, interdependence, and confidence.
Sample 8th grade problem I373
- 194a) 179b) 189
c) 221d) 289
94% correct
Sample 8th grade problem IIIIf John drives his car at an average speed of 50 miles per hour, how long will it take him to drive 275 miles?
a) 5 hours, 5 minutesb) 5 hours, 25 minutesc) 5 hours, 30 minutesd) 2 hours, 5 minutes
32% correct
Problem Solving Computational and Procedural Skills
Conceptual Understanding
“Where” the math works “How” the math works
“Why” the math works
Balance Defined
Memorization Procedureswithout connections to understanding, meaning or concepts
Procedureswith connections to understanding, meaning or concepts
Doing math
Cognitive Demand Spectrum
Tasks that require memorized procedures in routine ways
Tasks that require engagement with concepts, and stimulate students to make connections to meaning, representation, and other mathematical ideas
Smith, Stein, et al.
Enhancing Math Skills
• In both algebra and geometry, use different steps and the same colors to do the same steps in each problem. Alternatively, students can use different colored highlighters to identify the steps of problems.
• In word problems, box or highlight the words indicating the operation, underline the words indicating what the problem is looking for, and circle all numbers, whether in digits or in words. This will help the child translate the words into math.
Enhancing Math Skills
• 24. You can make this game yourself by taking square cards on which you put 4 numbers. The students are to make 24 by adding and subtracting all the number, and there may be more than one way to achieve the goal. More advanced students may have double-digit numbers and may use multiplication and division and division. Students in algebra can use the number as exponents.
Enhancing Math Skills
• PEMDAS. This exercise teaches students the importance of order of operations and is for middle school students. The numbers are place horizontally and the student is to use the order of operations without changing the positions of the numbers to make a true statement. You can use problems from the text by removing the operational signs.
Enhancing Math Skills
PEMDAS.
3 8 10 6 5
Enhancing Math Skills
PEMDAS.
(3+8) = (10+6-5)
(3+8-10) = (6-5)
Enhancing Math Skills
PEMDAS.
(3+8-10) = (6-5)
Enhancing Math Skills
School Statistics.
Graphing
Human Graphs
Enhancing Math Skills
Measurement
Length
Volume
Weight
Cinquains
• a short, simple poem.
• It is a creative outlet for reflecting on the meaning of a concept of information just learned.
Why use CINQUAINS in the classroom
• Synthesizes information for greater understanding
• An evaluation tool as an alternative to a regular assignment or quiz
• Creative expression—get students involved more
Guidelines
• The first line is a one word title (usually a noun).• The second line is a two word description of the topic
(usually two adjectives).• Line three is three words expressing action of the
topic (usually three “ing”• words).• The fourth line is a four word phrase showing feeling
for the topic.• The fifth line (last line) is a one word synonym that
restates the essence of the• topic.
6th Grade Cinquain
Integer
Positive, Negative
Opposing, Counting, Numbering
Whole and its opposite
Signed
7th Grade Cinquain
Mean
Central Number
Summing, Averaging, Middle
Sum divided by count
Statistic
Algebra 1 Cinquain
Slope
Steepness, direction
Rising before running
Measures rate of change
Pitch
Write your own Cinquain….
• Polygon
• Probability
• Circle
• Inequality
Prior Knowledge Shout Out
Assessing Your Current Knowledge Of How We Learn Mathematics
1. The brain comprehends numerals first as words, then as quantities.
2. Learning to multiply, like learning spoken language, is a natural ability.
3. It is easier to tell which is the greater of two larger numbers than of two smaller numbers.
4. Practicing mathematics procedures makes perfect.
5. Using technology for routine calculations leads to greater understanding and achievement in mathematics.
6. Symbolic number operations are strongly linked to the brain's language areas.
Is the statement generally True or False?
Each Brain is Unique
• We are products of genetics and experience
• The brain works better when facts and skills are embedded in real experiences
4
Multiple Learning Styles
HOW DO INDIVIDUALS PROCESS INFORMATION?
SIGHT - 75%
HEARING - 13%
TOUCH- 6%
TASTE - 3%
SMELL - 3%
Information Flow
• Eyes 10,000,000
Sensory System
TotalBandwith
(Bits/sec)
ConsciousBandwith
(Bits/sec)
• Ears 100,000
• Skin 1,000,000
• Taste 1,000
• Smell 100,000
40
30
5
1
1
Mind Map Guidelines
• Start in the center of the paper with an image or symbol.
• Make the branches closest to the center thicker, and make the attached branches farther from the center thinner.
• Each line must be connected, starting from the central image.
• Organize like things together on connected branches.
• Use at least 3 colors.• Use colors as your own special code to show
people, topics, themes or dates, to make the Mind Map more beautiful.
• Include symbols and pictures whenever possible.
• Develop your own personal style of Mind Mapping. Your Mind Maps should be unique.
Brain Compatible Strategies
HandshakeHandshake Handshake
Brain Compatible StrategiesHandshake – Handshake – Handshake
This activity can be used in many ways from introductions to reviewing content. Facing
each other pairs shake hands while repeating their life history, recent content, or content anticipated. It is not intended as a listening
activity as both people are talking at the same time. If they cannot think of what to say next
they repeat “hallelujah” over and over until the brain engages once again.
Thinking Outside the Box
Turn the Roman numeral IX into 6 by making only one line.
IX IX invert it XI top half is VI
SIX
I X 6 = 6
How many squares are there in the figure below?
Thinking Outside the Box
4 X 4 = 16 4 across and 4 down
Four more, 4 by 4 interior squares
4 X 4 = 16 + 4 = 20
4 X 4 = 16 + 4 = 20 + 2 = 22
Two more 4 by 4, vertical middle squares
Two more 4 by 4, horizontal middle squares
4 X 4 = 16 + 4 = 20 + 2 = 22 + 2 = 24
One more 4 by 4, middle square
4 X 4 = 16 + 4 = 20 + 2 = 22 + 2 = 24 + 1 = 25
3 by 3 squares
3 by 3
3 by 3
3 by 3
Four more, 3 by 3 squares
4 X 4 = 16 + 4 = 20 + 2 = 22 + 2 = 24 + 1 = 25 + 4 = 29 3 by 3
One more large square
4 X 4 = 16 + 4 = 20 + 2 = 22 + 2 = 24 + 1 = 25 + 4 = 29 + 1 = 30
What is Brain Based Learning?
• Taking what we know about the brain, about development and about learning and combining those factors in intelligent ways to connect and excite students’ desire to learn.
• Combining emotional, factual and skill knowledge into a cognitive tool.
Brain Basics
Parts of the Brain
Brain Functions
Brain Processing
• Frontal lobe - Cortex– Creativity - Judgment - Optimism - Context– Planning - Problem solving - Pattern making
• Upper temporal lobe - Wernicke’s Area– Comprehension - Relevancy - Link to past (experience) - Hearing - Memory
- Meaning• Lower frontal lobe - Cortex
– Speaking/language - Broca’s area• Occipital lobe - Spatial order
– Visual processing - Patterns - Discovery• Parietal lobe
– Motor - Primary Sensory Area - Insights - Language functions• Cerebellum
– Motor/motion - Novelty learning - cognition - balance - posture
Musical Activity
Choice (content, process, resource, environment)
Relevance (personal, in context)
Feedback (as immediate as possible)
Engagement (emotionally, physically, verbally)
“How Are You Today?”(choose one of these)
Passionate Very BlessedAmazing Nearly IllegalSlightly Irregular AwesomeExcellent Beautiful It’s a Long Story Highly
Underrated
Increase Student Motivation
Working in cooperative groups; encouraging teamwork
Providing challenging, exciting, and relevant curriculum
Extrinsically from rewards (or punishments) such as grades, promotion, jobs and opportunities
Providing an environment to develop ownership for learning
Teachers and Students Working Together!
• Intrinsic student motivation is powerful!
•Extrinsic motivations are less powerful in student learning
Differentiation by InterestMath
Sequence of Numbers –Real Number System Choice Board
Write a poem about the number groups or sequence of numbers
Sing a song/rap about the groups or sequence of numbers
Draw a picture that represents the grouping of numbers
Explain and describe the problem generated from a geometric representation of an irrational number using the Pythagorean Theorem
Construct a number line with only decimals and fractions with different denominators
Web search and report- if it’s not a Real Number, what is it?...how do we sequence non Real numbers
Write a paragraph about the importance of understanding the ordering of numbers in elation to Money/Finances and what number groups are associated with money
Emotion and Learning:Implications for Mathematics
Instruction
Emotion and Education
Emotion drives attention, which drives learning, memory and problem solving and almost everything else we do…by not exploring the role that emotion plays in learning and memory, our profession has fallen decades behind in devising useful instructional procedures that incorporate and enhance emotion. (Sylwester, 1998)
Mathematics Education Slope
• Standard 1: The student must be able to calculate the slope of a line that passes through any two points on the coordinate plane.
• Standard 2: The student must differentiate between positive/negative slope.
• Standard 3: The student must understand the meaning of slope in application problems.
Mathematics EducationSlope: Standard Approach
• Algebraically define slope
• Calculate slope for several pairs of points
• Describe positive and negative slope
Mathematics EducationSlope: Standard Approach
12
122211 ),(),(
xx
yymasdefinedisslopeyxandyxFor
5
4
5
4
23
488,3,4,2
m
righttoleftfromfallslinem
horizontalislinem
righttoleftfromriseslinem
0
0
0
Mathematics EducationSlope: Emotional Approach
Mathematics EducationSlope: Emotional Approach
Year Blue Book Value1997 $21,0651998 $18,0251999 $16,2402000 $14,4602001 $12,7052002 $11,0002003 $9,2502004 $7,600
Mathematics EducationSlope: Emotional Approach
Subaru Outback DepreciationSource: www.kellybluebook.com
$0
$5,000
$10,000
$15,000
$20,000
$25,000
1997 1999 2001 2003
Year
Valu
e
Mathematics EducationSlope: Emotional Approach
12
122211 ),(),(
xx
yymasdefinedisslopeyxandyxFor
5
4
5
4
23
488,3,4,2
m
righttoleftfromfallslinem
horizontalislinem
righttoleftfromriseslinem
0
0
0
Mathematics EducationSlope: Emotional Approach
• Interpretation: Depreciation of -1,923.57 $ per year• Follow Up:
– Which cars might have higher/lower rates of depreciation? Why?
– Do any cars have a positive appreciation? Why?
– Will depreciation always be linear? Why?
27.195320041997
600,7065,21
m
Mathematics EducationLines: Emotional Approach
The price of cable television has rapidly increased in the recent past. Mathematically model this trend and use your model to make several predictions regarding the future pricing of cable television.
Average Cable TV PricesSource: Paul Kagan Associates, Inc
1994 21.62
1995 23.07
1996 24.41
1997 26.48
1998 27.81
1999 28.92
Mathematics EducationLines: Emotional Approach
Cable TV Costsy = 1.5083x + 15.581
R2 = 0.9938
0
5
10
15
20
25
30
35
4 5 6 7 8 9
Year
Co
st
Mathematics EducationLines: Emotional Approach
Cable TV has steadily raised about $1.51 per year. A linear model was used because when a steady increase of something is used it is a viable way to predict the future. The linear model for this particular problem was y = 1.508x +15.581. The model tells us that the increase each year is about $1.51 and in 1990 the cost was about $15.58. If you plug in “x” … you can predict the cost … in the year 2010 the cost will be about $45.74.
(Student Response)
Mathematics EducationQuadratics: World Population
World population has been increasing rapidly in the last century. Accurately predicting world population is important from an economic, political and social standpoint.
World Population
(source: www.xist.org)
Year Pop (Billions)
1950 2.555
1955 2.780
1960 3.039
1965 3.346
1970 3.708
1975 4.088
1980 4.457
1985 4.855
1990 5.284
1995 5.691
Mathematics EducationQuadratics: World Population
Mathematics EducationQuadratics: World Population
World Population y = 0.0004x2 + 0.0526x + 2.5113
R2 = 0.9993
0
1
2
3
4
5
6
7
0 10 20 30 40 50
Year (1950=0)
Po
pu
lati
on
(B
illi
on
s)
Mathematics EducationQuadratics: World Population
I think the quadratic … (y = ax^2 +bx+c) better illustrates the world population. The past chart shows that the population [rate] rises nearly a full billion every 10 years. The quadratic form follows this pattern more closely. Notice that more of the points line up with the quadratic’s solution.
(Student Response)
Mathematics EducationQuadratics: World Population
Having an accurate count of future population makes it possible to make predictions for what social needs must be met in the future. It tells us how many people will be living in a voting district or estimated population of a city. It can tell us how much money we will need for social program like welfare, or how big to build roads leading to growing rural areas. With a linear growth model we will not have accurate numbers for determining social, political and economic needs. (Student Response)
Movement
• Physical activity helps oxygenate the blood, which in turn oxygenates the brain.
• Sensorimotor skills and practice encourage muscle memory.
• Muscle memory can tie to academics.
Problem-solving is promoted through individual and group physical activities.
Partners
Visual Representations are excellent for teaching math
Concrete to Representational to Abstract(CRA)
Three stages of learning
C = Learning through concrete hands-on manipulative objects
R = Learning through pictorial forms of the math skill
A = Learning through work with abstract (Arabic) notation
CRA Strategy
CRA Strategy
Fractions with popsicle sticks and how they translate into
adding and subtracting fractions
Small Manipulative Modeling Tips
Use transparent manipulative objects on an overhead projector
Apply magnetic adhesive to a teacher set of manipulative objects to use on magnetic white boards
Develop large posterboard renditions of the manipulative objects to use on table tops or walls
Use an evaluated table with an angled stand that can support manipulatives securely
Provide students with their own set of manipulatives as you modeled
Virtual Manipulatives
Virtual Library of Mathematics Manipulatives
http://nlvm.usu.edu/
Parallel Lines Cut by a Transversal
• Kinesthetic: Walk It Tape the diagram below on the floor with masking tape. Two players stand in assigned angles. As a team, they have to tell what they are called (ie: vertical angles) and their relationships (ie: congruent). Use all angle combinations, even if there is not a name or relationship. (ie: 2 and 7)
12 3
45
67
8
Circumference and Diameter of a Circle
Circumference and Diameter of a Circle
Create enough space for all students to form a circle.
Have one student walk heel to toe around the entire circle making sure to count his steps and finish at the same spot where he started.
At the original starting point have the same student turn toward the circle and walk a straight line from one side of the circle to the other, heel to toe and counting all of his steps.
That student has just marked both the circumference and diameter of a circle. The relationship is pi. The general equation is the circumference of a circle is a bit more than three times the diameter of the same circle (pi=3.141).
Doing this kinesthetically could produce and inexact results but will be close enough to teach the concept.
Triangle
Triangle
Have five students stand side by side, each about five feet apart.
Ask the student to the immediate left and right of center to take three steps forward.
Ask students on the outside to take six steps forward. The group has now formed a triangle.
ArabicComprehension
[8]
VerbalComprehension
[EIGHT]
MagnitudeComparison
5 < 8?
Prepare & ExecuteResponse
[right]
Identification Comparison Response Notation effect Distance effect Response-side effect(arabic vs. verbal) (close vs. far) (left vs. right)
(S. Dehaene, J. Cognitive Neuroscience, 8(1), p49, 1996)
Mapping Cognitive Functions onto Neural Structures
Dehaene & Cohen 1995
Comparison
139
The Neural Machinery of The Neural Machinery of MathematicsMathematics
EXECUTIVEDYSFUNCTION
• Selective Attention
• Planning Skills
BRAIN REGION
• Anterior Cingulate/• Subcortical
structures
• Dorsal-lateral PFC
MATH SKILL
• Procedure/algorithm knowledge impaired
• Poor attention to math operational signs
• Place value mis-aligned
• Poor estimation• Selection of math process
impaired• Difficulty determining
salient information in word problems
140
The Neural Machinery of The Neural Machinery of MathematicsMathematics
EXECUTIVEDYSFUNCTION
• Organization Skills
• Self-Monitoring
BRAIN REGION
• Dorsal-lateral PFC
• Dorsal-lateral PFC
MATH SKILL
• Inconsistent lining up math equations
• Frequent erasers• Difficulty setting up
problems
• Limited double-checking of work
• Unaware of plausibility to a response.
• Inability to transcode operations such as (4X9) = (4X10) – 4
The Neural Machinery of The Neural Machinery of MathematicsMathematics
EXECUTIVEDYSFUNCTION
• Retrieval Fluency
BRAIN REGION
• Orbital frontal PFC• Anterior Cingulate• Dorsolateral PFC(dictated by strategy
and effort)
MATH SKILL
• Slower retrieval of learned facts
• Accuracy of recall of learned facts is inconsistent
Working Memory In The BrainWorking Memory In The Brain
Working Memory System
• Phonological Loop
• Visual-Spatial Sketchpad
• Central Executive System
Mathematical Skill
• Retrieval of math facts
• Writing dictated numbers
• Mental math
• Magnitude comparisons
• Geometric Proofs
• Inhibiting distracting thoughts
• Modulating anxiety
• Regulating emotional distress.
Executive Functioning Skills: (frontal lobes)
Executive control mechanisms are a set of directive processes such as planning, self-monitoring, organizing, and allocating attention resources to effectively execute a goal directed task.
Executive functioning dictates “what to do when”, a critical process in solving word problems.
Executive functioning allows students to choose an appropriate algorithm when problem solving.
The Neural Machinery of The Neural Machinery of MathematicsMathematics
Executive Functioning Skills:
Dorsal-lateral cortex - helps to organize a behavioral response to solve complex problem solving tasks.
Orbitofrontal cortex - rich interconnections with limbic regions and helps modulate affective problem solving, judgement.
Anterior cingulate cortex - allocates attention resources and modulates motivation.
The Neural Machinery of The Neural Machinery of MathematicsMathematics
Discovery Neurogenesis
The discovery thatthe human brain cangrow new brain cells.
What downregulates neurogenesis:• Acute and chronic stress• Lack of physical activity• Poor nutrition
What enhances neurogenesis:• Low stress• 30-90 min./day of physical activity• Quality nutrition/Novelty
Neurogenesis:Neurogenesis: If you’re not enhancing it,
you’re working against yourself.
Runner Bingo
Discovery - Allostasis
“Allo” ….meaning “adjusted”
“Stasis” … meaning “stability”literally…
“a new stable baseline”
Breathing
Humor
Your Most Powerful Tool
Laughter!Physiological Effects –
Increases oxygen in the brain
Produces endorphins which boosts learning
Psychological Effects –
Bonding occurs when laughing together
Makes kids comfortable
Happily Ever Laughter
• Laughing releases the feel good hormone – endorphine- also a natural pain killer.
• Laughing lowers your blood pressure and heart rate.
• Laughing raises T-lymphocytes in your body.
• Increases your immune system .
• There is a 45 minutes residual chemical effect in your body after hearty laughter.
Laughter in the Classroom
• Increasing a feeling of hope --- taking the drudgery out of the time in class.
• Increasing retention
• Building rapport
• Relieving stress
How Humor Promotes RetentionHow Humor Promotes Retention
OxygenEndorphin Surge
Gets Attention!
Positive Climate
Increases Retention
Discipline
Mental Health
Function Junction
Function Junctionx f(x) Rule
Function Junctionx f(x) Rule
3x-4
Function Junctionx f(x) Rule
2x+3
• Demonstrate individual support for all students• Encourage and employ collaborative learning• Heighten student awareness about benefits of
nutrition, sleep, and fitness• Emphasize the importance of water for electrolytic
balance and hydration• Create opportunities for paired learning and peer
sharing • Provide experiences that target student interests
and concerns
Meet the intellectual, social, emotional, and physical developmental needs of
adolescents
•Tap prior knowledge to construct meaning
•Employ scaffolding strategies
•Teach study skills and learning how to learn
•Utilize both inductive and deductive reasoning
•Afford all students adequate processing time
Create atmosphere that is high in challenge and low in threat
• Provide choice of topics, ways of learning, and modes of expression
• Feature hands-on and touchable learning situations
• Increase motor activity, lab experiences, arts, music, and drama
• Build and nurture curiosity• Offer opportunities for self-assessment and
reflection • Give frequent and elaborative feedback
Emphasize an active approach to
instructional practice
• Seek patterns, relationships, and connections among the disciplines
• Reveal interconnectedness of concepts across multiple contexts
• Select essential concepts to investigate and explore deeply
• Plan and implement integrated units of study • Use project-based learning that connects
students’ effort with real life• Infuse the arts across the curriculum to build
and extend meaning
Adopt curricular models that support best practice
• Project-based and Authentic Learning Project-based and Authentic Learning OpportunitiesOpportunities
• Simulations and Role PlaysSimulations and Role Plays• Debates and Learner DiscourseDebates and Learner Discourse• Learning Strategies and Strategic Instruction Learning Strategies and Strategic Instruction
– Advance and Post organizersAdvance and Post organizers– Frequent checks for understandingFrequent checks for understanding– Elaboration and feedbackElaboration and feedback
Classroom Practices that Promote
Adolescent Learning
Content EnhancementsContent Enhancements• StorytellingStorytelling
• DrawingDrawing
• MusicMusic
• MnemonicsMnemonics• Concept maps; mind mappingConcept maps; mind mapping
Classroom Practices that Promote
Adolescent Learning
Batting Averages
Batting Averages – Middle School
Student teams will choose to follow one baseball team for ten games during the regular season. Each team will choose the most valuable member of that team for the season and be ready to defend the decision using appropriate statistics as evidence. The team will also be responsible for a broadcast in which they will report the analysis, categorization, and interpretation of the statistics. Team members will also create visual representations to demonstrate the results, including a Powerpoint presentation. In addition, they will submit individual journals in which each member has tracked the process of the broadcast development as well as daily team statistics.
Dealing with Uncertainty
Dealing with Uncertainty
Each student choose a partner, and together they take turns collecting and recording data.
The data recorder begins by guessing by guessing the sum of the last two digits of an imaginary zip code and writes the guess on data form.
Using the zip code page from a phone book, the other partner, with eyes closed randomly points to a zip code number on a page.
Dealing with Uncertainty Guess of sum of the last two digits________
Last two digits Sum of last two digits Matched the guess
Dealing with Uncertainty
The data collector then gives the actual sum of the last two digits of the number closest to his/her partner’s finger as well as the sum of the last two digits of the next number.
This is done for the next 13 consecutively listed numbers, bringing the total to 15 sums in all.
The recorder writes each of these sums and counts up and records the number of times his or her guess matched the sums.
How Do Genes Fit?
How Do Genes Fit? - Middle School
The goal of this unit, is to give the students hands-on experience in choosing appropriate strategies for problem solving. The students will find themselves drawing from past knowledge and experience as they progress in a logical and sequential manner toward their conclusions. They will begin to develop the critical thinking skills so vital today, skills necessary to create multiple strategies for multiple solutions.
How Do Genes Fit? - Middle School
Students will “marry” by picking matched numbers out of two baskets (one for girls and one for boys). They will organize and chart their hereditary history, and decide on the number of offspring they will have. Employing Mendel’s elementary probability models, they will attempt to predict the the probability of hair and eye color of these offspring. To do this, they must make their own probability models in a clear and easily understand manner.
How Do Genes Fit? - Middle School
Student Mother Father Sibling 1 Sibling 2 Sibling 3
Hair Color
Eye Color
How Do Genes Fit? - Middle School
In part 2, students with similar hair and eye color combinations will group together in sets of two pairs, and compare their model outcomes to see if similarities exist. They will then begin to look for any correlations, either positive or negative, to see if any outcomes are predictable. If a correlation is the extent to tow or more things are related to one another, would there be a positive or negative correlation between hair or eye color or parents and their off spring? Why?
Graphing
Graphing – Middle School
Students will work in groups (teams) of three to four. Each team will choose an original topic (name brand clothes) for use in the survey.
The teams will gather data from their sample populations, collate their data, and construct three graphs as a team; bar graph, a circle graph, and a pictograph, each using the identical set of data.
Graphing – Middle School
Each team will submit a written report describing the organization of their study, their data-gathering methods, and their conclusion.
The team will present their findings to the class in an oral report using visual supports (graph). The final oral presentation will also include an evaluation of the different graph types, listing positive and negative aspects as well as the appropriateness of each graph type in regard to the data.
Stock Market
Stock Market – Middle School
Students are each given a theoretical $500 to invest. They then pair up, “pool” their money, and select 6 or 7 stocks from a previously selected group of 20 choices.
These stocks are to be tracked over a fourteen week period on a spread sheet as well as a daily line graph (one graph per stock).
At the end of the four week period, students will combine the separate graphs into one long “multi-graph” showing the team’s total stock data.
Stock Market – Middle School
For the final graph, the teams will take the data directly from the spreadsheets so as to reduce the possibility of error due to carelessness.
Students as well as teacher will assess the final graphs as to their accuracy, readability, neatness, and ease of comprehension.
In addition, the students will draw inferences about their data.
Business Proposal
Business Proposal
Student groups will determine the business they might like to own and operate.
They will research the typical costs of running the business and the typical income made with the business.
They will perform a break-even analysis to study the results.
They will present their findings to the class in the form of a bank proposal.
Business Proposal
Students will create appropriate graphs using the research date and will demonstrate how they will arrive at and/or pass the break-even point.
The proposal should be persuasive.
Students will also reflect on their own presentations in a post-project reflection and in journal entries.
Problem Solving
Activities for Learning Problem Solving
Activity Description Process
Brainstorming Group Problem-solving technique that involves the spontaneous contribution of ideas from all members of the group
Students seated in small groups speaking simultaneously yet listening to each other’s ideas without pre-judging.
Make It Simple Analyze the problem Students working on problems in order to break them down into their basic components
Using Logical Reasoning Inductive reasoning moves from particular examples to general principle
Deductive reasoning move from general principle to specific examples
Inductive reasoning: If every bowl of soup served you was hot, you would assume all soup is served hot.
Deductive reasoning: If soup is always served hot, and you are eating soup, then what you are eating must be hot.
Activities for Learning Problem Solving
Activity Description Process
Work Backwards Begin with the problem solution and work backwards to identify the original problem
Have students start with a problem solution and have them work to create an original problem with a result that will match the original
Make a Picture or a Diagram A picture or a diagram gives visual learners an opportunity to better understand the problem
While reading the problem, students (or the teacher) are (is) encouraged to diagram or illustrate the problem as if it were a story board
Make a Chart or a Table A chart or table helps to present information in an organized way so that students can better visualize and understand the given information
While reading the problem, students (or the teacher) are (is) encouraged to create a chart or table to help organize thinking
Activities for Learning Problem Solving
Activity Description Process
Make an Organized List An organized list divides the information into groupings that help students understand and process the problem’s information
While reading the problem, students (or the teacher) are (is) encouraged to create an organized list to better visualize the problem’s information
Look for a Pattern The innately seeks to reorganize large groupings of information into smaller groupings that repeat in a fixed sequence it can comprehend
While reading the problem, students (or the teacher) are (is) encouraged to look for smaller subsets of information that seem to repeat throughout the problem
Guess and Check This method probable is the first strategy the students used when they were younger
Students guess the problem’s solution, and then check to see if their guess is correct
Home Improvement
Home Improvement
The class is divided into different teams for house painting.
Each team must submit a bid for painting several rooms in the principal’s home.
They must put together an estimate of what the job will run, both in labor and supplies as well as profit.
The job is to include painting the living room, dinning room, and family room walls, as well as the ceilings in all three areas.
A floor plan and elevation drawings are supplied.
Coaster Math Project
Coaster Math Project – High School
A large national amusement part is sponsoring a contest for design and construction of model roller coaster made from everyday, basic materials (e.g. toothpicks, straws, popsicle sticks, and metal tubing.) The coaster is to give the sensation of speed, acceleration, and thrill while keeping the ride safe. All of the completed models are to be functional so that designers will be able to demonstrate the capabilities of the various coaster models.
Data Analysis
Data Analysis – Grade 3 - 5
Have students discuss the number of brothers and sisters they have.
List the number in columns on the board.
Discuss that some students have same number or siblings and others have different numbers; some have more and some have less.
This activity introduces the concepts of range, mode, mean and median.
Brain Compatible Strategies
Truth – Truth – LieLike some of the others on this list you may have used this before. Remember that the bigger your repertoire the more
options you have “in the moment.” Have students in groups of three or more.
Each comes up with two truths and one lie. These can be personal or content
based. This is another way to check for understanding and review content. They
try to determine which is the lie.