Sunday, May 31, 2009

Calculator White Paper

Below appears the Calculator White Paper that I wrote for the NJDOE Math task force.  It was nearly unanimously supported by all members of the task force.

The calculator cannot think, and it cannot make decisions about what numbers or operations need to be used. The quality of the output of a calculator is wholly dependent on the input (Reys, Arbaugh, Joyner, 2001).

Calculators were first introduced into K-8 mathematics classrooms nearly 30 years ago, but their use in schools is still very controversial.  To achieve a balanced mathematics curriculum (a curriculum that focuses on conceptual understanding, computational and procedural fluency, and problem solving skills), we must develop students’ confidence and understanding of when and how to use these skills and tools.  The fundamental issue is about fluency with computation which is a requirement for higher order thinking.  Let us agree on the following: 

ª  calculators should not be used as a replacement for learning basic mathematical concepts;

ª  calculators can be beneficial to student learning;

ª  calculators can motivate students and give confidence to those anxious about mathematics;

ª  mastery of basic mathematical skills in the elementary grades is crucial to success in later grades.

Whether or not calculators should be used in grades K-6 is a shadow of the real issue.  That is, the issue is not if students should use calculators in the classroom, but rather how and when should calculators be used appropriately. Teachers and students must be taught to use the technology to enhance and support the students’ development of computational fluency. In fact, calculator use for some students may lead to computational fluency, while in others it may not.

ª  In third grade, having short timed multiplication drills/assessments without calculators is appropriate to develop the fluency, AND it is appropriate to use calculators to explore repeated addition to develop a conceptual understanding of multiplication.

ª  In sixth grade, doing addition with fractions with unlike denominators is a generalizable skill that should be practiced without technology is appropriate, AND using calculators to generalize operations with fractions (e.g., when is the quotient a/b greater than a?) is also appropriate.

ª  In eighth grade, it is appropriate to have students plotting points to explore non-linear growth patterns without technology, AND it is appropriate for students to use to graphing calculators to explore the impact of the parameters of variables on a function.

No one wants students to become dependent on calculators rather than their own thinking.  The best way to avoid this is for students and teachers to capitalize on such appropriate use of calculator technology to expand students’ mathematical understanding, not to replace it.

In order to support appropriate use in classroom, it is recommended that NJASK assessments have a balance between non-calculator items and calculator items in grades 3-6.  A simple 50-50 ratio of items (noncalculator to calculator) will demonstrate to NJ classroom teachers and students that appropriate use of calculators should be our goal.

Thursday, May 28, 2009

Combination Of Old And New Media Deepens Mathematical Understanding

ScienceDaily (2009-05-19) -- By combining the trusty old book, pen and paper with the possibilities offered by the computer and the interactive whiteboard, information and communication technologies can help to improve students' understanding in maths education. So conclude a team of researchers in the Netherlands.

Wednesday, May 27, 2009

Friday, May 22, 2009

Saxon Math

As the anti-reformists continue to post blogs, websites, and petition to rid schools of NSF reform minded programs, they say little about the programs they support.  Occasionally, anti-reformists point to Saxon Math as the answer to their prayers.  Consider below the comments from a mathematician, H. Wu, from Berkeley about Saxon Math:

But I think that what perhaps disturbs me the most about Saxon is to read through it. I myself do not get the feeling that I am reading something that when the children use it they would even have a remotely correct impression of what mathematics is about. It is extremely good at promoting procedural accuracy. And what David [Klein, of Cal State-Northridge] says about building everything up in small increments, that's correct, but the great pedagogy is devoted, is used, to serve only one purpose, which is to make sure that the procedures get memorized, get used correctly. And you would get the feeling that - I think of it as a logical analogy - you can see the skeleton presented with quite a bit of clarity, but you never see any methods, your never see any flesh, nothing - no connective tissue, you only see the bare stuff.  A little bit of this is okay, but when you read through a whole volume of it, really I am very, very, uneasy. . . . When I do this, I want to emphasize that I do not single out one or two examples. I am trying to describe through one or two examples the overall, the overriding, impression that I have. And when that happens, you get the feeling that, if my students use this, how could they not get the idea that mathematics is just a collection of techniques? If that is the case, what happens to them when they go on to middle school, and then to high school, and after that, God forbid, you might be facing them in your freshman calculus classes. And that is a frightening thought.

Wednesday, May 20, 2009

Future of High School Mathematics

Math is More (a group of mathematics educators, mathematicians, and concerned individuals committed to real and significant improvement in the performance of the complex system of mathematics education) has released a report: The Future of High School Mathematics.

A few important items from the report are listed below:

Traditional approaches to mathematics curricula, teaching, and assessment will not meet the challenge of preparing more students for the demands of a changing world. Too many adults have unpleasant memories of their high school mathematics, and they avoid using mathematical reasoning even when it would be useful. For far too long, many students have been relegated to mathematics courses that fail to prepare them for future study and work. As a result, there are unacceptable achievement gaps between students from different economic, social, and cultural sectors of society.  Every student— not just a select few—has the right to be mathematically prepared for the future.

To achieve success in college, the workplace and life, American students must not only master important content, they must also be adept problem-solvers and critical thinkers who can contribute and apply their knowledge and skills in novel contexts and unforeseen situations,  High school graduates must also be able to work collegially in teams and be keenly aware of the rapidly changing world around them.

The mathematics students need to learn today is not the same mathematics that their parents and grandparents needed to learn. When today’s students become adults, they will face new demands for mathematical proficiency that school mathematics should attempt to anticipate.  Moreover, mathematics is a realm no longer restricted to a select few. All young Americans must learn to think mathematically, and they must think mathematically to learn. 

Saturday, May 16, 2009

Teaching in Japan: What Can We Learn

Dr. James Stigler (Professor of Psychology at UCLA, Director of the TIMSS Video Studies) made a presentation to the California Board of Education entitled: Reflections on Mathematics Teaching and How to Improve It.

Dr. Stigler states that the most important things we have learned from this research and the implications that the research has for teaching (mathematics teaching in particular) are the following:

1. Teaching is a cultural activity. In Japan, teachers will give students problems to solve that they have never seen before. We actually don't do that in our country. In Japan, they have a high tolerance for what I would say is "confusion" in the classroom. They'll give a math problem and no one knows what to do. They sit there and they sweat and they look confused and they pull their hair out and they try to figure out what to do.  In the U.S., if you give a math problem and everyone looks confused, the teacher steps in right away to stop the confusion--clear it up, give a hint, tell them what to do-anything to stop the confusion.  Cultural activities are very hard to change. And one of the reasons for this is that there are all sorts of forces that work against changing cultural norms.  We have to recognize that fact as we try to figure out how to improve teaching.
2.  The second thing we've learned is that there are many ways to teach effectively…Teaching is contextual. There is no one best way.
3. Teaching quality needs to be defined not by what teachers DO but by the learning opportunities they create for students…. One finding: we marked every time a math problem started and stopped. And then we categorized the problems into one of three types: (a) Stating Concepts (recall a fact), (b) Using Procedures (e.g., do a worksheet practicing a taught procedure), or (c) Making Connections (very rich problems that connect students with core mathematical concepts). Teachers in every TIMSS country used some of all of these types. However, when looking at the videos, a pattern emerged. In the higher performing TIMSS countries such as Japan or Hong Kong, "Making Connections" problems remained as "Making Connections" problems. However, when these types of problems were given in the U.S., the problems were translated into "Using Procedures" problems by every teacher, which was related to the observation above about a cultural proclivity toward alleviating student confusion in the classroom.

He also stated that a review of research by Hiebert and Grouws identifies two features of classroom instruction that are associated with students' understanding of mathematics:
- Connections: Making mathematical relationships-among concepts, procedures, ideas-explicit in the lesson
- Struggle: Students spend at least some time struggling with important mathematics.

1. We need to shift the emphasis from "teachers" to "teaching" (need to change cultural routines to make them better over time)
2. We need to redefine "quality teaching." It's not a set of skills and strategies (e.g., lecturing vs. using small groups) but making informed judgments in the classrooms--how what teachers do creates opportunities for students to achieve important learning goals (i.e., for math, explicit ideas and struggle).

a.  Clearly define learning goals and understandings needed to achieve them (subject matter knowledge essential)

b. Design lessons that use strategies shown effective for achieving the goals and judged appropriate for specific content.
c.  Study effectiveness based on students' learning from instruction, and analyze teaching/learning as cause/effect (so teacher can learn from experience). Teachers need to analyze their students' thinking and reasoning.
d. Provide teachers with opportunities to learn the knowledge, skills, and judgment they will need for continuous improvement: stable settings to work with colleagues to learn what works better and share what is learned with the profession.
Moreover, Stigler, in a response to a question from Commissioner David Pearson, stated:

The goals of mathematics learning in the United States are very procedural, and teachers don't know how to talk about what it is that they want students to understand (apart from how to do procedures). It takes patience and fortitude to implement new teaching strategies, but it's possible to create a new classroom culture… Too many people think that mathematics isn't about thinking-it's about remembering. Once you get to algebra, you have a lot of steps to remember. However, if you can reason about what the steps are, it gives you a lot more power. In science, there's some real parallel findings. A lot of time is spent in lab activities, but not much time is being spent in the U.S. connecting the lab to science concepts.

Friday, May 15, 2009

AIR study

The American Institutes for Research (AIR) has released a study: Why Massachusetts Students, the Best in the U.S., Lag Behind 
Best-in-the-World Students of Hong Kong.  The comparison of the two assessments found that Hong Kong items differed from Massachusetts items in several important ways:

Hong Kong items were more concentrated in the number and measurement strands (75 percent), compared with Massachusetts (60 percent). A firm understanding of basic number concepts is essential for doing more-advanced work in fractions and algebra. And a solid understanding of measurement concepts is crucial to handling real-world math and learning geometry.

Hong Kong items were more likely to require students to construct a response (86 percent) than Massachusetts items (29 percent). Constructed-response items tend to be more demanding of students to generate the correct answer by working completely through the problem without the advantage of being able to select a correct answer from a list.

Items in the Hong Kong assessment were more likely to require more than low computational difficulty (37 percent), compared with Massachusetts items (3 percent). In the numbers domain, where computation is an integral component of the solution, 13 out of 15 (87 percent) of Hong Kong items were of higher computational difficulty, whereas only 1 out of 17 (6 percent) of the Massachusetts items in numbers required more than simple computational skills.

Hong Kong items were more likely to fall into the moderate or high cognitive complexity category (55 percent) compared with Massachusetts items (34 percent). Performance on higher cognitively complex items is an indicator of the ability to apply mathematical concepts to solving routine and non routine problems.

“Overall, the comparison revealed that each assessment covered similar mathematics topics. However, the Hong Kong assessment required a greater depth of mathematical understanding required to solve many items,” explained Leinwand. “This expectation of deep understanding of math concepts is a likely contributor to Hong Kong’s achievement as the highest performer on TIMMS in the early grades.”

Monday, May 11, 2009

NSF Programs

Anti-reformists expend tons of energy and rhetoric focusing on Everyday Math, Investigations and Connected Math.  If they only spent the same time and energy on supporting the programs and texts that they believe are better for students.  But instead these anti-reformist post blogs, websites, and petitions to rid districts of NSF programs without regard for what replaces it as if it doesn’t matter what replaces them as long as its not an NSF program. It is just as amazing how these critics consistently and conveniently ignore the facts that:

All three programs have been extensively studied and all have shown to have a positive impact on student achievement – particularly in the domains of conceptual understanding and problem solving; and

None of the four major brother-sister traditional programs (Houghton-Mifflin/McDougal-Littell, Harcourt/Holt, Scott Foresman/Prentice Hall, and McGraw Hill/Glencoe) have been subject to any kind of similar evaluation.

Robert Reys (Curators' Professor of Mathematics Education at the University of Missouri) writes:

It has been suggested that standards-based mathematics curricula don't have a research base of student mathematical performance to support their use. Clearly, much research remains to be done and reported. However, the statement implies that traditional programs, which still make up the overwhelming majority of the programs in use, have a sterling record of success in promoting mathematics learning. Moreover, it ignores decades of poor performance documented by the National Assessment of Educational Progress (NAEP) and by three international assessments, the latest being the Third International Mathematics and Science Study (TIMSS). Furthermore, the lack of knowledge and understanding of mathematics discussed by Liping Ma is the by-product of mathematics programs that were in place long before standards-based mathematics curricula existed.

People who demand research to document the effectiveness of reform curricula are either unaware of the history of student performance using the traditional curricula or choose to ignore more than 30 years of widely reported results. In fact, to assume that traditional mathematics programs have shown themselves to be successful is, according to James Hiebert, "ignoring the largest database we have." Hiebert goes on to say, "The evidence indicates that the traditional curriculum and instructional methods in the United States are not serving our students well.

In arguing against the use of standards-based, NSF-supported curricula, some have alleged that children were used as guinea pigs for untried programs. This argument has strong emotional appeal. Who wants a child to be used as a guinea pig? Critics have advocated "stricter controls to prevent schools from using untested programs without the informed consent of parents and students. This claim is ironic on at least two counts. First, the traditional mathematics curriculum supported by the critics has not been tested for effectiveness, unless international assessments are used as the measure, in which case these curricula fall far short. Second, there has been unprecedented field-testing of these NSF-supported curricula over the past decade. They have been piloted, revised, field-tested in real classrooms, and revised again prior to their commercial availability. Data continue to be systematically collected, and feedback from the field is reflected in later editions. Research reporting on student achievement in a variety of grades is beginning to emerge.  To criticize these curricula because of the philosophy they embody or the mathematical content of the materials is one thing. To suggest that they have not been extensively field-tested with teachers and students is blatantly untrue and irresponsible.

In fact, with very few exceptions, most widely used mathematics textbooks are not carefully researched and field-tested with children before they are sold to school districts. I base this assertion on my personal experience as well as on years of interacting with textbook authors. My own experience in co-authoring a textbook series was that, while every effort was made to create a product of the highest quality, publication deadlines made it impossible to do extensive field-testing. Lessons were written and edited and occasionally focus-tested with teachers. However, because of the market-driven nature of the textbook business, it was not possible to have teachers use the materials for several years and revise them based on student performance and feedback from teachers.

Calls for testing and documenting the impact of mathematics curricula will surely continue, and they should. However, the bar should be set at the same height for all publishers. If the developers of standards-based mathematics curricula are required to document the impact of those materials on student performance, then the same criterion should be applied to all companies producing mathematics textbooks for the same market. Of course, the prospects for such a requirement are dim.

Saturday, May 9, 2009

A Mathematician's View

Dr. Eve Torrence is a mathematician at Randolph-Macon.  She writes about her experiences with mathematics education in America and Europe below. 

I am a mathematician. I am a college professor. I am a mother. From all three perspectives I have been following with interest the controversy over the current mathematics education reform. Last year I had an experience that finally brought clarity. My husband, who is also a mathematician, and I had a sabbatical at the University of Utrecht in the Netherlands. We enrolled our eight year-old son, Robert, in a local Dutch school. In doing so we were unconsciously starting a very interesting experiment.  At home Robert had been experiencing a traditional mathematics curriculum where a great deal of time and effort is spent on learning the carrying and borrowing algorithms for addition and subtraction. The mathematics curriculum at his Dutch school was very different. The students were working on problems at the same level, but they were encouraged to develop their own techniques for doing the problems. They were not taught the carrying and borrowing algorithms. This approach has been used successfully in Holland for almost thirty years.At the same time Robert was adapting to a new curriculum, I was studying at the Freudenthal Institute at the University of Utrecht—a world-renowned center for research on mathematics education. I was learning that the curriculum he was experiencing is called Realistic Mathematics Education (RME). In RME, the mathematics is introduced in the context of a carefully chosen problem. In the process of trying to solve the problem the child develops mathematics. The teacher uses the method of guided reinvention, by which students are encouraged to develop their own informal methods for doing mathematics. Students exchange strategies in the classroom and learn from and adopt each other’s methods. I also learned that much research has been done on this approach, that it is based on what we know about child development and the development of numeracy, and that it is this body of research that is driving the math education reform in our country.

When we first arrived in the Netherlands and I began to learn about RME, I spent a little time quizzing Robert on how he would solve a few addition and subtraction problems. I was shocked by the rigid attitude he had developed at his school in the U.S. When asked to do any addition problem with summands larger than 20 he would always invoke the addition algorithm. He would sometimes make mistakes and then report an answer that made no sense. He was putting all his confidence in the procedure and little in his own ability to reason about what might be a sensible answer. When I suggested there was a simpler way he could think about the problem he became upset and told me, “You can’t do that!”

After a few months in Holland, I began to see an amazing difference in Robert’s number sense. He was able to do the same problems more quickly, more accurately, and with much more confidence. For example, I asked him to solve 702 minus 635. He explained, “700 minus 600 is 100. The difference between 2 and 35 is 33, and 100 minus 33 is 67.” When he tried using the algorithm he made a borrowing error and became very frustrated. I asked him to compute 23 times 12. He explained, “23 times 10 is 230, 23 times 2 is 46, 230 plus 46 is 276.” This multiplication problem was much harder than anything in the curriculum at home. I was very impressed with the flexibility and range of methods he had developed in only a few months.

What happened to Robert in those few months has had a profound effect on my perception of learning and on Robert’s understanding of mathematics. My child learned to think. He learned he could think. He was encouraged to think. He learned to see mathematics as creative and pleasurable. This independent attitude towards mathematics will remain with him forever and serve him well. It is this fact that has convinced me of the value of de-emphasizing algorithms in the elementary years.  Unfortunately, Robert is once again back in a school that focuses on the teaching of algorithms. The other day as we were driving to soccer, out of the blue Robert asked from the back seat, “Mommy, wouldn’t it be crazy to do 5000 minus 637 using borrowing?” I smiled proudly at him and said, “Yes, honey, it would.” 

Skills and Areas by employers

The Association of American Colleges & Universities has published a report entitled "Our Students Best Work: A Framework for Accountability Worthy of Our Mission."   The report has some very interesting data including a list of Skills and Areas of Knowledge a majority of employers would like schools to emphasize more.  The list is below and notice the emphasize on reform standards based skills.

Concepts and new developments in science and technology 

Teamwork skills and the ability to collaborate with others in diverse group settings

The ability to apply knowledge and skills to real-world settings through internships or other hands-on experiences

The ability to effectively communicate orally and in writing

Critical thinking and analytical reasoning skills

Global issues and developments and their implications for the future

The ability to locate, organize, and evaluate information from multiple sources

The ability to be innovative and think creatively

The ability to solve complex problems

The ability to work with numbers and understand statistics

The role of the United States in the world

A sense of integrity and ethics

Cultural values and traditions in America and other countries

Thursday, May 7, 2009

NAEP 2008

The 2008 NAEP long term mathematics assessment results have been released.  The assessment was designed to measure a student’s knowledge of basic mathematical facts, ability to carry out computations using paper and pencil, knowledge of basic formulas such as those applied in geometric settings, and ability to apply mathematics to daily-living skills such as those involving time and money. The complete 2008 mathematics assessment contained between 103 and 126 multiple-choice questions and between 30 and 36 constructed- response questions at each age. Unlike certain sections in the main NAEP assessment, students were not permitted to use a calculator in the long-term trend mathematics assessment.  The results show excellent gains in 9 years olds and 13 year olds but no changes in 17 year olds.  A summary is below.

The overall gain in mathematics since 2004 for 9-year-olds was also seen in increases for all but the lowest-performing students. While there was no significant change in the score for 9-year-olds performing at the 10th percentile from 2004 to 2008, the score in 2008 was 27 points higher than in 1978.  Scores were higher in 2008 than in all previous assessment years for students at the 25th, 50th, 75th, and 90th percentiles.

While the overall average score for 13-year-olds was higher in 2008 than in both 2004 and 1978, the results varied for students performing at different percentile levels. Scores increased since 2004 for students at the 10th and 50th percentiles, but there were no significant changes for students who scored at the 25th, 75th, and 90th percentiles over the same period. Students performing at all five percentile levels scored higher in 2008 compared to 1978.

As in the overall scale score results for 17-year-olds, there were no significant changes in scores from 2004 to 2008 for students at any of the five percentile levels. Scores for lower- and middle-performing 17-year-olds (at the 10th, 25th, and 50th percentiles) were higher in 2008 than in 1978.

Wednesday, May 6, 2009

NCTM Focal Points

At the last DOE Mathematics Task Force meeting, the afternoon was spent with subcommittees generating list of the most important standards in each grades.  In other words, time was spent generating lists of focus topics.  NCTM spent over one year doing this!  The NCTM Focal points has received positive reviews by many factions.  They are listed below.

Prekindergarten Curriculum Focal Points
 Number and Operations: Developing an understanding of whole numbers, including concepts of correspondence, counting, cardinality, and comparison.
 Geometry: Identifying shapes and describing spatial relationships.

 Measurement: Identifying measurable attributes and comparing objects by using these attributes.

Kindergarten Curriculum Focal Points
 Number and Operations: Representing, comparing, and ordering whole numbers and joining and separating sets

 Geometry: Describing shapes and space

 Measurement: Ordering objects by measurable attributes

Grade 1 Curriculum Focal Points
 Number and Operations and Algebra: Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts.
 Number and Operations: Developing an understanding of whole number relationships, including grouping in tens and ones.
 Geometry: Composing and decomposing geometric shapes.

Grade 2 Curriculum Focal Points
 Number and Operations: Developing an understanding of the base-ten numeration system and place-value concepts.
 Number and Operations and Algebra: Developing quick recall of addition facts and related subtraction facts and fluency with multidigit addition and subtraction.
 Measurement: Developing an understanding of linear measurement and facility in measuring lengths.

Grade 3 Curriculum Focal Points
 Number and Operations and Algebra: Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts.
 Number and Operations: Developing an understanding of fractions and fraction equivalence.

 Geometry: Describing and analyzing properties of two-dimensional shapes.

Grade 4 Curriculum Focal Points
 Number and Operations and Algebra: Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication.
 Number and Operations: Developing an understanding of decimals, including the connections between fractions and decimals.

 Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes.

Grade 5 Curriculum Focal Points
 Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers.
 Number and Operations: Developing an understanding of and fluency with addition and subtraction of fractions and decimals.

 Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area.

Grade 6 Curriculum Focal Points
 Number and Operations: Developing an understanding of and fluency with multiplication and division of fractions and decimals.
 Number and Operations: Connecting ratio and rate to multiplication and division.

 Algebra: Writing, interpreting, and using mathematical expressions and equations.

Grade 7 Curriculum Focal Points
 Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity.
 Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes.
 Number and Operations and Algebra: Developing an understanding of operations on all rational numbers and solving linear equations.

Grade 8 Curriculum Focal Points
 Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations.
 Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle.

 Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets.

Tuesday, May 5, 2009

CCSSO and NGA starting work on National Standards

The National Governors Association (NGA) and CCSSO have commissioned work on common standards to be done by Achieve, College Board and ACT. A small writing team (6 people, 2 from each organization) has been formed to begin work identifying mathematics learning expectations for end of high school.  The two Achieve writing team members for mathematics are Kaye Forgionne (Achieve staff) and Bill McCullom.  The standards are meant for all students, benchmarked internationally, aligned with college and work-ready needs, rigorous, and based on research and best evidence. By "college-ready" they mean ready for College Algebra.  The team is charged with completing their work by the end of the summer.  The current Achieve, CB and ACT documents will serve as a starting point for their work.  In the fall the next phase of work will begin -- developing K-12 common standards for mathematics and English/language arts. That work is supposed to be completed by the end of the 2009.  The third phase of work will be focused on common assessments.  The details are less clear about this.  States are being asked to sign a memorandum of agreement to be a partner in the standards work.

Is NJ one of these states?  We wait to hear from the Department of Education and Commissioner Davy who has remained silent on the issue.

Monday, May 4, 2009

Math in Everyday Life & Careers

Often, we hear that the poor performance of students on assessments will have a huge impact on America's ability to compete in the global economy.  However, there is little (if any) research to back such scare tactics.  In fact, the research tells a different story about mathematics in the real world and the traditional skill based computational curriculum that anti-reformists back.  Julie Gainsburg's research about mathematics in work and in life tells us a different story.  One of her studies, School mathematics in work and life: what we know and how we can learn more finds:

Ethnographic studies of child and adult math use in everyday life have generally all come to the same conclusion: there is a wide gap between school-taught mathematical methods and the math people use outside of school. Context plays a central role in everyday problem solving. Studying farmers, carpenters, and fishermen in Brazil, Nunes et al. showed that these adults relied on contextual aspects of the quantitative problems of their work to help them calculate in ways unlike school-taught algorithms. While school arithmetic is considered rule-based and divorced from meaning, these adults' ‘street arithmetic’ preserved situational meaning at each calculation step. In their study of grocery shoppers, Lave et al. demonstrated that the setting of the store (the layout of the aisles, the price labels, the appearance of products, etc.) was inextricably linked to the shoppers' arithmetic decision making, with the setting both constraining and supporting their solutions. The shoppers' arithmetic performance in shopping was far better than on a test of mathematically analogous, school-type operations. The shoppers' school backgrounds correlated only with their scores on the school-type test and not with their performance on, or willingness to use, arithmetic in shopping. This study also illuminated a key feature of real-life problem solving: gap closing. Typically, the shoppers would reshape their shopping calculation problems while simultaneously shaping the solutions, thereby moving problem and solution towards each other—an impossible strategy for most school math problems.

People avoid even well-known school-taught algorithms when solving everyday problems. Asked to find 3/4 of a 2/3-cup serving of cottage cheese, with the actual cheese and measuring cups available, adult dieters in de la Rocha's study ignored the standard algorithm for multiplying fractions and instead invented strategies that often involved manipulating the physical objects.

What about engineers, you ask?

Hutchins's (Cognition in the wild, The MIT Press, Cambridge, 1995) extensive ‘cognitive ethnography’ of the practice of navigation on a Navy ship illuminated several features of ‘cognition in the wild’. Navigation, Hutchins showed, is accomplished through a complicated, interconnected system of human tasks, methods of representation and communication, instruments, tools, charts, tables, codes, techniques, and traditions. The knowledge required to dock a huge ship is distributed among these system elements, too great for any one person or tool to possess entirely. The cognitive demands on any single crewmember are actually quite mundane and technological tools have supplanted most calculations. According to Hutchins, tools do not amplify the cognitive abilities of individuals in the system, as is often claimed, but replace their more difficult tasks (like calculating) with easier ones (like chart reading). Stressing the cultural diffusion of knowledge, Hutchins warned researchers not to infer the cognitive requirements of an individual from observations of the overall activity in which he participates. Hutchins's interest, in fact, was not the individual but the ‘environment for thinking’ that the culture of navigation has developed over its history. According to Hutchins:

The real power of human cognition lies in our ability to flexibly construct functional systems that accomplish our goals by bringing bits of structure into coordination… [A] proper understanding of human cognition must acknowledge the continual dynamic interconnectivity of functional elements inside with functional elements outside the boundary of the skin.

Hutchins's warning challenges the educational policy rhetoric that our highly mathematized world imposes high-level mathematical demands on each citizen. Today's technologies may indeed have replaced many of our mathematical tasks with button pushing and chart reading, thereby reducing our need to personally possess sophisticated calculation skills.

Analyzing the mathematical behavior of architects and structural engineers jointly engaged in a design session, Hall found that quantitative reasoning was prevalent but mostly involved only simple arithmetic with routine quantities. More interestingly, Hall showed how forms of mathematical representation are socially negotiated, both historically and on the spot. The diverse perspectives and accountabilities of the architects and engineers shaped their mathematical descriptions, models, and uses of quantity. Hall's portrayal of these workers' mathematical behavior—culturally and socially dependent, flexible, parochial, and open for reinvention and interpretation—clashes with the traditional image of mathematics as a static, objective, universal language that exists virtually independently of people, untainted by subjective perceptions and motivations.

Hoyles et al. investigated experienced nurses' use of proportional reasoning in calculating drug doses. Though trained to use a single computational routine for this purpose, in practice the nurses employed a wide range of effective proportional reasoning strategies. Hoyles et al. believe their findings confirm the existence of a large gap between school math and the math of the workplace, where abstract rules ‘fade into the background’, and culturally held and artifact-reliant procedures serve just as well or better. While rejecting the notion that the nurses used general, school-taught algorithms at work, Hoyles et al. proposed that the nurses had themselves ‘abstracted’ the theoretical concept of concentration (the proportion of drug to solvent) in the course of their years of practice.

Saturday, May 2, 2009

Singapore Math in US

While there is much be learned from examining Singapore mathematics syllabi (not standards) and also by looking at Singapore pedagogy, as noted in my last post, there is NO evidence that the Singapore approach can lead to higher achievement in America.  The US department of Education has now confirmed that no such evidence exists.  

No studies of Singapore Math that fall within the scope of the Middle School Math review protocol meet What Works Clearinghouse (WWC) evidence standards. The lack of studies meeting WWC evidence standards means that, at this time, the WWC is unable to draw any conclusions based on research about the effectiveness or ineffectiveness of Singapore Math.

Friday, May 1, 2009

Singapore Pedagogy

The anti-reformists often point to Singapore for as their hope for America.  However, they do so without any knowledge of the reform methodologies (that the anti-reformists hate) which are prevalent in Singapore. The article below highlights the constructivist nature of Singaporean mathematics classes.

Educational reform in Singapore: from quantity to quality by Pak Tee Ng

Knowledge Application—Students are to learn basic research skills, apply and transfer knowledge and skills learnt across disciplines and to make connections between them.

Communication—Students are to improve their ability to communicate ideas clearly and effectively in both written and oral modes.

Collaboration—Students are to develop and improve social skills in collaborating with others towards a common goal (students usually work in groups of 4–5).

 Independent learning—Students are to learn to take charge of and monitor their own learning as well as to develop a positive attitude and responsibility towards their work.

The learners are responsible for their own learning. They take charge of their learning and are self-regulated. They define learning goals and tackle issues that are meaningful to them. They know how the learning activities they undertake relate to the goals. They develop their own standards of excellence.

The learners are strategic in their learning process. They know how to learn, develop and refine their learning. They can apply and transfer the knowledge generated creatively.

The learners collaborate with others. They understand that learning is social. They recognise that different people can have different views about the same issue and the multiple points of view can enrich the learning process.

The learners are energised throughout the learning process. They derive excitement and pleasure from learning. They find learning fulfilling.

Construction of knowledge (not just transmission of knowledge): in the new teaching and learning processes, students are able to develop their own knowledge base, pulling information from many sources and making linkages, instead of waiting for the teachers to push information to them.

Understanding (not just memory): students know what they are doing rather than just memorising facts or applying methodologies that do not make sense to them.

Pedagogy (not just activity): in the new TLLM paradigm, teachers do not merely carry out activities for the sake of having activities, but each activity is a part of a well thought- through pedagogy that will bring students to a higher level of understanding or appreciation.