Math Grouping

Sort kids into homogeneous math sections by achievement, starting in 1st grade.

Arrange kids into homogeneous math sections by achievement. Don't worry about their self-esteem. Worry about their competency! 

Equal coverage of core math content is often confused with equal coverage of math content. While most kids can learn arithmetic basics and some algebra, many kids will not go on to learn trig or calculus. Let's face it; some kids are much better at math than others, and many are not being taught math they are capable of learning. 

Putting high achievers and low achievers in the same math class have been a recipe for underachievement and mediocrity. We have been mainstreaming kids for as long as I can remember. In a typical classroom, there is often a wide range of abilities or achievements in math. It means that the kids who learn math faster get bored and the kids who struggle stay behind. In my view, mainstreaming [inclusion] for math classes has led to underperformance at all levels. In short, the traditional system of heterogeneous classes for math is deeply flawed. Putting low achievers and high achievers in the same math class have hurt all our students because they are not challenged to learn the content they are capable of learning. Moreover, the "one-size-fits-all" common core reform math is not the answer. In addition, the popular instructional methods (minimal teacher guidance, inquiry, group work, etc.) are often less effective in teaching arithmetic and algebra. In my view, students underachieve for these reasons: the way students are grouped in math class, the math content taught (weak curriculum), the methods of instruction (ineffective), and teacher training (inadequate in math and science). [Note. In my view, common core is often driven by pedagogy leftover from the NCTM reform math it has replaced. More, later.]

Education creates inequalities. "In education, you increase differences. If someone's good at something, you try to develop his ability, which results in differences, or inequalities. So if education increases inequality, is this ethical?"  (Surely You're Joking, Mr. Feynman! by Richard P. Feynman.)  (First Draft. Please excuse errors. There are last-minute inserts.) 


Good education, says Feynman, should increase differences. The most realistic way to meet the diverse needs of most students in mathematics is homogeneous sectioning for weak students, average students, and advanced students. It is the most pragmatic means for upgrading all students to significantly higher levels of achievement. The sorting of students should be flexible and start early in the 1st grade. (Note. Kids enter school with differences, but US schools tend to ignore differences. In Singapore, differences are addressed, not ignored. In 1st grade, kids with weak number skills are pulled out for math periods and taught by a high-quality teacher. They are expected to learn core arithmetic just like the regular kids. The difference is these pull out kids are in a smaller class, instead of a class of 30, use different materials, and are assigned the best teacher. Most students catch up within a year because that is the goal, although the safety net program lasts through the 2nd grade.) 

The "many levels" in the same math class, an overemphasis on group work and inquiry learning, and minimal guidance methods have led to widespread underachievement and unintended consequences. In my view, the implementation of "many levels" in the same math class and a steady diet of group work do not define quality learning or excellence. Furthermore, educators seem caught up in a self-esteem mode rather than in a competency and achievement mode. Many students who get As and Bs in so-called "college prep" courses end up in remedial math courses at community college. We are told that common core math should eliminate the mushrooming remedial math problem, yet there is no evidence that it can. 

[Insert. Education is off the track because academic excellence has been replaced by a "fairness" mindset that equalizes downward and leads to widespread underachievement for all students. In addition, we need to switch from a "test culture" to an "achievement culture." The curriculum, which is taught in the classroom, is test-focused and narrow. In my view, the common core is an outcome of the fairness mindset and testing culture, which are interrelated. Common core math is not at the Asian level. Thus, US students start at a lower level and never catch up. Beginning in the 1st grade, math is not taught well. Nothing new. The math curriculum is weak, poorly thought out, and poorly taught. Standards (e.g., common core) are not the curriculum. The curriculum is what is actually taught in the classroom. Katherine Baird (Trapped in Mediocrity) says that too many teachers do not make good use of classroom time, which is one of the main themes of this post. One reason is that kids of uneven skill levels are in the same math class. She also states that most states define "proficiency" at low levels of competency. Professor Baird wonders why there are so few elementary schools that teach algebra and geometry? 

On ThinkAlgebra, I conclude that states should opt-out of the common core, adopt the Core Knowledge K-8 content and skills sequence, which is better than the common core, and allow individual schools to compose their own rigorous curriculum and achievement tests based on the students they serve. Note. Core Knowledge, not common core, sets up a coherent K-8 math sequence that prepares more students for Algebra 1 in 8th grade, which is the fundamental tenet of the National Mathematics Advisory Panel (2008) ]  

In reality, not all students will be high achievers. Kids vary widely in academic ability, motivation, persistence, effort, self-control, numbers, vocabulary, etc., yet we educators pretend that each student is the same, which is nonsense. We need to tackle the real world, not play with Utopian models. We create idealized models of reality (e.g., equality) then think they are a reality [William Byers], but every teacher knows some kids make little effort to learn things, many kids do enough to get by, and other kids just do not have the smarts.  Nevertheless, we bend over backward with money, time, and resources to make equality work. It just isn't going to work--it's not reality. We need a different recipe, one that works in the real world, not a fantasy.

The late Professor Feynman is right. In a sound education system, there will always be inequalities because education creates differences; however, the mixed group approach in place today is an unacceptable model because it has led to underachievement at all levels. And that's what many educators do not want to acknowledge. In short, our present system symbolizes [is code for] low-quality schooling. Kids need strong teacher guidance, a world-class math curriculum, a grouping that matches their achievement level, lots of practice to master math, and persistence. Instead, most kids get "minimal guidance" instructional methods (e.g., discovery, inquiry, group work, etc.), a weak math curriculum, mixed grouping in math class, and insufficient practice to automate fundamentals.

The road to equality is paved with good intentions, but it is easy to get stuck in the mud with good intentions because good intentions are not the same as good ideas that actually work. In fact, many of the ideas put into the classroom turn out to be bad ideas; e.g., kids must first have high self-esteem before they can learn. Wrong! Intellectual leaders in education have been wrong again and again. In education, inputs (differences like self-control, etc.) do affect outputs (learning). Kids come to school with sizable differences in academic ability and vast differences in vocabulary. You cannot equalize huge differences. The idea of "equalizing downward by lowering those at the top" [Thomas Sowell] is a "prevailing ideology" in education. It hurts kids. Unfortunately, says Sowell, high achievement is often equated with "privilege." Privilege, some say, is not fair! Sowell characterizes the progressive point of view this way: "Tests in school discriminate against students who did not study." Let's abolish tests and homework because some students, apparently the ones who study, delay gratification, and work hard, have an unfair advantage.] Note. Progressives, which have influenced education policies for decades, actually believe that utopia of equality is possible [Berezow & Campbell]. Get real.

Instead of giving each student the same, which is a fundamental premise of Common Core math [uniformity], we should bring students up to the level of mathematics they need to move forward and be successful in life. We are obligated as educators to give children opportunities as equal as possible and encourage students, regardless of background, to work hard to achieve and excel. We need more college-educated minorities and women, especially in the STEM fields. 

When content is taught explicitly, average kids can learn arithmetic and algebra at an acceptable level; however, weak-performing students should be in a math section that receives a double dose of instruction; e.g., KIPP 5th graders get 2 hours of math daily. Indeed, KIPP students spend more relevant time-on-task in mathematics in one day than some elementary students spend in one week. KIPP kids have longer school days, school on Saturdays, longer school year, and math homework. Advanced math students should be placed in a section that stresses depth, content acceleration, and rapid pace. Indeed, to move rapidly forward, the best math students, which often languish in the regular classroom, should be grouped together for math class. This is homogeneous sectioning for regular grades in elementary school and by course in middle school, e.g., pre-algebra or Algebra 1. It is time to bring back the "old school." Hey, I miss the chalkboards.

Homogeneous math sections should be taught by high-quality math teachers (not NCLB definition of highly qualified teachers). We don't have nearly enough high-quality math teachers. Furthermore, the grouping (low, average, high) should start in 1st grade. Weak kids in 1st grade should be pulled out for math class at the beginning of the school year. The best kids should be pulled out for math class, too. My algebra program helps identify young, mathematically able students.

Group the best 1st-grade math kids
for a daily pull-out class taught by
a high-quality math teacher.

Homogeneous Sectioning for ELEM Arithmetic; MS Pre-Algebra, Algebra 1 

1. Low performing students: (2C) double dose of core 

2. Average performing students: Core + 

3. High performing students: Core +++

Core denotes the knowledge and skills learned by average students in top-performing nations, e.g., Singapore. A good curriculum for US kids is Core Knowledge content and skills sequence, which is world-class and puts Algebra 1 in 8th grade. It does not refer to the common core. In short, Core means you don't dumb down the math. Elementary teachers must get better at teaching basic arithmetic. Moreover, the organizing of students should begin in 1st grade and be fluid up and down. Switching to homogeneous groupings and explicit teaching would be important progress, but it is not perfect. There is no perfect system, but there are systems that work much better than others. 

Students in three tracks end up at the same goal, which is to learn core. This would end "content incoherence," a term used by E.D. Hirsch, Jr. Kids need to learn core. The lower track learns core. The middle path moves faster and learns more than the core, and the advanced group soars way above grade level core. This is not equal coverage of math content; it is equal coverage of core.  

[Insert. Many middle schools offer some form of homogeneous groupings, such as advanced or honors-level classes in mathematics; however, this idea (honors math class) is rare in elementary schools because parents seem satisfied with math enrichment or with talented and gifted programs. The problem is that math enrichment does not move kids forward. Also, math enrichment is rarely taught by a high-quality math teacher. Consequently, some elementary school parents hire a private tutor or enroll their child in online courses from EPGY or Art of Problem Solving. In high school, students sort themselves; i.e., they can select from a range of math courses, including AP Calculus.] 

Decades ago, educators replaced the "old school" homogeneous sections (BAD) with a theory of equality (GOOD) that advocates mixed-level groups, self-esteem, group and project work activities, inquiry learning, grade inflation, less rigor, etc. Subsequently, there has been a steady decline in academic rigor in math, science, and other academic subjects. The outcome has been "massive underachievement" [Janine Bempechat], an epidemic of grade inflation, a glorification of "group work" that downplays individual achievement, and an explosion of remedial math classes at community colleges. In short, math achievement has stagnated over the past 30 years. What's more, memorization and practice, which are needed for the mastery of fundamentals of arithmetic and algebra in long-term memory, have fallen out of favor in many classrooms.

[Insert. There is a direct link between knowledge in long-term memory and the child's ability to solve math problems. Mathematicians have pointed out repeatedly that there is an intrinsic fusion between knowing the basics of mathematics in long-term memory (arithmetic, algebra, trig, etc.) and the quantitative reasoning skills needed to solve problems in mathematics. Prior knowledge is essential for problem solving and insight. Indeed, as Dr. Art Markman (Smart Thinking) points out, "Memory is all about connections." In mathematics, connections are vital because one idea builds on another. Everything fits together logically. To free "mental space" for problem-solving, math facts and efficient procedures need to be practiced, so they become automatic (mastery). There are no shortcuts. In spite of this, many US educators think the mastery of math facts, procedures, and skills are not that important.]

The replacement game plan of putting kids of mixed knowledge and skills in the same math classroom (inclusion policy) has not worked well. It does not make sense for a teacher to have several math levels in her elementary classroom. The teacher barely has enough time to plan for one good math lesson a day, much less several different math lessons, plus the reading groups and everything else. Consequently, quality instructional time-on-task at each math level has been limited and leads to underachievement at all levels. A lot of instructional time is wasted.

The idea of homogeneous sections is not perfect, but it is far better than what we have today, which are mixed-level classrooms and an almost impossible task of differentiating instruction in those classrooms. Consequently, many students, especially our best kids, go unchallenged and underachieve when compared to their peers in other nations. In a mixed group of students, while the teacher is working with one small group of students for 15 to 20 minutes (groups rotate), the teacher also has to classroom manage the other students, who are often distracted (talking, off-task behaviors, etc.) because they sit at desks in groups. In reality, students don't learn as much as they could or should and have less relevant time-on-task. FYI: In many of today's classrooms, the emphasis is more on improving group scores on the state math tests than on individual achievement. We are off-target.

[Insert. All children need challenging content, especially in math. For example, the content I introduce to little kids in Teach Kids Algebra (TKA) is difficult before it becomes easy. It is more difficult for some than for others. The "difficult" becomes easier a little at a time through memory, persistence, and practice--not group work. Indeed, success is a function of perseverance and hard work.] 

An attempt to make math classes [sections] more homogeneous is often met with harsh opposition because homogeneous grouping conflicts with [progressive] equality dogma. To "boost low-performing students," content has been weakened by subtracting core rigor.  States have lowered "proficiency" cut scores on NCLB math tests so that more students pass. Consequently, many students have been labeled "proficient" in state NCLB math tests, yet they do not meet the proficiency level in NAEP tests. 

Many kids, especially academically gifted students, go unchallenged in elementary and middle school and underachieve. It is caused not only by a weak math curriculum but also by mixed math classes. This "unthinking pursuit of equality" hurts all kids, explains Jacob Vigdor (Education Next, Winter 2013). The paradigm of subtracting rigor does not move students forward toward algebra in middle school. Indeed, it delays the math development of all students suggests Vigdor. My observation is the same. For example, 3rd-grade students who are not required to memorize multiplication facts for auto recall or practice the standard algorithm (x) for fluency are stalled. They cannot do long division, fractions, pre-algebra, or algebra. 

Subotnik, Olszewski-Kubilius, & Worrell (Scientific American Mind, November/December 2012) point out, "Today researchers, policymakers, and teachers pay little to no attention to high-achieving students ... Many such students spend their days in schools unchallenged--relearning materials they have already mastered." Students who are behind never catch up because that is not the goal. Our lower-skilled kids might get better in mixed classes because the focus is on them (NCLB), but most kids who are above average, especially our best students, according to Jacob Vigdor (Education Next, Winter 2013), are not challenged and underachieve because "instruction is not tailored to their varying needs." Like me, Vigdor wants to reorganize math classes via homogeneous sections because the sectioning works for most kids: weak, average, high.  

Homogeneous grouping across grade levels or by courses is not a new idea; it is not necessarily innovative, but it meets the needs of the vast majority of students substantially better than the mixed-group classes often found in our elementary and middle schools today. The practice of homogeneous grouping across grade levels was banished because it didn't fit the progressive concept of equality. On the other hand, Jacob Vigdor argues that mixed math classes hold all kids back, and he is right. Like me, Vigdor advocates differentiation via homogeneous groups, not "many groups" within the same math classroom, which impedes all kids.  

[Insert. Elementary teachers do not hesitate to place kids into several groups by ability for reading or for math within their individual classrooms, yet many balk at splitting all the kids at a particular grade level into math sections based on student knowledge and skills (homogeneous groups). The idea of establishing homogeneous sections for math, for example, conflicts with a progressive ideology of equality and self-esteem.]

[Insert. Dr. Janine Bempechat (Getting Our Kids Back on Track) says our children grossly underachieve. She asserts, "We need to worry less about self-esteem and more about competence ... We need to expect much more from our children and challenge our children to confront difficulty [and work hard]." Dr. Bempechat points out, "We have become so consumed with worry over our children's self-esteem that we take pains to manufacture it." Moreover, Bempechat insists, "We need to stop protecting children from hard work and sacrifice in the name of happiness and self-esteem." 

Also, Bempechat says that we need to teach children "critical academic and life skills, which are the ability to persist in the face of challenges, to delay gratification, and to endure boredom." It is indeed unfortunate that "many in our society [including elite educators] look down on academic excellence."]

Technology has been cast as the new panacea because, according to ardent supporters, kids can learn at their own pace. Sounds great, but it has never worked. For example, Individually Prescribed Instruction (IPI) in the early 70s was a total flop! Often in education, grand ideas that failed in the past are repackaged and pushed onto schools as innovative and transformative. They are not. Furthermore, adaptive software, such as Success Maker, is no match for a high-quality teacher, no matter the grade level. The equality dogmatists would have you think that the solution to our math woes is kids sitting at a computer learning math at their rates, which would be the ultimate in differentiation. Sounds great! But, it has not worked in the past. Unfortunately, some good ideas were banished by the "equality" dogmatists. One good idea was homogeneous grouping (weak, average, advanced).

I think the sorting of students should begin in early elementary school. For example, in Singapore, weak math students are pulled out for math class at the beginning of 1st grade and placed with a high-quality math teacher to catch them up. We don't do this. Moreover, weak math students in Singapore are expected to learn the same core arithmetic that is taught in the regular class. 

Notes1. Jacob L. Vigdor (Education Next, Winter 2013) makes a comparable proposal about grouping students in math, and Doug Lemov (Teach Like a Champion) thinks teachers should put desks in rows, so students face the board to enable attention during explicit teaching. Both ideas conflict with current practice and conventional thinking. Kids don't learn math by group; kids learn math [by inference and counterexample] from teachers who know math and use explicit teaching methods. 

Dr. Eric Hunushek says, "Schools do have a big influence on achievement." But, as Hanushek observes, "[Low income] schools...aren't geared to making sure that these kids [mostly minorities] get really high-quality teaching. They get average teachers, which, on average, doesn't make up for a family background [vocabulary gaps, etc.]." I am retired, but, as a guest teacher. I go to a Title 1 elementary school and teach algebra to little kids (grades 3 to 5), all minorities, to show teachers that many kids can learn content that leads to Algebra 1 in middle school. The explicit teaching of content is often a mismatch to school district policies of desks-in-groups, group work, and collaboration. Kids are novices, not experts. In the real world, experts collaborate, often by email. Frequently, schools do not live in the real world; they live in a fantasy world. If we want kids to become future innovators, then they must first "become an expert" [in a discipline] writes Evangelia G. Chrysikou (Scientific American Mind, July/August 2012). "A solid knowledge base will allow you to connect remote ideas and see their relevance to a problem." She points out, "Working alone is usually the best way to come up with creative solutions."  

[Insert. In Teach Kids Algebra (TKA), I want kids to master content to form a solid knowledge base because the kids who know stuff (facts, procedures, axioms, apps, and ideas) in long-term memory will be successful. Also, TKA is independent of the district's gifted programs; however, it does help identify students who learn math faster, have more insights, and handle complex, in-depth material. To stretch and inspire able students, I formed 4th grade and 5th grade Honors groups that met once a week. It isn't enough time, but it is a start. Moreover, "In academics, so far only in mathematics do we have reliable ways to detect potential talent early on," writes Subotnik, Olszewski-Kubilius & Worrell (Scientific American Mind, November/December 2012).]

[Insert. The content I introduce in elementary school classrooms is mostly algebra (variables, equality, true/false statements, writing and solving equations, x-y table building, graphing, functions, etc.) and pre-algebra stuff (integers, fractions, formulas, etc.). I do not teach specific items on the state NCLB math tests, although, at times, there is some overlap. By blending algebra ideas with arithmetic, students are more likely to make the conceptual leap from the specific to the general.]

Notes2. In the past couple of decades, [NCTM] reform math disciples have substituted "cooperative group work" for explicit teaching, and school districts, under NCLB, have focused instruction and resources on average and below-average students, often leaving the academically gifted, advanced, and even above-average students unchallenged. Reform math has downplayed the auto recall of number facts and the practice of standard procedures for mastery in long-term memory--both stall student achievement. Even students who are below average are frequently not challenged in this system. It is the wrong approach. We know that a lot of practice solidifies essential factual and procedural knowledge in long-term memory for use in problem-solving [prior knowledge is needed for problem-solving and critical thinking in math and science]. Educators need to reevaluate and challenge their assumptions, but they often don't. For example, kids who are not required to memorize multiplication facts in 3rd grade and work with the standard algorithm for fluency cannot do long division, fractions, or algebra. They are stalled. Group work and collaboration are championed in our schools, not individual achievement and academic excellence. We are in a test mode and not in an achievement mode. We are off-target. Under common core, this will not change much. 

The US math curriculum (i.e., content taught in the classroom) and instruction (i.e., methods of teaching math) are not a good model. If our curriculum and methods of instruction were an exemplary model, then most of our kids would score substantially higher on NAEP government tests and be near or at the top internationally rather than in the middle (TIMSS). Incidentally, Singapore teachers do not put kids on computers to learn basic math. I think much of the technology and software used in US classrooms by students is a distraction, not a viable solution.

Also, nearly 50% of the 8th graders in several Asian nations, including Singapore, score at the "advanced" level on international tests compared to only 7% of US 8th graders (TIMSS). This indicates that classroom teachers and cram school teachers in Asian nations not only teach core but way above the core for able students.  

Thinking Out Loud. Even though there has been some improvement, especially among less-skilled minority students, rapid growth, such as that found in many other nations, escapes us. We remain stuck in the mud [of mediocrity]. There are exceptions. For example, Massachusetts came in 6th with a score of 561 in 8th-grade math. South Korea was 1st at 613 on TIMSS. And, while the curriculum (e.g., algebra) has become more accessible to average students to promote equality, many students are not prepared academically to handle it because they are products of a weak elementary and middle school curriculum and inadequate instruction. Students should not take algebra if they are not prepared, yet many schools push kids into algebra, ready or not. It's an epic mistake. 

Jacob Vigdor writes, "America's lagging mathematics performance reflects a basic failure to understand the benefits of adapting the curriculum to meet the varying instructional needs of students." And, the adaptation Vigdor strongly suggests is differentiating via homogeneous math sections, starting early in elementary school, not the "many levels" in the same math class that we have endured for decades. Differentiating via homogeneous math sections is old school, and it works for almost all kids--weak, average, advanced.   

FYI. Regardless of the rhetoric from common core defenders, the hidden intention of the common core is to homogenize math content (equalize downward). In short, common core math is not designed to catch our kids up to international math benchmarks, which is a reason I classify it in the framework of equality/self-esteem [progressive] ideology. In my view, the common core is the latest manifestation of a "once size fits all" progressive dogma. Furthermore, our math textbooks and instructional methods often unduly focus on understanding, which is difficult to measure, at the expense of competency, which is easy to measure. (Peter Hanley, redefinED 12-31-12, writes, "Common Core standards seek to prepare students to achieve 1200 on the SAT.... [Nevertheless], the average score for America's teachers has been about 1000.") Schools of education lack academic standards. I think there are a lot of good teachers out there; we don’t have enough of them. Schools of education have not been graduating high-quality elementary and middle school math teachers.  

[Insert. Math, such as memorizing times tables, isn't much fun until you get good at it, which requires effort, study, and practice. Once you get good at something, you like it better. Parts of math can be hard and frustrating. It is a giant step to go from the specific (using numbers) to the general (using variables). 

In the US, math is often taught badly. We know that learning fractions and long division well in early elementary schools prepare students for algebra in middle school. Researcher Robert Siegler, Carnegie Mellon University, writes, "Early knowledge of fractions and long division predicts long-term math success." ]  

This document is an abridged version of the original (12-25-12) with additions. It is frequently updated, revised, and tweaked almost daily. 1-27-13

The document is not an essay. Please overlook disjoint parts, awkward sentence structure, incorrect grammar, spelling, and many inserts.

Comments may be addressed to ThinkAlgebra@cox.net.
Model Credit (top of page): Remi, 5th grade
Note. In this document, the term "progressive" is used as defined by Alex B. Berezow and Hank Campbell in their book Science Left Behind. 

Return to my main website. Click ThinkAlgebra
©2013 LT/ThinkAlgebra/MathNotes

K-12 Science Framework

This is in draft form and redundant. I wrote it in the summer (2011). Parts of this draft are found in Science in Elementary School and on my science page. 

The new K-12 Science [conceptual] framework from the National Research Council (July 2011) is a disappointment. The framework committee merged science with engineering, skimped over math needed to do science, stressed science practices over content, and required little content knowledge in elementary and middle school, which is the same problem we have had for decades. For instance, the word “atom” is not used until the 6-8 grade band. The idea that atoms are composed of electrons, protons, and neutrons (atomic structure) is reserved for grades 9-12. The sequence is off-target. But, in my view, the biggest blunder, in addition to a lack of math and mixing science and engineering, is combining chemistry and physics (total 24 pages). Life science content takes up 20 pages. Chemistry and physics are woefully underrepresented.

If we want students to understand the world, then we should teach them substantially more physics and mathematics early on. Furthermore, we should establish math and science standards that, at the least, match the benchmarks from the nations that excel in these academic disciplines. The new K-12 Science framework does not do this. The Science Framework is the latest version (vision) of science education, but it is off-target because it requires very little knowledge of math needed to do science and very little science content knowledge. The committee’s frame of mind in composing the framework is troublesome.

The Committee on a Conceptual Framework for New Science Education Standards was charged with developing a framework that articulates a broad set of expectations for students in science. The overarching goal of our framework for K-12 science education is to ensure that by the end of 12th grade, all students have some appreciation of the beauty and wonder of science; possess sufficient knowledge of science and engineering to engage in public discussions on related issues; are careful consumers of scientific and technological information related to their everyday lives; are able to continue to learn about science outside school; and have the skills to enter careers of their choice, including (but not limited to) careers in science, engineering, and technology. (p. 14)

It sounds great! The caveat is that the statement is packed with unclear ideas that cannot be measured. Indeed, unclear generalizations are commonplace in education stuff and, in this case, the opposite of what science is. What is sufficient knowledge or appreciation? Sure, I bet the average citizen is going to read and study science and scientific studies after they graduate from school. To imply this is nonsense. The Framework’s expectations are speculation. How can you have expectations that are nonspecific and not measurable? I guess the mostly “non-scientific” committee thought they could. For example, the word “atom” is not used until the 6-8 grade band, and the atomic structure (protons, electrons, and neutrons) is not introduced until high school. (Surely, you’re joking!) 

Table by LT, ThinkAlgebra. It is based on Chemistry/Physics.

We infer that quarks exist even though we have never seen an individual quark. And, we infer that electrons exist even though we have never seen an electron. These inferences are based on solid measurements. 

Larry Cuban writes that the Framework is a “science for living.”  A blogger’s reply rephrases Cuban by saying that the Framework represents “issues-orientated, inquiry-based science.” Ze’ve Wurman concludes, “The document simply teaches students science appreciation, rather than science.”  

I call it the document to nowhere. The science framework lacks sufficient chemistry and physics content and depth and states unclear goals; e.g., "all students have some appreciation of the beauty and wonder of science."

A few years ago I wrote that mathematics must be brought back into elementary and middle school science. Today’s science programs or textbooks seem to skimp on the math needed to do the science. In the Sputnik era, the United States produced superior, coherent science programs. For example, Science--A Process Approach (1967) stressed the process in the context of the content, along with the mathematics used to do science. For instance, in Part B (First Grade) four of the six science processes were math or math-related [Using Numbers (arithmetic), Measuring, Communicating (graphing), and Using Space/Time Relationships]. In short, the math needed to do science was a major part of the SAPA science program, starting in the 1st grade. Moreover, the math taught in the program was very specific and ahead of grade level. 

College professor James S. Walker (Physics) writes, “The goal of physics is to gain a deeper understanding of the world in which we live.”  Indeed, the goal of science is to gain a deeper understanding of the world. Richard Feynman says that students should study physics because it plays a basic role in all phenomena. But, the Framework does not specifically state this view as its main premise. In fact, the Framework stresses “practices” of scientists; however, learning the processes of scientists does not imply that the student is learning content. Critical thinking requires considerable content knowledge. Learning what a scientist does is not the same as learning content. Kids are novices, not experts. They need to learn content, lots of it. In math class, I do not expect students to learn what mathematician do. I expect them to learn how mathematics works--how one idea links to or builds on another idea. I want students to learn essential content and skills so they can work math problems from different disciplines, including physics.   

Richard Feynman writes, “Physics is the most fundamental and all-inclusive of the sciences, and has had a profound effect on all scientific development.” To Feynman, the scientific method is “observation, reason, and experiment.” This is what we should teach kids. Feynman refers to rules of the game, which scientists guess and check by experiment. Feynman stresses, “The sole test of the validity of any idea is an experiment.” Untestable ideas do not make sense in science. Ian Stewart, a mathematician, writes, “Mathematics has played a central role in the physical sciences for hundreds of years.” The framework does not emphasize the intrinsic link between science and mathematics.  

What about biology? There are plenty of numerical patterns in biology (e.g., Fibonacci numbers, golden number, etc.), but Ian Stewart (The Mathematics of Life), points out that “Mathematics is being used not just to help biologists manage their data [e.g., enormous DNA genome databases] or improve their instruments, but on a deeper level: to provide significant insights into the science itself, to help explain how life work. Over the past ten years, there has been a massive growth in biomathematics--mathematical biology.”

Physics is the fundamental science, but it is mistreated in the Framework. First, it is lumped together with chemistry. If life science is treated as a separate topic, then chemistry and physics should be separated and expanded. Life science content takes up about 20 pages, while physics and chemistry combined take up 24 pages. The choices made by the Framework’s committee show a bias--life science content is more important than chemistry or physics. The committee writers say that chemistry and physics have too much in common to be treated separately, but I can make the case that chemistry and life science have much in common, too.   

The framework committee tries to justify lumping science with engineering rather than with mathematics. This is the new vision: the committee skimps over math. 

The laws of thermodynamics are nowhere to be found. The word “atom” is not used until 6-8. The idea that atoms are composed of electrons, protons, and neutrons (atomic structure) is reserved for grades 9-12. The sequence is off-target. 

The committee justifies its decisions on content by saying that the document is broad-based and for all students. It uses very general statements. It is merely a structure to composed standards. In my view, the framework falls woefully short because it leaves out important ideas in both chemistry and physics--from thermodynamics to relativity. 

The new science framework from the National Research Council (Framework for K-12 Science Education) has never been tested. Ironically, experimental testability is a fundamental principle in science. Yet, educators are asked to accept the framework “on authority,” something Galileo Galilei argued against. The framework is a guide for states and schools to write new science standards. It is not Common Core, which is [or will be] working on its own set of science standards, presumably using this framework. The new science framework, oddly enough, was written by the Division of Behavioral and Social Sciences and Education and its committee, which is made up of mostly of educators, not real scientists. How good is the framework? Don’t ask. (I think I hear the late Richard Feynman grumbling, “If it disagrees with experiment, then it is wrong.”) Accepting something on authority takes us back to the days before Galileo. 

The writers insist that the framework is for all students, broad-based, and not a grade-by-grade or course description. Its function is to develop a new set of science standards. And, it lumps science with engineering right for the start.

The science framework lacks sufficient chemistry and physics content and depth and states unclear goals; e.g., "all students have some appreciation of the beauty and wonder of science."

The framework writers “anticipate” that all students will be able to “to engage in public discussions on science-related issues, to be critical consumers of scientific information related to their everyday lives, and to continue to learn about science throughout their lives.” Lastly, the writers write, “We hope that a science education based on the Framework will motivate and inspire a greater number of people [to go into the science fields]” and its allied subjects, such as psychology, computer science, and economics.” Let me point out that these well-intended expectations are assumptions and assumptions are just that. There is absolutely no evidence that the Framework's new vision will produce better science standards or better science students. But, this is the committee’s “hope,” which means that there is no supportive evidence. Moreover, the Framework committee says that students should continue to take honors and AP courses in the sciences. But, this may not be possible because students have not learned enough content. 

The college-educated citizen, much less the average citizen, does not have the expertise needed to understand many of the scientific issues or studies that arise. This will not change. Often, I have trouble comprehending some of the articles in Scientific American or parts of M-theory (The Grand Design, Hawking, Mlodinow). 

What is troublesome is that the design teams (content experts in the sciences) were excluded from the committee’s final decisions. The document states (p. 17), “No members of the design teams participated in the discussions during which the committee reached consensus on the content of the final draft.” This alone makes the framework suspect. 

Ze’ve Wurman writes, “I noticed something odd. The Framework does not expect students to use any kind of analytical mathematics while studying science.” In short, kids do not use mathematics to solve science problems. No algebra, no trig, no calculus. Wurman notes, “There is nothing about actually being able to model a system by equations, or solve it using mathematical techniques.”  Wurman searched for words like algebra in the 280 pages of “lofty prose.” Nothing, well almost. 

Wurman did find a reference to one equation, which starts as a word equation (distance traveled = velocity multiplied by time elapsed) and is then symbolized as s = vt. I am not sure students understand what velocity means in science, because students are seldom taught vectors and do not learn how to “resolve” the components of a vector to solve physics problems. 

The Framework does not require students to use mathematics to model systems. Wurman points out, “Only statistics and computer applications (e.g., simulations, spreadsheets) seem to have a place in this strange document.” Wurman concludes, "The document simply teaches students science appreciation, rather than science.” 

End
Draft 1
Needs revision.
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Science in Elementary School

Kids are left behind in science.

What has happened to science in elementary school? 
With an emphasis on math and reading, science has been pushed to the side in many elementary school classrooms [1]. There is no time and little equipment. Often, K-8 teachers do not know enough science to teach it well. Furthermore, the math needed to do science is seldom taught or introduced. Science is highly mathematical, yet K-8 science textbooks and teachers seem to limit the math. None of this is new. Science has been neglected for decades. 

The thinking required in science, however, is different from the thinking done in school math. Mathematical statements are shown to be true or false by following a set of rules called number properties (axioms) of operations, equality, etc. [2]. One true statement forms the basis for another true statement and so on. This is the way math knowledge builds: one idea builds on another. In short, the rules (properties of numbers) in math do not change. The definitions in our number system do not change either. A fraction (rational number) will always be represented by the quotient of two integers (a/b, b ≠ 0). Equivalents like 3/4 = .75 = 75% will not change; 7 is always 6 + 1, etc. Equivalency and substitution are important ideas in mathematics. 

In science, however, there is no "true or false" like in school math. Instead, there are observations and inferences. There are no absolutes in science. The rules in science can change based on "partitioning by scale." In short, Newton's laws of motion still work, but not at the atomic (very tiny) scale.

There are facts, such as the number of protons in the hydrogen atom, etc. Kids must know facts (background knowledge). In addition, they also must know how to measure (make observations), how to draw valid inferences (conclusions) based on observations (data), how to minimize confirmation bias and errors in experiments, how to do the required math, how to communicate results with charts and graphs, and how to distinguish between correlation and cause-effect. Above all, students must learn to distinguish between observation and inference. But, this is not the organizing principle of science textbooks. Too often, TV programs, documentaries, news programs, textbooks, and other materials blend the two. Often, students interpret an inference as fact. 

Students should do science projects that have clearly defined independent and dependent variables and control.  Moreover, students should be taught what the late Richard Feynman calls intellectual honesty in science, something that is often lacking. In real science, we bend over backward to prove our conjectures wrong through experiments. We also present data that does not support our conjectures. We do not fudge data. And, we do not extrapolate beyond known data, i.e., make an inference based on an inference. An inference based on another inference is a misguided conclusion. Such extrapolation of data is more common than most people think. A common example would be a computer model of a complex system [stock market, weather, etc.] that attempts to forecast the future [based on the past] and makes unproven claims. This is not science; it is speculation


Feynman states, in a lecture, that physicists guess theory, then they test it. "If [we guess a theory that] disagrees with experiment, then it is wrong." Science is not based on, authority, opinion, consensus, or political agenda. It is based on an experiment. Furthermore, experiments must be repeatable and peer-reviewed. Lisa Randall, a particle physicist, writes, "People too often confuse evolving scientific knowledge with no knowledge at all and mistake a situation in which we are discovering new physical laws with a total absence of reliable rules." 


Dr. Randall clarifies, "Science evolves as old ideas get incorporated into more fundamental theories. The old ideas still apply. The wisdom and methods we acquired in the past survive. Today's methodology began in the seventeenth century." Kids should study Newton's laws of motion because they still apply. Randall says that we can still measure pressure, temperature, and volume because they are real quantities. In short, fundamental scientific knowledge is important and should be stressed in school. 


Regrettably, many of the elementary school science textbooks I have seen are incomplete and often perpetuate misconceptions. They are almost math-less, which misleads students. The real world is explained (modeled) through equations. The textbooks do not teach what science really is. One fundamental idea is that scientists try to prove ideas wrong, not right, by carefully crafted experiments; i.e., science does not prove anything right. Scientists seek out counterexamples and correct itself by getting rid of false ideas. According to Karl Popper, every theory must be falsifiable.

New Science Framework

The new science Framework from the National Research Council, oddly enough, was written by the Division of Behavioral and Social Sciences and Education and its committee, which is made up of mostly of educators, not real scientists. In fact, the "science content" experts (i.e., the design team) were not allowed in meetings in which the final decisions (consensus) regarding content were made (p. 17). Surely, You're Joking. No! Also, read the K-12 Science Framework.

I was disappointed after reading parts of the new K-12 science framework from the National Research Council (July 19, 2011).  The Framework committee merges science with engineering and technology, skimps over math needed to do science, stresses scientific processes (called "practices" in the document) over content, requires little content knowledge in elementary and middle school, and combines chemistry and physics, leaving important content out. 

In my view, the new Science Framework from the National Research Council is flat. It does not challenge children, and it does not paint a true picture of what science is all about. The fundamental idea, that science does not prove anything right, is missing. The Framework also lacks a historical perspective. Missing are the great scientists and how they changed the focus of science, e.g., Galileo, Dalton, Maxwell, Bohr, Heisenberg, Plank, Dirac, Einstein, Feynman, Higgs, etc. Moreover, the Framework lumps technology and engineering together; however, they are applications or products of science, not science. The framework skimps on chemistry and physics and the math needed to do the science. Is this the best we can do? It is disappointing! 

Note. I wrote an analysis of the Framework last summer (July 2011). It is very long. Here is a snippet: The Framework writers insist that a hypothesis (or theory) is not a guess. It is. This is what scientists do--they guess or make conjectures, then they test to see if the guess can be shown false. If an idea (guess) is not testable, then it is not science. We need to teach the testability principle by experiment to kids learning science. David Deutsch (The Beginning of Infinity, 2011) writes that conjecture (making a guess) is the real source of all our theories. Theories must be testable. He writes, “Knowledge must be first conjectured and then tested.” In science, we do not rely on authority or opinion. We have “a tradition of criticism,” says Deutsch. The bottom line, according to physicist Richard Feynman is, “If it does not agree with experiment, then it is wrong.” In other words, real science self-corrects itself over time. Ideology does not. Science does not prove ideas right; it eliminates wrong ideas. 

Many old ideas (e.g., Newton's laws of motion) are correct but incomplete. The laws work well at one scale, but not at another scale (e.g., atomic). We should not toss out Newton because his "laws" are incomplete. Lisa Randall, a particle physicist, says that many of the old ideas apply and have practical applications at the right scale ("appropriate conditions"). This [the scales] is what we should teach kids. Scales are an organizing principle in science. 

Elementary and middle school kids should learn Newton's laws of motion and the historical contributions of scientists like Galileo. The radical methods pioneered by Galileo in the 17th century are still used today: proof by experimentation (not authority, opinion, or consensus), thought experiments, and the use of technology to extend our senses to make better observations. For Galileo, the technology was the telescope. Technology plays an important role in science, but it is not science. by LT, ThinkAlgebra, July 2011 

If we want students to understand the world, then we should teach them substantially more physics and mathematics early on. Furthermore, we should establish math and science standards that, at the least, match the benchmarks from nations that excel in these academic disciplines. In my view, the new K-12 Science Framework does not do this. The Science Framework is the latest version of science education written by a committee made up of mostly nonscientists. It is off-target because it requires very little knowledge of math needed to do science and very little science content knowledge. The committee's makeup and its frame of mind in composing the framework are troublesome. It is not the best we can do.  It is not even close. And, as ZE"ve Wurman, a critic of Common Core math standards, explains, the conceptual science Framework is "science appreciation" all over again.

Also, read  Most Kids Don't Understand Science by ThinkAlgebra

Endnotes
[1] A report supporting my observations was released at the end of October (Strengthening Science Education in California). The report states the obvious: little science is taught in elementary school. But, neglecting science in grade school is not new. In my experience, not much science has been taught in elementary school for decades. Middle school science has gone downhill, too. There is not enough stress on basic science content, reading science, and learning the math needed to do science. Furthermore, many teachers are ill-prepared to teach science. 10-28-11


[2] There are not that many properties. A few of the basic properties [axioms] of numbers that should be learned in first grade in the first month or two of school are: add zero [identity] property, add one property, commutative property of addition (2 + 3 = 3 + 2), equality [or equivalency] property (2 + 3 = 1 + 4), "add in any order" property (3 + 4 + 7 is 10 + 4 or 14), etc. The idea that teaching arithmetic to 1st graders should use a framework based on number properties, rather than counting, is absent in American programs. Morris Kline writes, "Axioms are suggested by experience and observation. Kline also writes, "Operations on numbers [must] give a result that fits our experience." He states that "axioms are useful when our experience fails us or leaves us in doubt." Indeed, axioms (number properties) come in handy as kids learn arithmetic. For example, 3 + 5 = 10 - 2 is a true statement because of the transitive property of equality. In "little kids" talk, both 3 + 5 and 10 - 2 name the same point on the number line and, therefore, are equal to each other (equivalent). 


Mathematician Morris Kline (Mathematics for the Nonmathematician) states that operations (let's say, fractions) are designed to "fit experience." Arithmetic facts and operations are learned mostly by rote, but students should also be aware of the axioms or properties (e.g., commutative property of addition and multiplication) that govern operations to determine whether or not the mathematics is correct. For example, I can explain why 1/2 of 1/3 is 1/6 on the number line, but this type of understanding does not come into play when students are multiplying fractions (e.g., 2/3 x 3/4). In short, when applying the multiplication of fractions algorithm, students do not think in terms of marking off 3/4 of one whole on a number line, then dividing each fourth into thirds, which gives 12ths (but from 0 to 3/4, there are nine equal parts or ninths. Converting 2/3 to 9ths = 6/9). Counting over 6 tick marks, you end up at 6/12 or 1/2, etc. Sounds confusing? It is to many kids. 


Furthermore, making a number line model for fractions with larger numerators or denominators becomes a total mess. The multiplication of fractions algorithm can be inferred from a number line demonstration. Students should be taught to depend on efficient methods (algorithms, step-by-step procedures, operations on numbers, or recipes) that produce correct answers fast. 


The multiplication of fractions algorithm can be "formulated" by the number line idea and other clues by the 3rd or 4th grade. For example, ½ of a number (e.g., 1/2 of 10) produces a smaller number, not a larger number (½ of 10 is 5; it means ½ x 10 = 5). We know this by experience and develop a multiplication of fractions algorithm so that the answer is always correct. (See Example 2 below)


In division, 5 oranges divided into halves is 10 (halves). In arithmetic, this is 5 ÷ ½ = 10. To divide by ½ gives the same result as multiplying by 2/1 (the reciprocal of the divisor. This is invert and multiply). The algorithm for the division of fractions is formulated to fit experience. In short, 5 ÷ 1/2 = 5 x 2/1. Thus, to divide by any number, multiply the number by the reciprocal of the divisor and then apply the multiplication of fractions algorithm. In short, students change division to multiplication. 

In Example (1), adding the fractions should produce a larger fraction. Thus, the idea of adding the numerators and adding the denominators does not work because it does not fit our experience. Adding fractions can be represented by adding lengths on the number line. The answer is greater than one, not less than 1. The algorithm does not work. 

In Example (2), multiplying the numerators and multiplying the denominators works 

(fits our experience). It is the algorithm that kids are taught to use when multiplying fractions. A fractional part of any number (fractional part must be less than 1) produces a smaller number. (But, 3/2 x 7/5 will produce a larger number because 3/2 is 1 + 1/2 and 7/5 is 1 + 2/5. This is consistent with our experience when both factors are greater than 1.)

Algorithms (operations on numbers) must fit experience, be efficient, and produce the correct answer. 
10-24-11, 10-28-11, 11-1-11, 12-223-11


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