Tuesday, March 25, 2014
Friday, August 9, 2013
Tuesday, March 26, 2013
Early Number Sense Plays Role in Later Math Skills
From www.teachhub.com
Early Number Sense Plays Role in Later Math Skills
WASHINGTON (AP) — We know a lot about how babies learn to talk, and youngsters learn to read. Now scientists are unraveling the earliest building blocks of math — and what children know about numbers as they begin first grade seems to play a big role in how well they do everyday calculations later on.
The findings have specialists considering steps that parents might take to spur math abilities, just like they do to try to raise a good reader.
This isn't only about trying to improve the nation's math scores and attract kids to become engineers. It's far more basic.
Consider: How rapidly can you calculate a tip? Do the fractions to double a recipe? Know how many quarters and dimes the cashier should hand back as your change?
About 1 in 5 adults in the U.S. lacks the math competence expected of a middle-schooler, meaning they have trouble with those ordinary tasks and aren't qualified for many of today's jobs.
"It's not just, can you do well in school? It's how well can you do in your life," says Dr. Kathy Mann Koepke of the National Institutes of Health, which is funding much of this research into math cognition. "We are in the midst of math all the time."
A new study shows trouble can start early.
University of Missouri researchers tested 180 seventh-graders. Those who lagged behind their peers in a test of core math skills needed to function as adults were the same kids who'd had the least number sense or fluency way back when they started first grade.
"The gap they started with, they don't close it," says Dr. David Geary, a cognitive psychologist who leads the study that is tracking children from kindergarten to high school in the Columbia, Mo., school system. "They're not catching up" to the kids who started ahead.
If first grade sounds pretty young to be predicting math ability, well, no one expects tots to be scribbling sums. But this number sense, or what Geary more precisely terms "number system knowledge," turns out to be a fundamental skill that students continually build on, much more than the simple ability to count.
What's involved? Understanding that numbers represent different quantities — that three dots is the same as the numeral "3'' or the word "three." Grasping magnitude — that 23 is bigger than 17. Getting the concept that numbers can be broken into parts — that 5 is the same as 2 and 3, or 4 and 1. Showing on a number line that the difference between 10 and 12 is the same as the difference between 20 and 22.
Factors such as IQ and attention span didn't explain why some first-graders did better than others. Now Geary is studying if something that youngsters learn in preschool offers an advantage.
There's other evidence that math matters early in life. Numerous studies with young babies and a variety of animals show that a related ability — to estimate numbers without counting — is intuitive, sort of hard-wired in the brain, says Mann Koepke, of NIH's National Institute of Child Health and Human Development. That's the ability that lets you choose the shortest grocery check-out line at a glance, or that guides a bird to the bush with the most berries.
Number system knowledge is more sophisticated, and the Missouri study shows children who start elementary school without those concepts "seem to struggle enormously," says Mann Koepke, who wasn't part of that research.
While schools tend to focus on math problems around third grade, and math learning disabilities often are diagnosed by fifth grade, the new findings suggest "the need to intervene is much earlier than we ever used to think," she adds.
Exactly how to intervene still is being studied, sure to be a topic when NIH brings experts together this spring to assess what's known about math cognition.
But Geary sees a strong parallel with reading. Scientists have long known that preschoolers who know the names of letters and can better distinguish what sounds those letters make go on to read more easily. So parents today are advised to read to their children from birth, and many youngsters' books use rhyming to focus on sounds.
Likewise for math, "kids need to know number words" early on, he says.
NIH's Mann Koepke agrees, and offers some tips:
___
Early Number Sense Plays Role in Later Math Skills
Scientists unravel how kids learn math, 1st grade abilities key to skills later on
WASHINGTON (AP) — We know a lot about how babies learn to talk, and youngsters learn to read. Now scientists are unraveling the earliest building blocks of math — and what children know about numbers as they begin first grade seems to play a big role in how well they do everyday calculations later on.The findings have specialists considering steps that parents might take to spur math abilities, just like they do to try to raise a good reader.
This isn't only about trying to improve the nation's math scores and attract kids to become engineers. It's far more basic.
Consider: How rapidly can you calculate a tip? Do the fractions to double a recipe? Know how many quarters and dimes the cashier should hand back as your change?
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About 1 in 5 adults in the U.S. lacks the math competence expected of a middle-schooler, meaning they have trouble with those ordinary tasks and aren't qualified for many of today's jobs.
"It's not just, can you do well in school? It's how well can you do in your life," says Dr. Kathy Mann Koepke of the National Institutes of Health, which is funding much of this research into math cognition. "We are in the midst of math all the time."
A new study shows trouble can start early.
University of Missouri researchers tested 180 seventh-graders. Those who lagged behind their peers in a test of core math skills needed to function as adults were the same kids who'd had the least number sense or fluency way back when they started first grade.
"The gap they started with, they don't close it," says Dr. David Geary, a cognitive psychologist who leads the study that is tracking children from kindergarten to high school in the Columbia, Mo., school system. "They're not catching up" to the kids who started ahead.
If first grade sounds pretty young to be predicting math ability, well, no one expects tots to be scribbling sums. But this number sense, or what Geary more precisely terms "number system knowledge," turns out to be a fundamental skill that students continually build on, much more than the simple ability to count.
What's involved? Understanding that numbers represent different quantities — that three dots is the same as the numeral "3'' or the word "three." Grasping magnitude — that 23 is bigger than 17. Getting the concept that numbers can be broken into parts — that 5 is the same as 2 and 3, or 4 and 1. Showing on a number line that the difference between 10 and 12 is the same as the difference between 20 and 22.
Factors such as IQ and attention span didn't explain why some first-graders did better than others. Now Geary is studying if something that youngsters learn in preschool offers an advantage.
There's other evidence that math matters early in life. Numerous studies with young babies and a variety of animals show that a related ability — to estimate numbers without counting — is intuitive, sort of hard-wired in the brain, says Mann Koepke, of NIH's National Institute of Child Health and Human Development. That's the ability that lets you choose the shortest grocery check-out line at a glance, or that guides a bird to the bush with the most berries.
Number system knowledge is more sophisticated, and the Missouri study shows children who start elementary school without those concepts "seem to struggle enormously," says Mann Koepke, who wasn't part of that research.
While schools tend to focus on math problems around third grade, and math learning disabilities often are diagnosed by fifth grade, the new findings suggest "the need to intervene is much earlier than we ever used to think," she adds.
Exactly how to intervene still is being studied, sure to be a topic when NIH brings experts together this spring to assess what's known about math cognition.
But Geary sees a strong parallel with reading. Scientists have long known that preschoolers who know the names of letters and can better distinguish what sounds those letters make go on to read more easily. So parents today are advised to read to their children from birth, and many youngsters' books use rhyming to focus on sounds.
Likewise for math, "kids need to know number words" early on, he says.
NIH's Mann Koepke agrees, and offers some tips:
- Don't teach your toddler to count solely by reciting numbers. Attach numbers to a noun — "Here are five crayons: One crayon, two crayons..." or say "I need to buy two yogurts" as you pick them from the store shelf — so they'll absorb the quantity concept.
- Talk about distance: How many steps to your ball? The swing is farther away; it takes more steps.
- Describe shapes: The ellipse is round like a circle but flatter.
- As they grow, show children how math is part of daily life, as you make change, or measure ingredients, or decide how soon to leave for a destination 10 miles away,
___
Wednesday, August 8, 2012
OVERVIEW
We have taken several months to research and investigate so that we might discover the best approaches through research that will support math learning for second language learners. The main purpose of the research is to determine Esperanza's goals and find the strategies necessary to help all scholars achieve those goals. Some of our initial thoughts are . . .
- Do what it takes to serve second language learners
- Find ways to make math culturally relevant
- Make decisions based on research and philosophy, not programs
- Remember that one program is not sufficient to do the job
- Effective teachers draw upon multiple strategies and understand good math pedagogy to make instructional decisions
- The teacher is the determining factor in student achievement
- Professional development and coaching are necessary to implement effective math instruction
According to Michael Schmoker the most important decisions come to what we teach, how we teach, and the use of authentic literature. He states that there is a greater need for focus (simplicity, clarity and priority) in education and that a dedication to these principles over time will create the school culture and climate that will create necessary change. (Schmoker, chapter 1).
The hope is through the blog to create a simple and clear plan to bring about the highest achievement possible for all students at Esperanza and help us prioritize an appropriate timeline to watch this progression and make the best decisions.
The following topics must be considered to implement such a plan:
A Teacher Education
B Teacher Support
C Staying tightly aligned to the core
D Pre- / Post- Assessments
E. Performance Tasks / Assessments
F. Benchmark Assessments
G. :Formative Assessments
H. Student Self-Assessments
I. Progress Monitoring
J. Tight Alignment of Curriculum, Instruction, and Assessment
K. Daily Use of the 8 Mathematical Practice Standards
L. Vertical / Horizontal Alignment of Curriculum (teacher to teacher, year to year, K-6th math concepts progressions.)
M. The Upkeep of Basic Skills
N. Tier I, II, III Plan
O. Team / Individual Lesson / Unit Planning
P. Instructional Decisions Based on Student Data
Q. Complete Student Understanding (CPR: Conceptual, Procedural, Representational Understanding)
R. PLC Cycles w/ Math
S. Change in Pedagogy = Focus on Theory (the Why of math) and then the Application and Integration, Not just the Procedure of mathematics
T. Other Dual Immersion Support
U. Other Support for Children in Poverty
V. Bridging to English / Vocabulary Support
Thursday, August 2, 2012
Literary References / Bibliography
The following is a list of great books that teach the principles to create the best environment for kids to participate in a totally radical math program. Topics include everything from, What is the best way to assess a student? to How can I engage my kids in deeper mathematical discourse? I have included them so that I might reference the authors in the blog entries without having to list them over and over again.
I could see the following list of books used in a plethora of amazing ways from book clubs for teachers to education / endorsement like classes. I would hope that we could brainstorm on ways to get teachers at Esperanza to continue their education in the ways of mathematical pedagogy to personal mathematical study. I would hope that we would choose to adopt a few and begin a teacher education library as well. So here goes.
Archer, Anita L.
Explicit Instruction: Effective and Efficient Teaching
c. 2011 The Guilford Press
ASCD
Educating Everybody's Children
c. 1995 Association for Supervision and Curriculum Development
Brinkman, Annette (Forlini, Gary & Williams, Ellen)
Class Acts: Every Teachers Guide to Activate Learning
c. 2010 Lavender Hill Press
Brinkman, Annette (Forlini, Gary & Williams, Ellen)
Help Teachers Engage Students: Action Tools for Administrators
c. 2009 Eye on Education, Inc.
Carr, John & amp; (Carroll, Catherine & Cremer, Sarah & Gale, Mardi & Langunoff, Rachel & Sexton, Ursula)
Making Mathematics Accessible to English Learners: A Guidebook for Teachers
c. 2009 WestEd
Chamot, Anna Uhl & O'Malley, J. Michael
The CALLA Handbook: Implementing the Cognative Academic Language Learning Approach
c. 1994 Addison-Wesley Publishing Company
Clarke, David
Constructive Assessment in Mathematics
c. 1997 Key Curriculum Press
Crawford, James
Bilingual Education: History, Politics, Theory & Practice
c. 1995 Crane Publishing Company
Danielson, Charlotte
Talk About Teaching
c. 2009 Corwin Press
Gonzales, Dr. Linda Northcutt
Sheltered Instruction Handbook
c. 1994 AM Graphics & Printing
Haberman, Dr. Martin
Star Teachers
c. 2004 The Haberman Education Foundation
Hattie, John
Visible Learning For Teachers
c. 2012 Routledge
Lemov, Doug
Teach Like a Champion
c. 2010 Josey-Bass
McNamara, Julie & Shaughnessy, Meghan M.
Beyond Puzzas & Pies
c. 2010 Scholastic, Inc.
Meyers. Mary
Teaching to Diversity: Teaching & Learning in the Multi-Ethnic Classroom
c. 1993 Irwin Publishing
Mokros, Jan (Russell, Susan Jo & Economopoulos, Karen)
Beyond Arithmetic: Changing Mathematics in the Elementary Classroom
c. 1995 Dale Seymour Publications
NCTM
Involving Families in Mathematics Education
c. 2000 National Council of Teachers of Mathematics
NCTM
Mathemativs Assessment: A Practical Handbook
c. 2003 The National Council of Teachers of Mathematics, Inc.
NCTM
5 Practices for Orchestrating Productive Mathematics Discussions
c. 2011 NCTM
NCTM
The Roles of Representation in Mathematics: 2001 Yearbook
c. 2001 NCTM
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Number & Operations (Part 1): Building a System of Tens Casebook
c. 1999 Dale Seymour Publications
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Number & Operations (Part 2): Making Meaning for Operations Casebook
c. 1999 Dale Seymour Publications
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Geometry: Measuring Spaace in One, Two & Three Dimensions Casebook
c. 2002 Dale Seymour Publications
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Geometry: Examining Features of Shape Casebook
c. 2002 Dale Seymour Publications
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Algebra: Patterns, Functions & Change Casebook
c. 2008 Dale Seymour Publications
Schmoker, Mike
Focus
c. 2011 ASCD
Seeley, Cathy L.
Faster Isn't Smarter
c. 2009 Scholastic, Inc.
Sherman, Helene J. (Richardson, Lloyd L. & Yard, George J.)
Teaching Children Who Struggle with Mathematics
c. 2005 Pearson Education Inc.
Van de Walle, John A.
Elementary & Middle School Mathematics: Teaching Developmentally
c. 2010 Pearson Education, Inc
Various
Leader: Spring 2011
Published by Utah Association of Elementary School Principals
I could see the following list of books used in a plethora of amazing ways from book clubs for teachers to education / endorsement like classes. I would hope that we could brainstorm on ways to get teachers at Esperanza to continue their education in the ways of mathematical pedagogy to personal mathematical study. I would hope that we would choose to adopt a few and begin a teacher education library as well. So here goes.
Archer, Anita L.
Explicit Instruction: Effective and Efficient Teaching
c. 2011 The Guilford Press
ASCD
Educating Everybody's Children
c. 1995 Association for Supervision and Curriculum Development
Brinkman, Annette (Forlini, Gary & Williams, Ellen)
Class Acts: Every Teachers Guide to Activate Learning
c. 2010 Lavender Hill Press
Brinkman, Annette (Forlini, Gary & Williams, Ellen)
Help Teachers Engage Students: Action Tools for Administrators
c. 2009 Eye on Education, Inc.
Carr, John & amp; (Carroll, Catherine & Cremer, Sarah & Gale, Mardi & Langunoff, Rachel & Sexton, Ursula)
Making Mathematics Accessible to English Learners: A Guidebook for Teachers
c. 2009 WestEd
Chamot, Anna Uhl & O'Malley, J. Michael
The CALLA Handbook: Implementing the Cognative Academic Language Learning Approach
c. 1994 Addison-Wesley Publishing Company
Clarke, David
Constructive Assessment in Mathematics
c. 1997 Key Curriculum Press
Crawford, James
Bilingual Education: History, Politics, Theory & Practice
c. 1995 Crane Publishing Company
Danielson, Charlotte
Talk About Teaching
c. 2009 Corwin Press
Gonzales, Dr. Linda Northcutt
Sheltered Instruction Handbook
c. 1994 AM Graphics & Printing
Haberman, Dr. Martin
Star Teachers
c. 2004 The Haberman Education Foundation
Hattie, John
Visible Learning For Teachers
c. 2012 Routledge
Lemov, Doug
Teach Like a Champion
c. 2010 Josey-Bass
McNamara, Julie & Shaughnessy, Meghan M.
Beyond Puzzas & Pies
c. 2010 Scholastic, Inc.
Meyers. Mary
Teaching to Diversity: Teaching & Learning in the Multi-Ethnic Classroom
c. 1993 Irwin Publishing
Mokros, Jan (Russell, Susan Jo & Economopoulos, Karen)
Beyond Arithmetic: Changing Mathematics in the Elementary Classroom
c. 1995 Dale Seymour Publications
NCTM
Involving Families in Mathematics Education
c. 2000 National Council of Teachers of Mathematics
NCTM
Mathemativs Assessment: A Practical Handbook
c. 2003 The National Council of Teachers of Mathematics, Inc.
NCTM
5 Practices for Orchestrating Productive Mathematics Discussions
c. 2011 NCTM
NCTM
The Roles of Representation in Mathematics: 2001 Yearbook
c. 2001 NCTM
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Number & Operations (Part 1): Building a System of Tens Casebook
c. 1999 Dale Seymour Publications
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Number & Operations (Part 2): Making Meaning for Operations Casebook
c. 1999 Dale Seymour Publications
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Geometry: Measuring Spaace in One, Two & Three Dimensions Casebook
c. 2002 Dale Seymour Publications
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Geometry: Examining Features of Shape Casebook
c. 2002 Dale Seymour Publications
Schifter, Deborah (Bastable, Virginia & Russell, Susan Jo)
DMI Series: Algebra: Patterns, Functions & Change Casebook
c. 2008 Dale Seymour Publications
Schmoker, Mike
Focus
c. 2011 ASCD
Seeley, Cathy L.
Faster Isn't Smarter
c. 2009 Scholastic, Inc.
Sherman, Helene J. (Richardson, Lloyd L. & Yard, George J.)
Teaching Children Who Struggle with Mathematics
c. 2005 Pearson Education Inc.
Van de Walle, John A.
Elementary & Middle School Mathematics: Teaching Developmentally
c. 2010 Pearson Education, Inc
Various
Leader: Spring 2011
Published by Utah Association of Elementary School Principals
Friday, May 4, 2012
Algebra
Harvard Education Letter
Introducing letters to represent quantities, as in these first-grade math problems, helps students prepare for algebra. (Source: Curriculum Research and Development Group, Univ. of Hawaii)
Volume 28, Number 3
May/June 2012
May/June 2012
The Algebra Problem
How to elicit algebraic thinking in students before eighth grade
The Algebra Problem, continued
It’s Crazy Hair Day at Marshall Elementary School in Boston’s Dorchester neighborhood—which is perfect, because Tufts University researcher Bárbara Brizuela has brought a hat.
In the stovepipe style and made from oaktag paper, the hat is one foot tall. Brizuela then asks, “If I’m five and a half feet tall, how tall will I be with the hat on?” Second-grader Jasmine, smiley in a pink sweatsuit, answers, “Six and a half feet.” Rather than say, “Right!” Brizuela offers another question: “How do you know?”
Thus begins a math conversation that researchers like Brizuela believe may hold the key to tackling one of our biggest school bugaboos: algebra. As they talk, Jasmine uses words, bar graphs, and a table to describe how tall each person they discuss will be if they put on the hat. Jasmine creates a rule—“add one foot to the number you already had”—and applies it to an imaginary person 100 feet tall.
Brizuela even throws out a variable. “So, to show someone whose height I don’t know, I will use z feet,” she says, adding a z to Jasmine’s table. “What should I do now?” Jasmine pauses. “This is kind of hard,” she says. Brizuela, whose pilot study explores mathematical thinking among children in grades K–2, understands. “Would you like to use a different letter?” she asks, erasing the z and replacing it with a y. Jasmine smiles. She picks up her pencil and easily jots down the rule: y + 1 = z feet.
A Dreaded, Scary Subject
It may seem adorable that young children are stumped if asked to add 1 to z but not if asked to add 1 to y, but to Brizuela, director of the Mathematics, Science, Technology, and Engineering Education Program in Tuft’s education department, it reveals the reasoning capacity of young minds and the need to engage them in algebraic thinking long before it becomes a dreaded and scary subject.
To many, algebra is about the first or last three letters of the alphabet, and it provokes groaning, trash talk (think Forever 21’s “Allergic to Algebra” T-shirt), and heated debate. Should it be mandated? At what grade? Algebra’s status as a “gatekeeper course” has made it a touchstone on matters of access and equity. As a result, in many places it’s become a graduation requirement.
Back in the early 1980s, one-quarter of high school graduates never even took algebra, says Daniel Chazen, director of the Center for Mathematics Education at the University of Maryland. Today, educators are pushing students to take algebra even before high school. According to the National Assessment of Educational Progress (NAEP), the number of students taking Algebra I in eighth grade more than doubled between 1986 and 2011, from 16 to 34 percent. Strikingly, eighth-grade NAEP math test scores have edged up too, with 43 percent scoring advanced or proficient in 2011, compared with 27 percent in 1996.
But amid the good news is a troubling reality: Many kids are failing algebra. In California, where standards call for Algebra I in grade 8, a 2011 EdSource report shows that nearly one-third of those who took the course—or 80,000 students—scored “below basic” or “far below basic.” In districts across the country, failure rates for Algebra I vary but run as high as 40 or 50 percent, raising questions about how students are prepared—and how the subject is taught.
Starting Algebra Early
Why is algebra so hard? For many students, math experts say, it is a dramatic leap to go from the concrete world of computation-focused grade school math to the abstract world of algebra, which requires work with variables and changing quantitative relationships. It is not just the shock of seeing letters where numbers have been but also the type of thinking those letters represent.
“In arithmetic, you are dealing with explicit numbers,” says Hung-Hsi Wu, a professor emeritus of mathematics at the University of California, Berkeley. “Algebra says, ‘I have a number; I don’t know what it is, but three times it and subtract three is 15.’ You have a number floating out there, and you have to catch it. It is the thinking behind catching the number that baffles students.”
While some argue that children must be developmentally ready to learn algebra—around ages 11–13, when they can grasp abstract thought—Brizuela and others say it’s critical to introduce it earlier. “Kids need to develop some comfort with these tools,” she says. “Babies are exposed to written and spoken language, and after six years we expect them to become somewhat fluent with that. In math, we just drop it on them like a bomb.”
Brizuela’s research spans more than a dozen years and seeks to find out if explicitly teaching algebraic thinking, including a comfort with letter variables and the ability to express mathematical values in multiple forms (Jasmine’s words, table, and bar graph), might be helpful later on.
In a study to be published in October in Recherches en Didactique des Mathématiques, a French math education journal, Brizuela and her colleagues tracked 19 students in Boston Public Schools in grades 3, 4, and 5 who received weekly algebra lessons plus homework, as compared with a control group, and followed them through middle school. Results showed that those students outperformed their peers on algebra assessments given in grades 5, 7, and 8 and drawn from NAEP, Massachusetts state tests, and the Trends in International Mathematics and Science Study, or TIMMS.
In the stovepipe style and made from oaktag paper, the hat is one foot tall. Brizuela then asks, “If I’m five and a half feet tall, how tall will I be with the hat on?” Second-grader Jasmine, smiley in a pink sweatsuit, answers, “Six and a half feet.” Rather than say, “Right!” Brizuela offers another question: “How do you know?”
Thus begins a math conversation that researchers like Brizuela believe may hold the key to tackling one of our biggest school bugaboos: algebra. As they talk, Jasmine uses words, bar graphs, and a table to describe how tall each person they discuss will be if they put on the hat. Jasmine creates a rule—“add one foot to the number you already had”—and applies it to an imaginary person 100 feet tall.
Brizuela even throws out a variable. “So, to show someone whose height I don’t know, I will use z feet,” she says, adding a z to Jasmine’s table. “What should I do now?” Jasmine pauses. “This is kind of hard,” she says. Brizuela, whose pilot study explores mathematical thinking among children in grades K–2, understands. “Would you like to use a different letter?” she asks, erasing the z and replacing it with a y. Jasmine smiles. She picks up her pencil and easily jots down the rule: y + 1 = z feet.
A Dreaded, Scary Subject
It may seem adorable that young children are stumped if asked to add 1 to z but not if asked to add 1 to y, but to Brizuela, director of the Mathematics, Science, Technology, and Engineering Education Program in Tuft’s education department, it reveals the reasoning capacity of young minds and the need to engage them in algebraic thinking long before it becomes a dreaded and scary subject.
To many, algebra is about the first or last three letters of the alphabet, and it provokes groaning, trash talk (think Forever 21’s “Allergic to Algebra” T-shirt), and heated debate. Should it be mandated? At what grade? Algebra’s status as a “gatekeeper course” has made it a touchstone on matters of access and equity. As a result, in many places it’s become a graduation requirement.
Back in the early 1980s, one-quarter of high school graduates never even took algebra, says Daniel Chazen, director of the Center for Mathematics Education at the University of Maryland. Today, educators are pushing students to take algebra even before high school. According to the National Assessment of Educational Progress (NAEP), the number of students taking Algebra I in eighth grade more than doubled between 1986 and 2011, from 16 to 34 percent. Strikingly, eighth-grade NAEP math test scores have edged up too, with 43 percent scoring advanced or proficient in 2011, compared with 27 percent in 1996.
But amid the good news is a troubling reality: Many kids are failing algebra. In California, where standards call for Algebra I in grade 8, a 2011 EdSource report shows that nearly one-third of those who took the course—or 80,000 students—scored “below basic” or “far below basic.” In districts across the country, failure rates for Algebra I vary but run as high as 40 or 50 percent, raising questions about how students are prepared—and how the subject is taught.
Starting Algebra Early
Why is algebra so hard? For many students, math experts say, it is a dramatic leap to go from the concrete world of computation-focused grade school math to the abstract world of algebra, which requires work with variables and changing quantitative relationships. It is not just the shock of seeing letters where numbers have been but also the type of thinking those letters represent.
“In arithmetic, you are dealing with explicit numbers,” says Hung-Hsi Wu, a professor emeritus of mathematics at the University of California, Berkeley. “Algebra says, ‘I have a number; I don’t know what it is, but three times it and subtract three is 15.’ You have a number floating out there, and you have to catch it. It is the thinking behind catching the number that baffles students.”
While some argue that children must be developmentally ready to learn algebra—around ages 11–13, when they can grasp abstract thought—Brizuela and others say it’s critical to introduce it earlier. “Kids need to develop some comfort with these tools,” she says. “Babies are exposed to written and spoken language, and after six years we expect them to become somewhat fluent with that. In math, we just drop it on them like a bomb.”
Brizuela’s research spans more than a dozen years and seeks to find out if explicitly teaching algebraic thinking, including a comfort with letter variables and the ability to express mathematical values in multiple forms (Jasmine’s words, table, and bar graph), might be helpful later on.
In a study to be published in October in Recherches en Didactique des Mathématiques, a French math education journal, Brizuela and her colleagues tracked 19 students in Boston Public Schools in grades 3, 4, and 5 who received weekly algebra lessons plus homework, as compared with a control group, and followed them through middle school. Results showed that those students outperformed their peers on algebra assessments given in grades 5, 7, and 8 and drawn from NAEP, Massachusetts state tests, and the Trends in International Mathematics and Science Study, or TIMMS.
Building Math Minds
Central to Brizuela’s work is a striking idea: Rather than pushing eighth-grade or high school algebra down to elementary school, she begins with what children already tend to do, such as generalizing. For example, when children hear the word “hundred,” they know to add two zeros. Brizuela uses that natural ability to lure children into thinking about quantitative relationships that then become algebraic rules. This exercises their natural mathematical reasoning, which is often pushed aside in favor of getting the “right” answer or learning to memorize or compute (see sidebar “Laying the Groundwork for Algebra”).
Close SidebarLaying the Groundwork for Algebra
Here are three things that teachers can do to encourage algebraic thinking, according to researchers:
• Broaden your definition of the equal sign. Children should be trained to view an equal sign (=) as balancing an equation, not as a command to produce an answer, says Cathy L. Seeley, a senior fellow at the Charles A. Dana Center at the University of Texas at Austin. “If you help them be fluid with what the equal sign is, it starts helping children to grasp algebra.”
• Introduce letters, carefully. Including letters in math problems early on can help children grow comfortable with seeing and working with them, but they can also be misleading. Some young children can correlate a letter with its order in the alphabet, like a (first) or z (last). Tufts University researcher Bárbara Brizuela does not use x as a variable because children view it as “crossing out.”
• Talk about math. So much of grade school math is “what you do with paper,” but paper work is typically about computation and answers, not mathematical reasoning, says former math teacher Paul Goldenberg of the Educational Development Center in Waltham, Mass. Presenting problems orally and framing them as a continuation of earlier ideas, rather than a “frightening new language,” can help, he says.
Similarly, Barbara J. Dougherty, Richard G. Miller Chair of Mathematics Education at the University of Missouri, observes that first-graders naturally compare, often to be sure they have the same amount (of whatever is in question) as somebody else.
“In starting with children at six, rather than starting with numbers, we ask, ‘How do you know if you have more than somebody else or less?’” says Dougherty. She and her colleagues use measurement as a vehicle for discussing comparisons of, say, the height of a cereal box to the length of a pencil. Then, instead of writing down “the height of the cereal box” and “the length of the pencil,” she says, “we’ll say, ‘Let b represent the height of the cereal box and l be the length of the pencil.’ It sounds pretty simple, but it is actually pretty powerful.” Dougherty, who has been following a cohort of students at the University Laboratory School in Honolulu, Hawaii, since 2001, says that by the time the students reach high school, they consistently outperform peers in their understanding of algebraic concepts like variables and quantitative relationships.
In the Lab School, whose student population reflects the state’s socioeconomic and racial composition, first-grade teacher Maria DaSilva says that rather than presenting the students with, say, a number line right off, she lets the class puzzle through a problem—sometimes over the course of days—until they realize that having a number line will help them in their work (see sidebar “Algebra in First Grade?”).
Close SidebarAlgebra in First Grade?
At the University Laboratory School in Honolulu, Hawaii, first-graders solve algebraic problems disguised as real-life dilemmas. One such problem involves figuring out how much growth hormone a doctor must give a population of shrimp for them to reach a certain size, given that over time they need different amounts because previous doses have made them grow. When the doctor “gets confused” about how much growth hormone to give, the children must find a way to keep track.
Teacher Maria DaSilva has students measure out liquid “doses” to “feed” the growing shrimp by marking on masking tape placed along the side of a container. Later, she removes the tape and places it horizontally on a piece of paper to become a number line. This exercise gets students thinking about changing variables as opposed to fixed amounts and demonstrates that between whole units there exist partial units—or fractions—which experts say is absolutely critical to understanding and solving algebraic equations. A common reason students get tripped up in algebra is that they don’t understand what fractions really represent and how to manipulate them, experts say.
The Teaching Challenge
The drive to improve U.S. math performance among students has focused on two main worries: (1) Are students well enough prepared, and (2) are teachers prepared enough to teach math well?
William Schmidt, professor and codirector of the Education Policy Center at Michigan State University, says the new Common Core standards likely to be adopted by most states for 2013–2014, “capture the logic of mathematics,”—an upgrade from the seemingly unrelated lessons that have made learning math “like reading the phone book.”
But he wonders: Will teachers be able to teach it? In a 2010 study, Breaking the Cycle: An International Comparison of U.S. Mathematics Teacher Preparation, comparing U.S. primary and middle school teachers with peers in 16 countries, Schmidt and his colleagues found that American teachers had “weak training mathematically” and less math coursework than teachers in high-performing nations. “We have this new demanding curriculum in the middle grades and teachers who are ill prepared to teach it,” he warns.
Meanwhile, excitement over raised standards has been met with a worry: What about the kids who are struggling now? Math researchers, like James J. Lynn at the University of Illinois at Chicago, with colleagues in New York and Seattle, are in the third year of a four-year National Science Foundation–funded project to study 17,000 high school students who struggle with algebra. Their approach is to promote sense-making, which they say has been lacking in many students’ earlier algebra experiences.
Along with work aimed at bolstering students’ sense of how quantities relate—including filling deficits as they go rather than undertaking long periods of “re-teaching”—the project also seeks to change the mindset around algebra. Instead of viewing algebra as insurmountable, students learn that applying effort and wrestling with problems can grow brain connections and make them smarter and better at math. “We try to shape their attitudes of themselves as capable learners,” says Lynn. The program is showing some gain, with about half the students scoring “high mastery” after the course (most students scored “low mastery” prior to the course).
Given such difficulty, one has to wonder: Why even learn algebra?
According to Jon R. Star, associate professor at the Harvard Graduate School of Education, that’s like asking: “Why are they reading Wuthering Heights?” Star says the answer is that—like literature—algebra tells us something about human nature and understanding. Algebra, he says, “is our students’ first exposure to what mathematics is.” It offers students the sort of critical thinking about mathematical ideas that simply doesn’t come with the computation skills of early school math. Instead, he argues, we should simply point out that, when we get to algebra, “we are here to learn some mathematics.” Not computation. Not calculation. But real math.
Freelance education writer and author Laura Pappano is a frequent contributor to the Harvard Education Letter.
Central to Brizuela’s work is a striking idea: Rather than pushing eighth-grade or high school algebra down to elementary school, she begins with what children already tend to do, such as generalizing. For example, when children hear the word “hundred,” they know to add two zeros. Brizuela uses that natural ability to lure children into thinking about quantitative relationships that then become algebraic rules. This exercises their natural mathematical reasoning, which is often pushed aside in favor of getting the “right” answer or learning to memorize or compute (see sidebar “Laying the Groundwork for Algebra”).
Close SidebarLaying the Groundwork for Algebra
Here are three things that teachers can do to encourage algebraic thinking, according to researchers:
• Broaden your definition of the equal sign. Children should be trained to view an equal sign (=) as balancing an equation, not as a command to produce an answer, says Cathy L. Seeley, a senior fellow at the Charles A. Dana Center at the University of Texas at Austin. “If you help them be fluid with what the equal sign is, it starts helping children to grasp algebra.”
• Introduce letters, carefully. Including letters in math problems early on can help children grow comfortable with seeing and working with them, but they can also be misleading. Some young children can correlate a letter with its order in the alphabet, like a (first) or z (last). Tufts University researcher Bárbara Brizuela does not use x as a variable because children view it as “crossing out.”
• Talk about math. So much of grade school math is “what you do with paper,” but paper work is typically about computation and answers, not mathematical reasoning, says former math teacher Paul Goldenberg of the Educational Development Center in Waltham, Mass. Presenting problems orally and framing them as a continuation of earlier ideas, rather than a “frightening new language,” can help, he says.
Similarly, Barbara J. Dougherty, Richard G. Miller Chair of Mathematics Education at the University of Missouri, observes that first-graders naturally compare, often to be sure they have the same amount (of whatever is in question) as somebody else.
“In starting with children at six, rather than starting with numbers, we ask, ‘How do you know if you have more than somebody else or less?’” says Dougherty. She and her colleagues use measurement as a vehicle for discussing comparisons of, say, the height of a cereal box to the length of a pencil. Then, instead of writing down “the height of the cereal box” and “the length of the pencil,” she says, “we’ll say, ‘Let b represent the height of the cereal box and l be the length of the pencil.’ It sounds pretty simple, but it is actually pretty powerful.” Dougherty, who has been following a cohort of students at the University Laboratory School in Honolulu, Hawaii, since 2001, says that by the time the students reach high school, they consistently outperform peers in their understanding of algebraic concepts like variables and quantitative relationships.
In the Lab School, whose student population reflects the state’s socioeconomic and racial composition, first-grade teacher Maria DaSilva says that rather than presenting the students with, say, a number line right off, she lets the class puzzle through a problem—sometimes over the course of days—until they realize that having a number line will help them in their work (see sidebar “Algebra in First Grade?”).
Close SidebarAlgebra in First Grade?
At the University Laboratory School in Honolulu, Hawaii, first-graders solve algebraic problems disguised as real-life dilemmas. One such problem involves figuring out how much growth hormone a doctor must give a population of shrimp for them to reach a certain size, given that over time they need different amounts because previous doses have made them grow. When the doctor “gets confused” about how much growth hormone to give, the children must find a way to keep track.
Teacher Maria DaSilva has students measure out liquid “doses” to “feed” the growing shrimp by marking on masking tape placed along the side of a container. Later, she removes the tape and places it horizontally on a piece of paper to become a number line. This exercise gets students thinking about changing variables as opposed to fixed amounts and demonstrates that between whole units there exist partial units—or fractions—which experts say is absolutely critical to understanding and solving algebraic equations. A common reason students get tripped up in algebra is that they don’t understand what fractions really represent and how to manipulate them, experts say.
The Teaching Challenge
The drive to improve U.S. math performance among students has focused on two main worries: (1) Are students well enough prepared, and (2) are teachers prepared enough to teach math well?
William Schmidt, professor and codirector of the Education Policy Center at Michigan State University, says the new Common Core standards likely to be adopted by most states for 2013–2014, “capture the logic of mathematics,”—an upgrade from the seemingly unrelated lessons that have made learning math “like reading the phone book.”
But he wonders: Will teachers be able to teach it? In a 2010 study, Breaking the Cycle: An International Comparison of U.S. Mathematics Teacher Preparation, comparing U.S. primary and middle school teachers with peers in 16 countries, Schmidt and his colleagues found that American teachers had “weak training mathematically” and less math coursework than teachers in high-performing nations. “We have this new demanding curriculum in the middle grades and teachers who are ill prepared to teach it,” he warns.
Meanwhile, excitement over raised standards has been met with a worry: What about the kids who are struggling now? Math researchers, like James J. Lynn at the University of Illinois at Chicago, with colleagues in New York and Seattle, are in the third year of a four-year National Science Foundation–funded project to study 17,000 high school students who struggle with algebra. Their approach is to promote sense-making, which they say has been lacking in many students’ earlier algebra experiences.
Along with work aimed at bolstering students’ sense of how quantities relate—including filling deficits as they go rather than undertaking long periods of “re-teaching”—the project also seeks to change the mindset around algebra. Instead of viewing algebra as insurmountable, students learn that applying effort and wrestling with problems can grow brain connections and make them smarter and better at math. “We try to shape their attitudes of themselves as capable learners,” says Lynn. The program is showing some gain, with about half the students scoring “high mastery” after the course (most students scored “low mastery” prior to the course).
Given such difficulty, one has to wonder: Why even learn algebra?
According to Jon R. Star, associate professor at the Harvard Graduate School of Education, that’s like asking: “Why are they reading Wuthering Heights?” Star says the answer is that—like literature—algebra tells us something about human nature and understanding. Algebra, he says, “is our students’ first exposure to what mathematics is.” It offers students the sort of critical thinking about mathematical ideas that simply doesn’t come with the computation skills of early school math. Instead, he argues, we should simply point out that, when we get to algebra, “we are here to learn some mathematics.” Not computation. Not calculation. But real math.
Freelance education writer and author Laura Pappano is a frequent contributor to the Harvard Education Letter.
It’s Crazy Hair Day at Marshall Elementary School in Boston’s Dorchester neighborhood—which is perfect, because Tufts University researcher Bárbara Brizuela has brought a hat.
In the stovepipe style and made from oaktag paper, the hat is one foot tall. Brizuela then asks, “If I’m five and a half feet tall, how tall will I be with the hat on?” Second-grader Jasmine, smiley in a pink sweatsuit, answers, “Six and a half feet.” Rather than say, “Right!” Brizuela offers another question: “How do you know?”
Thus begins a math conversation that researchers like Brizuela believe may hold the key to tackling one of our biggest school bugaboos: algebra. As they talk, Jasmine uses words, bar graphs, and a table to describe how tall each person they discuss will be if they put on the hat. Jasmine creates a rule—“add one foot to the number you already had”—and applies it to an imaginary person 100 feet tall.
Brizuela even throws out a variable. “So, to show someone whose height I don’t know, I will use z feet,” she says, adding a z to Jasmine’s table. “What should I do now?” Jasmine pauses. “This is kind of hard,” she says. Brizuela, whose pilot study explores mathematical thinking among children in grades K–2, understands. “Would you like to use a different letter?” she asks, erasing the z and replacing it with a y. Jasmine smiles. She picks up her pencil and easily jots down the rule: y + 1 = z feet.
A Dreaded, Scary Subject
It may seem adorable that young children are stumped if asked to add 1 to z but not if asked to add 1 to y, but to Brizuela, director of the Mathematics, Science, Technology, and Engineering Education Program in Tuft’s education department, it reveals the reasoning capacity of young minds and the need to engage them in algebraic thinking long before it becomes a dreaded and scary subject.
To many, algebra is about the first or last three letters of the alphabet, and it provokes groaning, trash talk (think Forever 21’s “Allergic to Algebra” T-shirt), and heated debate. Should it be mandated? At what grade? Algebra’s status as a “gatekeeper course” has made it a touchstone on matters of access and equity. As a result, in many places it’s become a graduation requirement.
Back in the early 1980s, one-quarter of high school graduates never even took algebra, says Daniel Chazen, director of the Center for Mathematics Education at the University of Maryland. Today, educators are pushing students to take algebra even before high school. According to the National Assessment of Educational Progress (NAEP), the number of students taking Algebra I in eighth grade more than doubled between 1986 and 2011, from 16 to 34 percent. Strikingly, eighth-grade NAEP math test scores have edged up too, with 43 percent scoring advanced or proficient in 2011, compared with 27 percent in 1996.
But amid the good news is a troubling reality: Many kids are failing algebra. In California, where standards call for Algebra I in grade 8, a 2011 EdSource report shows that nearly one-third of those who took the course—or 80,000 students—scored “below basic” or “far below basic.” In districts across the country, failure rates for Algebra I vary but run as high as 40 or 50 percent, raising questions about how students are prepared—and how the subject is taught.
Starting Algebra Early
Why is algebra so hard? For many students, math experts say, it is a dramatic leap to go from the concrete world of computation-focused grade school math to the abstract world of algebra, which requires work with variables and changing quantitative relationships. It is not just the shock of seeing letters where numbers have been but also the type of thinking those letters represent.
“In arithmetic, you are dealing with explicit numbers,” says Hung-Hsi Wu, a professor emeritus of mathematics at the University of California, Berkeley. “Algebra says, ‘I have a number; I don’t know what it is, but three times it and subtract three is 15.’ You have a number floating out there, and you have to catch it. It is the thinking behind catching the number that baffles students.”
While some argue that children must be developmentally ready to learn algebra—around ages 11–13, when they can grasp abstract thought—Brizuela and others say it’s critical to introduce it earlier. “Kids need to develop some comfort with these tools,” she says. “Babies are exposed to written and spoken language, and after six years we expect them to become somewhat fluent with that. In math, we just drop it on them like a bomb.”
Brizuela’s research spans more than a dozen years and seeks to find out if explicitly teaching algebraic thinking, including a comfort with letter variables and the ability to express mathematical values in multiple forms (Jasmine’s words, table, and bar graph), might be helpful later on.
In a study to be published in October in Recherches en Didactique des Mathématiques, a French math education journal, Brizuela and her colleagues tracked 19 students in Boston Public Schools in grades 3, 4, and 5 who received weekly algebra lessons plus homework, as compared with a control group, and followed them through middle school. Results showed that those students outperformed their peers on algebra assessments given in grades 5, 7, and 8 and drawn from NAEP, Massachusetts state tests, and the Trends in International Mathematics and Science Study, or TIMMS.
In the stovepipe style and made from oaktag paper, the hat is one foot tall. Brizuela then asks, “If I’m five and a half feet tall, how tall will I be with the hat on?” Second-grader Jasmine, smiley in a pink sweatsuit, answers, “Six and a half feet.” Rather than say, “Right!” Brizuela offers another question: “How do you know?”
Thus begins a math conversation that researchers like Brizuela believe may hold the key to tackling one of our biggest school bugaboos: algebra. As they talk, Jasmine uses words, bar graphs, and a table to describe how tall each person they discuss will be if they put on the hat. Jasmine creates a rule—“add one foot to the number you already had”—and applies it to an imaginary person 100 feet tall.
Brizuela even throws out a variable. “So, to show someone whose height I don’t know, I will use z feet,” she says, adding a z to Jasmine’s table. “What should I do now?” Jasmine pauses. “This is kind of hard,” she says. Brizuela, whose pilot study explores mathematical thinking among children in grades K–2, understands. “Would you like to use a different letter?” she asks, erasing the z and replacing it with a y. Jasmine smiles. She picks up her pencil and easily jots down the rule: y + 1 = z feet.
A Dreaded, Scary Subject
It may seem adorable that young children are stumped if asked to add 1 to z but not if asked to add 1 to y, but to Brizuela, director of the Mathematics, Science, Technology, and Engineering Education Program in Tuft’s education department, it reveals the reasoning capacity of young minds and the need to engage them in algebraic thinking long before it becomes a dreaded and scary subject.
To many, algebra is about the first or last three letters of the alphabet, and it provokes groaning, trash talk (think Forever 21’s “Allergic to Algebra” T-shirt), and heated debate. Should it be mandated? At what grade? Algebra’s status as a “gatekeeper course” has made it a touchstone on matters of access and equity. As a result, in many places it’s become a graduation requirement.
Back in the early 1980s, one-quarter of high school graduates never even took algebra, says Daniel Chazen, director of the Center for Mathematics Education at the University of Maryland. Today, educators are pushing students to take algebra even before high school. According to the National Assessment of Educational Progress (NAEP), the number of students taking Algebra I in eighth grade more than doubled between 1986 and 2011, from 16 to 34 percent. Strikingly, eighth-grade NAEP math test scores have edged up too, with 43 percent scoring advanced or proficient in 2011, compared with 27 percent in 1996.
But amid the good news is a troubling reality: Many kids are failing algebra. In California, where standards call for Algebra I in grade 8, a 2011 EdSource report shows that nearly one-third of those who took the course—or 80,000 students—scored “below basic” or “far below basic.” In districts across the country, failure rates for Algebra I vary but run as high as 40 or 50 percent, raising questions about how students are prepared—and how the subject is taught.
Starting Algebra Early
Why is algebra so hard? For many students, math experts say, it is a dramatic leap to go from the concrete world of computation-focused grade school math to the abstract world of algebra, which requires work with variables and changing quantitative relationships. It is not just the shock of seeing letters where numbers have been but also the type of thinking those letters represent.
“In arithmetic, you are dealing with explicit numbers,” says Hung-Hsi Wu, a professor emeritus of mathematics at the University of California, Berkeley. “Algebra says, ‘I have a number; I don’t know what it is, but three times it and subtract three is 15.’ You have a number floating out there, and you have to catch it. It is the thinking behind catching the number that baffles students.”
While some argue that children must be developmentally ready to learn algebra—around ages 11–13, when they can grasp abstract thought—Brizuela and others say it’s critical to introduce it earlier. “Kids need to develop some comfort with these tools,” she says. “Babies are exposed to written and spoken language, and after six years we expect them to become somewhat fluent with that. In math, we just drop it on them like a bomb.”
Brizuela’s research spans more than a dozen years and seeks to find out if explicitly teaching algebraic thinking, including a comfort with letter variables and the ability to express mathematical values in multiple forms (Jasmine’s words, table, and bar graph), might be helpful later on.
In a study to be published in October in Recherches en Didactique des Mathématiques, a French math education journal, Brizuela and her colleagues tracked 19 students in Boston Public Schools in grades 3, 4, and 5 who received weekly algebra lessons plus homework, as compared with a control group, and followed them through middle school. Results showed that those students outperformed their peers on algebra assessments given in grades 5, 7, and 8 and drawn from NAEP, Massachusetts state tests, and the Trends in International Mathematics and Science Study, or TIMMS.
Building Math Minds
Central to Brizuela’s work is a striking idea: Rather than pushing eighth-grade or high school algebra down to elementary school, she begins with what children already tend to do, such as generalizing. For example, when children hear the word “hundred,” they know to add two zeros. Brizuela uses that natural ability to lure children into thinking about quantitative relationships that then become algebraic rules. This exercises their natural mathematical reasoning, which is often pushed aside in favor of getting the “right” answer or learning to memorize or compute (see sidebar “Laying the Groundwork for Algebra”).
Close SidebarLaying the Groundwork for Algebra
Here are three things that teachers can do to encourage algebraic thinking, according to researchers:
• Broaden your definition of the equal sign. Children should be trained to view an equal sign (=) as balancing an equation, not as a command to produce an answer, says Cathy L. Seeley, a senior fellow at the Charles A. Dana Center at the University of Texas at Austin. “If you help them be fluid with what the equal sign is, it starts helping children to grasp algebra.”
• Introduce letters, carefully. Including letters in math problems early on can help children grow comfortable with seeing and working with them, but they can also be misleading. Some young children can correlate a letter with its order in the alphabet, like a (first) or z (last). Tufts University researcher Bárbara Brizuela does not use x as a variable because children view it as “crossing out.”
• Talk about math. So much of grade school math is “what you do with paper,” but paper work is typically about computation and answers, not mathematical reasoning, says former math teacher Paul Goldenberg of the Educational Development Center in Waltham, Mass. Presenting problems orally and framing them as a continuation of earlier ideas, rather than a “frightening new language,” can help, he says.
Similarly, Barbara J. Dougherty, Richard G. Miller Chair of Mathematics Education at the University of Missouri, observes that first-graders naturally compare, often to be sure they have the same amount (of whatever is in question) as somebody else.
“In starting with children at six, rather than starting with numbers, we ask, ‘How do you know if you have more than somebody else or less?’” says Dougherty. She and her colleagues use measurement as a vehicle for discussing comparisons of, say, the height of a cereal box to the length of a pencil. Then, instead of writing down “the height of the cereal box” and “the length of the pencil,” she says, “we’ll say, ‘Let b represent the height of the cereal box and l be the length of the pencil.’ It sounds pretty simple, but it is actually pretty powerful.” Dougherty, who has been following a cohort of students at the University Laboratory School in Honolulu, Hawaii, since 2001, says that by the time the students reach high school, they consistently outperform peers in their understanding of algebraic concepts like variables and quantitative relationships.
In the Lab School, whose student population reflects the state’s socioeconomic and racial composition, first-grade teacher Maria DaSilva says that rather than presenting the students with, say, a number line right off, she lets the class puzzle through a problem—sometimes over the course of days—until they realize that having a number line will help them in their work (see sidebar “Algebra in First Grade?”).
Close SidebarAlgebra in First Grade?
At the University Laboratory School in Honolulu, Hawaii, first-graders solve algebraic problems disguised as real-life dilemmas. One such problem involves figuring out how much growth hormone a doctor must give a population of shrimp for them to reach a certain size, given that over time they need different amounts because previous doses have made them grow. When the doctor “gets confused” about how much growth hormone to give, the children must find a way to keep track.
Teacher Maria DaSilva has students measure out liquid “doses” to “feed” the growing shrimp by marking on masking tape placed along the side of a container. Later, she removes the tape and places it horizontally on a piece of paper to become a number line. This exercise gets students thinking about changing variables as opposed to fixed amounts and demonstrates that between whole units there exist partial units—or fractions—which experts say is absolutely critical to understanding and solving algebraic equations. A common reason students get tripped up in algebra is that they don’t understand what fractions really represent and how to manipulate them, experts say.
The Teaching Challenge
The drive to improve U.S. math performance among students has focused on two main worries: (1) Are students well enough prepared, and (2) are teachers prepared enough to teach math well?
William Schmidt, professor and codirector of the Education Policy Center at Michigan State University, says the new Common Core standards likely to be adopted by most states for 2013–2014, “capture the logic of mathematics,”—an upgrade from the seemingly unrelated lessons that have made learning math “like reading the phone book.”
But he wonders: Will teachers be able to teach it? In a 2010 study, Breaking the Cycle: An International Comparison of U.S. Mathematics Teacher Preparation, comparing U.S. primary and middle school teachers with peers in 16 countries, Schmidt and his colleagues found that American teachers had “weak training mathematically” and less math coursework than teachers in high-performing nations. “We have this new demanding curriculum in the middle grades and teachers who are ill prepared to teach it,” he warns.
Meanwhile, excitement over raised standards has been met with a worry: What about the kids who are struggling now? Math researchers, like James J. Lynn at the University of Illinois at Chicago, with colleagues in New York and Seattle, are in the third year of a four-year National Science Foundation–funded project to study 17,000 high school students who struggle with algebra. Their approach is to promote sense-making, which they say has been lacking in many students’ earlier algebra experiences.
Along with work aimed at bolstering students’ sense of how quantities relate—including filling deficits as they go rather than undertaking long periods of “re-teaching”—the project also seeks to change the mindset around algebra. Instead of viewing algebra as insurmountable, students learn that applying effort and wrestling with problems can grow brain connections and make them smarter and better at math. “We try to shape their attitudes of themselves as capable learners,” says Lynn. The program is showing some gain, with about half the students scoring “high mastery” after the course (most students scored “low mastery” prior to the course).
Given such difficulty, one has to wonder: Why even learn algebra?
According to Jon R. Star, associate professor at the Harvard Graduate School of Education, that’s like asking: “Why are they reading Wuthering Heights?” Star says the answer is that—like literature—algebra tells us something about human nature and understanding. Algebra, he says, “is our students’ first exposure to what mathematics is.” It offers students the sort of critical thinking about mathematical ideas that simply doesn’t come with the computation skills of early school math. Instead, he argues, we should simply point out that, when we get to algebra, “we are here to learn some mathematics.” Not computation. Not calculation. But real math.
Freelance education writer and author Laura Pappano is a frequent contributor to the Harvard Education Letter.
Central to Brizuela’s work is a striking idea: Rather than pushing eighth-grade or high school algebra down to elementary school, she begins with what children already tend to do, such as generalizing. For example, when children hear the word “hundred,” they know to add two zeros. Brizuela uses that natural ability to lure children into thinking about quantitative relationships that then become algebraic rules. This exercises their natural mathematical reasoning, which is often pushed aside in favor of getting the “right” answer or learning to memorize or compute (see sidebar “Laying the Groundwork for Algebra”).
Close SidebarLaying the Groundwork for Algebra
Here are three things that teachers can do to encourage algebraic thinking, according to researchers:
• Broaden your definition of the equal sign. Children should be trained to view an equal sign (=) as balancing an equation, not as a command to produce an answer, says Cathy L. Seeley, a senior fellow at the Charles A. Dana Center at the University of Texas at Austin. “If you help them be fluid with what the equal sign is, it starts helping children to grasp algebra.”
• Introduce letters, carefully. Including letters in math problems early on can help children grow comfortable with seeing and working with them, but they can also be misleading. Some young children can correlate a letter with its order in the alphabet, like a (first) or z (last). Tufts University researcher Bárbara Brizuela does not use x as a variable because children view it as “crossing out.”
• Talk about math. So much of grade school math is “what you do with paper,” but paper work is typically about computation and answers, not mathematical reasoning, says former math teacher Paul Goldenberg of the Educational Development Center in Waltham, Mass. Presenting problems orally and framing them as a continuation of earlier ideas, rather than a “frightening new language,” can help, he says.
Similarly, Barbara J. Dougherty, Richard G. Miller Chair of Mathematics Education at the University of Missouri, observes that first-graders naturally compare, often to be sure they have the same amount (of whatever is in question) as somebody else.
“In starting with children at six, rather than starting with numbers, we ask, ‘How do you know if you have more than somebody else or less?’” says Dougherty. She and her colleagues use measurement as a vehicle for discussing comparisons of, say, the height of a cereal box to the length of a pencil. Then, instead of writing down “the height of the cereal box” and “the length of the pencil,” she says, “we’ll say, ‘Let b represent the height of the cereal box and l be the length of the pencil.’ It sounds pretty simple, but it is actually pretty powerful.” Dougherty, who has been following a cohort of students at the University Laboratory School in Honolulu, Hawaii, since 2001, says that by the time the students reach high school, they consistently outperform peers in their understanding of algebraic concepts like variables and quantitative relationships.
In the Lab School, whose student population reflects the state’s socioeconomic and racial composition, first-grade teacher Maria DaSilva says that rather than presenting the students with, say, a number line right off, she lets the class puzzle through a problem—sometimes over the course of days—until they realize that having a number line will help them in their work (see sidebar “Algebra in First Grade?”).
Close SidebarAlgebra in First Grade?
At the University Laboratory School in Honolulu, Hawaii, first-graders solve algebraic problems disguised as real-life dilemmas. One such problem involves figuring out how much growth hormone a doctor must give a population of shrimp for them to reach a certain size, given that over time they need different amounts because previous doses have made them grow. When the doctor “gets confused” about how much growth hormone to give, the children must find a way to keep track.
Teacher Maria DaSilva has students measure out liquid “doses” to “feed” the growing shrimp by marking on masking tape placed along the side of a container. Later, she removes the tape and places it horizontally on a piece of paper to become a number line. This exercise gets students thinking about changing variables as opposed to fixed amounts and demonstrates that between whole units there exist partial units—or fractions—which experts say is absolutely critical to understanding and solving algebraic equations. A common reason students get tripped up in algebra is that they don’t understand what fractions really represent and how to manipulate them, experts say.
The Teaching Challenge
The drive to improve U.S. math performance among students has focused on two main worries: (1) Are students well enough prepared, and (2) are teachers prepared enough to teach math well?
William Schmidt, professor and codirector of the Education Policy Center at Michigan State University, says the new Common Core standards likely to be adopted by most states for 2013–2014, “capture the logic of mathematics,”—an upgrade from the seemingly unrelated lessons that have made learning math “like reading the phone book.”
But he wonders: Will teachers be able to teach it? In a 2010 study, Breaking the Cycle: An International Comparison of U.S. Mathematics Teacher Preparation, comparing U.S. primary and middle school teachers with peers in 16 countries, Schmidt and his colleagues found that American teachers had “weak training mathematically” and less math coursework than teachers in high-performing nations. “We have this new demanding curriculum in the middle grades and teachers who are ill prepared to teach it,” he warns.
Meanwhile, excitement over raised standards has been met with a worry: What about the kids who are struggling now? Math researchers, like James J. Lynn at the University of Illinois at Chicago, with colleagues in New York and Seattle, are in the third year of a four-year National Science Foundation–funded project to study 17,000 high school students who struggle with algebra. Their approach is to promote sense-making, which they say has been lacking in many students’ earlier algebra experiences.
Along with work aimed at bolstering students’ sense of how quantities relate—including filling deficits as they go rather than undertaking long periods of “re-teaching”—the project also seeks to change the mindset around algebra. Instead of viewing algebra as insurmountable, students learn that applying effort and wrestling with problems can grow brain connections and make them smarter and better at math. “We try to shape their attitudes of themselves as capable learners,” says Lynn. The program is showing some gain, with about half the students scoring “high mastery” after the course (most students scored “low mastery” prior to the course).
Given such difficulty, one has to wonder: Why even learn algebra?
According to Jon R. Star, associate professor at the Harvard Graduate School of Education, that’s like asking: “Why are they reading Wuthering Heights?” Star says the answer is that—like literature—algebra tells us something about human nature and understanding. Algebra, he says, “is our students’ first exposure to what mathematics is.” It offers students the sort of critical thinking about mathematical ideas that simply doesn’t come with the computation skills of early school math. Instead, he argues, we should simply point out that, when we get to algebra, “we are here to learn some mathematics.” Not computation. Not calculation. But real math.
Freelance education writer and author Laura Pappano is a frequent contributor to the Harvard Education Letter.
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