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ThinkAlgebra: LT's Mathematics Page
More Math, Faster, Earlier
ThinkAlgebra is an information source for parents and educators who value schooling and are concerned about math education in the United States. It is also for parents who want their children well prepared for college. This is the issue: American 15-year olds rank 24th out of 29 countries in mathematics in the latest PISA results. The quandary originates in early elementary school (1st grade) with inadequate teaching of arithmetic and continues up the grades (NAEP). I want teachers to teach and curriculum and teach above it to challenge students. Kids thrive on challenging mathematics. Moreover, schools must focus on academics and restructure spending ($) and instructional time-on-task priorities to optimize student achievement in math, starting in 1st grade.
1. Algebra starts with arithmetic. Arithmetic knowledge is cumulative, so a structured, systematic learning of arithmetic to fluency, starting in first grade, is essential for student success in algebra and higher mathematics. For example, in K-6 Singapore math, number facts are memorized, algorithms mastered, axioms used, and reasoning learned. No calculators. [FYI: Singapore is a global leader in math education.] Unlike most math programs in the United States, Singapore's national curriculum is focused, rigorous, and coherent. Furthermore, Singapore's world-class standards are taught by teachers who know math and know how to teach it. In short, Singapore's K-6 math program concentrates on mastery of arithmetic and geometry basics. (Perspective: Read Dr. William H Schmidt's article, "What Missing from Math Standards? Focus, Rigor, and Coherence).
2. State standards and tests often foster mediocrity. In addition to having deficient math standards, "The states have made a mockery of that provision [NCLB accountability], using weak tests, setting passing scores low or rewriting tests from year to year, making it impossible to compare progress — or its absence — over time. The country will have difficulty moving ahead educationally until that changes. Most states that report strong performances on their own tests do poorly on the more rigorous and respected National Assessment of Educational Progress, which is often referred to as NAEP...." (Perspective: Read NY Times, August 11, 2008 Editorial.)
3. Typically, my best math students are my best readers. These students have higher than average academic ability. Charles Murray (Real Education, 2008) argues rightly that academic ability varies considerably and is normally distributed [normal curve statistics]. This means that half of the students in K-12 are below average in academic ability, which is a reason some children have limits and cannot learn much more than basic reading and math. That said, the idea that students who are below average cannot learn school math is simplistic. There is no substitute for focused, rigorous, coherent instruction in math ; however, too often, good math instruction is hard to achieve because many elementary teachers are weak in math, even math phobic. The curriculum content must be fixed, and teachers must be rigorously trained to teach it well.
➠ Students deserve better than slow-paced, low-content instruction. Young children, when taught well, are capable of learning significantly more arithmetic in a much shorter time frame than the chronological grade curriculum taught in most elementary school classrooms today. This has been known since the 1960's when elementary school mathematics and science programs, which were exceptional products of the Sputnik era, successfully shifted math and science content down to lower grades: more math, faster, earlier. Indeed, Science--A Process Approach (K-6) put math back into elementary school science. Read more below at ➠ ThinkInnovative Curricula from the 1960's.
➠ If we upgrade to academic standards on par with Singapore K-6 math curriculum, then there will be some kids who won't get it. Starting in first grade, the kids who don't get it need immediate academic support. Pushing kids into 2nd grade without knowing basics of reading and math is misguided education. On the other hand, most kids can learn upgraded content with good instruction, effort, and practice. Indeed, average kids can do well when challenged. Furthermore, superior kids should be given opportunities to take advanced math. Unfortunately, we rarely teach kids advanced mathematics (e.g., algebra) in elementary school--not in the United States.
➠ The idea that kids can have academic limits is often disregarded by educators. Indeed, Murray comments, "When the facts get in the way we ignore them." He asserts, "The educational system is living a lie. The lie is that every child can be anything he or she wants to be. No one really believes it, but we approach education's problems as if we did. We are phobic about saying out loud that children differ in their ability to learn the things that schools teach."
➠ Algebra is for students who plan to go to college or community college, but it may not be for all high school students as many claim. Do all students need to prepare for college or learn algebra when many jobs require only good basic skills, an excellent work ethic, and on-the-job training--not algebra or a postsecondary degree? In my contrarian view, the popular mantra "algebra for all" does not have a solid evidential base. By algebra, I mean Algebra I and Algebra II.
Requiring all students to "prepare for college" as prerequisite for a high school diploma is not supported by DOL data. According to the U.S. Department of Labor (DOL), of the 164,540,000 total employment projected for 2014, almost half of the jobs (45.9%) will go to those who have only a high school diploma or less, 28.4% will go to those who have acquired some college education, and 25.7% will go to those who hold a bachelor's degree or higher. And, I quote, "Among the three groups [high school or less, some college, and bachelor's or above], jobs projected to go to those with a high school degree or less will predominate--accounting for 45.8 percent of all jobs in 2014 and 36.6 percent of new jobs created between 2004 and 2014. However, the fastest percentage growth in new jobs, 19.0 percent, will go to those with a bachelor's degree or higher." Given this information, requiring all students to take algebra as preparation for college does not make sense.
That said, students who plan to go to college must pass the college math entrance test, which is mostly Algebra II, or end up taking remedial math courses. If students want to get into college, then they must master algebra, even if algebra is never used in their coursework, major, or future career. Also, getting into college is the easy part; graduating with a bachelor's degree in a field of choice in 4 years is the difficult part. (The drop out rate in college is roughly half.) Many incoming college students aren't prepared for the rigor of academics or the study discipline needed at the university level. High schools should place less emphasis on getting into college and more on graduating from college with a degree. Students need to take tough courses (precalculus and physics) in high school to prepare for the academics required in college.
⎮ Algebra I --> Algebra II --> College --> Bachelor's Degree (40%)
⎮ Algebra I --> Algebra II --> Precalculus --> College --> Bachelor's (74%)
⎮ Algebra I --> Algebra II --> Community College --> Two-Year Degree (? %)
⎮ Resource: What You Need To Do in High School If You Want To Graduate from College
On the other hand, if your child covets a bachelor's degree, then it makes sense to get your child on the "algebra/precalculus" tract in middle and high school because nearly 75% of students who complete precalculus in high school go on to earn a bachelor's degree in college. (Like colleges and universities, community colleges require algebra.) Even though all kids must master basic arithmetic, critical reading, concise writing, and computer skills for future employment, there is credible evidence that many schools fall short on teaching students basics. TO Be Revised
➠ The most neglected kids in our educational system--especially in elementary and middle schools--are students high in academic ability, i.e., kids that have IQ's of 120 or above. Too little is expected from the best students. Most programs for the academically gifted boil down to oversimplified enrichment programs that do not have a knowledge component. The nominal, pull-out enrichment programs are meant to appease parents of precocious students.
For additional perspective read Academic Ability at the bottom of the page.
➠ ThinkAlgebra What do elementary and middle school students need to know to be successful in Algebra I? The short answer is arithmetic. Many view arithmetic as finding sums and products, and while sums and products are essential, arithmetic is significantly more. It also includes decimals, fractions, percentages, ratios, and proportions. Furthermore, several important ideas in algebra can be taught to well-prepared elementary school students through arithmetic, starting in first grade. For case in point, my first grade students not only learned arithmetic comparable to Singapore students, they also learned integers, algebra, plane geometry using LOGO, a computer programming language, and measurement. Students measured mass in grams (equal arm balances), liquid volume in milliliters (graduated cylinders), and lengths of line segments and polygonal curves and perimeters of squares, rectangles, and other polygons in centimeters (metric rulers). In algebra, students learned variables and rule for substituting, equations, integers, function ideas, and mathematical reasoning. |
➠ Think Arithmetic!
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➠ Think Teacher Quality
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➠ Think World Class
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➠ Think No Calculators K-6
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➠ Think Cognitive World
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➠ Think Long-Term Memory
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➠ The Silent Crisis & A Stagnant Nation"American 15-year olds rank 24th in math out of 29 countries. There is a silent crisis in American education that puts our economic competitiveness at serious long-term risk, yet the American public seems unaware of the severity of the situation. As a nation, we are not producing enough educated, highly skilled workers to compete in global markets." [Quote: Solutions Through Higher Education] Most of our best high schools do not meet international standards in advanced mathematics and physics. (Click for 2008 TIMSS-A Framework) The United States did not participate in the 2008 TIMSS-A (A for advanced) international assessment. According to NCES (US Department of Education), the 2008 TIMSS-A cost too much money. The last time the TIMSS-A was administered, our "advanced" students ranked second to last in math and last in physics. These were students who studied advanced mathematics and physics; i.e., our best high school students. Read A Stangant Nation, Why American Students Are Still At Risk, pdf file |
➠ Report from the National Math Advisory PanelFinal Report: National Mathematics Advisory Panel The report states, "International and domestic comparisons show that American students have not been succeeding in the mathematical part of their education at anything like a level expected of an international leader. Particularly disturbing is the consistency of findings that American students achieve in mathematics at a mediocre level by comparison to peers worldwide." ✔ Observation: Excellent standards are no guarantee of high achievement. What happens in the classroom is the key to excellent achievement. What counts most is a teacher who knows content and knows how to teach content well. |
➠ Lesson Ideas: K-8 K-8 Lesson Ideas: Real Algebra |
➠ ThinkInnovative Curricula from the 1960's.Many cutting-edge programs shifted math topics down to lower grades. The idea that kids can learn more math content, earlier and faster, if given proper instruction, has been the motivation for some of my teaching and my e-book, Teaching Kids Algebra. The innovative programs from the 60's are instructive. One program dates back to 1957 when Dr. Robert B. Davis of Syracuse University taught 3rd and 4th graders how to work with quadratic equations. Later, his lessons on key math topics, which were never taught in elementary school, became known as The Madison Project. Another program developed in the 1960's called Science--A Process Approach introduced the math needed to do science. The math was often a couple years ahead of the standard math curriculum and was taught as part of the science course. Today, the science taught in K-8 avoids mathematics. (The one exception is a good 8th grade physical science course.) In the 1950's and 1960's, there were many elementary school math and science experimental programs, e.g., The Madison Project (math), School Mathematics Study Group (math), Science--A Process Approach (SAPA) (science with mathematics used in science), and many others. Unlike math programs and texts today, these experimental materials were written, tested in classrooms, and revised several times before they were commercially available to schools. Some of the conclusions based on these programs were:
The experimental programs were initially taught to young children by the mathematicians, themselves, or by teachers directly trained by the mathematicians. Additionally, the experiments showed that the 7th and 8th grade general math curriculum can be shifted down to upper elementary school and that first graders can learn mutliplication as repeated addition: 3 x 5 = _____ + _____ + _____ = _____ and intuitive algebra (below) and a lot more. There is only one answer, {2}, because the algebraic rule for substitution must be followed when working with variables. If you place a 2 in the first square [variable], for example, then you must place a 2 in all the squares in the same equation. Substituting a 3 in the first square, would result in 3 + 3 + 3 = 9. The substitution is correct, but "3" leads to a false number sentence [equation]. The substitution 3 + 2 + 1 = 6 does not follow the substitution rule and is discounted as a solution to the linear equation, x + x + x = 6. FYI. Algebra Summary: x + x + x = 6 is 3x = 6; therefore x = 2. The corresponding multiplication fact is written 3 x 2 = 6, which is 3 sets of 2. |
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Early Math in Science: 1960's Science--A Process Approach (SAPA) introduced math topics earlier than the traditional math curriculum. For example, the distributive property is introduced in a 2nd grade SAPA activity. SAPA was K-6 science curriculum that had been repeatedly tested, rewritten, and fine tuned for average classroom use: Experimental SAPA Editions: 1963, 1964, 1965; Commercial Edition: 1967) Math is important in science, but today elementary school science isn't taught this way. Of the six SAPA science processes taught in first grade, for example, four were math or math related: Using Numbers, Measuring, Using Space/Time Relationships, Communicating (Graphs). Part C [Grade 2] Using Numbers 9 (Multiplication) Editor: On a personal note, I taught SAPA K-6 at a private school in the late 60's. In SAPA, integers were introduced in 1st grade (Part B Using Numbers 5) and integer sums in 2nd grade (Part C Using Numbers 8). In the early 80's, I used many SAPA ideas. For example, I taught integers and integer sums (-5 + 8) using a number line approach to first graders. My first grade students easily learned integer sums at an understandable and applicable operational level.
References: Note: Teach Academic Skills Early |
➠ Quotes for Thought1. Niels Bohr: “Prediction is very difficult, especially if it's about the future.” 2. Dr. David Ruelle (The Mathematician's Brain, 2007): "What is mathematical intuition? When we study a mathematical topic, we develop an intuition for it. We put in our memory [long-term memory] a large number of facts that we can access readily and even unconsciously. Since part of our mathematical thinking is unconscious and part nonverbal it is convenient to say that we proceed intuitively. This means that processes of mathematical thought are difficult to analyze.... Mathematics is a matter of knowledge, not of opinion." 3. Dr. Ian Stewart (Letters to a Young Mathematician, 2006): "An awful lot of what is now called "mathematics" at [elementary and middle] school is really arithmetic. Unfortunately, it's almost impossible to progress to more interesting regions of the subject [algebra, trig, coordinate geometry, calculus] if you don't know how to do sums and get them right, how to solve basic equations, or what an ellipse is. The highest levels of very human activity demand a solid grasps of basics; think of tennis or playing the violin. Mathematics happens to require rather a lot of basic knowledge and technique." |
➠ Academic Ability Reading Comprehension Readability Analysis: Gunning FOG [Reading] Index
From http://www.writing-information-and-tips.com/fog-index.html: According to Charles Murrary (Real Education 2008), "The child who knows all the answers in math class [logical-mathematical ability] has a high probability of reading above grade level as well [linguistic ability] and, what's more, a higher than average chance of being industrious and determined. Conversely, children who are at the bottom of the math class usually have trouble with reading as well, and also have a higher than average chance of showing problems with thinking ahead and disciplining thesmelves. Many exceptions exist." When I read books written by mathematicians and scientists, I am astonished at how well they write. Those who are lucky enough to have high logical-mathematical ability normally have good linguistic ability [academic ability]. According to Murray, "The three components of academic ability [spacial ability, logical-mathematical ability, and linguistic ability] are interchangeable in groups but not in individuals. You probably think of yourself as better in one of the components than in the other two." |
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