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Studying Young Minds, and How to Teach Them

Susan Notes:

Ohmygod, just when you were thinking it was safe to let your baby out of the house.

I picked the Week 14 letter from the product extolled below. This letter goes home--in English and Spanish--to parents of pre-schoolers enrolled in this systematic, explicit math program offered by McGraw-Hill/SRA.

The letter begins with pictures of hexagon, triangle, rectangle, trapezoid, square, rhombos (diamond)
sphere, cone, rectangular prism, triangular prism, cylinder.

Then we get the Help-at-Home Math Tips

  • Look everywhere--stores, buildings, parks--for the chance to discuss shapes. Help your child identify and name shapes that are two-dimensional (for example, a speed limit sign is a rectangle) and those that are three dimensional (for example, a ball is a sphere, and a can is a cylinder).
    Benefit: This activity provides practice with shape recognition.

  • Let your child compare clean, empty plastic food containers and pots and pans. Discuss their shapes and sizes, noting which fit inside of others and which can be safely stacked on each other.
    Benefit: This activity helps develop awareness of spatial relationships.

  • Ask your child to draw something in the room that is a rectangle.

  • I'm scared to look at any more letters. Let's move on to the products a school buys to implement this program. Please note: I think it's a very good idea to introduce math concepts to young children. But I am of the David Hawkins school of thought--and experience. In "Messing About in Science," an essay that transformed my professional career, Hawkins says, "When learning is at the most fundamental level. . . with all the abstractions of Newtonian mechanics just around the corner, don't rush! When the mind is evolving the abstractions which will lead to physical comprehension, all of us must cross the line between ignorance and insight many times before we truly understand. Little facts, "discoveries" without the growth of insight, are not what we should seek to harvest. Such facts are only seedlings and should sometimes be let alone to grow into. . . ."

    Yes, that's how Hawkins ends the sentence--sometimes be let alone to grow into. . . ." Open Ended. He does not decree what 4-year-olds will be doing during the 14th week of school.

    But on to the program, touted in the New York Times article below, an article that takes the notion of a child "quickly falling behind in kindergarten" for granted and not an invention of the corporate politicos who dominate the education conversation these days, thugs who have turned what used to be kindergarten into a high-pressure skill zone, demanding of children what used to be taught in first and second grades. The New York Times is in awe of a pre-K program that takes math to a whole new stratosphere, not asking just what you've got when you've got a 4-year-old who can recite the definition of a rhomboid.

    Features for Building Blocks Building Blocks Pre-K Complete Curriculum

    * Teacher's Edition: The Teacher's Edition provides systematic, explicit instruction built on a foundation of successful mathematics research. A wealth of background information and strategies help you provide quality instruction.[emphasis added]

    * Teacher's Resource Guide: Offers key resources that help in delivering the curriculum including Family Letters for each week, English Learner Support, Counting Cards, Puzzles and Patterns, Shape Sets and the Shape Flip Book. [emphasis added]

    * Assessment: A comprehensive research-based guide to assessing preschool children's math proficiencies..

    * Big Books: Four big books provide excellent math related literature! Trade literature specifically chosen for preschoolers to help you teach math concepts include: Bat Jamboree by Kathi Appelt; The Shape of Things by Dayle Ann Dodds; I Spy Two Eyes: Numbers in Art by Lucy Miucklethwait; and The Right Number of Elephants by Jeff Sheppard.

    * Building Blocks Technology: Students develop and practice skills through engaging activities.
    o Building Blocks Software: The engaging activities are essential to the curriculum. Each activity addresses a specific developmental level of the math learning trajectories. Available online (free trial at www.review.realmath.com) with a dynamic teacher management system or activities only on CD-ROM. [emphasis added]
    o A complete Online eSuite is available for teachers and students including ePlanner, eAssess, Building Blocks Software, and eMath Tools. Visit review.realmath.com for a free trial!

    * Teacher Manipulative Kit: A variety of items to demonstrate math concepts includes: 0-5 Number Cubes (25); 5-10 Number Cubes (25); Unifex Cubes (5) Pkg. of 100; 0-5 "Tens" Number Cubes (25); 5-10 "Tens" Number Cubes (25); Dice with Dots (2); Pattern Blocks (1) Pkg. of 250; Dot Card Set (1); Attribute Blocks (1) Set of 60; Counters (1) Pkg. of 200; Numeral Cards (1) Set; Counting Cards (1) Set; Platonic Solids (1) 5-Piece Set; Dice with Numeral (2) Sets; Dinosaur Counters (4) Pkgs. of 100; Shape Counters (1) Pkg. of 500; Foam Shapes (2) Sets; Blank Wooden Number Cubes (10); Counting Board with Game Pieces (1), and Puppet (1).

    Price List
    Real Math Building Blocks - Teacher Edition - Pre-K
    ISBN: 9780076037087
    Price: $224.88
    McGraw-Hill School Education: SRA

    Student Materials
    Building Blocks Online - One Year, Per Student
    ISBN: 9780076179589
    Price: $11.22

    Real Math - Building Blocks CD-ROM
    ISBN: 9780076146819
    Price: $78.00

    Building Blocks Online - One Year, Per Building
    ISBN: 9780076059379
    Price: $1,999.98

    Real Math Building Blocks - Teacher Manipulative Kit - Pre-K
    ISBN: 9780076037865
    Price: $323.31

    Real Math Building Blocks - Shape Sets - Pre-K
    ISBN: 9780076182176
    Price: $21.84

    Building Blocks Technology (PreK-6)

    Building Blocks Online - One Year, Per Student
    ISBN: 9780076179589
    Price: $11.22

    NW - Building Blocks CD-ROM
    ISBN: 9780076145317

    Building Blocks Online - One Year, Per Building
    ISBN: 9780076059379
    Price: $1,999.98

    Research Base--from researchers who developed the product
    Effects of a Preschool Mathematics Curriculum:
    Research on the NSF-funded Building Blocks Project Douglas H. Clements and Julie Sarama
    University at Buffalo, State University of New York

    Building Blocks is a NSF-funded PreK to grade 2 software-based mathematics curriculum development project, designed to comprehensively address the most recent mathematics standards. Building Blocks materials were created upon explicit design principles and a nine-phase formative model--hey are truly research-based (details are provided in Clements, 2002a; Clements, 2002b; Sarama & Clements, in press). More here

    The favorite phrase here seems to be activities for every step in each learning trajectory for different core mathematical topics. God forbid that young children would "mess around" in math the way noted philosopher of education David Hawkins advocated they do in science. As Hawkins is quick to point out, such messing around is not without structure but The structure is vastly different.

    Truth in Disclosure: I am the author of Garbage Pizza, Patchwork Quilts, and Math Magic, Math as a Way of Knowing, Math at a Glance: A Month-by-Month Celebration of the Numbers Around Us, Day-By-Day Math: Activities for Grade 3-6, and co-author of Lessons for Introducing Division, Grades 3 & 4.

    And this may come as a surprise: I am also the author of the SRA product Math CrossSections, an 8-volume set revealing the kid-fun math in places like The White House, Barnum & Bailey Circus, The Denver Mint, and so on. Although incorporating the first NCTM Standards, the product definitely does not call for systematic, explicit instruction. It calls for provoking student curiosity. The product was definitely an anomaly for SRA and was removed from the catalogue almost as soon as it was put in.

    Notice how the journalist uses "experts say" to buttress the point that SRA has come up with a wonderful new product to revolutionize mathematics instruction. Why didn't the journalist seek out experts who have different views about children's development and what their school day should look like? Here's someone who commented online who shares a worry I have:

    What's missing? Well-- play. After all, these are very little children we're talking about. I can't stand seeing all this discussion of how much they can handle (i.e., work) with no mention of the fundamental developmental tasks served by play. Constructive play (which teaches some of the concepts needed for mathematics!), pretend play, even starter board games (which also include lots of math) are important. Let's raise bright kids who can work and also know how to live a good life-- what they're working for.

    And another:

    All very interesting but...so what? Our 2 oldest children attended a Waldorf school in MD through the end of middle school where academics, math in particular, aren't focused on until about 1st grade. In accordance with the teachings of founder Rudolph Steiner, still valid and effective after almost 100 years, early childhood education is approached via storytelling, dance, singing and hands on arts & crafts. Once they are exposed to math, reading, etc, Waldorf students invariably catch and frequently surpass mainstream students quickly. Maybe science should take a break from the stampede to justify ramming irrelevant academics down children's throats and focus on what really produces strong, confident, valuable citizens which, after all, is the purpose of education. Teaching a child math at 4 is usually about the parent's ego; it's a quaint pony trick and does nothing but distract from the far, far more important job of a 4 year old - enjoying being a child.

    And one final comment. You can find over 200 more at the url below.

    This is very interesting research. I'm all in favor of low-key math games for kids. However, I am concerned that this will lead to an increase in pressure on very young children to be academic achievers. It's already rampant in elementary school. My young children have a classmate in the second grade who is a perfectly smart boy. He spends all day in our very good school, then in the afternoon goes to a Kumon learning center and then has a tutor as well. Not surprisingly, this child hates school, Kumon, and his tutor! In my youth, it was rare for children to go to preschool or even kindergarten. Yet the children in my school managed to learn and to go on to become doctors, university professors, architects, and newspaper editors. I believe that encouraging an active imagination and allowing children in participate in adult activities (counting while shopping, measuring while cooking, and so forth) are more important to preschoolers than drills and flash cards.

    Another truth in disclosure seems in order: Benedict Carey graduated from college with a degree in mathematics. His reporting for the New York Times has not been free of controversy.

    By Benedict Carey

    BUFFALO--Many 4-year-olds cannot count up to their own age when they arrive at preschool, and those at the Stanley M. Makowski Early Childhood Center are hardly prodigies. Most live in this city's poorer districts and begin their academic life well behind the curve.

    But there they were on a recent Wednesday morning, three months into the school year, counting up to seven and higher, even doing some elementary addition and subtraction. At recess, one boy, Joshua, used a pointer to illustrate a math concept known as cardinality, by completing place settings on a whiteboard.

    "You just put one plate there, and one there, and one here," he explained, stepping aside as two other students ambled by, one wearing a pair of clown pants as a headscarf. "That's it. See?"

    For much of the last century, educators and many scientists believed that children could not learn math at all before the age of five, that their brains simply were not ready.

    But recent research has turned that assumption on its head --that, and a host of other conventional wisdom about geometry, reading, language and self-control in class. The findings, mostly from a branch of research called cognitive neuroscience, are helping to clarify when young brains are best able to grasp fundamental concepts.

    In one recent study, for instance, researchers found that most entering preschoolers could perform rudimentary division, by distributing candies among two or three play animals. In another, scientists found that the brain's ability to link letter combinations with sounds may not be fully developed until age 11 -- much later than many have assumed.

    The teaching of basic academic skills, until now largely the realm of tradition and guesswork, is giving way to approaches based on cognitive science. In several cities, including Boston, Washington and Nashville, schools have been experimenting with new curriculums to improve math skills in preschoolers. In others, teachers have used techniques developed by brain scientists to help children overcome dyslexia.

    And schools in about a dozen states have begun to use a program intended to accelerate the development of young students' frontal lobes, improving self-control in class.

    "Teaching is an ancient craft, and yet we really have had no idea how it affected the developing brain," said Kurt Fischer, director of the Mind, Brain and Education program at Harvard. "Well, that is beginning to change, and for the first time we are seeing the fields of brain science and education work together."

    This relationship is new and still awkward, experts say, and there is more hyperbole than evidence surrounding many "brain-based" commercial products on the market. But there are others, like an early math program taught in Buffalo schools, that have a track record. If these and similar efforts find traction in schools, experts say, they could transform teaching from the bottom up -- giving the ancient craft a modern scientific compass.

    Beyond Counting

    In a typical preschool class, children do very little math. They may practice counting, and occasionally look at books about numbers, but that is about it. Many classes devote mere minutes a day to math instruction or no time at all, recent studies have found-- far less than most children can handle, and not nearly enough to prepare those who, deprived of math-related games at home, quickly fall behind in kindergarten. [emphasis added]

    "Once that happens, it can be very hard to catch up," said Julie Sarama, a researcher in the graduate school of education at the University at Buffalo who, with her colleague and husband, Doug Clements, a professor in the same department, developed a program called Building Blocks to enrich early math education.

    "They decide they're no good at math -- 'I'm not a math person,' they say -- and pretty soon the school agrees, the parents agree," Dr. Clements said.

    "Everyone agrees."

    In a Building Blocks classroom, numbers are in artwork, on computer games and in lessons, sharing equal time with letters. Like "Sesame Street," Building Blocks has children play creative counting games; but it also focuses on other number skills, including cardinality (how many objects are in a set) and one-to-one correspondence (matching groups of objects, like cups and saucers). Teachers can tailor the Building Block lesson to a student's individual ability.

    On a recent Wednesday afternoon at the Makowski center, Buffalo's Public School 99, Pat Andzel asked her preschool class a question:

    "How many did you count?"

    She had drilled them on the number seven. She held up a sign with "7" and asked her students what number they saw ("seven!"); had the group jump seven times, counting; then had them touch their nose seven times. As the class finished counting seven objects on a poster, she asked again:

    "How many?"

    "I never used to ask that," Ms. Andzel said in an interview after the lesson. She asks it all the time now, she said, because it drives home a subtle but crucial idea: that the last number they said in counting is the quantity; it is the answer.

    "Many of these kids don't understand that yet," she said.

    The curriculum includes a variety of math-based lessons and activities, as well as software programs, all drawing on findings from cognitive science. When it comes to understanding numbers, for example, recent research suggests that infants can distinguish one object from two, and two from three.

    By preschool, the brain can handle larger numbers and is struggling to link three crucial concepts: physical quantities (seven marbles, seven inches) with abstract digit symbols ("7"), with the corresponding number words ("seven"). Lessons like the one Ms. Andzel taught are meant to fuse this numeric trinity, which is crucial for understanding basic math in kindergarten.

    Children begin recognizing geometric shapes as early as 18 months, studies find; by preschool, the brain can begin to grasp informal geometric definitions.

    It can when taught properly, that is. Many books use a pizza slice to illustrate a triangle, for example, even though slices are rounded at one end. Once a child has fused the word triangle with a specific shape (triangle = pizza slice), it is hard to break that association later on.

    "The definition," Dr. Clements said, "is a three-angled shape. Period." Building Blocks teaches this definition, illustrating it with triangles skinny and fat, squat and tall.

    In all, this curriculum and others link numbers to objects, to rhythms, to the chairs and plates around a table -- to the physical world.

    "If children have games and activities that demonstrate the relationship between numbers, then quantity becomes a physical experience," said Sharon Griffin, a psychologist at Clark University in Worcester, Mass., who found in a series of careful studies that a curriculum she devised, called Number Worlds, raised the scores of children who lagged in math. "Counting, by contrast, is very abstract." What a coincidence: Also a McGraw-Hill/SRA product. Building Blocks software is incorporated in the Number Worlds program.

    In a study published last year, scientists at Carnegie Mellon University reported that playing what seems a simple childhood game, similar to Chutes and Ladders (sometimes called Snakes and Slides), accelerates the understanding of numbers for low-income preschoolers.

    "Being told 8 is 2 times 4 is one thing," said Robert S. Siegler, a psychologist who is one of the authors. "It's another to see that it's twice as far to the number 8, and that it takes twice as long to get there."

    The Number Instinct

    "Use your eyes like cameras," said Lara Lazo, one of the teachers at P.S. 99, after the midmorning break. "Get ready to take a snapshot."

    The children bracketed their eyes with their hands, making "cameras," and Ms. Lazo showed them a paper plate with three dots on it -- then quickly covered the plate.

    "What number did you see?"

    A cacophony of "threes" and "fours" erupted.

    "O.K., she said. "Let's try it again."

    The lesson is intended to teach a skill called subitizing. "The idea," Dr. Sarama said, "is to get them to recognize quantity -- to say, "I see three" -- not by counting, but by instantly recognizing how many are there by sight."

    A crude "number instinct" is hard-wired into the anatomy of the brain, recent research has found. Mammals can quickly recognize differences in quantity, choosing the tree or bush with the most fruit. Human beings, even if they live in remote cultures with no formal math education, have a general grasp of quantities as well, anthropologists have found.

    In a series of recent imaging studies, scientists have discovered that a sliver of the parietal cortex, on the surface of the brain about an inch above the ears, is particularly active when the brain judges quantity. In this area, called the intraparietal sulcus, clusters of neurons are sensitive to the sight of specific quantities, research suggests. Some fire vigorously at the sight of five objects, for instance, less so at the sight of four or six, and not at all at two or nine. Others are most active in response to one, two, three, and so on.

    When engaged in a lesson or exercise, these regions actively communicate with areas of the frontal lobe, where planning and critical thinking are centered.

    "This is what we believe focused math education does: It sharpens the firing of these quantity neurons," said Stanislas Dehaene, a cognitive neuroscientist at the College de France in Paris and author of the books The Number Sense and Reading and the Brain. The firing of the number neurons becomes increasingly more selective to single quantities, he said; and these cells apparently begin to communicate with neurons across the brain in language areas, connecting precise quantities to words: "two," "ten," "five."

    A similar honing process is thought to occur when young children begin to link letter shapes and their associated sounds. Cells in the visual cortex wired to recognize shapes specialize in recognizing letters; these cells communicate with neurons in the auditory cortex as the letters are associated with sounds.

    The process may take longer to develop than many assume. A study published in March by neuroscientists at Maastricht University in the Netherlands suggested that the brain does not fully fuse letters and sounds until about age 11.

    "As these kinds of findings come in, they will have implications not only for teaching, but also education policy," said Daniel Ansari, an assistant professor in developmental cognitive neuroscience at Western Ontario University.

    Explaining Five

    In math, there is no faking it. Children either know that five is more than three, or they do not. Either they can put number symbols in exactly the right order, or they cannot. In their studies, Dr. Clements and Dr. Sarama test children one on one and videotape the results for comparisons.

    Over the past four years, the couple has tested Building Blocks in more than 400 classrooms in Buffalo, Boston and Nashville, comparing the progress of children in the program with that of peers in classes offering another math curriculum or none at all. On tests of addition, subtraction and number recognition after one school year, children who had the program scored in the 76th percentile on average, and those who did not scored in the 50th percentile.

    By the end of kindergarten, a year after the program has ended, those who had had it sustained their gains, scoring in the 71st percentile, on average.

    Many hurdles remain for this and similar curriculums based in cognitive science, experts say. Schools may move away from the curriculum; teachers move around, as do students; and in later grades there is always the risk that children who have mastered basic math will not get the attention they need to advance even further.

    But for now at least, education based on brain science has helped hundreds of Buffalo children refine their native abilities in math. In one videotaped exam, a 4-year-old boy in a FUBU jersey and long dreadlocks who entered P.S. 99 in 2006 was unable to count or match cards with 3, 5, 2, 1 and 4 on them to cards with equivalent numbers of grapes.

    In a video of his post-Building Blocks exam, six months later, he instantly says there are 10 pennies placed in front of him, without counting. He easily matches the number cards to their corresponding grape cards -- and puts the mixed-up numerals in the correct order.

    "What's the biggest, nine or seven or five?" asks the teacher giving the exam.

    The boy thinks for a moment. "Nine," he says. "Five is the littlest." Then he holds one palm above the other and says: "Five is like this. See?"

    "Do you see what he's doing?" Dr. Clements said, interrupting the video. "Right there. He wants to explain. He wants to explain five."

    — Benedict Carey
    New York Times


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