Robinson, Matthew. "National Issue: Does Good Practice Make Perfect? Six In 10 Educators Say No; Researchers Say Yes." Investors Business Daily, Inc., April 24, 1998.
Nothing attracts more ire from modern educators than asking children to memorize and practice, whether it be their spelling words or multiplication tables. When polled last year, some six of 10 education professors objected to having kids memorize material. These educators, who teach K-12 teachers, warn that practice, homework and direct, systematic instruction turn kids into automatons, stifling their creativity and ultimately dooming their ability to learn. They even have a term for it: “Drill and kill.”
The result? American kids spend less time working under instruction, and do less homework than their global peers. But new findings in cognitive science and psychology support drills and practice. “Nothing flies more in the face of the last 20 years of research than the assertion that practice is bad,” asserted Professors John Anderson, Lynne Reder and Herbert Simon of Carnegie-Mellon University. “All evidence . . . indicates that real competence only comes with extensive practice. By denying the critical role of practice, one is denying children the very thing they need to achieve competence,” they wrote in a recent study. “The instructional problem is not to kill motivation by demanding drill, but to find tasks that provide practice - while at the same time sustaining interest.”
That idea is causing a stir in the education world. Anderson, Reder and Simon are applying their findings in cognitive psychology to challenge the education status quo. Kids, they argue, can learn better through “deliberate practice” - through hard work and constant feedback to master knowledge and tasks. Of course, this is what most Americans think of when they are asked about education. They think of learning core knowledge that will be useful later in life.
“Nobody expects someone to be great without a great deal of practice and time in sports or music,” Anderson said. “But it still seems that in the area of education, there is the notion that all we have to do is give a child a critical insight or inspiration and everything else will fall into place. “Intellectual competence has to build up with the same kind of deliberate practice as musical talent or athletic ability,” he added.
The education establishment’s views on practice and memorization go back to the beginning of the century. They were popularized by progressive educators such as John Dewey, William Heard Kilpatrick and Carl Rogers. Ideas in education drawn from these thinkers go by many names: rationalist or romantic theories of learning, constructivism, situated learning, project-style learning and discovery learning. The theories have in common the belief that kids need only look inside themselves for knowledge. In this view, education is all about letting kids discover what they need to learn. In fact, some theorists claim that kids are actually hurt by direct, systematic instruction.
And such ideas get broad support in schools of education. Fully 92% of education professors say that “teachers should see themselves as facilitators of learning, who enable their students to learn on their own.” Only 7% think teachers should be “conveyors of knowledge who enlighten their students with what they know.” They also believe that direct instruction leads to “routinization.” This, they think, drives out understanding.
“These valid objections to purely verbal, fragmented and passive education have . . . been used as a blunt instrument to attack all emphasis on factual knowledge and vocabulary,” wrote E.D. Hirsch in The Schools We Need. California learned the hard way that romantic theories of education don’t work as well as direct instruction. The State Board of Education reversed its position in 1995 on whole language reading theory. Whole language says that kids learn to read the same way they learn to speak - by absorbing new words in the reading situation. Instead of teaching kids to sound out words and break down harder words, they would learn to guess words’ meaning from context.
In other words, kids were largely left on their own to “discover” how to read and spell. After almost a decade of practicing this method, the state scored at the bottom of national reading tests. The nation’s biggest state now backs using direct instruction - including phonics instead of whole language to teach reading. “Children vary in the amount of practice that is required for automaticity and fluency in reading to occur,” the state’s Comprehensive Reading Leadership Program found. “Some need to read a word only once to recognize it again with greater speed; others need more than 20 exposures. . . . Therefore, it is vital that students read a large amount of text at their independent reading level, and that the text provide specific practice in the skills being learned.”
Translation: Learning requires reinforcement — practice and memorization — to master a subject. “It’s the missing link in American education,” said Arthur Bornstein, a memory training expert in Los Angeles. “Schools tell kids what to learn, but not how to do it.” Bornstein coaches schools on how to help students remember what they’re taught. Bornstein has watched schools slowly move away from stressing drills. “It’s a tragedy,” he said, “but too many schools now assume kids will just pick up things as they go along.”
Some experts argue that the education establishment’s downplaying of deliberate practice helps explain American students’ lower test scores. In the Third International Mathematics and Science Study, American students were near the bottom. U.S. high school seniors scored 19th out of 21 nations in math. In science, they scored 16th. American kids get nearly an hour less homework a day than the foreign average - 1.7 hours compared with 2.6 hours.
A study by James W. Stigler, professor of education at the University of California, Los Angeles, found that the average Japanese student gets instruction 90% of the time spent in the classroom. American kids get it only 46% of the time. But cognitive research shows practice isn't just important for students. Teachers need it, too. Asian teachers spend much more time preparing to teach lessons. And although American teachers have smaller classes, they get less prep time.
But do drills stifle creativity, as many educators charge? No, says Temple University psychology professor Robert W. Weisberg. Weisberg has studied the link between creativity and knowledge in artists such as Mozart, Picasso and Jackson Pollack. What he’s found is surprising. “It’s a paradox,” Weisberg said. “There is evidence that deep immersion is required in a discipline before you produce anything of great novelty. Before you look at significant achievement, expect to see 10 years of deep immersion to gain knowledge.” But, he noted, “There is this concept that genius has leaps of insight way beyond everybody else. If you look at the backgrounds of these people, there is much more of a progression. They don’t make leaps — they build in small pieces.” Studies show that the brain actually changes with deliberate practice. A report in the journal Science shows that the cortical areas of the brain devoted to controlling the fingers actually expand for expert violinists.
Of course, repetitive drilling has its skeptics. “If deliberate practice just meant rote memorization, then I wouldn’t like it,” said Boston College psychology professor Ellen Winner. “If (it) means working at something until you got it just right, like in a play, then I am all for it.”
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Thursday, December 6, 2007
Does Good Practice Make Perfect?
Rationales for Cursive First Penmanship Instruction
Fitzgerald, Elizabeth. "Rationales for Cursive First Penmanship Instruction." swrtraining.com. http://www.swrtraining.com/id29.html.
Introduction
Although the United States today surpasses other nations in economics, manufacturing, technology, scientific research, and political power, her citizens suffer from a dangerous and crippling deficit: we are an illiterate people. Even the briefest survey of our national history reveals that this was not always the case. While the youth of the past was given the tools to reach his goals, today’s children flounder in an educational system which increasingly fails to provide even the most basic of skills. America is a nation at risk, and something must be done to revive the teaching methods that once produced a well-educated and literate society.
The purpose of this chapter is to discuss the rationales for teaching children cursive as the primary method for penmanship instruction. Topics to be discussed include today’s literacy and penmanship failures, the history of handwriting, how our educational system changed with regard to the teaching of reading and writing, and how returning to the method of cursive first will be a step toward solving some of these problems. The necessity for handwriting instruction in our computer age will also be addressed. The goal is that the teacher will be better equipped to understand and explain the rationale of this superior method.
Our Nation's Literacy Failures
Our population today is poorly equipped to handle the typical reading and writing requirements in our society. Today’s failing literacy rates contrast sharply with those of previous centuries. Not only has the literacy rate dropped substantially, but so has the definition of literacy itself (Gatto, 2001). Lack of handwriting skills also contributes to our literacy failures. Platt Spencer, creator of the Spencerian method, wrote of the importance of penmanship:Writing is almost as important as speaking, as a medium for communicating thought. For this reason it is said that writing is a secondary power of speech, and they who cannot write are in part dumb. Scrawls that cannot be read may be compared to talking that cannot be understood; and writing difficult to decipher, to stammering speech (Spencer, [n.d.]).
Our nation’s declining handwriting skills signify a deplorable failure in our communicative abilities.
A Brief History of Handwriting
In order to understand where we are and how we got here, we must look to the past and see from whence we came. Betty Edwards, in Drawing from the Right Side of the Brain, recounts: “In past centuries, handwriting was considered an art....in the nineteenth century much time and attention was consumed in perfecting the extravagant loops and swirls of Copperplate script” (Edwards, 1989).
By the late 1930s, the ball-and-stick method for printing lettering was introduced with an intended transitional period around fourth grade to “real” writing or cursive. The shift from cursive to manuscript was also based on the philosophies of the progressive education movement. Typical of the modern movements of the day was a sweeping attitude of “out with the old and in with the new,” regardless of logic or prior success.
Philosophical Changes in Reading Instruction
Prior to the 20th century, reading was always taught by way of phonics, since English is an alphabetic writing system. In the 1930s, phonics was replaced with the whole-word method (also called look-say or sight-word) as if English were an ideographic writing system like Chinese. Gallaudet’s [whole-word] ideas were touted by John Dewey, the leading philosopher of the progressive education movement, and were given the needed scientific sanction by psychologist Edmund Burke Huey’s book The Psychology and Pedagogy of Reading (1908). Through such literature, teachers and principals were convinced that the new and radical methods were educationally sound (Blumenfeld, 1995).
The motivation for changing the learning foundation was based on a shift in basic ideologies regarding the value of individual thought and intelligence. The plan was to delay formal reading instruction until the child was eight or nine years of age, the same age by which researchers today are warning us a child must have mastered this skill or face severe challenges (Lyon, 1998).
At the time, some voiced direct warnings about this change in direction, but were not only ignored as having valuable insight, but were denounced by the teaching profession, the university professors, and the book publishers who backed the International Reading Association, the world’s most powerful professional lobby for the advocacy of the look-say method.
Over the years, despite repeated warnings, the proponents of the whole-word method were blind to the faulty methodology. Samuel Blumenfeld explains, “In many ways, the whole-language juggernaut is like a computer virus wiping out all past knowledge and practice of traditional, time-tested reading instruction" (Blumenfeld, 1995). Fortunately, educators and researchers are now rediscovering the direct relationship between instructional methodology and success.
The Introduction of Manuscript
In the same way that reading methodology changed, directly affecting the ability of children to learn to read, handwriting methodology was altered, with its resultant decline in skills. Unfortunately, few people today remember the way penmanship was taught prior to the progressive education movement, and the misguided belief is that manuscript-first is the method of choice because that is the method that has always been used.
The first rationale proposed for primarily teaching ball-and-stick print was the idea that the first grade child did not have the fine motor skills or muscular dexterity in his fingers to be able to write elaborate cursive (Blumenfeld, 1997; Joss, 2001). It was believed that manuscript would be developmentally best for young children. Contrary to this theory, children prior to the 1930s never had a problem learning cursive in first grade, their manual dexterity never interfered, and they were well trained in cursive penmanship (Blumenfeld, 1997). In fact, experts in the field of dyslexia are now contending that cursive is actually more developmentally appropriate and easier for the child to learn.
Educators turned to teaching manuscript first for another reason directly related to reading. The goal was to teach children to write in the same style which they would read in books, thus reducing the confusion between the two alphabets and enhancing their ability to read. The belief that teaching manuscript first would help reading proved defective for several reasons.
Thinking that children cannot recognize a cursive letter “a” because it looks different than a manuscript or bookface letter a is as ridiculous as thinking the child could not recognize the color blue outside of his initial learning context.
Other educators have also noted that teaching cursive does not impair a child's ability to read. McInnis further attested to children's ability to recognize letters formed within various fonts and stated that as long as both cursive and manuscript forms of the letter are taught, children have no confusion in learning to read them. He further proposed:Conventional wisdom says that children who write in cursive would have difficulty reading manuscript. There is no evidence to support this opinion! In fact, the only country in Europe which does not teach cursive first is Great Britain. The literacy rate in the European countries does not suffer because of the introduction of cursive. In the 1950's French educators attempted to improve their educational system by shifting to manuscript. Within three years they realized their error and immediately changed back to cursive (McInnis, 1995; emphasis original).
Problems Related to the Transitional Process
In spite of the theory that manuscript should be taught to children first, it was never intended to be an adult hand, and a transition to cursive around the third or fourth grade became necessary. Requiring a child to transition to another writing style once the primary system has been automated creates many problems. Following are several reasons why this transitional process prohibits the development of a good cursive hand.
The transition from one writing style to another constitutes flawed teaching. As Gladstone aptly points out, “We do not allow this artificial split to be created in any other area of education. It would be considered ridiculous to teach math entirely in Roman numerals up into third grade, then drop it all for modern Arabic numerals....and we don’t teach English by starting off with Chinese” (Gladstone, 2000).
Experienced educators understand that instruction should build upon itself rather than working against future teaching. Blumenfeld asserts,
That is why it is so important to teach the basics in the right manner from the very beginning, and why I advise parents that there are two very important no-no’s in primary education. Do not teach anything that later has to be unlearned, and do not let the child develop a bad habit. Instruct the learner to do it right from the start (Blumenfeld, 1997).
Blumenfeld further warns, “Children will only make the effort to learn one primary way of writing which they will use for the rest of their lives. They don’t need to be taught three ways, two of which will be discarded” (ibid.). A Beka Books agrees, “By starting with cursive writing rather than manuscript printing, we help the child develop good writing habits from the very beginning. This means that habits acquired from manuscript printing do not need to be unlearned” (A Beka, 2003).
Children struggle through the transition process because the grip used for manuscript writing differs significantly from that used for cursive. Manuscript writing requires more of a vertical hold, which children do not change when they attempt cursive. Blumenfeld attributes the failure of many to successfully transition from manuscript to cursive to this problem with the pencil grip (Blumenfeld, 1997).
Another problem with the transitional process is that it discredits the work children have done for three to four years to develop their writing. More importantly, the transition process interrupts the child’s ability to write at all, just at the time he is expected to put expanded thoughts into writing (Joss, 2001). A ten-year study on handwriting by Berninger and Graham revealed that the speed and legibility of third grade children reverted to first-grade levels. The authors credited this regression to the transitional process (Gladstone, 2000).
The transitional process fails because third and fourth grade teachers do not have the time to devote to handwriting instruction that teachers in kindergarten and first grade have.
Parents often question the idea of not teaching children to print, citing job applications as examples where print is needed. The student does need to learn to print, though not as his primary hand. Traditionally, print was taught as an art form in high school. There is plenty of time for a student to learn print before he applies for his first part-time job.
For reasons already mentioned, the student often fails to complete the transition process, which in turn can affect his ability to read cursive. The experience among secondary teachers at the school is that students, particularly those with special educational needs, have difficulty deciphering teachers’ handwriting.
Children asked to transition from manuscript to cursive often develop a writing style which is more of a hybrid of the two forms.
Lack of Instruction
American children’s penmanship has also suffered from inadequate, inconsistent, and in some cases a complete lack of instruction by our nation’s teachers.
Advantages of Cursive-First Instruction
Discussion thus far has addressed the history of reading and writing instruction, the reasoning behind the manuscript-first methodology, the faults with the transitional process, and the lack of teacher instruction. These weaknesses are resolved by returning to the cursive-first approach, the advantages of which will be discussed below.
Cursive writing involves a flowing, uninterrupted movement which reinforces the left-to-right directionality of our written language. The connected writing allows for continuous flow of thought and thinking ahead while writing. It also reinforces the beginning and ending of words, with proper spacing of letters, unlike manuscript.
The flowing nature of cursive virtually eliminates reversals, a common occurrence in manuscript writing. McInnis stated, “Each time the writing implement is picked up from the paper the potential for error is increased” (McInnis, 1995). To illustrate, a musician might make an error in the middle of a performance were he to move his hands too far from the instrument’s frets or keys. Mosse cited that confusion over similarly formed lower-case letters is eliminated with cursive (Mosse, 1982). Blumenfeld explained that with cursive, children do not confuse the letters because the movements of the hands make it impossible. He also affirmed, “...this knowledge is transferred to the reading process. Thus, by teaching children the distinctive differences between letters, learning to write cursive helps learning to read print” (Blumenfeld, 1997).
As Blumenfeld has pointed out, several educators and publishers are now recognizing that teaching cursive writing from the beginning actually facilitates the reading and spelling processes. ABeka Books states, “We also strengthen the child’s reading skills. By joining letters, cursive writing reinforces the blending of sounds within words” (ABeka, 2003). Blumenfeld credits cursive with aiding the spelling process.[Cursive] helps the child learn to spell correctly since the hand acquires knowledge of spelling patterns through repeated hand movements. This is the same phenomenon that occurs when pianists or typists learn patterns of hand movements through continued repetition (Blumenfeld, 1997).
As already noted, children who are taught manuscript only and who struggle with the transition process often fail in their ability to read the cursive writing of others. On the other hand, when cursive is taught first, the student is able to read what others write, limited only by his own reading ability rather than the handwriting style of the writer.
The advantages of cursive writing also extend into other areas of the curriculum. Thomas found that French teachers find a direct connection between handwriting instruction and art, PE, and music (Thomas, 1997).
Experts in the field of dyslexia and reading disorders are recognizing the advantages of cursive instruction. As McInnis began examining the handwriting problems of students, he contended, “It became apparent that the sequence utilized was incorrect. It became increasingly clear that cursive was the preferred pattern.” According to his research, cursive-first is the preferred method for remedial students (McInnis, 1995). Nelson also notes, “There are a number of reading specialists who are now convinced that cursive should be taught in the beginning. They believe that it offers advantages over printing for reading skill development” (Nelson, [n.d.]). Finally, the experts are returning to the time-tested and solid methodology of cursive first. Unfortunately, more educators need to understand the value of this approach and teach it first, before the reading and writing disabilities develop.
Penmanship Instruction in the Computer Age
The question often arises as to the need for penmanship instruction at all in view of our current “computer age”. The rationales for continuing to teach penmanship, despite technological advances of the day, are numerous. Of primary importance is the sensory input involved in the writing process and its crucial relationship to learning to read. Blumenfeld propounded:[The child] must get used to the idea that reading and writing are inseparable skills. One goes with the other.... Thus, reading and writing, or decoding and encoding as the linguists prefer to call it, are two parts of one skill, and should be learned simultaneously. The learning of one reinforces the other (Blumenfeld, 1973).
It is obvious that penmanship instruction is vital to the student’s ability to learn to read and write, regardless of his computer skills.
The ability to write remains critical in our computer age. The need to write by hand will always be with us. Everyday tasks such as sending cards, writing “to do” lists, adding names to an address book, taking notes from a phone message, etc., require the quick use of a pen or pencil, not a computer. Pens and pencils are affordable to the average person whereas the cost of computers prohibits their use by everyone. There are still schools that cannot afford computers for student use. The pencil does not require electricity or backup battery power. Perhaps most noticeable of all the needs for penmanship is how it provides a personal expression of the individual.
Call to Action
America is a nation at risk for failing communicative abilities. History reveals that this was not always the case. Philosophical changes in reading and writing instruction have led to a deplorable decline in the abilities of this nation’s school children. The methodologies thought to be developmentally appropriate for children have served only to handicap them and cause learning disabilities in shocking proportions. The transitional process required with the manuscript-first approach has resulted in the failure for most to successfully master the cursive hand. Adequate instruction when children are expected to learn cursive has been found wanting. Despite the promises of the computer age, penmanship skills remain a high priority for all well educated individuals. The cursive-first methodology of penmanship instruction and explicit phonics for reading instruction offer the beginning of a solution to our nation’s literacy crisis. It is my hope that those embarking on the awesome responsibility of teaching the most basic skills to our children would be encouraged to lay a worthy foundation, beginning by teaching cursive first, and to use the ideologies and methods that once made this nation great.
References
A Beka, 2003. www.abeka.com/Resources/Cursive.html
Beall, Ruth Ann, The Phonogram Fun Packet. This and other Bealls Learning Games are described on our Games page.
Bibeau, Simone, Developing the Early Learner: The Complete Perceptual Growth Program. Perception Publications, 1982. Available from Sonlight Curriculum, Ltd. (303) 730-6292. Note: If you’re ordering from Sonlight’s online catalog, they call this ?Readiness Skills Vol. I-IV? and do not refer to it by the books? name; however, it is listed by name in the paper catalog.
Blumenfeld, Samuel, Homeschooling: A Parents Guide to Teaching Children, Boise, Idaho: The Paradigm Company, 1997.
Blumenfeld, Samuel, How to Tutor, Boise, Idaho: The Paradigm Company, 1973.
Blumenfeld, Samuel, The Whole Language/OBE Fraud, Boise, Idaho: The Paradigm Company, 1995.
Corcoran, John, with Carole Carlson. The Teacher Who Couldn't Read. Colorado Springs, CO: Focus on the Family, 1994.
Douglass, Frederick. Narratives of the Life of Frederick Douglass, an American Slave. New York: Penguin Books, 1986.
Dunlap, Joy Marie. "How to Teach Your Child to Read: Preparing Little Ones to Read." Teaching Home Magazine, Sept/Oct 1998. An excellent article on working with pre-readers. You can order this and other back copies from the Teaching Home magazine.
Edwards, Betty, Drawing from the Right Side of the Brain. Los Angeles, CA: Jeremy P Tarcher, Inc., 1989.
Gatto, John Taylor, The Underground History of American Education. New York: Oxford Village Press, 2001.
Gladstone, Kate, “Handwriting Repair” www.global2000.net, 2000. Ms. Gladstone works nationwide to help people improve their penmanship. Like me, she is a firm believer in the one-tiered method for penmanship instruction--as opposed to teaching one style and then a different one later. Our application of this instructional methodology differs, however, in that Ms. Gladstone is a proponent of the Italic method of penmanship.
Hall, Susan. Straight Talk about Reading. Chicago, IL: Contemporary Books, 1999.
Huey, Edmund Burke, The Psychology and Pedagogy of Reading. New York: Macmillan, 1908; Cambridge, MA: The MIT press, 1968.
Joss, Jan, “BJU PreCursive and Cursive Handwriting.” Teacher to Teacher, Vol. 5, Num. 1, April 2001.
Kozol, Jonathan. Illiterate America. New York: Plume, 1985.
Lyon, G. Reid. Ph.D., Chief, Child Development and Behavior Branch of the National Institute of Child Health and Human Development, National Institutes of Health, in a presentation before the U.S. Senate entitled ?Overview of Reading and Literacy Initiatives,? April 1998.
McCrum, Robert with William Cran and Robert MacNeil. The Story of English. New York: Penguin Books, 1987.
McGinness, Diane. Why Our Children Can't Read and What We Can Do About It. New York: Simon & Schuster: 1997.
McInnis, Philip, “Simplifying the Writing Process.” NATHHAN NEWS, 1995. www.NATHHAN.com
Mosse, Hilde. You Can Prevent or Correct Learning Disorders.Beaverton, OR: Riggs Institute, 1982.
Myers, Edwin C. and Nellie M. Myers, ReadyWriter. Providence Project, 1989. Also called Learning Vitamins for Stylus Skills and Beginning Penmanship. Available from the publishers: Providence Project,14566 NW 110th Street, Whitewater, KS 67154, 1-888-776-8776.
National Advisory Council on Adult Education. Illiteracy in America: Extent, Causes, and Suggested Solutions. Washington, D.C.: U.S. Government Printing Office, [n.d.]
Nelson, Rand H., “What is it about Cursive?” Greensburg, PA: Peterson Directed Handwriting, [n.d.].
Spalding, Romalda. The Writing Road to Reading. New York: William Morrow, 1990.
Stanovich, Keith E. "Romance and Reality." The Reading Teacher 47, no. 4, December 1993/January 1994.
Thomas, Fiona, “Une question de writing?” A research project commissioned by the Teacher Training Agency as part of the Teacher Research Grant Scheme, England, 1997.
Yopp, Hallie Kay, and Ruth Helen Yopp, Oo-pples and Boo-noo-noos: Songs and Activities for Phonemic Awareness. Harcourt Brace & Company, 1996.
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A Critique of Learning Styles
BY STEVEN A. STAHL
Steven A. Stahl is professor of reading education at the University of Georgia and co-director of the Center for Improvement of Early Reading Achievement. His research interests are in beginning reading and vocabulary instruction. The original source can be found by clicking here.
I WORK WITH a lot of different schools and listen to a lot of teachers talk. Nowhere have I seen a greater conflict between “craft knowledge” or what teachers know (or at least think they know) and “academic knowledge” or what researchers know (or at least think they know) than in the area of learning styles. Over the years, my experience has told me to trust teachers; it has also taught me that teachers’ craft knowledge is generally on target. I don’t mean to say that teachers are always right, but they have learned a great deal from their thousands of observations of children learning in classrooms.
So, when teachers talk about the need to take into account children’s learning styles when teaching, and researchers roll their eyes at the sound of the term “learning styles,” there is more to it than meets the eye.
The whole notion seems fairly intuitive. People are different. Certainly different people might learn differently from each other. It makes sense. Consider the following from the Web site of the National Reading Styles Institute, a major proponent of the application of learning styles to the teaching of reading:
We all have personal styles that influence the way we work, play, and make decisions. Some people are very analytical, and they think in a logical, sequential way. Some students are visual or auditory learners; they learn best by seeing or hearing. These students are likely to conform well to traditional methods of study.
Some people (we call them “global learners”) need an idea of the whole picture before they can understand it,while “analytic learners” proceed more easily from the parts to the whole. Global learners also tend to learn best when they can touch what they are learning or move around while they learn. We call these styles of learning “tactile”and “kinesthetic.” In a strictly traditional classroom, these students are often a problem for the teacher. She has trouble keeping them still or quiet. They seem unable to learn to read. (http://www.nrsi.com/about.html)
This all seems reasonable, but it isn’t.
Research and Learning Styles
The reason researchers roll their eyes at learning styles is the utter failure to find that assessing children’s learning styles and matching to instructional methods has any effect on their learning. The area with the most research has been the global and analytic styles referred to in the NRSI blurb above. Over the past 30 years, the names of these styles have changed—from “visual” to “global” and from “auditory” to “analytic”—but the research results have not changed.
In 1978, Tarver and Dawson reviewed 15 studies that matched visual learners to sight word approaches and auditory learners to phonics. Thirteen of the studies failed to find an effect, and the two that found the effect used unusual methodology. They concluded:Modality preference has not been demonstrated to interact significantly with the method of teaching reading.1
One year later, Arter and Jenkins reviewed 14 studies (some of these are overlapping), all of which failed to find that matching children to reading methods by preferred modalities did any good. They concluded:[The assumption that one can improve instruction by matching materials to children’s modality strengths] appears to lack even minimal empirical support.2
Kampwirth and Bates, in 1980, found 24 studies that looked at this issue. Again, they concluded:
Matching children’s modality strengths to reading materials has not been found to be effective.3
In 1987, Kavale and Forness reviewed 39 studies, using a meta-analysis technique that would be more sensitive to these effects. They found that matching children by reading styles had nearly no effect on achievement. They concluded:Although the presumption of matching instructional strategies to individual modality preferences has great intuitive appeal, little empirical support for this proposition was found.... Neither modality testing nor modality teaching were shown to be [effective].4
A fifth review, in 1992, by Snider found difficulties in reliably assessing learning styles and a lack of convincing research that such assessment leads to improvement in reading.
Recognition of individuals’ strengths and weaknesses is good practice; using this information, however, to categorize children and prescribe methods can be detrimental to low-performing students. Although the idea of reading style is superficially appealing, critical examination should cause educators to be skeptical of this current educational fad.5
These five research reviews, all published in well-regarded journals, found the same thing: One cannot reliably measure children’s reading styles and even if one could, matching children to reading programs by learning styles does not improve their learning. In other words, it is difficult to accurately identify children who are “global” and “analytic.” So-called global children do not do better in whole language programs than they would in more phonics-based programs. And so-called analytic children do not do better in phonics programs than they do in whole language programs. In short, time after time, this notion of reading styles does not work.
This is an area that has been well researched. Many other approaches to matching teaching approaches to learning styles have not been well researched, if at all. I could not find studies in refereed journals, for example, documenting whether the use of Howard Gardner’s Multiple Intelligences Model6 improved instruction. This does not mean, of course, that the use of the model does not improve achievement, only that I could not find studies validating its use. The same is true of other learning style models.
One cannot prove a negative. Even if all of these studies failed to find that matching children by learning styles helps them read better, it is always possible that another study or another measure or another something will find that matching children to their preferred learning modality will produce results. But in the meantime, we have other things that we know will improve children’s reading achievement. We should look elsewhere for solutions to reading problems.
Yet, the notion of reading styles (or learning styles) lingers on. This is true not only in my talks with teachers, but also in the literature that teachers read. The most recent issue of Educational Leadership included, as part of a themed issue on innovations, several articles on learning styles. Phi Delta Kappan also regularly contains articles on learning styles, as do other publications intended for
teachers.
Research into Learning Styles
Among others, Marie Carbo claims that her learning styles work is based on research. [I discuss Carbo because she publishes extensively on her model and is very prominent on the workshop circuit. In the references for this article, I cite a few examples of her numerous writings on the topic.7] But given the overwhelmingly negative findings in the published research, I wondered what she was citing, and about a decade ago, I thought it would be interesting to take a look. Reviewing her articles, I found that out of 17 studies she had cited, only one was published.8 Fifteen were doctoral dissertations and 13 of these came out of one university—St. John’s University in New York, Carbo’s alma mater.
None of these had been in a peer-refereed journal. When I looked closely at the dissertations and other materials, I found that 13 of the 17 studies that supposedly support her claim had to do with learning styles based on something other than modality. In 1997, I found 11 additional citations. None of these was published, eight were dissertations, and six of these came from St. John’s.
In short, the research cited would not cause anyone to change his or her mind about learning styles.
What Do People Mean by Learning Styles?
Modality refers to one of the main avenues of sensation such as vision and hearing. I have only talked about modality-based reading styles because these are both the best researched and the most heavily promoted. The National Reading Styles Institute claims that it has worked with “over 150,000 teachers,” and its advertisements seem to be everywhere. Furthermore, these notions of “visual” and “auditory” learners or “global” and “analytic” learners have been around for a long time and have found their way into a number of different programs, not just the NRSI programs.
There are other ways of looking at learning styles. People have proposed that children vary not only in perceptual styles, but on a host of different dimensions. To name a few, people have suggested that children are either two-dimensional/three-dimensional, simultaneous/sequential, connecting/compartmentalizing, inventing/reproducing, reflective/impulsive, field dependent/field independent, and so on.
Some of these are learning preferences, or how an individual chooses to work. These might include whether a person prefers to work in silence or with music playing, in bright light or dim light, with a partner or alone, in a warm room or a cool room, etc.
Some of these are cognitive styles, such as whether a person tends to reflect before making a choice or makes it impulsively, or whether a person tends to focus on details or sees the big picture. Some of these are personality types, such as whether a person is introverted or extroverted.
Some of these are aptitudes, like many of Howard Gardner’s multiple intelligences. Gardner suggests that people vary along at least seven different imensions—linguistic or the ability to use language, logico-mathematical or the ability to use reasoning especially in mathematics, spatial or the ability to use images or pictures, bodily-kinesthetic or the ability to control movement, musical, interpersonal or the ability to work with people, and intrapersonal or the thinking done inside oneself. The last two are more like personality types, rather than aptitudes or even learning styles. The others are Gardner’s attempt to expand the notion of what we think is intelligent behavior to people who are skilled in music, or dance, or even in interpersonal relations. In contrast to the traditional vision of learning styles as either/or categories (either a person is visual or he or she is auditory), multiple intelligences are put forth by Gardner as separate abilities. A child may be strong in a few of these areas, or none of these areas.
What is a teacher to do with all this? If there are children who prefer to work with music, then the teacher might either provide Walkmans for those who prefer music or play music openly and provide earplugs for those who don’t. If there are children who prefer to work in bright light, the teacher might seat those children over by the window. Children who like to snack while reading can be allowed to eat during class (healthy foods, of course). It would be easy to see how accommodating all of these preferences in a class could lead to chaos. How would a teacher lecture, give assignments, or even call to order a class in which a sizable proportion of the students was wearing earplugs? Or how does one regulate the temperature so part of the room is warm and part cool?
Others have used learning styles theory as a way of making sure that all the needs of diverse learners are being met. Marguerite Radenich used Gardner’s model to examine literature study guides.9 Her ideal was one that incorporated all of these ways of knowing into an integrated whole to be used to study adolescent literature. Thus, Gardner’s model was used here to create more multidimensional instruction. This is very different from using these different styles to segregate children into groups where they would receive fairly one-dimensional instruction.
Thoughtful educators have tried to make this work, and perhaps it is workable, but trying to meet all of the preferences of a group of children would seem to take energy that would be better spent on other things. This is especially true since no one has proven that it works.
Learning Styles and Fortune Telling
Why does the notion of “learning styles” have such enduring popularity—despite the lack of supporting evidence? I believe that this phenomenon has a lot in common with fortune telling.
You go to see a fortune teller at a circus. She looks you over and makes some quick judgments—how young or old you are, how nicely you are dressed, whether you appear anxious or sad or lonely—and based on these judgments, tells your fortune. The fortune she tells may be full of simple and ambiguous statements—“you will be successful at your next venture,” “you will be lucky at love,” or may be more complex—“you are successful at home, but someone is jealous; make sure you watch yourself.” Either way, the statements are specific enough so that they sound predictive, but ambiguous enough that they could apply to a number of situations.
When we read the statements on a Learning Style Inventory, they sound enough like us that we have a flash of recognition. These inventories typically consist of a series of forced choices, such as these from Marie Carbo’s Reading Style Inventory, Intermediate, 1995.
A) I always like to be told exactly how I should do my reading work.
B) Sometimes I like to be told exactly how I should do my reading work.
C) I like to decide how to do my reading work by myself.
Or
A) I like to read in the morning.
B) I don’t like to read in the morning.
A) I like to read after lunch.
B) I don’t like to read after lunch.
A) I like to read at night.
B) I don’t like to read at night.
Or
A) I read best where it’s quiet with no music playing.
B) I read best where there is music playing.
C) I read about the same where it’s quiet or where there is music playing.
Since all of us have some preferences (my experience is that adults have clear preferences about music during reading, especially), these items tend to ring true. Like the fortunes told by the fortune teller, these statements at first light seem specific enough to capture real distinctions among people. But the problem with choices like these is that people tend to make the same choices. Nearly everybody would prefer a demonstration in science class to an uninterrupted lecture. This does not mean that such individuals have a visual style, but that good science teaching involves demonstrations. Similarly, nearly everybody would agree that one learns more about playing tennis from playing than from watching someone else play. Again, this does not mean that people are tactile/kinesthetic, but that this is how one learns to play sports. Many of these “learning styles” are not really choices, since common sense would suggest that there would not be much variance among people. In the class sample provided with the Reading Style Inventory above, for example, 96 percent of the fifth-graders assessed preferred quiet to working while other people were talking, 88 percent preferred quiet to music, 79 percent picked at least two times of day when they preferred to work, 71 percent had no preference about temperature, and so on. Virtually all of the questions had one answer preferred by a majority of the students.
The questions are just specific enough to sound like they mean something, but vague enough to allow different interpretations. For example, does “music” refer to Mozart or Rap? Obviously, one’s choices would be different for different types of music. A more serious question would arise over the “teacher direction” item. Doesn’t the amount of teacher direction needed depend on the difficulty of the assignment? There are some assignments that are self-evident and do not need much teacher direction, but when work gets complex, students need more direction. This is not a matter of preference.
The other major problem with these inventories is that there are no questions about a child’s reading ability. So children with reading problems are given the same measure as children who are doing well in reading. This has two effects. First, there is a bias on some items for children with different abilities. Consider these two items, also from the Carbo inventory:
A) It’s easy for me to remember rules about sounding out words.
B) It’s hard for me to remember rules about sounding out words.
Or
A) When I write words, I sometimes mix up the letters.
B) When I write words, I almost never mix up the letters.
Children with reading problems are more likely to answer that they do not remember phonics rules and that they sometimes mix up the letters. According to the learning styles research reports, such children are likely to be considered as having a global (or visual) preference.11 Actually, this may not be a preference at all, but a reflection of the child’s current level of reading ability. The potential for harm occurs when children with reading problems are classified as “global” (visual) learners and thereby miss out on important instruction in decoding, or are classified as “analytic” (auditory) learners and miss out on opportunities to practice reading in connected text.
Not including information about reading ability also leads to some strange prescriptions. Adults attending learning styles workshops often get prescriptions for beginning reading instruction methods, such as the language experience approach or phonics/linguistic approaches, certainly not needed by competent readers. And for children, too, some of the approaches may be inappropriate.
The language experience approach, for example, is best suited for children at the emergent literacy stage, when they need to learn about basic print concepts, one-to-one matching, letter identification, and so on.12 For a second-grader, or even a newly literate adult, language experience may be appropriate (if they still have not mastered basic print concepts) or highly inappropriate (if they are already reading fluently). It depends on the readers’ skill, not their learning styles.
Reliability
If you are to use a test, even an inventory like the one cited above, it should be reliable. If a test is reliable, that means you are going to get the same (or close to the same) results every time you administer it. If a test is 100 percent reliable (or has a reliability coefficient of 1.0), then a person will score exactly the same on Thursday as on Tuesday. Perfection is tough to come by, so we generally want a reliability coefficient to be .90 or higher.13 If a test is not reliable, or trustworthy, then it is difficult to believe the results. This is a problem, not only with inventories, but with any measure that asks subjects to report about themselves.
Reliabilities of these measures are relatively low. The self-reported reliabilities of Carbo’s Reading Style Inventory and Dunn and Dunn’s Learning Style Inventories are moderate, especially for a measure of this kind—in the neighborhood of the .60s and the .70s. Similar reliabilities are reported for the Myers-Briggs Inventory, another learning styles assessment.14 These are lower than one would want for a diagnostic measure. And, these scores are inflated, since for many items there is generally one answer that nearly everybody chooses. This would tend to make the reliabilities higher.
The vagueness in the items may tend to make the reliabilities low. Again, how a child interprets each item will influence how it is answered, as with the “teacher direction” and “music” examples discussed earlier.
Test-retest reliabilities are particularly important for a measure of learning styles. These moderate reliabilities could be interpreted in two ways. The test itself may not be a reliable measure of what it is supposed to measure—that is, a person has a stable learning style, but the test is not getting at it. If the test is not reliable, then the information it gives is not trustworthy.
The other possibility is that learning styles may change, from month to month, or even week to week. This is also problematic. If we are talking about matching a person to a situation using this instrument, this is a relatively long-term (semester or academic year) matching. If a person’s style changes, then one either must measure learning styles frequently, or allow for more flexible assignments.
How Reading Develops
The Learning Style model assumes that different children need different approaches to learn to read. Children are different. They come to us with different personalities, preferences, ways of doing things. However, the research so far shows that this has little to do with how successful they will be as readers and writers. Children also come to us with different amounts of exposure to written text, with different skills and abilities, with different exposure to oral language. The research shows that these differences are important.
Rather than different methods being appropriate for different children, we ought to think about different methods being appropriate for children at different stages in their development. Children differ in their phonemic abilities, in their ability to recognize words automatically, in their ability to comprehend and learn from text, and in their motivation and appreciation of literature.15 Different methods are appropriate for different goals. For example, approaches that involve the children in reading books of their own choice are important to develop motivated readers.16 But whole language approaches, which rely largely on children to choose the materials they read, tend not to be as effective as more teacher-directed approaches for developing children’s word recognition or comprehension.17
A language experience approach may be appropriate to help a kindergarten child learn basic print concepts. The child may learn some words using visual cues, such as might be taught through a whole word method. With some degree of phonological awareness, the child is ready to learn letters and sounds, as through a phonic approach. Learning about letters and sounds, in combination with practice with increasingly challenging texts, will develop children’s ability to use phonetic cues in reading, and to cross-check using context. With additional practice in wide reading, children will develop fluent and automatic word recognition. None of this has anything to do with learning styles; it has to do with the children’s current abilities and the demands of the task they have to master next.
What Do Teachers Get out of Learning Styles Workshops?
I have interviewed a number of teachers who have attended learning styles workshops. These were meetings of 200 to 300 teachers and principals, who paid $129 or so to attend a one-day workshop or up to $500 to attend a longer conference. They have found them to be pleasant experiences, with professional presenters. The teachers also feel that they learned something from the workshops.
After I pressed them, what it seemed that they learned is a wide variety of reading methods, a respect for individual differences among children, and a sense of possibilities of how to teach reading. This is no small thing. However, the same information, and much more, can be gotten from a graduate class in the teaching of reading.
These teachers have another thing in common—after one year, they had all stopped trying to match children by learning styles.
REFERENCES
1 Tarver, Sara, and M.M. Dawson. 1978. Modality preference and the teaching of reading. Journal of Learning Disabilities 11:17-29.
2 Arter, J.A., and Joseph A. Jenkins. 1979. Differential diagnosis-prescriptive teaching: A critical appraisal. Review of Educational Research 49:517-555.
3 Kampwirth, T.J., and M. Bates. 1980. Modality preference and teaching method. A review of the research. Academic Therapy 15:597-605.
4 Kavale,Kenneth,A.,and Steven R.Forness.1987.Substance over style: Assessing the efficacy of modality testing and teaching. Exceptional Children 54:228-239.
5 Snider,Vicki. E. 1992. Learning styles and learning to read: A critique. Remedial and Special Education 13:6-18.
6 Gardner,Howard.1993. Frames of mind: The theory of multiple intelligences. New York: Basic Books.
7 For example, Carbo, Marie. 1997. Reading styles times twenty. Educational Leadership 54 (6):38-42; Carbo, Marie, Rita Dunn, and Kenneth Dunn. 1986. Teaching students to read through their individual learning styles. Englewood Cliffs, N.J.: Prentice-Hall.
8 See Stahl, Steven A. 1988. Is there evidence to support matching reading styles and initial reading methods? A reply to Carbo. Phi Delta Kappan 70 (4):317-322.
9 Radenich, Marguerite Cogorno. 1997. Separating the wheat from the chaff in middle school literature study guides. Journal of Adolescent and Adult Literacy 41 (1):46-57.
10 All examples are from Carbo, Marie. 1995. Reading Style Inventory Intermediate (RSI-I): Author.
11 Carbo, M. 1988. Debunking the great phonics myth. Phi Delta Kappan 70:226-240.
12 Stahl, Steven A., and Patricia D. Miller. 1989.Whole language and language experience approaches for beginning reading: A quantitative research synthesis. Review of Educational Research 59 (1):87-116.
13 Harris,Albert J.,and Edward Sipay.1990. How to increase reading ability. 10th ed. White Plains, N.Y.: Longman.
14 Pittenger, David, J. 1993.The utility of the Myers-Briggs Type Indicator. Review of Educational Research 63:467-488.
15 Stahl, Steven A. 1998. Understanding shifts in reading and its instruction, Peabody Journal of Education 73(3-4):31-67.
16 Morrow, Lesley M., and Diane Tracey. 1998. Motivating contexts for young children’s literacy development: Implications for word recognition development. In Word recognition in beginning literacy, edited by J. Metsala and L. Ehri. Mahwah, N.J.: Erlbaum; Turner, Julianne, and Scott G. Paris. 1995. How literacy tasks influence children’s motivation for literacy. The Reading Teacher 48:662-673.
17 Stahl and Miller, op cit., Stahl, Steven A., C. William Suttles, and Joan R. Pagnucco. 1996.The effects of traditional and process literacy instruction on first-graders’ reading and writing achievement and orientation toward reading. Journal of Educational Research. 89:131-144.
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Is this Math fuzzy?
BY CARALEE ADAMS
Caralee is the editor of Parent Power! magazine. Parent Power! is published eight times a year by The Center for Education Reform. This article appeared in the August 2001 edition.
http://www.edreform.com/_upload/01august.pdf
Fuzzy Math is one of the phrases used to refer to the many newer mathematics programs that are designed around current fads in mathematics education and often are inadequate for effective learning. The word fuzzy is used in the sense of warm and fuzzy or fuzzy thinking, and does not refer to advanced topics like fuzzy logic.
Fuzzy Math programs are those that emphasize process over content (and even correct answers), and are deficient in mathematical breadth and depth.
Fuzzy Math takes many forms, but here are some of the varieties.
No text book - Publishers are selling instructional materials, including overheads, software, dice, spinners, and blocks instead of textbooks, which have clear explanations, worked out examples, and practice problems.
MTV - Characterized by lots of color and pictures that often have no relationship to the mathematics and do not contribute to learning, parents are right to worry that students cannot hear the mathematics amid all of the noise.
PC - These programs often include politically correct lessons in math class, such as writing essays about how to save the trees or the whales. Grades may be biased on political correctness, not math skills and knowledge.
Discover-It-Yourself - Based on the idea that children learn what they discover on their own, this approach avoids giving direct instruction to students. It takes longer for students to learn this way, so less material is covered, and great demands are placed on teachers and their mathematical knowledge.
Guesswork - Many of these programs promote guessing (rather than knowing) and spend considerable time on this process. They encourage students to work on problems that they have no idea how to solve and discourage practicing methods to mastery.
Anti-Algorithm – These books may talk about “many ways to multiply” but fail to cover the traditional method. Often, they encourage the use of calculators and pay little attention to manual computation.
Project and Investigation - Students often spend great deal of time on non-mathematical aspects of working on projects. Sometimes, a group of four students may spend a week on a project that requires just one student to solve a few simple problems.
Group Learning and Group Testing - In groups, the quicker students often do the work and the slower ones go along for the ride. Group learning and group testing help to equalize grades and thus students who fail to learn can pass their classes anyway.
Integrated Content - Some argue that mathematical topics, such as algebra and geometry, should not be taught in isolation. In practice, the mathematical development becomes so mixed up that it is impossible to tell what children should learn at any given time. Accountability becomes impractical, and achievement can to suffer.
Write About It – Based on the idea that students should be able to explain their math, precious time is spent writing essays that have very little mathematics content. In some cases, most of the math they do involves figuring out margins and spacing on the word processor.
If fuzzy math has invaded your world, the first thing to do is to figure out what your children should be learning. See the grade level standards in California (http://www.cde.ca.gov/ci/math.html) for help or go to http://www.mathematicallycorrect. com. Next, look for less fuzzy materials children can use at home and at school to meet these goals. Stay involved with your children’s math and their progress toward the goals for each grade level. This is no easy task, but it is the only way to be sure of real success.
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Video: A University View
BY CLIFF MASS
Dr. Mass is a professor of atmospheric science at the university of Washington. In this video he shares his experiences with students, including his son, who were taught with an integrated math approach.
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Video: Math Education - An Inconvenient Truth
BY M.J. MCDERMOTT
M.J. is a meteorologist with a degree in atmospheric sciences from the University of Washington. She also hosts the weather for the morning show on channel 13 in Seattle. In this video she explains Washington's current state of math education in 4th and 5th grades.
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Parent Review: "Math Trailblazers"
BY REBECCA P.
The following review of the Kendall/Hunt "Math Trailblazers" program was written by a parent in
A couple of notes about the abbreviations used: In the original of this article, the author referred to this program as "KHM" -- it is changed here to "MTB" to match the way the vendor refers to it. Also, the author frequently offers comparisons to Houghton-Mifflin Mathematics, which she refers to here as "HMM." --IllinoisLoop.org
A review of
Kendall/Hunt’s Math Trailblazers
Second Grade, Unit 11:
"Ways of Subtracting with Larger Numbers"
By Rebecca P., April 2002
I. Introduction
Parents are ultimately responsible for the quality of their children’s education. Ten years from now a parent shouldn’t say "But the teacher promised this series would lay a solid foundation for my child’s conceptual understanding of mathematics…."
My twin boys will start First Grade this fall at a public elementary school in
I spent one weekend studying Unit 11, "Ways of Subtracting Larger Numbers" from Kendall/Hunt’s Math Trailblazers (MTB) series for the Second Grade. I had the Teacher’s Implementation Guide (TIG), the Student Activity Book, and the Unit Resource Guide (URG) in front of me. There is no textbook for children, just simple word problems in an activity book. Thus I had to read the suggested teacher scripts from the Unit 11 URG if I wanted to find out how MTB is taught. As Kendall/Hunt will tell you in their promotional literature, this math series has been years in the making, and has been extensively field-tested at hundreds of schools. I am sure that they are sure that the teacher script in the URG is the ideal script.
My goal was to hear and see for myself the difference between how a whole number operation, such as Subtraction, is taught in the MTB series, compared to how it is taught in the Houghton-Mifflin Mathematics (HMM) series. I was concerned because our school staff proposes to split the math curriculum between MTB in grades K-2 and HMM in grades 3-5. I had no idea why they would do that. It sounded like they were going to cut the baby, my children’s education, in half.
Some would call HMM a more traditional approach to teaching arithmetic. Being an old math major, I thought it looked sufficiently rigorous for my boys, who are going to spend all day at school, five days a week. HMM has all the bells and whistles, all the games and manipulatives, that are supposed to make it fun for kids today. What was wrong with it?
You might think I was kidding if I told you that what was wrong with HMM is that it teaches children to "start with the ones" when they add or subtract two-digit numbers. What educators think is right with MTB is that it teaches children to "start with the tens."
Educators believe that starting with the tens is a natural way for Second Grade children to compute with two-digit numbers, and therefore it is good. Specifically, it is good enough. At least it is better than what happens to children when teachers teach the standard written arithmetic procedures by rote.
I think there is some high ground in the middle.
Teachers could tell students about starting with the tens and about starting with the ones. The former is our preferred approach when we’re standing in front of the shelves at the grocery store and the dollar amounts are low. The latter is our preferred approach when we sit down with a pencil and paper and numbers of any size. To save time, we start with the ones. But our children also need to know why we can start with the ones.
Teachers should teach standard written arithmetic procedures in the primary grades, and our teachers should try to do a very good job, because:
- Taught well, they improve our children’s understanding of whole numbers, and prepare our children for algebra and higher mathematics.
- Used correctly, they are guaranteed to work with whole numbers of any size.
II. Kendall/Hunt Math Trailblazers
It is necessary to write out, in detail, the teacher script for the lessons in Unit 11 in order to explain what is wrong with MTB. I’m sorry. It does no good to look at the student activity book. My summary of the text from MTB is in italics, and quoted where exact.
Lesson 1: "Subtraction Seminar," 2 days, TIG suggests one hour per day
The teacher writes several two-digit subtraction problems on the board. The class is to share ideas as to which ones are easy or hard. "Explain that it is acceptable to have different opinions." There is a journaling opportunity.
This seems like a great opportunity for children to put on their thinking caps, and for the teacher to gauge where the children are at with respect to subtraction of larger numbers. This Unit is laid out like a previous unit on two-digit addition.
Will the children think the problems with the larger numbers are harder? Will they recognize that they can use simple strategies such as counting if the numbers are large but close, or if the number to be subtracted is very small?
The teacher is not supposed to say which is easy or hard. Looking ahead to the end of this unit, will the children have an opportunity to reflect on which problems really were easy, and which really were hard?
The problem for the whole class to discuss is: what happens if you use $1 to buy a 39-cent item? What change should you get back? The teacher can expect strategies such as
- "I thought, 1 dollar is the same as 10 dimes. I took away 3 dimes. To take away 9 pennies, I had to trade 1 of the dimes for 10 pennies. I had 6 dimes and 1 penny left, which equals 61 cents." (100 hash marks are shown on the page with 39 hash marks circled)
- "I thought 49, 59, 69, 79, 89, 99 on my fingers, plus 1 equals 61 cents."
- "I used a 200 chart…"
- "I made a pile of 100 counters, then I took away 39. That left me with 61."
These counting strategies seem richly varied. They exhibit a wide developmental range.
The fourth strategy is to count up from 1 to 100 with the aid of counters, then to count up from 1 to 39 and remove 39 counters, and then to count up from 1 to 61.
The third strategy is to count down by 39 with the aid of a chart: 100 — 39 = 100, 90, 80, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61. The teacher could point out that this child made use of the fact that 39 decomposes into 3 tens and 9 ones, and the strategy could be written:
100 — 39 = 100 — 30 — 9 = 100 — 10 — 10 — 10 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 = 61.
The second strategy is to count up by tens then ones with the aid of fingers. This child understands that a — b = c is the same as c + b = a, and then makes use of the fact that 61 decomposes into 6 tens and 1 one. It could be written:
39 + 10 + 10 + 10 + 10 + 10 + 10 + 1 = 100.
There is no guidance in the URG or the TIG as to whether the teacher should stop now and lead a short discussion about the algebraic concept behind her strategy.
The first strategy is not just a counting strategy, and it is possibly the most advanced in the sense that the child stops to trade one ten for ten ones before counting further. Given the time, the teacher could see if that child could express her strategy in simple mathematical sentences. The teacher could write out what the child had assumed so that the rest of the class could follow the reasoning:
1 dollar = 10 dimes (and 39 cents = 3 dimes + 9 pennies.)
10 dimes — 3 dimes (= 7 dimes.)
(7 dimes = 6 dimes + 1 dime and) 1 dime = 10 pennies.
(6 dimes + 10 pennies — 9 pennies =) 6 dimes + 1 penny = 61 cents.
That is a lot to write, and each step is not properly justified. I have no idea if this child is constructing her own mathematical understanding right there in the classroom, or if her mother has been working on subtraction problems with her at home, or if her grandma had a chat with her about money and making change, and one of them spilled the beans about trading one ten for ten ones.
In the hour that is allotted for math each day, I have no sense that the teacher could do more than acknowledge each of 20 different strategies, and reassure each child that if they got the number "61," they have a valid strategy.
It appears that from both the URG and the TIG it is not okay for the teacher to single out the first or second strategies as any better or more interesting and worthy of discussion than the others. There is no direction from the URG that the teacher might have children write their strategies in complete mathematical sentences.
Homework: At the Playground. N children are playing at the playground, if X are boys, how many are girls? If Y are swinging, how many are not swinging? Etcetera.
The letter home to the parents says "It is important that your child understands that the traditional approach for solving problems is just one of many acceptable methods of finding the answer. Other problem-solving strategies your child will explore include estimating, drawing pictures, and using tools such as base-ten pieces and calculators."
That sounds like a gentle warning to parents: Don’t spill the beans about writing down the problem columnwise on paper, decomposing the numbers, and regrouping when necessary. That must be what they mean by the "traditional approach". The message is that your child needs to discover subtraction for herself. You shouldn’t say qualitative things such as how you’ve always started with the ones and "borrowed from the tens" since you were in Second Grade. You would ruin it for your child.
Lesson 2: "Is It Reasonable?" 1 hour
The problem for the whole class to discuss is: I used $1 to buy a 29-cent pretzel, and I got 2 quarters and 1 nickel back. My change should have been about 70 cents. How do I know they gave me the wrong change?
The teacher can expect strategies (that I have simplified) such as
- 29 is almost 30, and 10 — 3 is 7…
- 29 is almost 30, and I can count from 30 to 100 on a 200 chart…
- 29 is almost 30, and 100 — 30 is 70…
- I should have gotten a penny in change.
Class work consists of five problems.
My favorite answer is the fourth answer, although the strategy is not robust.
I was amazed that MTB put estimation into the unit right here. It took me a very long time to wrap my mind around it. The simple answer is that this MTB unit on two-digit subtraction is laid out like the unit on two-digit addition.
The more complex answer is that MTB is trying to get children to consider "the tens" before "the ones." The standard written procedure for subtraction that starts with "the ones" has not been presented to the children. This estimation exercise forces children to consider "the tens" first. The teacher is reminded to discourage children from solving the equations exactly. Children are told not to worry about "the ones."
I looked at the class work, and none of the problems gets the children into deep water, forcing them to decide what, say, 24, 25, or 26 are near to.
A quick look at the HMM text shows the lesson on estimation comes after lessons on tens subtraction (why 100 — 30 is like 10 — 3) and after exact subtraction, where the children do worry about "the ones." That seems more logical to me. I thought estimation is what you do when you are trying to change an exact tens and ones problem to its nearest tens subtraction problem, which you then solve with single digits.
To me, it’s a sophisticated thing to simplify a problem appropriately, to know what numbers in the ones column you can safely ignore and still get a good estimate. It requires discretion. Which is why neither MTB nor HMM have the children estimate something like 56 — 24. A child should not leave this unit thinking they can estimate all two-digit subtraction problems.
Lesson 3: "Base-ten Subtraction," 2 hours
The teacher has the children work in pairs and use manipulatives (a base-ten board) to solve the problem 78 — 42. She is to ask, "Will the answer be more or less than 70? Estimate a reasonable answer."
"Give students one or two other problems that involve regrouping. Before subtracting, they should again make an estimate of the difference. Ask the students to share how they might have solved the subtraction problem without base-ten pieces. (It remains important to continue to validate other methods.)"
Class work consists of 4 problems. First they must estimate. Then they must do it "with base-ten pieces" and "one other way."
Using proportionate manipulatives is a compelling way to visually demonstrate number composition, decomposition, and regrouping from the tens.
The teacher directs children to estimate, which means she is directing them to think about the tens first, and therefore the reasonableness of their final answer.
In the class work, I’ll assume the "one other way" must be a counting strategy, like those sample strategies given in Lesson 1, and that each child will write it down somehow, using some English.
Second graders who still have a lot to learn about writing English are forced to use a lot of English words to describe what they are doing as they work with manipulatives and invent counting strategies, rather than being encouraged to represent their work with numbers in proper mathematical sentences.
A quick look forward and back through the Student Activity Book comes up with very few pages of problems. Presumably MTB reduces the quantity of problems to focus children on the quality of their problem solving. However, if MTB thought to rid itself of tedious pages of problems, they are substituting a lot of tedious writing. Apparently, much of the time for Lesson 3 is spent on English literacy. I think more time should be spent on Mathematical numeracy. Teachers who are trying to read and respond constructively to the mathematics represented in the children’s activity books and journals may have little time to comment on the quality of the English. It might be counterproductive to have our children write more English under these circumstances.
Consider an exceptional case. I recall my brother’s experience in school. He is dyslexic; he wasn’t diagnosed until the Fifth Grade. He can read "in" the words and numbers, his difficulty is in writing "out" the words and numbers. Obviously, in today’s schools, he would be diagnosed much sooner. I can imagine his IEP would include a note from the learning specialist, "Please encourage
Lesson 4: "Paper-and-Pencil Subtraction," 2 hours
As a warm-up, the teacher puts the problem 35 + 36 on the overhead, and asks for ideas to solve it. "Emphasize the idea that no particular problem-solving strategy is better than another and that students should use whatever method they find most helpful."
Then, the teacher puts the problem 60 — 25 on the overhead, and she asks for suggestions on how to solve it. If nobody suggests base-ten pieces, the teacher suggests base-ten pieces and demonstrates regrouping.
Next, the teacher has student pairs solve the problem 42 — 18. The children review their answers as a group.
Finally, the teacher puts the problem 53 — 29 = 24 on the overhead, in vertical form. The compact notation to describe the regrouping in our standard, right to left subtraction procedure, is written: 5 crossed out and a small 4 above, and 3 crossed out with a small 13 above. The teacher asks the children what they think happened. Then the teacher demonstrates what happened, using base-ten pieces.
Class work consists of 8 problems, 4 of which are word problems. There is no requirement to use the compact paper-and-pencil description of our standard procedure. The children need to "draw a picture or write to explain your thinking."
A suggested game is "Difference Wars" where the children play in pairs. Each child is dealt a hand of four single-digit number cards, and the child who constructs the largest two-digit subtraction difference wins the hand. The teacher is told to "Observe their methods and correct any errors you notice." The children are then to write in their journals about their experience playing this game.
The assessment page for this lesson is "Many Ways to find the Answer". The children are to find 3 ways to solve one subtraction problem.
I don’t know whether to laugh or to cry. I am amazed that the URG suggests that this lesson will take 2 hours. I would expect many, many more hours to do it justice.
The brief appearance of our standard written subtraction procedure is apparently made only to satisfy the letter, but not the spirit, of our state laws that set forth standards and benchmarks for Second Grade mathematics content that include teaching children how to start with "the ones."
Let me make sure we are all understand the semantics. For the purposes of adults discussing how children do arithmetic, a "procedure" is a "strategy" is an "algorithm." A procedure is a sequence of steps to solve an arithmetic problem. "Strategy" sounds a whole lot better. It sounds very creative. Parents love to hear that their child is strategizing to solve math problems in the classroom. A more accurate term would be "tactic." "Algorithm" sounds a whole lot worse than "strategy," doesn’t it? "Algorithm" sounds very robotic. It sounds like something a computer does, not what a human child should be doing in the classroom. But "algorithm" is the term mathematicians use to describe a sequence of steps. In this review I will keep writing about "procedures," in an effort to stay neutral. Our standard written subtraction procedure belongs to all of us. I don’t think of it as a cultural artifact. It is an intellectual treasure held in common.
The title of this lesson, "Paper-and-Pencil Subtraction", is barely civil. It is pejorative. Perhaps MTB was trying to make paper and pencil sound less scary to children. But at the start of each day’s literacy block, does the teacher say, "Okay, kids, let’s do some paper-and-pencil English?"
I would prefer the teacher said, "Let’s write down, in mathematical sentences, what we’re doing with the base-ten pieces." There are rules for writing mathematics. "53 — 29 = 24" is written very well. I want my boys to see and hear lots of mathematics, and I want them to learn to read and write in mathematical sentences. I don’t care if they are considering the tens first, or the ones. I want them to write numbers. They will then be numerate.
Suppose the children at our school had a Greetings Seminar in their Spanish class. If the teacher asked a child, "
Regarding the class work in Lesson 4, notice that children are not required to try out our standard procedure with paper and pencil, starting with the ones. They can use any of their previous counting strategies. They can count up by ones. They can count down by ones. They can count up by three and down by two. They can start with the tens or start with the ones. They can use hash marks and they can draw pictures.
The URG urges the teacher to emphasize to children that no strategy is better than another. This urgent message is included in every lesson. But my personal experience tells me that some strategies are better than others because of their speed or reliability, or because they are a basis for further conceptual or procedural work. A child’s use of a particular strategy tells the teacher exactly where the child is at on their mathematical journey from point A, the concrete, to point B, the abstract. It’s not clear when, if ever, MTB has a teacher apply this information to help a child advance along the path.
I think MTB finds itself in a bind. The teacher assiduously collects the children’s strategies, and then just as assiduously she must avoid making quality distinctions among the strategies. Perhaps the children make quality distinctions. The URG does not suggest a class vote on the best strategy for solving the day’s subtraction problem.
It’s really quite noble that the folks at Kendall/Hunt want our children to feel very equal to each other. "All men are created equal." One child is not better than another child. But their equality in the sight of God should not blind us to the fact that children arrive at the start of Second Grade with unequal mathematical experiences. Each child is at a different point on his or her mathematical journey, and our only compassionate action must be to help every child move further down the path to mathematical mastery. Isn’t that mastery written in black and white in our state standards and benchmarks for Second Grade?
Do the good folks at Kendall/Hunt fear that if we order and compare the children’s strategies by talking about good, better, and best strategies, that we will also order and compare our children, and that will silence our children?
I fear that there are damaging consequences if we teach our children to believe that mathematics is subjective, that it is not outside of ourselves, that mathematics is an individual expression made within the confines of a peer group. Mathematics is expressed by individuals, but it is not a form of individual expression.
Of course we want our children to independently discover as much as they can about math, but we don’t just throw them all into the pool and watch to see which children swim to the side at the end of the school year. We must act responsibly. We must give them small, manageable problems in an incrementally optimal order. Which reminds me of a specific episode in preschool art of developmentally appropriate practice: Let the children explore the process of using three different paint colors to make a relief print with bubble wrap; let the teacher choose the three different paint colors.
One of the consequences of spending so much time on so many different strategies is that some children are paralyzed by so many choices, as at a strategy buffet. Still other children put pieces and parts of strategies together, from what their peers suggest in the classroom, but the cobbled strategy does not work. There are children who learn more effectively if they are presented with fewer choices.
If I may suggest one beautiful thing about our standard subtraction procedure: once a child masters it, it is hers forever, and forever she will be on an equal footing with her classmates, with her mother, her brother, and her grandmother. It doesn’t matter how big the numbers get.
Returning to the teacher script for Lesson 4, there is no suggestion that now might be a good time to revisit the strategies suggested by the children in Lesson 1. A teacher could point out connections between children’s strategies, and how they do or do not relate to our standard arithmetic procedures. Now would be a good time to go back and decide which of the problems at the beginning of the unit were "easy" and which were "hard."
Perhaps the main difficulty with the MTB approach is that a teacher is responsible for twenty or more children in her classroom, not just two or three. MTB may be best implemented in a small group setting, with one teacher and just a few creative children.
The Difference War game sounds fantastic. It sounds like the most powerful teaching tool in Unit 11. To be dealt the numbers 7, 8, 6, and 1, and to have to start thinking about the tens AND the ones… I guess I’d pick 87 — 16 = 71. Two things concern me about the game’s implementation, though. First, with a classroom of twenty children there will be ten pairs, and the teacher is told to "observe their methods and correct any errors you notice." I think a teacher would be well occupied to keep track of the work of one pair. Secondly, the children are supposed to privately journal about this activity. Why not facilitate a class discussion to verify who got the biggest difference? Who got the smallest difference? This game could easily occupy the whole hour, especially if the children play long enough to allow the teacher to observe each child’s method.
Lesson 5: "Snack Shop Addition and Subtraction," 1 hour
A menu for a snack shop is provided. There is a warm-up problem. "Read the problem and solve it as a class. Have students explain their thinking and demonstrate how they would solve the problem. Some may solve it in their heads, while others choose to use counters, the 200 chart, base-ten pieces, or paper and pencil. Some students may suggest the use of the calculator. Distribute calculators and introduce keystrokes for subtraction. Tell the students to pretend they each have $2."
I hope they are not learning those keystrokes by rote! I hope the kids use their calculators with some understanding of the electronics!
I was excited when I started to read about the Snack Shop; I thought this would be the opportunity for individual children to discover if their individual counting strategy was efficient, by being presented with more than three or four problems at a time. Children could experience the logical consequences of their strategy choice. But the teacher hands out calculators. It is true that the purpose of this unit was to discover subtraction, to explore ways of subtracting, but not to practice subtracting or develop an efficient subtraction strategy.
Calculators certainly do level the playing field for all children. But instead of leveling the playing field by handing out calculators, why not help each child develop an efficient subtraction strategy, or maybe even master our standard written subtraction procedure? That’s an old-fashioned way to level the playing field.
Back in the MTB Teacher Implementation Guide, p. 182, I unearthed this description of how MTB teaches subtraction:
"Later in grade 2, systematic work begins on paper-and-pencil methods for subtracting two-digit numbers. Students are asked to solve two-digit subtraction problems using their own methods and to record their solution on paper. The class examines and discusses the various procedures that students devise. At this time, if no student introduces the standard subtraction algorithm, then the teacher does so, explaining that it is a subtraction method that many people use. The standard method is examined and discussed, just as the invented methods were. Students who do not have an effective method of their own are urged to adopt the standard method."
Unfortunately,
- The work with paper and pencil is not systematic. Did they mean "more frequent that usual?" And it is not clear exactly what children learn by recording their invented strategies on paper in bad English.
- The teacher has barely enough time to acknowledge the existence of up to twenty different invented strategies (if she is careful not to overlook a quiet child), let alone examine and discuss them all. It would be difficult to thoroughly discuss how they work, and the teacher is not allowed to discuss efficiency or reliability.
- How can a child introduce our standard written subtraction procedure if all of the adults have kept quiet about it? And when the teacher does introduce it in Lesson 4, there is no support in MTB for the teacher to teach it well. How can the quiet child, the child without an effective method, adopt it? By rote?
- In the end, children without an effective method are urged to adopt a calculator.
III. How could a teacher teach our standard written subtraction procedure very well?
Take a look at how children learn about starting with "the ones" in HMM Chapter 6. These topics are covered in this order:
- Subtract by Tens (e.g. 70 — 30 = 40)
- Subtract without having to Regroup (e.g. 77 — 32 = 43)
- How to Regroup Tens (using base-ten manipulatives)
- Deciding When to Regroup
- Subtracting a One-Digit Number from a Two-Digit Number
- Subtracting Two-Digit Numbers
- Problem Solving (lots of well-posed word problems)
- Practice Regrouping with 10 or 11
- Practice Regrouping with 12, 13, or 14
- Practice Regrouping with 15, 16, 17, or 18
- Estimating Differences (after firm knowledge of exact differences)
- Horizontal Subtraction (rewriting as vertical)
- Money!
- Algebra Readiness (if a — b = c, check: does b + c = a?)
- Problem Solving (word problems are "scientific", using data tables)
The Teacher’s Guide suggests a warm-up activity before each lesson, and each lesson plan has two different discovery activities to choose from. One activity is a cooperative kinesthetic/tactile activity for pairs, small groups, or the whole class, and the other activity is a class discussion of a subtraction problem. That’s where the games are.
Each child has a "consumable" workbook that is really a textbook. Each lesson of their textbook starts with a simple concept statement at the top of the page, written at a Second Grade level, and there is a pictorial or symbolic demonstration of the concept.
Through the use of "ten-frames," every subtraction problem can be set up column-wise with the tens and the ones named, to help a child remember place value during the move to this symbolic representation. If a child brings home the HMM textbook, the parent can know exactly what is going on in the classroom, and the parent can help teach it.
Each lesson ends with guided practice so that children learn to write correct mathematical sentences. There is no journaling. After practice, children can solve an interesting word problem. For example, Lesson 8 ends with "data" on birds that children use to create their own subtraction problem.
For more ideas on how to teach a child how to start with the ones, see Appendix A.
There are many things in Houghton-Mifflin Mathematics for a mother (or a father) to love. HMM is not unique in this sense, there are many other fine mathematics series available for K-6 that have the following features:
- HMM helps your teacher teach our standard arithmetic procedures very well.
- Liping Ma is the content reviewer for HMM K-2. Children construct their understanding of mathematical concepts using conceptual materials and procedural practice that are provided in an incrementally optimal fashion.
- HMM is fair. All of the children are let in on the same conceptual and procedural secrets at the same time.
- HMM assessments find out exactly where your child is at on her mathematical journey. There are interventions for both emergent and advanced learners.
- HMM provides enough opportunities for individual work so that your child can demonstrate she owns both the conceptual understanding and the procedural skill.
- HMM is concerned with mathematical numeracy.
- HMM is considerate. The written English in its Second Grade textbook is accessible to your 8 year-old.
- HMM helps you work with your child at home.
IV. Two parting thoughts about Kendall/Hunt Math Trailblazers Unit 11
What is the point of radically de-emphasizing our standard written procedures? Where did this bright idea come from? Rousseau. See E.D. Hirsch’s article in Education Next, http://www.educationnext.org/2001sp/34.html. But closer to home, Constance Kamii has been pushing a constructivist ideology in primary mathematics education for many years. Many teachers graduating from schools of education today know only a little math, but a lot about Kamii’s work with children. Kamii takes it as an article of her faith in Piaget that children can reinvent arithmetic, that standard written procedures are neither required nor desired in the primary classroom. See Appendix D.
MTB is paradoxical. It is an example of extreme differentiation. One child might know how all the mental arithmetic strategies that start with the tens work, but she still prefers to use our standard procedure with paper and pencil. Another child never feels comfortable with the mental arithmetic strategies proposed by his classmates, and instead he prefers to count by ones all the way through this unit. These children coexist in the same classroom, without quality distinctions. They could be chosen as a pair to play the Difference War game. That’s a differentiated classroom. Each child’s preference is accommodated all the way to the bitter end.
Which means that MTB is completely undifferentiated. The child who understands and prefers to use paper and pencil must listen to, and perform, other subtraction strategies. I guess that is meant to teach tolerance. But how about teaching tolerance by asking all of the children to tolerate starting from the ones?
In MTB, it’s not okay for the teacher to directly instruct the seated children in how our standard strategy works, but it is okay for children to sit through instruction from their peers. The child who prefers to use counters to count by ones, is left quite alone with his thoughts. How could a teacher intervene, if his strategy is as valid as any other?
A child is not allowed to say, "My way is fast." She is only told that her strategy is "valid." A child is not allowed to find out, "My way is pretty damn slow." He is only told that his strategy is "valid." I hope the girl using paper and pencil doesn’t say something like "You are going too slow" to her partner, the boy who is counting by ones in the Difference War game. That might be one of my boys, and the teacher might not notice her impatience, what with all ten pairs of children to oversee.
I do not want a child who resists using a better strategy to be punished or made fun of. His current location on the path to mathematical mastery is a private matter. Discussing it in front of other children is not helpful. What might help is for the teacher to actively teach him. Unfortunately, it must be easier and cheaper for schools to adopt curricula that, for ideological reasons, do not recognize children who would benefit from intervention. We have reading specialists in our schools. Why not math specialists?
V. If wishes were horses, the poor would ride
- I wish teachers were prepared to teach our standard written arithmetic procedures very well to all of the children in the classroom.
- I wish teachers were responsible for communicating to parents exactly where teachers get the idea that they should not teach our standard written arithmetic procedures in the classroom.
- I wish our neighborhood school would offer a more rigorous mathematical experience for young children.
- I wish I could sign a waiver form to release the school from liability for any conceptual or developmental damage that might occur if my children were shown why we consider "the ones" before "the tens" when we add and subtract on paper.
- I wish MTB would rename "Math Trailblazers" "Science Trailblazers" and market it as an enrichment program.
Appendix A: What you need to know about rote arithmetic
Arithmetic is not rote.
The word rote describes the process of memorizing, using repetition without full comprehension, e.g. "to learn by rote".
An arithmetic procedure can be taught by rote, and learned by rote, but the procedure itself is not rote. Any procedure, even those invented by children, can be taught by rote. And it can be taught so badly that children would be better off having never seen it.
Is learning by rote always bad? No. In fact, we learn to count by rote. We learn to count by ones, tens, twos, and fives….
We count collections of objects for our children, e.g. "One school bus, two school bus, three school bus." My toddler understands "one", and "more than one", so that if you hold up three toy school buses, he will say "Two Bus!" His understanding of numbers is not complete. However, I am cheered by what he does understand, and I will continue to teach him to count by rote without worrying about his full comprehension. The words "One, Two, Three" were a gift given to me, and I will give them to my toddler. I will continue to sing "Over in the Meadow" right along with Raffi.
Addition and subtraction are essentially counting up or down "by ones." We can take advantage of our base-ten number system to reduce the effort of counting up or down by ones. The numbers considered in Second Grade are composed from, and therefore can be decomposed into, ones and tens. If a child understands that 16 is 10 + 6, and that 10 + 6 is 16, they are freed to count the tens apart from the ones. Now, there is a choice: start with the ones or start with the tens. Or try it both ways.
If you read no other web article, please read "Basic Skills vs. Conceptual Understanding: A Bogus Dichotomy in Mathematics Education" by H. Wu at the American Federation of Teachers web site, http://www.aft.org/pubs-reports/american_educator/fall99/. While you are there, you can check out Richard Askey’s review of Liping Ma’s book, Knowing and Teaching Elementary Mathematics.
Wu suggests an ideal script for teaching our standard written addition procedure, which could easily be translated for teaching subtraction. There are two main features of his approach. The first is to have children write out the numbers decomposed into tens and ones. Children will be especially ready for this if they have been decomposing numbers to count up or down by tens in their mental arithmetic. The second feature of Wu’s approach is to give children enough problems to solve on paper that they are receptive to any suggestions that save labor. Children respond to logical consequences.
Wu also suggests scripts for teaching multiplication and the division of fractions. It is very helpful for a parent whose child does not get this same instruction at school.
Another excellent discussion of the value of teaching standard arithmetic procedures is available on the AMS web site, http://www.ams.org/notices/199802/comm-amsarg.pdf.
To summarize, there are two good reasons to do a good job teaching standard arithmetic procedures that "start with the ones:"
- Taught well, they improve our children’s understanding of whole numbers, and prepare our children for algebra and higher mathematics.
- Used correctly, they are guaranteed to work with whole numbers of any size.
For some comic relief, the following selections are from the MTB Teacher Implementation Guide. Do they understand the difference between what is rote and what is arithmetic?
TIG p. 98 "Flexible thinking and mathematical power are our goals, not rote facility with a handful of standard algorithms." Which teacher in what school is teaching our children by rote? Learning math through science experiments with marshmallows in MTB may not lead to all of the flexible thinking and mathematical power that we might have hoped for. First, there is the potential for children to confuse uncertainty in scientific data and interpretation with uncertainty in mathematics. Instead of flexible thinking, you get uncertain thinking. Second, there is the temptation to do only enough mathematics to support the scientific experiment, and no more. That is not true mathematical power. That is called "slamming the door on your way out."
TIG p.177 "The pre-eminence of arithmetic in (an elementary mathematics) curriculum has faded, both because of the availability of calculators and because of the realization that a curriculum focused on rote arithmetic will not meet the needs of students…." Arithmetic develops a child’s number sense. Calculators take the place of a child’s developing number sense. Calculators do not meet the needs of students in Second Grade.
Proficiency with whole number operations is articulated across grade levels. How a child spends her time in Second Grade affects what she will have time to learn in Fifth Grade. Proficiency with a calculator is not a vertically integrated skill. In Fifth Grade, it’ll take about five minutes for a child to learn all she needs to know about using a calculator. In Second Grade, our children should put down the calculators. Better to play number games. Or do science experiments with marshmallows. Or read a good book.
TIG p. 178 "The hundreds of hours devoted to arithmetic computation in elementary school leave too little time for other important topics. The traditional approach to rote computation undermines higher-level thinking. Worst of all, the long hours of computational drudgery teach children that mathematics is a most unpleasant business." I agree that teaching computation by rote would be a sad waste of time. Computation is senseless drudgery if our children are never shown why standard procedures work, and why they work so well.
However, I take exception to the assertion that the time spent developing arithmetic proficiency would be better spent on other important topics. It’s hard for a child to go on to other important topics if he always has to stop and strategize about the subtraction, or count counters. Proficiency with arithmetic frees your mind for other important thoughts.
If critics of our standard arithmetic procedures would only look more closely, they would see that high-level thinking is employed. It is high-order thinking to understand our so-called low-order arithmetic. But it requires the teacher or the parent to take the time to ask high-order questions on behalf of the children, such as "Why can we start with the ones?" and "What happens if we always start with the tens?"
TIG p. 194 "Pages of problems on the basic facts are not only unnecessary, they can be counterproductive. Students may come to regard mathematics as mostly memorization and may perceive it as meaningless and unconnected to their everyday lives."
My children are a joy to me and to others, but I don’t yet trust them to know what is meaningful and connected.
For instance, all 8 year-olds think practicing the piano is meaningless and unconnected to their everyday lives, but that doesn’t stop their parents from driving them to piano lessons once a week. Can you imagine a piano teacher who didn’t like scales or chords or musical notes printed on a page? A teacher who told you your child was going to invent music?
Many 8 year-olds think broccoli is meaningless and unconnected to their everyday lives, but that doesn’t stop parents from serving green vegetables with dinner every night. Have you seen the commercial with the little girl who doesn’t want to eat her peas? She is so sweetly sad, her mom can’t help but pop open a chocolate nutritional shake, marketed just for kids. My pediatrician told me we were supposed to offer the boys a banana!
Some 8 year-olds think soccer drills are meaningless and unconnected to the game. Why not just play soccer? Truly, if your child has a lot of time and natural talent, that might work. If your child is an average athlete, a few drills might level the playing field for her. Can you imagine a soccer coach whose idea of practice was only to play games? Don’t we want coaches who are tough but fair? Coaches that take an interest in each child’s ball-handling skills?
I had my boys memorize the Lord’s Prayer last year. They do not fully comprehend everything in it. But that’s okay. They are proud of their ability to recite it. With it they gain entrance into a spiritual community. And they understand enough of the spiritual language so that is has some meaning and connection to their everyday lives. They could not have constructed something so beautiful, nor could I.
Appendix B: What you need to know about the word "conceptual."
Have you been told over and over again that Math Trailblazers is conceptual? I have. The word "conceptual" is an attractive way to describe a learning environment that keeps the arithmetic inside your child’s head. To be conceptual is to not get hung up on any one procedure written down on paper with a pencil. Parents should not worry; their child is busy learning mathematical concepts without any distractions printed on pieces of paper.
Advocates of programs like MTB imply that a rigorous approach to written arithmetic in the primary grades is not conceptual. When they say MTB is conceptual, they don’t say outright that HMM is not conceptual, they merely imply that it is not. They imply that 8 year-olds cannot develop a complete conceptual understanding of subtraction if we teach them why our standard written procedure for subtraction always works, and why it works so well. It is, after all, printed on a piece of paper.
A concept is a general idea that is derived from and considered apart from what is observed by the senses. The mental process by which this idea is obtained is called abstraction. My toddler is deriving the general idea of "bus" having seen many buses on the road and in picture books and at school and in our toybox. Buses are not always yellow, but they always have many windows. It took him awhile to figure out that taxis are not buses. He still wonders if Maybelle the Cable Car is a bus.
A teacher at our school offered the following example for why she did not like the Houghton-Mifflin approach to teaching mathematics. There is a lesson for First Grade that has all the children in the class look at a number line at the same time, to count up and down on it to add and subtract. The teacher truly believes that number lines are not conceptual. She added that different children might need to use fingers or counters or blocks.
I submit to you that number lines are conceptual. They help a child form the general idea of continuous numbers, that is to say, numbers in between other numbers. You recall the joke about wanting a house with a white picket fence and 2.2 children? There is no such thing as 2.2 children. But there is such a thing as 2.2 candy bars. How about if our children see a number line as an abstraction from candy bars laid end to end?
To help our children move beyond conceiving of numbers as discrete counters, we might show them number lines. A ruler would do the trick. On a ruler, there is a fixed distance between any two adjacent numbers. Counting up and down on a ruler is a geometric way to add and subtract.
Wu asserts in his tutorial, http://www.math.berkeley.edu/~wu/EMI1b.pdf,
"The introduction of the number line is not just a matter of convenience or pedagogy. The number line is one of the most fundamental objects in mathematics. It is, for example, the first step in Descartes’ introduction of coordinates in space. Its importance cannot be exaggerated."
Appendix C: What you need to know about the word "developmental."
Have you been told over and over that Math Trailblazers is developmental? I have. The word is used to get parents, the pushy parents who know a little mathematics as well as the worried parents who know a lot of mathematics, to back off. To wait, then wait some more. It becomes a contest to see who can wait the longest. No parent wants to push too hard. But all parents do worry. We want to do enough for our children.
Advocates of programs like MTB imply that a rigorous approach to written arithmetic in the primary grades is not developmental. When they say MTB is developmental, they don’t say outright that HMM is not developmental, they merely imply that it is not. They imply that it is not developmentally appropriate to teach 8 year-old children why our standard written procedure for subtraction always works, and why it works so well.
If we understand what is developmentally appropriate practice, we can decide for ourselves if written arithmetic is developmentally inappropriate.
A great place to become acquainted with developmentally appropriate practice is at the National Association for the Education of Young Children (NAEYC) web site, www.naeyc.org. I am a real NAEYC supporter, having enrolled my children in NAEYC-accredited preschools in
What follows are two selections from the brochure, "Developmentally Appropriate Practice in Early Childhood Programs Serving Children from Birth through 8." I’ve simplified the language, emphases are mine, and I’ve added editorial comments.
Principles of developmentally appropriate practice:
- Domains of children’s development — physical, social, emotional, and cognitive — are interrelated
- Development occurs in a relatively orderly sequence, with later abilities, skills, and knowledge building on those already acquired If a child hasn’t acquired certain arithmetic skills by the time she is 8 years old, she can’t build on those certain arithmetic skills when she is 9 years old.
- Development proceeds at varying rates from child to child, and within a child, from domain to domain One child is ready for written arithmetic when another child is not, and for a particular child, she may not know how to ride a bike yet, but she does know how to decompose a number into tens and ones.
- Early experiences have both cumulative and delayed effects on a child’s development. We want to avoid bad early experiences. But in general, a more rich and stimulating environment will be to our children’s cumulative benefit. In short, the more good early experiences, the better. Try, then try again.
- Optimal periods exist for certain types of development and learning. We know our children are little scientists, and we also know that they are little sponges.
- Development proceeds in predictable directions toward greater complexity, organization, and internalization.
- Children are active learners, drawing on direct physical and social experiences, as well as culturally transmitted knowledge to construct their own understandings of the world around them. In some areas, only development makes learning possible, and in other areas, only learning will spur development. Adults need to present children with something to learn, such as our standard written procedure for subtraction. It’s really okay to say to our children, "Hey, have you seen how this works when you start with the ones?"
- The child interacts with his world
- Play is a vehicle for development and reflects their development.
- Development advances when children have opportunities to practice newly acquired skills as well as when they experience a challenge just beyond the level of their present mastery. "Confronted by repeated failure, most children will simply stop trying. So most of the time, teachers should give young children tasks that, with effort, they can accomplish, and present them with content that is accessible at their level of understanding. At the same time, children continually gravitate to situations and stimuli that give them the chance to work at their ‘growing edge’. Moreover, in a task just beyond the child’s independent reach, the adult and more-competent peers contribute significantly to development by providing the supportive ‘scaffolding’ that allows the child to take the next step. Development and learning are dynamic processes requiring that adults understand the continuum, observe children closely to match curriculum and teaching to children’s emerging competencies, needs, and interests, and then help children move forward by targeting educational experiences to the edge of children’s changing capacities so as to challenge but not frustrate them." Nowhere does it say that we should not challenge children, to move from mental arithmetic to written arithmetic, to move from their own natural strategies to our standard strategies. In fact, not challenging children would ultimately be the most frustrating thing to do to them. When a teacher has found what is challenging to half the class, she is only halfway done.
- Children demonstrate different modes of learning, knowing, and different ways of representing what they know. "The principle of diverse modalities implies that teachers should provide not only opportunities for individual children to use their preferred modes of learning to capitalize on their strengths, but also opportunities to help children develop in the modes or intelligences in which they may not be as strong." Children may find their own counting strategies, considering the tens first, to be the most natural, and they may think that is their only strength, but they need to be shown how they can also consider the ones first, and do arithmetic on paper with numbers of any size. It will only make them stronger, mathematically.
- Children develop and learn best when they are safe and feel psychologically secure. Children may not feel psychologically secure in an academic environment without quality distinctions. 8 year-olds are very concerned to know better from worse, faster from slower, to know what the rules of the game are. A classroom where every child does his own thing, mathematically speaking, may make some children very insecure. When no one strategy is any better than another, no one strategy is very important, is it?
In summary, NAEYC urges us to move from either/or thinking to both/and thinking, with some examples as follows:
- Children construct their own understanding of concepts and they benefit from instruction by more competent peers and adults.
- Children benefit from opportunities to see connections across disciplines through integration of curriculum and from opportunities to engage in in-depth study within a content area.
- Children benefit from situations that challenge them to work at the edge of their developing capacities, and from ample opportunities to practice newly acquired skills and to acquire the disposition to persist.
Imagine my surprise when I read further about selecting developmentally appropriate curricula, and NAEYC gave as an example the rejection of any math curriculum that attempts to teach place value concepts to children in the First, Second, or Third grade:
"First, Second, and Third Grade children cannot comprehend place value; they spend hours trying to teach this abstract concept, and children either become frustrated or resort to memorizing meaningless tricks. This is an example of an unrealistic objective that could be attained much more easily later on."
As noted before, one of the excesses of developmentally appropriate practice is that some adults use it to scare other adults into waiting. Wait, then wait some more. Nobody gets to do place value until everybody is ready for place value.
Quite frankly, in an effort to preserve some developmental space for my children, I’m more concerned with sheltering them from pernicious messages in our consumer culture to EAT! to BUY! to have SEX! than with sheltering them from the formal study of place value in Second Grade. Somebody get me a WAIVER!
I believe NAEYC borrowed their torrid little speech about place value from Constance Kamii; see Appendix D.
The National Council of Teachers of Mathematics, www.nctm.org, a "big-tent" organization that has published Kamii, came to a different conclusion than Kamii when they put forth their revised Standards in 2000:
"It is absolutely essential that students develop a solid understanding of the base-ten numeration system and place-value concepts by the end of grade 2."
Appendix D: Constance Kamii
The Reason for the Season of Constructivism in primary mathematics education.
"The teaching of algorithms is based on the erroneous assumption that mathematics is a cultural heritage that must be transmitted to the next generation."
Constance Kamii is a professor of education at the
Although I have not purchased Kamii’s books on how children reinvent arithmetic, I have read her paper, "Teaching Place Value and Double-Column Addition" available at http://www.enc.org/professional/learn/research/journal/math/document.shtm?input=ACQ-104912-4912_48#3. Some interesting features of Kamii’s teaching emerge:
- The only arithmetic allowed in her classroom is mental arithmetic.
- Mental arithmetic never starts with the ones.
- Teachers must ask the right questions at the right time, and teachers must avoid saying that an answer is right or wrong.
- Teachers must speak correctly and demand the same of the children.
The only arithmetic allowed in her classroom is mental arithmetic.
In a Kamii classroom, the children read and solve problems that the teacher writes in correct mathematical sentences on the chalkboard. But the children do not write anything down on paper with a pencil. Textbooks and workbooks are absent from her classroom. Even manipulatives are off-limits, such as the base-ten blocks that are popular in MTB. Kamii teaches that these manipulatives are external to the child, and cannot be relied upon as a means for the child to internalize concepts of number and place value. The child must solve the problems in her head and give the answers verbally.
Increasing a child’s capacity for mental arithmetic with numbers up to 100 is laudable.
They think it, they speak it. The teacher writes it. It reminds me of language immersion. But it does come at the expense of having children start learning to write mathematical sentences by themselves that can be understood by anyone.
Imagine that we didn’t teach writing in K-2. No "writing road to reading." Not even invented spellings. The children only played games with letters and words in Kindergarten and First Grade, and then played games and had class discussions about what the teacher writes on the chalkboard in Second Grade.
Kamii’s emphasis on mental math alone has enabled me to see most clearly the difference between MTB and HMM. I was trying to compare the writing of mathematics between MTB and HMM, and I couldn’t figure out why MTB set great store by journaling in bad, but authentic, English. Now I understand that they were trying to get away from having the children write correct mathematical sentences because they want to keep the emphasis on mental arithmetic. Of course, MTB employs manipulatives and 200 charts and calculators. But now that I have read Kamii’s conception of the K-2 classroom I can understand what MTB attempts to do and why they fail to do it.
Mental arithmetic never starts with the ones.
What’s the advantage of Left to Right Arithmetic? Certainly, when we’re looking at numbers between one and nine, there is no left or right. It’s when we go past nine that there is suddenly a left and a right. The left is the "tens" and the right is the "ones".
Kamii gives some good examples of how children can perform their mental addition. She notes that all of the children in her classroom start with the tens.
In mental arithmetic, often easier for us to add starting from the left, as long as the number remains small. Think of yourself at the shoe store. You’ve just picked out a pair of shoes that cost $36. You’d like to take advantage of a sale and get the second pair of the same shoes for half price, but you left your checkbook and credit cards at home. You have three 20s and two 1s in your purse, and you promised you’d stop for fast food on the way home. It takes about $10 to feed your family. You do the arithmetic in your head, and you start with the tens because you have nothing to write with. If it was me, I would calculate thus: 36 + 18 = 46 + 8 = 50 + 4 = 54. And 62 — 54 = 2 + 6 = 8. Not enough money for fast food.
The subject of mental strategies has been studied extensively, and there is a lot of research published on it. I used an "N10" strategy to find the cost of two pairs of shoes. In the
Teachers must ask the right questions at the right time, and teachers must avoid saying that an answer is right or wrong.
Although Kamii claims that children can construct their own logico-mathematical relationships, teachers are essential to ask children to consider the right arithmetic problem at the right time. Without such teachers, our children would have few opportunities to reinvent arithmetic.
Teachers are also encouraged to disagree or agree with a child’s explanation of their mental arithmetic, but they should do it politely and to explain themselves:
"The teacher is careful not to be the omniscient authority who decrees what is right or wrong. She or he only agrees or disagrees and tries to present another point of view on an equal footing."
"The teacher calls on individual children and writes all the answers that are given by them. Being careful not to say that an answer is right or wrong, the teacher then asks for an explanation of each answer or procedure invented by the children."
I think that is the best way to correct children — let them do it themselves. However, the teacher needs to commit to this corrective procedure on behalf of all twenty children in the classroom. The Socratic method requires teachers to continue asking questions until the truth comes out. The AMS, in their discussion with NCTM on the value of mathematical algorithms, kindly refers to this as "mathematical closure."
Teachers must speak correctly and demand the same of the children.
The premise of Kamii’s academic career is that children do not understand place value when they are taught by "traditional," "rote" methods that try to "put" place value in a child’s head. To prove that children do not understand place value, she interviews them and asks them about the number 16. You can read the interview details at www.enc.org.
Her interview is unforgiving; a child is supposed to offer the units without prompting, and the child fails to convey any understanding of place value if the child fails to offer the units. It is well worth your time to conduct her interview with your own child, and also to extend the interview to get more complete information about your child’s understanding of place value.
Kamii claims that no Kindergarteners from "traditional" classrooms say that the 1 means ten. Kamii claims that a Third of Second Graders and Half of Fourth Graders say that the 1 means ten. As any pollster can tell you, you get the result you want by carefully choosing the words in the question.
When the interviewer circles the number 1 in the number 16, isolating it, and asks the child "what does this part mean? Can you show me with your chips?" I assert that a child is absolutely correct to say "one". "One" is the coefficient of the tens. Let me put it this way. If I write down the name, "Becky Doe" and I circle "Becky" and I ask you "what does this part mean?" and you say it means "Becky," you are absolutely right. If I ask you "what does this part of the name mean?" you might even think to say, "It means her first name is Becky".
I’m extremely disappointed that Kamii never asks children "what does this part of the number mean?"
I’d like to borrow an example that Kamii gives from her classroom. The children are considering the problem 36 + 46. A child says "three and four are seventy." The child does not nor ever will believe that 3 + 4 = 70. The child merely left out the units, the word "tens." It is entirely natural to omit the units when you are sure of what you are doing — that you can treat addition of tens as single digit addition. Of course, it’s best for the teacher to remind the child to say "three tens" and "four tens" or "thirty and forty" for the benefit of every child in the classroom.
In a Kamii classroom, the teacher doesn’t let a child get sloppy and say "three and four are seventy." The teacher always asks the child to keep track of the units.
On the first Monday of Spring Break, I sat down with Tom and Luke at the kitchen table. I poured out a pile of Cheerios and asked Tom to not say anything. I wrote 27 on a piece of paper and asked Luke to count out that many Cheerios. He did. I asked him "what does this part mean?" and he said 7. I asked him to count out that many. Then I asked him "what does this part mean?" and he said 2. Tom immediately interjected "2 Tens!" and Luke corrected himself and said "2 Tens." I scolded Tom. In educator-speak, Tom "elicited the multidigit context" for Luke.
Then I set about to see where Tom’s knowledge of place value would fail him. I wrote down 14 and 23 columnwise as to add them. I asked Tom to count out this many and that many, and he did. I asked him what this part meant and what that part meant. He got it all right. So I tried to get a little tricky. I asked him, what do I get if I put this part (the 1 from 14) and that part (the 3 from 23) together. He said "1 and 3 is four" and then he immediately corrected himself and said "1 ten and 3 is … ten, twenty, thirty, forty!" and I said "think about it." He then said "1 ten and 3 is thirty!" and I asked him to think about it some more and he said "thirteen!" In educator-speak, that means he had 13 "available as his fourth meaning." Then I asked Tom about the other cross parts (the 2 from 23 and the 4 from 14) and he correctly said "2 tens and 4 is twenty-four!"
I know the sample size is extremely small, but fully Half of the Kindergarteners at my house understand place value. My boys have had no special mathematics instruction. They celebrated the 100th day of Kindergarten; their teacher taught them to count by tens to 100. We do a little bit of single-digit cookie math at the kitchen table.
Karen Fuson, a researcher from
Parting thoughts about Constance Kamii
In her paper "The Harmful Effects of Algorithms in Grades 1 — 4," Kamii ends with this thought:
"Teaching algorithms to (struggling) students… sent them the message that ‘the logic of this procedure is too much for you; so just follow these steps and you’ll get the right answer.’"
Constance Kamii and constructivist curricula such as Kendall/Hunt’s Math Trailblazers have sent a clear message to teachers that the logic of our standard written procedures is too much for Second Grade children. Don’t even try to teach it. All a teacher can do is put a worked subtraction problem on the overhead and see if it causes any child to construct a private logico-mathematical understanding of "starting with the ones."
I wish Kamii would apply her formidable talents to teaching teachers how to speak correctly and ask the right questions, to instruct children in the logic of our standard written procedures. Her insistence on construction blinds her to the possibilities of instruction.
Must teachers reinvent instruction, or does Kamii think they can be directly taught how to teach? What happens if a teacher learns a constructivist approach, such as that in MTB, "by rote?" How many teachers have taught from constructivist curricula without ever asking high-order questions, such as "what happens if we don’t tell children about starting with the ones?" and "what if we never start with the ones?"
If you read more of Kamii’s teachings, as well as educational research that compares good new methods with the bad old methods, you should be aware of a phenomenon called the
I ran across a reference to the
"Working with specially employed teachers permitted a faithful implementation of an alternative instructional approach but confounded teachers with treatment. This means that the