This is a work in progress. It will undergo frequent changes and updates.
Is your child in a math sequence that prepares her for college mathematics or for remedial math? Will she be ready for formal algebra in middle school?
Approximately 80% of high school students want some form of post-secondary education, either an associate’s degree at a community college or a bachelor’s degree at a four-year college or university; however, according to Achieve, "Only a subset end up taking a curriculum that prepares them for college."
Reality Check: Most students are not fluent in arithmetic needed for algebra. Algebra courses are watered down. State standards lack coherence, rigor, and focus. In NCLB, “mediocre” performance is labeled “proficient.” Students lag behind their peers in high-achieving nations, starting in elementary school.
ThinkAlgebra outlines high school math pathways (common core algebra courses and assessments that make teachers, students, and schools accountable), reiterates the contention that K-8 math should be fixed first, advocates that schools use good programs that are already available (e.g., Singapore Math, ADP Algebra), believes that four-year college-bound students should take at least one college-level math course in high school, stipulates that not all students need to take algebra, and champions the retraining of teachers to teach world-class math.
I COLLEGE PREP MATH SEQUENCES
High School Math Pathways by ThinkAlgebra is a response [alternative] to the college-career readiness draft from the Common Core State Standards Initiative (CCSSI). Click the chart for a larger view.
The chart, while incomplete and in first draft form, represents the academic realities of college preparation. It shows real algebra courses supported by levels of achievement as defined by the American Diploma Project (ADP) Assessment Consortium. (Note. I named Calculus Prep and [College] Algebra Prep pathways after the two University of Arizona math placement tests.) While the ADP algebra exams are not perfect, they can be refined and polished over years of use.
College-career readiness standards should focus on the "80% of high school students" who want to go to college. Furthermore, they should have specific common assessments to determine mastery along the pathway to guide students into informed choices. But, this is not how CCSSI approaches the problem.
College-career readiness standards should focus on the "80% of high school students" who want to go to college. Furthermore, they should have specific common assessments to determine mastery along the pathway to guide students into informed choices. But, this is not how CCSSI approaches the problem.
Pathway Assessments in Algebra
We can use programs already in play, such as Algebra 1 standards, Algebra 2 standards, and their assessments (see exam links below) from the American Diploma Project (ADP) by Achieve. They are used by several states and illustrate that students coming into algebra are not prepared.
The ADP topics match well with most of the algebra topics advocated by the National Mathematics Advisory Panel.
From the ADP Assessment Consortium (Achieve)
Algebra II Exam
Students need authentic algebra courses, not diluted courses. The challenging, ADP algebra courses and exams would be the same for all students who take Algebra I and II. Students must work hard and study more to achieve the proper levels of mastery. The scores on ADP Algebra I exam translate to Below Basic, Basic, Proficient, and Advanced, while the Algebra II exam scores translate to Needs Preparation, Prepared, and Well Prepared [for College]. STEM and non-STEM students who need calculus are expected to do much better on the ADP algebra exams: Algebra I ( Advanced) and Algebra II (Well Prepared).
Students need authentic algebra courses, not diluted courses. The challenging, ADP algebra courses and exams would be the same for all students who take Algebra I and II. Students must work hard and study more to achieve the proper levels of mastery. The scores on ADP Algebra I exam translate to Below Basic, Basic, Proficient, and Advanced, while the Algebra II exam scores translate to Needs Preparation, Prepared, and Well Prepared [for College]. STEM and non-STEM students who need calculus are expected to do much better on the ADP algebra exams: Algebra I ( Advanced) and Algebra II (Well Prepared).
Students who plan to go to a four-year college or university to earn a bachelor's degree should take math beyond Algebra 1 and 2, such as precalculus, which includes trig, and college-level courses, e.g., College Algebra, AP Calculus, AP Statistics. AP exams are benchmarked to college students. A score of 5 is about an A in college; 4, a B in college; and 3, a C in college. (Note. Unfortunately, the College Board does not have AP exams for College Algebra or Precalculus, only CLEP exams. Many colleges won't give students college credit for AP courses unless they score 4s or 5s, especially in math and science AP courses.)
According to a report from the National Governors Association (Raising Rigor, Getting Results, David Wakelyn, August 2009), "Students who score well on the exam [AP] are more likely to persist in college and earn a degree." The report points out, "AP is not just for the elite; it's for the prepared." Dr. Wakelyn explains, "Whether a student earns a college degree depends foremost on the intensity of the high school curriculum, especially if that student takes at least one Advanced Placement course. Merely enrolling in an AP course is not enough; high school students must score well on the exam to do well in college." All students who seek a bachelor's degree should take at least one AP exam in high school.
According to a report from the National Governors Association (Raising Rigor, Getting Results, David Wakelyn, August 2009), "Students who score well on the exam [AP] are more likely to persist in college and earn a degree." The report points out, "AP is not just for the elite; it's for the prepared." Dr. Wakelyn explains, "Whether a student earns a college degree depends foremost on the intensity of the high school curriculum, especially if that student takes at least one Advanced Placement course. Merely enrolling in an AP course is not enough; high school students must score well on the exam to do well in college." All students who seek a bachelor's degree should take at least one AP exam in high school.
Calculus Prep students should take a rigorous precalculus course, which is a combination of College Algebra and Trig, to prepare for AP Calculus AB. Algebra Prep students should consider taking a rigorous precalculus course, too. I envision high schools partnering with community colleges or local colleges and universities to provide Algebra Prep students with a legitimate College Algebra course. In this way, many students who successfully complete College Algebra can fulfill "college mathematics requirements" in high school. According to ACT, "English Composition, College Algebra, and Biology courses are the first credit-bearing courses most commonly taken by first-year college students." Also, there are Pre-AP strategies for middle school students, Algebra I, and Algebra II classes. (See Pathway Assessments Beyond High School Algebra below.)
Technology Transforming Education: Educating Our Best Math/Science Students
According to Paul Peterson (Education Next), "The United States needs to begin growing its own creative talent by educating the best of our young people in science, math, and cognitive science skills from an early age. Nothing is more tragic than the virtual abdication by the American high school of its responsibility for the mathematical and scientific education of the next generation, leaving U. S. 15-year-olds below the industrial world average on math and science tests." Because high schools can't seem to attract good math and science teachers, Peterson thinks the solution for our best students will be top caliber, online math and science courses taught by expert content teachers. Why stop with our most advanced students? High-quality, online instruction can be produced for ordinary students who need rigorous math and science courses to prepare for college-level work.
Technology, once in place, can transform high schools into hybrid schools: some classes will be online (even at home) and others in the classroom. If technology is used to deliver high-quality instruction, then fewer classroom teachers will be needed and costs will be cut significantly. Of course, this transformation won't please teacher unions, but it is coming. There are K-12 virtual schools already in place in almost every state. For example, the K-12 Arizona Virtual Academy has four levels of core high school subjects (Math, English, Science, and History): Core, Comprehensive, Honors, and AP. It is a public charter school, so there is no tuition. All instruction is online, at home. Also, there are online courses from the Education Program for Gifted Youth (EPGY, Stanford University); however, EPGY courses are expensive and students must meet qualifications for admission. The EPGY math courses start with Accelerated K-2 Mathematics. A major shift in instructional delivery--from the classroom to an online computer at home--has started.
Technology Transforming Education: Educating Our Best Math/Science Students
According to Paul Peterson (Education Next), "The United States needs to begin growing its own creative talent by educating the best of our young people in science, math, and cognitive science skills from an early age. Nothing is more tragic than the virtual abdication by the American high school of its responsibility for the mathematical and scientific education of the next generation, leaving U. S. 15-year-olds below the industrial world average on math and science tests." Because high schools can't seem to attract good math and science teachers, Peterson thinks the solution for our best students will be top caliber, online math and science courses taught by expert content teachers. Why stop with our most advanced students? High-quality, online instruction can be produced for ordinary students who need rigorous math and science courses to prepare for college-level work.
Technology, once in place, can transform high schools into hybrid schools: some classes will be online (even at home) and others in the classroom. If technology is used to deliver high-quality instruction, then fewer classroom teachers will be needed and costs will be cut significantly. Of course, this transformation won't please teacher unions, but it is coming. There are K-12 virtual schools already in place in almost every state. For example, the K-12 Arizona Virtual Academy has four levels of core high school subjects (Math, English, Science, and History): Core, Comprehensive, Honors, and AP. It is a public charter school, so there is no tuition. All instruction is online, at home. Also, there are online courses from the Education Program for Gifted Youth (EPGY, Stanford University); however, EPGY courses are expensive and students must meet qualifications for admission. The EPGY math courses start with Accelerated K-2 Mathematics. A major shift in instructional delivery--from the classroom to an online computer at home--has started.
Note on Geometry
There is no logical reason to separate Algebra I and Algebra II with a year of geometry. Geometry can be taken concurrently with Algebra II (The student would take two math courses at the same time), or in summer school, or through a K12 virtual school available in most states. (For example, the Arizona Virtual Academy is a full-time, tuition-free online public school for students in grades K-12.) Also, regular high schools should accommodate double courses by eliminating some of its non-academic electives. Note. The geometry needed for ADP algebra is taught in a thorough 7th-grade pre-algebra course. Thus, elementary and middle school (K-7) preparation in math is key for success in algebra.
Two major trends in colleges have been rapidly rising remediation and drop out rates.
K-12 math is off-target when it comes to college preparation. Students come to college and community college underprepared in math. At Pima Community College (Tucson), for example, there are 315 remedial math classes listed in the Spring 2010 Schedule of Classes. Remediation is a national trend and illustrates that our K-12 system of math education is deeply flawed for many students.
Moreover, high school guidance counselors often encourage students to chase after bachelor’s degrees when career degrees are a much better fit for many students, e.g., an associate’s degree from a community college. Students need better academic and career guidance, starting in middle school.
The idea is, students should pursue a math sequence that squares with both their goals and capacity to handle mathematics. Effort or persistence is another factor. (Note. K-12 schools do no favor to students when they don't teach enough arithmetic, dilute algebra courses, and inflate grades, which is a rampant K-12 practice, even in some undergraduate courses in college.)
The idea is, students should pursue a math sequence that squares with both their goals and capacity to handle mathematics. Effort or persistence is another factor. (Note. K-12 schools do no favor to students when they don't teach enough arithmetic, dilute algebra courses, and inflate grades, which is a rampant K-12 practice, even in some undergraduate courses in college.)
Common Core State Standards Initiative (CCSSI or Common Core)
In contrast to ADP, the college and career readiness standards (September 2009 Draft) from CCSSI do not use single subject courses (e.g., Algebra I, Algebra II). Instead, the CCSSI outlines a maze of 10 math strands, 5 levels of understanding and/or proficiencies, and 6 core math practices. Furthermore, CCSSI makes a weak case for a focus on exponential functions, downplaying quadratic functions. Apparently, CCSSI didn’t consult with college physics professors or even high school teachers of algebra-based physics.
Unlike ADP, Common Core has no pathway assessments to indicate a mastery level, and its standards draft ignores STEM and non-STEM majors who need calculus.
According to Sandra Stotsky, “The effort [CCSSI], which is being pushed very quickly, seems determined to do an end-run around the country’s mathematical and scientific organizations and the panel’s [National Mathematics Advisory Panel] recommendation on the major topics for school algebra.”
Additionally, I think CCSSI started at the wrong end; they should have started with K-6 and worked up the grades toward high school. In my view, to fix high school math, we should fix K-6 math first, not Visa Versa. In an e-mail to ThinkAlgebra (12-1-09), Dr. Stotsky agrees, "Your point about the backward way in which math standards have been created is right on target." CCSSI should have focused first on K-6 standards that lead to pre-algebra in grade 7 and a rigorous Algebra I course in grade 8, and so on. They didn't.
If states adopt a common set of standards, then each state would write its own interpretation or version of the standards (i.e., a curriculum), assessments, and definitions of proficiency levels (cut scores). Isn't this what we have now--50 different sets of standards, assessments, and definitions of proficiency. Furthermore, CCSSI does not address teacher education. It is one thing to have better standards, but another to have well-trained teachers able to teach them well, especially in elementary school. Frankly, I doubt that common core standards will work well.
Additionally, I think CCSSI started at the wrong end; they should have started with K-6 and worked up the grades toward high school. In my view, to fix high school math, we should fix K-6 math first, not Visa Versa. In an e-mail to ThinkAlgebra (12-1-09), Dr. Stotsky agrees, "Your point about the backward way in which math standards have been created is right on target." CCSSI should have focused first on K-6 standards that lead to pre-algebra in grade 7 and a rigorous Algebra I course in grade 8, and so on. They didn't.
If states adopt a common set of standards, then each state would write its own interpretation or version of the standards (i.e., a curriculum), assessments, and definitions of proficiency levels (cut scores). Isn't this what we have now--50 different sets of standards, assessments, and definitions of proficiency. Furthermore, CCSSI does not address teacher education. It is one thing to have better standards, but another to have well-trained teachers able to teach them well, especially in elementary school. Frankly, I doubt that common core standards will work well.
Pathway Assessments Beyond High School Algebra
It seems logical, at least to me, that students who aim for college should take at least one college-level course like AP Calculus or College Algebra in high school for college credit.
Many high schools have AP Calculus courses, but, to get college credit for Calculus AB, most colleges accept only scores of 4 or 5. Also, high schools don’t have a separate college-credit course called College Algebra, which is aimed at students who won’t be taking calculus in high school or college. Schools, however, can partner with local community colleges, colleges, and universities to provide a legitimate College Algebra course for students in the Algebra Prep track.
Taking harder math courses in high school correlates well with getting a bachelor's degree. For example, many independent schools require students to take precalculus because 70% of students who take precalculus in high school go on to earn a bachelor's degree. Until College Algebra is offered in high schools for academically qualified, college-bound students who don't need calculus, then precalculus is the next best option. (FYI: Many colleges and universities accept CLEP and IB exams for college credit.)
ADP Algebra II Exam Is Designed As a College Math Readiness Exam
The ADP Algebra 2 Exam determines whether students are ready for college-level mathematics, typically, a course called College Algebra. According to ADP, “The Algebra II exam includes more advanced algebra than college admissions exams [ACT and SAT]...and are internationally competitive.” If students take the ADP Algebra II ADP exam, then they will know where they stand academically and can make informed decisions.
The University of Arizona has two math placements tests:
Here’s the catch. Teachers should teach rigorous algebra, but the level is usually beyond the student’s preparation. Scores will stay low until students in K-8 catch up. For example, the Algebra 1 and Algebra 2 ADP exam results from participating states are dismal. This illustrates that ordinary students who want to attend college are not properly prepared to take high school-level algebra.
The root of the problem, however, does not start in high school or with algebra. It starts in K-6 with inadequate instruction in arithmetic (e.g., fractions). K-6 are critical years that prepare students for pre-algebra in seventh grade and algebra in eighth grade. In my view, K-6 should be the main focus of common core, but it isn't, which is the reason I think common core's approach is misguided. To fix high school math, we need to fix K-6 math first, not visa versa. Most of the high school math solutions are makeshift until incoming students are properly prepared for success in algebra. See StandardsUpdate from ThinkAlgebra.
II NOT COLLEGE PREP
Not all students need a college-prep (algebra-based, or calculus-based, or community-college-based) curriculum. Some students plan to go directly into the workforce or work minimum wage jobs, while others plan to work in fields that don’t require a college degree, but do require specialized training and/or apprenticeships? Students who have good math skills have an advantage. Here is a suggested non-college mathcore, but it needs work.
Many states, like Arizona, are making Algebra II a requirement for high school graduation to align its standards with college. This is okay for students who plan to enroll in post-secondary studies because they need at least two years of solid algebra. But, what about students who have no such plans, or students who plan to go from high school to the workforce, or students who don't have the capacity to learn algebra at this level? In my view, mandatory Algebra II means diluting the course so that almost every student can pass. This is a misguided approach to upgrading math achievement.
The message should be that students who plan to go to college or community college must do well in a rigorous Algebra II course. Well-informed students who want to attend a four-year college have always taken a rigorous Algebra II course without it being mandated as a graduation requirement. What has changed over the years is that Algebra II is an essential course for all students who want to attend community college. The catch is, many "so-called" Algebra II courses are not rigorous enough and do not meet the specifications outlined by the National Mathematics Advisory Panel.
Real College is Hard
According to Charles Murray, Real Education, "We need to redefine educational success." There is nothing wrong with being below the median because half of the students are. And, there is nothing wrong with attending community college to get technical training for a career, which is a wise move for most high school students. Murray makes a good case that too many students are going to four-year colleges instead of two-year colleges. He writes, "So few [about 9 to 12%] can do well in real colleges because real college material is hard. This is obvious for engineering and most of the natural sciences, where students cannot get a degree unless they can handle the math. 'Handle the math' means being able to pass courses in at least advanced calculus and statistics, a requirement that immediately makes 10 percent estimate plausible."
Murray continues, "In the humanities and most of the social sciences, the difference between high school work and college-level work is fuzzier." Murray indicates that students with average reading ability won't understand much of what they read in college textbooks because of long sentences (average of 26 words) and advanced vocabulary, e.g., Western History, Art, Economics, Psychology, Philosophy, English Literature, etc. Students might muddle through and get a degree, but, as Murray points out, "They [students with average reading ability] take away a mishmash of half-understood information and outright misunderstanding that probably leave them under the illusion they know something they do not."
One negative trend is that many college-level courses are often downgraded academically so students pass. Grade inflation is commonplace in the humanities and social science, especially in introductory undergraduate courses. As one student told me, "I can write junk and still get an A in psychology."
World Class Math?
When the K-12 standards from CCSSI are released in December (release date is delayed to January), I will read them with eagerness, but, in the back of my mind, I don't believe K-8 math will be equivalent to what is taught and learned [mastered] at the different grade levels in Singapore and Hong Kong. In short, I doubt the standards will be world-class in the lower grades.
We should have a simple and straightforward plan. We don't need to reinvent the wheel, which, in my view, is what Common Core is doing. And, while the top-achieving math nations have national curricula and assessments, they also have highly respected, knowledgeable teachers who teach math in a logical sequence, without fads, and spend more time-on-task because of a longer school year. Additionally, the math programs are teacher-directed (strong teacher guidance method) and the goal is to master arithmetic and algebra. The prime objective of curriculum and instruction should be coherent content mastery and skill fluency like high-performing nations. Mastery takes practice, lots of it, along with good teaching that implements solid cognitive science. Sadly, content mastery and skill fluency have not been the goals of NCTM reform math programs that dominate math education in the United States.
Sandra Stotsky, a member of the National Mathematics Advisory Panel, observes, “High-math achievement countries teach arithmetic in the elementary grades in a coherent curriculum leading, step by step, to formal algebra and geometry in middle school.”
Will students be ready for real algebra in middle school?
The existing education system in almost all states is ill-equipped to boost math achievement to world-class levels, such as preparing most students for rigorous pre-algebra and Algebra I courses in middle school. The math problem starts in K-6 math. What should we do in K-6 math to prepare most students for a legitimate algebra course in middle school?
We need drastic changes in what we teach (content and skills must be world-class), beginning in K-6, how we teach, and how we select and prepare teachers, especially in the K-6 pivotal years.
More coming soon...
Draft I: Last Tweaked on December 15, 2009
To Be Revised
Credits: MariaB, ChloeM
Thanks, Kids!
Thanks, Kids!
©2009-2010 LT/ThinkAlgebra
