Tuesday, February 13, 2018

Old School

Poor math achievement starts in early elementary school, even in the best schools.  

The paper-pencil standard algorithms are the best tools for beginners to do basic arithmetic.  So, why are they not the primary focus in elementary school arithmetic? It seems that standard algorithms, which require the memorization and auto recall of single-digit number facts, have been displaced by "standards of mathematical practice" and "alternative algorithms" (i.e., reform math). In short, students are taught reform math, not standard arithmetic. 

We need to apply statistical methods with care, especially when negotiating big data, says Stephen M. Stigler. In education, we depend too much on averages. "Averaging is a radical idea: you can actually gain information by throwing information away." writes Stigler. In averaging, the identities of the test takers are tossed, which means that no test score holds more weight. 

But, a school's average (arithmetic mean) in math is influenced by individuals. Indeed, math ability and performance of students vary widely. In averaging, however, all the test scores are treated the same. Mary's score of 100 is treated the same as Billy's score of 50. Consequently, several very high scores can cover up the lower scores of many of the other students and skew or distort the data's average up to an "acceptable level." In short, poor math achievement is covered up by averaging.  

Some reasons for high graduation rates in many high schools include online credit recovery, grade inflation, and watered-down courses. But these reasons hurt students. Richard N. Haass summarizes the problem: "Students are leaving school without the math and science skills needed for jobs in modern industry." Furthermore, they lack the math and reading skills for college. Up to 88% of students enrolling at a community college are placed in remedial math (algebra). Clearly, the K-12 reform math curriculum does not prepare students for college-level math courses. 

An Algebra-2 course in one school may be different in content compared with an Algebra-2 course at another school, even if the same textbook was used.  The teachers are different, the grading is different, and the expectations are different, etc. Making an accurate comparison is difficult. An A in one school could be a C in another school. 

We should never leave judgment to a computer software program. 

©2018 LT/ThinkAlgebra

Friday, February 9, 2018

Fact or Opinion

Old School: Desks in a Row.
Desks in a row may be old fashioned, but it was highly effective.
Kids learned and listened to the teacher. 

Photo: 46 students at a Catholic school in 1950. Students sat down, got quiet, and paid attention to the teacher. No Common Core. No federal regulations. No unions. No bureaucracy. Nuns were the teachers and disciplinarians.

Opinion is Not Evidence
Knowledge Supports Thinking

Educators rely too much on opinion or anecdotal claims, which have led to incorrect assumptions, false beliefs, ineffective programs, and bad policies. Education is a road littered with failed policies, questionable programs, and theories that don't work. 

Paying attention in class and practicing the fundamentals for mastery promotes better learning (not group work, discovery learning, or tech). Learning is remembering from long-term memory. Knowledge in long-term memory supports thinking, not vice versa. (You can't solve an algebra problem easily unless you know some algebra.)
"The more math I know, the more I can learn, the faster I can learn it, and the better I can think, i.e., solve problems." Factual and procedural knowledge in long-term memory enables thinking. In the cognitive science of learning, "knowing and applying" supports problem-solving, which is higher-level thinking.

In OLD School, teachers were respected, which may no longer be true in some schools today.

Critical Thinking Should Not Replace Knowledge
Chester E. Finn, Jr. thinks that the "emphasis on thinking skills over facts contributes to students' inability to identify disinformation and misleading information." Clear thinking requires the separation of observations from inferences or facts from opinion. Finn says that critical thinking can go awry in at least two ways. One way is when it replaces knowledge. The second is when interpretation is everything, i.e., everyone's opinion is as valid as anyone else's opinion. If you have an opinion, you can make up the facts that support it. It is unfortunate that "thinking gets detached from knowledge."

Disrupting Education: Tossing Out the Old School Stuff (Wrong Approach)
Christian Madsbjerg (Sensemaking) observes, "Everything has become a disruption: a clean break from the past leaning far forward into the future. The [postmodern] culture has upended the way we educate our children." Indeed, reformers have disrupted education by tossing out the Old School stuff for a clean break, including memorization, drill to develop skill, desks in a row, cursive writing, and so on. In short, reformers advocate "out with the old," even if it worked well, and "in the new such as tech, "even if it has no basis in evidence. Reformers seem to ignore scientific evidence and the cognitive science of learning.

Group Work --> Inattentiveness
Walk into elementary school classrooms and notice that the students are seated at tables or in pods (groups) of 3 or 4 desks facing each other. How stupid! I have suggested to teachers not to set up group work pods with desks. It leads to disruptions and inattentiveness. Kids talk. Starting in K, they are conditioned not to pay attention to the teacher and explanations. It is a flawed practice. In contrast, desks in a row worked much better (Old School). The reasons often given for pod seating and group work are that students need to collaborate and do discovery activities. Really? Since when has student collaboration become the primary focus in school? Moreover, the so-called minimal guidance methods such as discovery or inquiry methods are ineffective in learning arithmetic.

Critical Thinking Without Knowledge Is Pointless and Shallow
In a postmodern world, educators have substituted critical thinking for knowledge. We are told that learning facts is not that important. Individual interpretation and opinion are much more critical. Really? First of all, critical thinking is not knowledge; it requires knowledge. Immanuel Kant (1724 - 1804) once wrote that "thought [critical thinking] without content knowledge [facts] is empty." And he was right! It is a basic premise of the cognitive science of learning.

Hard for Humans: Separating Observation From Inference
Guy P. Harrison (Think: Why You Should Question Everything) writes, "The brutal truth is that human brains do a poor job of separating truth from fiction. This leads to many false beliefs." Factual proof (i.e., scientific evidence) should come before opinion, belief, or assumptions, but too often opinion is valued more than the facts. Our brains have difficulty separating observation from inference. We should think like a scientist, but we do not. Also, computers are good at logic, but human brains are not.

Opinion Is Not Fact
The problem has been that opinion is often disguised as fact. One opinion seems as good as another. Opinion and gossip are considered news. Newspapers and the media are packed with opinions. Comments on YouTube, news articles, and social media are packed with opinions. Most of our beliefs are wrong. Belief and anecdotal "evidence," which are often found in education, are not evidence of anything. Still, many educators rely on anecdotal evidence, not scientific proof.

In the U. S., Sorting Has Been Anathema.
In top-performing nations, it is acceptable.
Putting high achievers with low achievers in the same math class has been a recipe for underachievement and mediocrity. Kids need strong teacher guidance, a world-class math curriculum, a grouping that matches their achievement level, lots of practice to master fundamentals, and persistence to get things done right. Homogeneous sectioning by achievement level is not equal coverage of math content, but it is coverage of the math fundamentals. And, it does not equalize downward.

Old School Fundamentals Are Missing From Eureka Reform Math
Fundamentals for 3rd grade, for example, should include the cumulative memorization and continual review of single-digit multiplication facts for instant recall, along with performing the standard multiplication algorithm correctly (1st Semester) and the long-division standard algorithm correctly (2nd Semester). These Old School fundamentals are absent from Eureka Math, which is a typical reform math program. Eureka Math does focus on single-digit areas, such as the area of a 5 by 7 rectangle by counting square units inside the rectangle. Still, students won't be able to compute the space of a 173 cm x 60 cm rectangle as 3rd graders did decades ago using the standard multiplication algorithm. Cumulative and continual fluency practice should be a primary goal of arithmetic in the 3rd grade, but, too often, it is not.

In Common Core, fluency does not always mean auto recall of number facts or learning the mechanics of standard algorithms. Instead of memorizing single-digit number facts, reform math advocates want students to calculate them. Instead of standard algorithms, reform math zealots want students to learn many different ways to add or multiply. The alternative methods (i.e., reform math) found in Eureka Math clutter the curriculum, confuse students, and waste instructional time.

Here is an example of "calculating" from 1st Grade Eureka Math. Instead of learning the standard algorithm and the place value system of adding ones to ones and tens to tens, students are directed to number bonds.  Who adds numbers using cumbersome number bonds? 

Say What?
There are Seven Modules in 3rd Grade Eureka Math, a reform math Common Core curriculum. The Teacher Edition of Module 1 is over 300 pages long. The Teacher Edition of Module 7 is over 500 pages long. It's math education run amok!

©2018 LT/ThinkAlgebra

Tuesday, February 6, 2018

Talented and Gifted

I love polynomials!

Finding Talent

Students who are better at math should be put together and accelerated, which rarely happens at the elementary school levels. Bright kids need to shine, but we give them grade-level learning rather than advanced learning, i.e., acceleration. Placing an outstanding 4th-grade math student in a 5th-grade textbook is not acceleration. The 5th-grade textbook is for average students, not advanced learning. Note: Not all kids who are good at math want to be accelerated. Likewise, some good readers don't like to read.

Many gifted programs are enrichment programs with different names such as TAG, GATE, etc. 
Students must show above-average ability (acumen) on a test such as the Otis Lennon, especially in the math and verbal areas. 

Students should also demonstrate task commitment (persistence), which is just as important as intellect. 

The third component of an enrichment model is creativity, but I discounted it because it is difficult to measure. According to Kevin Ashton (How To Fly A Horse), every student is creative, but not all students are equally creative, just as we are not all similarly gifted athletically, musically, etc.

Ashton and others point out that thinking is domain-specific and requires knowledge, lots of it. "Having ideas is not the same as being creative. Creation is execution, not inspiration." Being "clever" does not mean that a student is gifted. Persistence (task commitment) is needed for self-study and progress. The more math a student knows, the better thinker the student is in math. Enrichment keeps students where they are, but true acceleration moves them forward. 

Central to thinking is the mastery of core academic content. The thinking is domain-specific. Critical thinking in mathematics, which is called problem-solving, is different from thinking in science or literature, etc. E. D. Hirsch, Jr. explains, "Critical thinking does not exist as an independent skill." Still, the assumption persists in many gifted programs. 

Mathematics is not a matter of opinion or argument; it is a matter of fact and the deductive reasoning that links one idea to another, i.e., a hierarchical sequence. In mathematics, everything fits together logically, but this is not true for science, literature, history, and other academic domains. 

The best students cluster at the top of the grade-level tests. The problem is that there is no separation between the very good students and the truly gifted and talented students, which is the reason that Johns Hopkins Center for Talented Youth (CTY) tests kids above grade level in verbal and math using the SCAT (School and College Ability Test). If an elementary school student's score meets an established CTY benchmark, then the student qualifies for gifted and talented programs through CTY, including summer programs, online programs, internships and research opportunities, academic competitions, early college, etc. "The SCAT is used by CTY to identify gifted youth who demonstrate high academic potential. CTY offers various programs for gifted youth." 

But, many students in so-called gifted "enrichment programs" established within school districts probably won't qualify for CTY programs. Enrichment is not acceleration. 

"The School and College Ability Test (SCAT) is a standardized test conducted in the United States that measures math and verbal reasoning abilities in gifted children. CTY uses three levels of the SCAT:
1. Students in grades 2-3 take the Elementary SCAT designed for students in grades 4-5.
2. Students in grades 4-5 take the Intermediate SCAT designed for students in grades 6-8.
3. Students in grades 6 and above take the Advanced SCAT designed for students in grades 9-12."
Students who are outstanding at math, for example, often take online or video courses in the 4th or 5th grade starting with PreAlgebra and Algebra-1 (Chalk Dust Video Learning, the Art of Problem Solving, etc.)

Many advanced students study with tutors!

The Art of Problem Solving texts and courses are for outstanding math students. Students should have a firm grasp of basic arithmetic, including whole numbers, fractions, percentages, proportions, along with parts of algebra, measurement, and geometry no later than the 3rd or 4th grade. Most elementary schools do not accelerate students, so parents of advanced kids need to look elsewhere for opportunities, including hiring private math tutors. (Also, taking Algebra-1 in 8th grade is not acceleration. It is a course for average students who are prepared.)    

Note: The PreAlgebra text from the Art of Problem Solving is written to challenge students at a deeper level than a traditional middle school prealgebra course.

Aside: At a private school, many of my prealgebra 7th-grade students qualified for CTY summer programs by scoring exceptionally high on the College Board (SAT) mathematics section. (Students took the SAT by invitation.) There was no test prep. These same students were on my math league team, which took 2nd place in the State of Delaware. 

Many advanced kids study outside of regular school, which requires task commitment. Some students have math tutors. Also, some students who are good at math are not interested in acceleration. Likewise, there are children with high verbal ability who don't like reading that much.  

©2018 LT/ThinkAlgebra