Math Teaching 2

Running in Place & Fallacy of Fairness
The abstract goes to Concrete
The How always preceded The Why

No matter how hard Alice ran, she stayed in the same place, so has U.S. math achievement. "Faster! Faster! Now here you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!?" (The Queen to Alice, Through the Looking-Glass) 

Running in Place
Math Achievement Stagnation

For many years, students seem to be running in place in math achievement. It's stagnation. In short, our kids are not getting better at math. They are staying in the same place. Kids are not learning much because the fundamentals are not taught for mastery. Indeed, most teachers do not want to admit that they teach math poorly. Teachers get good evaluations when they teach math the Common Core reform math way. It's group work and test prep. Teachers focus on test-based proficiency not the mastery of content. Even if a student scores at the proficient level on the state test, it should not imply that the student has mastered grade-level content.

The constructivist approach of reform math hasn't worked. Scores on national and international tests have been low. Students are not required to practice arithmetic basics for mastery. 

The only way to judge whether or not the instruction is working is through performance testing. Can the student do standard arithmetic efficiently?  Zig Engelmann writes, "How successful is your version of the constructivist approach?"  

Petrilli: Lost Decade of Educational Progress: 2007-2017

After all the reforms and billions and billions ($) spent over the years, our 4th-grade and 8th-grade students are no better in mathematics than they were 10 years ago. Not only are the math scores below expectations; they are also flat. 

Michael J. Petrilli says the NAEP scores in math from 2007 to 2017 indicate a "lost decade of educational progress." Also, Bill Gates spent "hundreds of millions to improve teaching," but the initiative "failed to improve student achievement." 

What is wrong with education? More money, various reforms, so-called innovative programs, and professional development haven't worked. We have been unable to improve the "teaching" in the classroom, that is, the teaching that substantially boosts real student achievement. 

In schools of education, teachers are taught inferior (minimal guidance) instructional methods, unreliable theory, flawed equity and diversity guidelines, and a reform math curriculum that is not world-class. Moreover, K-8 teachers are weak in math and science content because they majored in education, which is not an academic subject. How can they teach subjects they don't know well? I am convinced that some elementary school teachers, today, do not know how to teach math or reading effectively. 

The progressive scheme dumbs down math for equity. High achievers and low achievers are mixed in the same classroom or in the same math class (inclusion for fairness). Furthermore, all students get the same instruction (e.g., Common Core or its state rebrand) regardless of achievement level or ability (equalizing for fairness). Indeed, "equalizing downward by lowering those at the top," says Thomas Sowell, is "a crazy idea," It is a "fallacy of fairness" that hurts all children, especially children of color. In American schools, it has been taboo to sort kids by achievement for math. In contrast, Singapore sorts kids in math in the 1st. grade. All kids should be given opportunities in schooling but not always the same opportunities or the same instruction. Kids who are advanced in arithmetic, for example, need an entirely different math curriculum from other kids. Instead of advanced math, they are given grade-level material. It is called equity, but it is false equity. 

Also, the priority should be given to learning the compact, standard algorithms first, but this is not the case in the reform math as taught in our schools. Starting in 1st grade, we don't stress the mastery of fundamentals. Memorization and drill-to-develop-skill have fallen out of favor in modern classrooms. Group work, discovery learning, and other fads are much more important than automating the fundamentals of arithmetic, progressives (aka liberals) say. They are wrong!

Help

We fell down the Rabbit Hole decades ago! 
Let's say we needed to calculate 6 x 1584. In Mock Turtle's 3rd-grade "Uglification" class, students were taught a wide range of alternative methods or strategies to multiply: lattice, repeated addition, array, area, partial products, make a drawing, calculator, distributive. (Which method should I use?) But, learning all these alternative methods clutters the curriculum and the minds of children (cognitive load). Who multiplies using the area model? The standard algorithm has been pushed aside. Some kids never learn it. Progressive education strikes again. We spend a lot of money, time, and effort on progressive ideas and theories that don't work well. Why make arithmetic harder than it is? 

The Real World: To move forward, students must know the compact, standard algorithms, not complicated alternatives.


Part I Abstraction
Why isn't math taught this way?
We think concrete goes to the abstract because that's what everyone says. But, what if the "experts" were wrong? The idea is that children develop number concepts only from concrete representations, then, later, as abstract ideas (Piaget's theory), but it doesn't work that way. Indeed, 5 is 5 and can be represented in hundreds of different ways such as 5 people, 5 pencils, 5 cubes, 5 dots, etc., but the number 5, itself, is an abstract idea. It is a concept in our mind. You don't need to go from concrete to abstract to understand the meaning of 5. The number 5 is already abstract, which is the starting point. 

The number line helps novices visualize what 5 is. It helps beginners with the idea of magnitude. The number 5 is a point on the number line that is one more than 4 and two less than 7, and so on. 

The meaning of 5 is derived from its connections with other numbers. 

What about 3 + 4? You don't need to show 3 + 4 is 7 in several concrete ways, or make a drawing, or write an explanation. If need be, the number line can verify that 3 + 4 is 7, which is obvious. But the number line is not a proof or a why. Also, showing different ways to represent 3 + 4 is not a verification of "deep understanding" or why 3 and 4 add to 7.

The number line is essential arithmetic, but it is seldom used in the primary grades, which is a huge mistake as children try to grasp magnitude and how numbers behave. 

The Number Line Is Important Arithmetic

Numbers are invented in our minds. They are abstract. Children start with the whole numbers as concepts and should learn connections between numbers on a number line, such as "add 1" (4 + 1 = 5), or 3 + 4 = 7, or 7 - 3 = 4, or 3 x 4 as the sum of three fours: 4 + 4 + 4 = 12. 

So, why do we ask novices to explain the obvious several different ways? We are told that it shows understanding. Really? I don't buy it. If needed, the number line verifies it. That said, 3 + 4 = 7 should be memorized early in the 1st grade, but, unfortunately, the memorization of number facts has fallen out of favor in progressive classrooms. 

In short, arithmetic is invented in our minds. In every word problem, no matter how simple or complicated, the student needs to pull out the numbers and figure out what to do with them (execute an operation) to find an answer. You see, arithmetic is about numbers that are abstract ideas. It's not about 5 pencils. It's about the 5 and how it relates to other numbers. We use straightforward arithmetic to solve word problems. 

If arithmetic is abstract, then why isn't it taught that way?

Indeed, the equation 3 + 4 = 7 is the solution to hundreds of word problems. In 2nd grade, I would write an equation on the board and ask students to make up a story (word) problem. Some students surprised me by turning 3 + 4 = 7 into a missing addend problem (a subtraction problem) such as, "I have 7 apples and give 3 of them to Bill. How many apples do I have left? Or, 7 - 3 = 4. The problem isn't about apples; it is about the numbers and how they are related. The basic relationship between addition and subtraction is abstract.

Part II
The HOW always preceded the WHY.
Why isn't math taught this way?
Tobias Dantzig (Number: The Language of Science), a book highly praised by Albert Einstein, writes, "In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy. This is particularly true of arithmetic. The strength of arithmetic lies in its absolute generality such as a + b = b + a. Its rules admit of no exceptions: they apply to all numbers. Every number has a successor [add one]. There is an infinity of numbers." Teaching novices arithmetic should be the "how," not the "why." Kids are novices, not little mathematicians. 

Ian Stewart (Letters to a Young Mathematician) clarifies, "One of the most significant differences between school math and university math is proof. At school we learn how to solve equations or find areas of a triangle; at university, we learn why those methods work and prove that they do." We need to stop teaching kids as if they are little mathematicians, which they are not. 

American teachers are hung up on the "why" of everything, but in arithmetic, it is proper to learn the "how" first and the "why" later, which is what happens in Asian nations. Learn the technique first and get it right (i,e., automate it). Learn the "how" first and don't worry about why it works.

Note: Standard arithmetic is simple and compact, but it is not easy to learn without the memorization of single-digit number facts and practice-practice-practice. 

National Mathematics Advisory Panel (2008):
"Computational proficiency with whole number operations is dependent on sufficient and appropriate practice to develop automatic recall of addition and related subtraction facts and of multiplication and related division facts. It also requires fluency with the standard algorithms for addition, subtraction, multiplication, and division." I do not recall teaching arithmetic without some level of student understanding. Understanding develops slowly over the years. Also, automating the mechanics of arithmetic for fast access frees working memory space for problem-solving. Arithmetic is a tool for solving problems.  

First-Grade Subtraction: 
Go Vertical! Let's Borrow! Missing Addend Equations!
The standard algorithm of subtraction is a key math skill that can be taught in the 1st grade. The vertical format is a place value system. Start with problems that don't involve borrowing, such as 47 - 25, but use the vertical format to stress place value.

Teach borrowing. Borrowing shifts a ten back into the ones place, that is, one ten equals 10 ones. The 16 can mean 16 ones. It is simple place value and reverse engineering. Students can compose missing addend equations in their heads and recall memorized addition facts to solve them. What number plus 9 is 16? {7 + 9 = 16} Or, x + 9 = 16 . It is a missing addend equation. Children can use memorized addition facts to do subtraction: 
addend1 + addend2 = sum. It is important to relate subtraction to addition.

Like it or not, Arithmetic is based on Rules
I often hear teachers say that they don't want kids memorizing a bunch of rules. But, arithmetic and algebra are governed by rules, and students must know the rules and be able to apply them to do correct mathematics. For example, the commutative rule should be learned in the first full week of 1st grade: 2 + 3 and 3 + 2 give the same answer. Arithmetic is based on rules.  There are not that many, but they should be learned as early as possible, such as the "zero rule" for addition: 3 + 0 = 3. The properties of numbers that children learn are rules. They are also mini procedures.

Learn the procedure first (i.e., the "how," such the procedure for solving 1/2 of 1/2, which is a multiplication of fractions 1/2 x 1/2 = 1/4.)

The stress on understanding "why" something works is misplaced. For example, Isaac Newton invented a calculating method (calculus). He knew it worked because the results of his calculus agreed with the experimental results, but he didn't know why his calculating method worked. The "why" wouldn't come for another 200 years with "limits." But, that didn't dissuade Newton from using his calculating method to figure out physics. It always worked. He worked out the how, but not the why. Still, he was a brilliant mathematician and physicist. 

So, when we teach kids (novices) an operation, such as addition in 1st grade, we should first make sure they can do the procedure well (the "how"), which requires practice-practice-practice. Focus on the how not the why. 

Teachers should teach the procedures (aka algorithms) that are the most efficient and compact and work all the time. These are the standard algorithms. 

Understanding comes out of doing. Why are parents concerned about their children's math education? Kids can't do simple arithmetic. 

Students Learn Math Inductively
Mathematics uses deductive reasoning and logic to create new math. But, students--who are novices, not experts--don't learn math that way. They learn math inductively. That is, the teacher explains several examples and then tells students that the method always works. It is impossible to test every possible case or present a formal proof. The teacher should also explain the counterexample.

Students don't need to draw pictures or count objects to understand math.
enVision Problem Solving
Kara found 5 shells. Then she found 3 more. How many shells did Kara find? Draw a picture. Write the number. 


The problem-solving idea of enVision Math, a popular reform math program, is for the student to draw little pictures then count the objects depicted. The memorizing of critical single-digit number facts is not part of the curriculum in 1st-grade state standards. In my opinion, 5 + 3 = 8 should be memorized in 1st grade. It can be verified on a number line if need be. Moreover, the answer to the question is "Kara found 8 shells," not the number 8.

Teachers teach math for test-based proficiency, not mastery, and that is what is wrong with math instruction today. Proficiency in state math tests should not imply that the student has mastered grade-level math content. Under Common Core, state standards, test-based accountability, and "teach to the test items" mentality, our curriculum has become fragmented, and our instructional methods have been inferior.

Let me repeat. Educators focus on low achievers. It is called "equalizing downward by lowering those at the top," which is a "fallacy of fairness," says, Thomas Sowell. Instead, teachers should focus on equalizing upward to challenge high achievers by tracking them in at least one of their best subjects, such as mathematics, but not all their subjects. It is tracking-by-subject, but there should always be some flexibility in grouping along with way. 

To Be Revised
Updated: 7-7-18, 7-10-18, 7-11-18, 7-13-18, 7-16-18, 7-17-18, 8-31-18
Model Credit: Gabby, Hannah, Shayna
The NAEP chart is from an article by Michael J. Petrilli (Fordham)

©2018 LT/ThinkAlgebra.org

Math Teaching

Progressive (aka Liberal) reformers are promoting the de-tracking of high school math in San Francisco [1], that is, having heterogeneous math classes for "social justice." The scheme is to dumb down the math for equity. A one-size-fits-all reform, such as Common Core, has not changed outcomes. Also, Bill Gates' funding of teacher effectiveness has not changed student achievement outcomes either. It has been a flop! 

Kids are asked to solve math tasks in groups and talk a lot. Really? How has it worked in K-8, that is, the idea of mixing high achieving kids in math with low achieving kids in the same math class (inclusion) with plenty of group work? I am not sure how students can solve math problems without knowing the necessary math content and skills.
[1] Reference: Stephen Sawchuck in Education Week  

Tom Loveless ("High Achievers, Tracking, and Common Core," Brookings report, 2015) describes the effect of de-tracking math in middle schools such as in the San Mateo-Foster City School District. He writes, "The changes were brought about by the Common Core State Standards (CCSS).  Under previous policies, most eighth graders in the district took Algebra I. Some very sharp math students, who had already completed Algebra I in seventh grade, took Geometry in eighth grade. The new CCSS-aligned math program will reduce eighth-grade enrollments in Algebra I and eliminate Geometry altogether as a middle school course." In 2018, many school districts no longer offer Algebra-1 in 8th grade. State standards based on Common Core push Algebra I to high school. The detracking move from Algebra I in middle school is contrary to the recommendation by the National Mathematics Advisory Panel (2008) that advocated more students should take Algebra I in middle school, not fewer. Schools (K-7) must upgrade the math curriculum to prepare more students for Algebra I in 8th grade. In California, Common Core was a downgrade. 

For decades, the de-tracking in elementary school, for example, has been a recipe for mediocrity. Lackluster achievement climbs up the grades. Thomas Sowell writes, "Equalizing downward by lowering those at the top is a crazy idea--a fallacy of fairness--taught in the schools of education."

Phil Daro, one of the architects of the de-track plan in San Francisco, writes, "Tracking is an evil." Really? What is evil is equalizing downward. The liberal ideology that students are the same and, therefore, should get the same instruction in math is folly. Moreover, achievement is not privilege. Achievement is achievement. Branding it as privilege subtracts from the hard work, effort, accomplishment, and success of children, including children of color. Contrary to the liberal de-tracking ideology, "Academic talent, like musical or athletic ability, needs to be assessed, developed, and cheered onward." (Center for Talented Youth, Johns Hopkins)

If you want to know why our children are not learning much, then look into the classrooms of the 21st-century. Tech has replaced knowledge as the holy grail. The state test has warped and fragmented the curriculum. Not good!

Evgeny Morozov (To Save Everything, Click Here) writes about the folly of technological solutionism. He also points out, "Schools concentrate all their efforts on improving test scores [metrics], even if children learn much less as a result."

Students are taught reform math, not standard arithmetic straightforwardly for mastery. Educators implement a substandard curriculum based on standards that are not world class. They often use inefficient minimal-guidance methods, test-prep, and group work. Unfortunately, learning knowledge is no longer the bedrock of schooling, even though knowledge in long-term memory and its application are fundamental for higher-level thinking within a domain. In short, we have taught math badly. It boils down to teachers, curriculum, methods of instruction, and bad progressive ideas. Good teaching, memorization of basics, and ample practice for mastery are often absent from many classrooms. The paper-pencil standard algorithms are the best tools for beginners to do basic arithmetic.

Many teachers aren't teaching the right arithmetic, which should focus on the memorization of single-digit number facts that support the standard algorithms. Moreover, they use an inferior curriculum and inefficient methods of instruction. Common Core and its state rebrands are below world-class benchmarks.  

Moreover, our best students are not challenged and underachieve. Kids who are advanced need advanced material. The typical talented and gifted programs found in many school districts don't identify these kids, but the Johns Hopkins Center for Talented Youth (CTY) does that for students in grades 2 to 8 by using the School and College Ability Test or SCAT that is above the student's grade level. It is the only way to sort the most advanced performers  (verbal and math) from the rest of the good students that cluster at the top in grade-level standardized tests. In math, talented youth need textbooks that are written explicitly by math geeks, such as books from the Art of Problem Solving. Beginning in early elementary school, advanced math kids need an entirely different math curriculum taught by an algebra teacher. A recent report from Johns Hopkins shows a widespread lack of support for high-ability, low-income students. CTY starts with the Talent Search (SCAT). "Academic talent, like musical or athletic ability, needs to be assessed, developed, and cheered onward."

NAEP National Test 2007-2017
4th-Grade Math Scores Are Flat

We teach math poorly.
The proficiency standards of the National Assessment of Educational Progress (2017 NAEP)--the Nation's Report Card--show what students “should know and be able to do.” To me, if 60% of the 4th graders, 66% of 8th graders, and 75% of 12th graders can't do math well (not proficient or above), then we have a major problem. Indeed, flat scores in math and reading have been a difficulty in American education for years. Even though many changes have been attempted, they have done little to alter the lackluster NAEP scores. Also, Michael J. Petrilli says the NAEP 2017 indicates a "lost decade of educational progress." In addition to inadequate teaching, the NAEP test scores show that state math standards and curriculum are not world class. Indeed, we teach math poorly. (Incidentally, both math and reading NAEP scores are stagnated.) 

Improving State Test Scores is the primary focus of education today.  
Political issues seem to dominate the education landscape, but little is said about teaching, itself. Good teaching and plenty of practice are absent in many classrooms. Kids are not learning much because the fundamentals are not taught for mastery. It seems simple enough, but what should teachers do when they are required to follow a reform math curriculum based on Common-Core-laced state standards? The standards are not world class. Reform math is not standard arithmetic. It is an alternative arithmetic and does not emphasize content mastery. Unfortunately, many teachers and schools are judged on their students' test scores.

Teaching is no longer about teaching valued content; it's about improving state test scores (metrics), which is a mistake. "It is difficult to make large-scale improvements in education," explains Andrew Ho, a Harvard University professor. For decades, huge reforms haven't worked. The crux of the matter is that teachers aren't teaching straightforward core arithmetic. (And by core, I do not mean Common Core.) Our education system needs international benchmarks of excellence and knowledgeable teachers who can adhere to and teach those standards. We don't have that today. "By international standards, our 8th-grade students are exposed to 6th-grade content" (Schmidt-Cogan-McKnight).  

(Note: Improving state math scores has had little effect on national and international scores. Proficiency on state tests does not correlate well to NAEP. It seems that standards-based accountability (e.g., Common Core) has led to controversial reforms that don't work well. Learning is flat. For example, test prep for state tests has not impacted NAEP scores. Kids should be taught the fundamentals of standard arithmetic for mastery. They are not. Instead of standard arithmetic, kids are taught reform math, which has had little effect on achievement as measured by the NAEP.)

Algebra
According to mathematician Steven Strogatz, all the symbols, definitions, and procedures of algebra boil down to two activities: solving for x and working with formulae. Young children need to "think about numbers and the
relationships between numbers" Moreover, the "relationships are harder because they much more abstract than numbers," but "they are also much more powerful." 

The formula I taught 1st graders for the perimeter of a simple rectangle is P = L + L + W + W (e.g., P = 5 + 5 + 2 + 2 = 14 cm). It made sense that perimeter is a sum of the distances around the rectangle. "To understand mathematics means to be able to do mathematics"   (G. Polya). In short, if you can't calculate perimeters (doing the math), then you don't understand perimeters (math). 


Strogatz writes (The Joy of x), "Numbers and all mathematical ideas have lives of their own. They obey certain laws and have certain properties, personalities, and ways of combining with one another, and there's nothing we can do about it except watch and try to understand." 

1877

Premise
Very young children can learn much more math content than teachers were prepared to teach. It's about learning prerequisites. U.S. students were taught basic arithmetic at earlier ages in the 19th century. Kids in the 1800s learned much more fundamental arithmetic than students learn today, according to Ray's New Intellectual Arithmetic book 1877. 

3rd-4th Grade Traditional Arithmetic 
(Ray's New Intellectual Arithmetic, 1877)
(1) 3/4 of 24 are 6 more than 2/3 of what number? 
(2) Find the interest on $50 for 6 months, at 6%. 

Progressive educators of the 20th century followed Piaget, which was a significant blunder in American education. Educators should have followed Bruner, who wrote, "We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development." 

Thus, we have substantially underestimated what children can learn by saying some content is developmentally inappropriate, which is nonsense. Also, we hide behind the fallacies of learning styles, fairness, averages, and other misconceptions that disregard the cognitive science of learning. 

Learning algebra in elementary school started in the 1950s with the Madison Project for grades 3 to 5. I wrote my own algebra curriculum for grades 1 to 3 and added 4th and 5th grade later. In my "early algebra" project (Teach Kids Algebra - TKA), which I started in the spring of 2011 for grades 1 to 3, I contradicted Piaget's theory of cognition every time I gave an algebra lesson, starting with two first-grade classes, two second-grade classes, and one third-grade class.

1st-Grade TKA Student - Spring 2011.

Very young children can deal with abstractions and do simple reasoning. Numbers, operations, axioms (e.g., a + b = b + a), and equations (e.g., y = x + x + 2) are all abstractions. Writing symbols on paper makes math visual. To learn math, children need to learn symbolics.

The Number Line is important mathematics! It starts at ZERO.
Unfortunately, it is seldom used in 1st-grade reform math.   

Number Line showing 3 + 4 = 7

Numbers are invented in our minds. They are abstract. Children start with the  whole numbers as concepts and should learn connections between numbers on a number line, such as "add 1" (4 + 1 = 5), or 3 + 4 = 7, or 7 - 3 = 4, or 3 x 4 as the sum of three fours: 4 + 4 + 4 = 12. All of these ideas are basic 1st-grade arithmetic. The next step is to memorize the single-digit number facts and use the standard algorithm. Unfortunately, the number line is seldom found in the early grades, even though it is basic arithmetic. Unfortunately, most 1st-grade students are not required to memorize the addition facts or learn the standard algorithm for larger numbers.  

First Grade: The Standard Algorithm

First-grade students can learn to add "ones to ones" and "tens to tens" with the standard algorithm, which is based on the place value system. (37 means 3 tens + 7 ones by place value or 3t +7.) Children need to memorize the single-digit addition facts to efficiently use the standard algorithm place value system. Learning the addition facts means remembering them from long-term memory, which requires daily practice and continual review. Also, the mechanics of the algorithm should be taught first with the explanation later. Children will start with a functional or practical understanding. A more in-depth understanding will come only with continued practice.

The four whole-number operations should be taught by the 3rd grade. It rarely happens in our schools because the curriculum focuses on reform math, not the mastery of basic arithmetic.

Students need continual practice to get good at the mechanics of the standard algorithms. Single-digit number facts must be memorized for immediate recall from long-term memory. Average students should learn the standard algorithms for both multiplication and long division no later than the 3rd grade. It is not advanced content.

Young children learn by imitation (of the teacher) and through practice drills. Repetition and review are important for learning essentials. Teachers should spend most of their time on high-value content. Select 30% of the key material that will make a 70% impact and spend most of math class time mastering those essentials through practice and review. Unfortunately, teachers blindly follow the Common Core reform math curriculum, such as Eureka math. Not knowing the single-digit number facts for instant recall or the standard algorithms may create a cognitive load in working memory that interferes with solving problems and learning (John Sweller).   

In math, the learning goal should be the mastery of fundamental content (factual and efficient procedural knowledge) in long-term memory, not proficiency on state tests or cumbersome, unorthodox algorithms.  Also, a curriculum that is warped to fit test items is a fragmented curriculum that perpetuates lackluster achievement. Common Core and state standards based on Common Core are not world class.  



Kids are novices, not experts or pint-sized mathematicians. They must master standard arithmetic to advance, but many do not. Kids are stuck in reform math programs that don't work. Knowledge matters in long-term memory. Kids are novices who need to memorize single-digit facts, learn rules and formulae, recognize problem types, and master the mechanics of basic operations to perform standard arithmetic and solve problems.

You can't teach arithmetic like you teach social studies. But, isn't this what most teachers do? Group work! Group work! Group work! Kids aren't going to discover critical mathematical ideas that took geniuses like Euclid, Gauss, Euler, Newton, and many others to figure out. 

The primary cognitive building block is knowledge and the skills that are embedded in knowledge. It is factual and procedural knowledge. "Knowledge is critical to thought," says Daniel Willingham, a cognitive scientist. 


Knowledge is the foundation that enables a higher level of skills.

Knowledge is critical to thought (Daniel Willingham). 

Many schools emphasize higher-level thinking skills, but not the content knowledge that enables higher-level thinking. Higher-level thinking is domain specific. Problem-solving or critical thinking (thought) that is independent of content is empty. Kids who are advanced need advanced content, but they rarely receive it in elementary or middle school. Richard Rusczyk (the Art of Problem Solving), high school calculus is for average students who are prepared. It is too easy for gifted kids in math. The advanced math kids in elementary and middle school should be sorted for math class and taught by an algebra-precalculus teacher. Furthermore, the advanced kids should compete in various math contests (e.g., math league, MathCounts, etc.). Rusczyk's textbooks were written specifically for advanced math kids starting with prealgebra.

Children are easily distracted with gadgets.

They need to focus when they do homework, or little gets done. (And little is learned.) 

Attention
Students facing each other in small groups of 3 or 4 or at tables are conditioned not to pay attention. Still, paying attention in class is critical for learning. Any interruption or distraction in the classroom [or at home] and there are many, diminishes the working memory and restricts learning. Students must pay attention in class to start the learning process.

Learning arithmetic is hard work. Learning is remembering. If you can't remember something, then you haven't learned it. Learning requires regular practice and frequent review. Forgetting is easy; learning is hard. Thus, learning is hard work, and it is not always fun. If you don't have instant recall of 4 x 8 = 32 from long-term memory, then you haven't learned the fact, and more practice is needed to develop the skill. Indeed, much is taught, but little is learned. 

End

©2018 LT/ThinkAlgebra

Novice

I am a novice, not a pint-sized mathematician.
[Cognitive Load]

Teaching Math

You can't teach math like you teach other subjects such as social studies. There are prerequisites to arithmetic content. Students must know the previous lesson to go on to the new lesson. You can't skip around and ignore context. Kids cannot "discover" context. Indeed, math is cumulative and sequential because one idea builds on another. "Everything fits together logically," says mathematician Ian Stewart. One skill is needed for another, and so on. Regarding group work, frankly, I don't want students reinventing arithmetic as in reform math. They are novices, not pint-sized mathematicians. Teachers should focus on mastery of essential content, not test-based proficiency, but often they have little choice. 

David Didau (The Learning Spy) summarizes novice learners. Novices know little relevant background knowledge, rely on working memory, not long-term memory, and have not automated necessary procedural knowledge. For novices,  problem-solving requires following clear steps. They learn little when exposed to new information, are prone to cognitive overload, and learn best through explicit instruction and worked examples. He states that most students, most of the time are novice learners, not experts.   

Kids are novices. We should stop teaching them as if they were little mathematicians. The best pedagogy for novices is explicit instruction and practice-practice-practice for retention of essential factual and procedural knowledge in long-term memory. Novices need to master traditional arithmetic, not the many alternative algorithms of reform math. Also, the minimal guidance methods of reform math (discovery learning, project learning, etc.) do not work, as Kirschner-Sweller-Clark pointed out in their research: Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. Furthermore, be skeptical of great sounding new stuff such as personalized learning, blended learning, social-emotional learning, 21st Century, digital learning, and other manifestations. The hype is there, but, as usual, the evidence isn't.

David Didau (What if everything you knew about education was wrong?) challenges a "status quo" inference that learning is invisible, that is, "All we can see is what students can do, and from that, we infer what they might have learned." I think the inference is not consistent with the cognitive science of learning. Should we infer learning from a performance? Being able to do math is important, but retention in long-term memory is what learning is all about, and retention is paramount. Learning requires a change in long-term memory. If you "learn" the XYZ procedure one day, but forget it the next day, then you haven't learned it. I am defining learning as remembering. If you can't remember something, then you haven't learned it. Learning requires continual practice and regular review. Forgetting is easy, but remembering is hard work. Learning is hard work. If you don't have instant recall of 4 x 8 = 32, then you haven't learned the fact, and more practice is needed to develop the skill. Indeed, much is taught, but little is learned.]

Teachers should teach kids as novices, which they are, not as little mathematicians, which they are not. The best pedagogy for novices is explicit instruction and practice-practice-practice for retention of necessary factual and procedural knowledge in long-term memory. By essential procedural knowledge, I mean standard algorithms. Also, there are facts in math that need to be learned such as single-digit number facts, axioms and formulae, pattern recognition for problem-solving, mechanics of standard algorithms, place value system, and others. Incidentally, all of these start with 1st-grade addition.

The traditional method of teaching arithmetic and math is called direct instruction or explicit instruction, which is the pedagogy most suitable for novices. Traditional education in math includes giving clear explanations of worked examples, conducting demonstrations, asking questions of students during the lesson presentation, providing adequate practice and feedback, and testing students "to see if they have learned the content and skills." In short, the traditional method is straightforward, and it works. In contrast, the popular minimal guidance instructional methods (e.g., project-based learning, discovery/inquiry, group work, etc.), which are glorified by the progressive reformers, are inferior (Kirschner-Sweller-Clark: Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching).

Kids are novices, not experts. 

Still, so-called "math educators," trained in schools of education, insist that kids should think like a mathematician, which they are not. It is obvious that regular students do not have the same background knowledge, experience, or wisdom as experts in a field of study. Even though students are not little mathematicians, scientists, or historians, it doesn't mean that they are not interested in math, chemistry, physics, history, computer science, literature or many other subjects, including the fine arts. Kids should dream! A student can do an intriguing physics experiment, but it doesn't mean the student thinks like an expert physicist such as the late Richard Feynman or Stephen Hawking. Maybe the student will grow up and become a physicist, so it is important that we upgrade the math for future STEMers. Also, students should be told that it takes years of hard study to be an expert in any field. Even college students are novices when learning new academic content. Unfortunately, Common Core pushes many complicated, alternative algorithms and controversial "standards for mathematical practice" into K-8 classrooms in which critical thinking is more important than the factual and procedural knowledge that enables the high-level thinking. The math curriculum is crowded with reform math junk. Many students are being denied the math education they need to move forward. 

It is important to beef up the math for future STEMers.

Even though novices and experts think differently, progressive educators seem to ignore the difference. Novices need fully guided instruction and worked examples to help them "acquire relevant schema" (John Sweller) in long-term memory. As students progress, mixing problem-solving with guided instruction provides "practice at accessing schema" (Sweller's Intermediate Level). Skills should be automated before applied to problem types. 

David Didau writes, "Novices have not automated the necessary factual and procedural knowledge." Automation is an essential process for students to move forward in math. Many students come to 4th grade with only a partial recall of multiplication facts if that. These students will not advance much until they automate the x-facts. Also, the incoming 4th-grade students don't know the mechanics of the standard multiplication and long-division algorithms, which depend on the automatic recall of single-digit multiplication facts. Why is that? The fundamentals of standard arithmetic are not taught for mastery.  

At first, problem-solving types should be routine and straightforward then gradually build in difficulty. To develop problem-solving skills, students should follow clear steps. Also, novices learn best through explicit instruction and worked examples. Learning is remembering and requires lots of practice. Instruction should be unambiguous, that is, extra information should not be included in word problems. Word problems should be straightforward. Be aware that cognitive overload is commonplace as working memory is "swamped by new information." 

Prerequisites

In math, the next lesson depends on mastery of the previous lessons. Frequent reviews are vital to keeping up skills. Genuinely advanced math kids in elementary and middle school need different curriculum and instruction. However, we don’t sort kids in elementary school according to performance in math. Instead, we place high achieving math students with low achieving students in math, which has been a counterproductive approach and a blueprint for mediocrity. 

Real Mastery

The meaning of expert varies, but for Anders Ericsson, an expert in math is an individual who has achieved true mastery of the subject (e.g., Ph.D. in mathematics and above) and has "probably spent at least ten years in deliberate practice" and doing research at the highest levels. If an 8-year-old scores 760 on the SAT math section, then the student is genuinely advanced on the math tested and likely gifted in mathematics, but the student is not an expert in math until after years and years of study. He was an expert among his peers, so expertise is relative. This student and others like him should be given instruction that is different from novices.)

Gifted & Center for Talented Youth

In many school districts, the selection process for advanced or gifted students doesn’t work, which is the reason that Johns Hopkins Center for Talented Youth gives the SCAT (School & College Ability Test). CTY says that the best kids cluster at the top in grade level tests, so there is no way to spread out the students. Thus, to separate the very best students from the very good students, CTY does not give the regular grade level SCAT, but a couple of grade levels above. The test consists of two specific parts: verbal and quantitative. 

When do school children become experts? 

They don’t. Students get better with explicit instruction, practice, feedback, and review, but better is far from Ericsson's concept of an expert. However, some students will achieve an advanced level in certain math topics. And, according to John Sweller (Cognitive Load Theory), students who are advanced need different instruction. (Sweller: Novice --> Intermediate --> Advanced). In short, you can't treat your best students like average students. They need acceleration (i.e., advanced content). Also, when advanced students hit new content, they are novices, but the pace is faster and the content more difficult. Also, there are other factors that influence the way teachers teach, such as state and district policies, teacher education, school environment, and mandatory testing, as Larry Cuban points out.

Kids who are advanced need advanced content. It is a simple idea, but advanced math kids in elementary school rarely receive advanced content. Gifted programs are geared toward enrichment, not acceleration. In the early 90s, I treated my fast moving Honors 7th-grade math class as novices because much of the 7th-grade content was similar to algebra-one. (I wish I had the Art of Problem Solving textbooks by Rusczyk, but they weren't written until 2007.) Also, six students from the Honors group made up the school's 7th-grade math league team, which placed 2nd in the state. One student was the top 7th grader in the state. I used a more traditional approach when teaching outstanding math students because the focus was on learning new content.

Richard Rusczyk (the Art of Problem Solving) wrote, "We believe that students learn best when they are challenged with hard problems that at first they may not know how to do." But, he adds that "other students would learn better from a more traditional approach to new material." The math books from the Art of Problem Solving were written for outstanding math students, not average or even good math students. Some of the problems are from different math contests such as MATHCOUNTS, American Mathematics Competitions, Harvard-MIT Mathematics Tournament, and so on. See Comment below.   

(Comment: The students in my 7th-grade Honors math were invited to take the College Board SAT and scored high enough to be accepted into programs from the Johns Hopkins Center for Talented Youth or CTY. Still, I treated the 7th-grade advanced students as novices much of the time because the content taught was new, e.g., trig and inverse trig functions, linear and quadratic functions, exponential functions, polynomials, negative and zero exponents, simplifying square roots, permutations and combinations, and other topics.) 

Cognitive Load During Problem Solving (John Sweller)

“First, we might expect the cognitive load to be correlated with the number of statements in working memory. We know that human short-term memory is severely limited and any problem that requires a large number of items to be stored in short-term memory may contribute to an excessive cognitive load. In so far as short-term memory corresponds to a production system’s working memory, it is reasonable to suppose that an increased number of statements in working memory increases cognitive load."

It is important not to clutter working memory with extras that are not needed. 

The Kirschner-Sweller-Clark equation:
Minimal Guidance = Minimal Learning

Source: Kirschner-Sweller-Clark: Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. 

Note: Novices need to learn Standard Algorithms and have lots of practice for mastery, not reform math.

241 ÷ 7
Novices should learn the mechanics of the long-division algorithm in the 3rd grade and not waste time with complicated alternative algorithms and other reform math junk.

Standard algorithms are traditional arithmetic, and kids must be fluent in them to advance.   




Below: 241 ÷ 7 (Reform Math)

Look at what reform math did to a simple division: 241 ÷ 7.

Why break down long-division problem into component parts using a cluster as suggested in 5th-Grade Investigations, a reform math program? In fact, 241 ÷ 7 is a simple 3rd-grade calculation, not a 5th-grade problem. It makes no sense! Is it any wonder that our children do math poorly? 

In my opinion, reform math methods or strategies clutter the working memory and the curriculum and increase cognitive load. 

I would hope that teachers don't use lessons from Investigations such as the one below.

Reform Math: Investigations Grade 5
No one uses this method to do division, so why teach it?
It is a complicated, alternative method--a product of reform math.

Dr. W. Stephen Wilson, a research mathematician, testified that Investigations, Grade 5 was "pre-arithmetic and consisted of about 1/3 of a course each year." Kids won't learn what they need to know. Wilson said there is so much material that it is impossible to cover it all in one year. Also, Professor Wilson explained that the Investigations curriculum is "about how to solve math problems without knowing the math, i.e., think critical thinking." Investigations wants students to just use critical thinking. Of course, math doesn't work that way. The Investigations curriculum deviates from standard arithmetic and its well-known standard algorithms that students must know. 

Eureka Math, which is another Common Core reform math program, is similar in that there is much more material than you can teach. The standard algorithms are not stressed. For example, there are seven modules for 3rd grade. If the Teacher Editions were stacked, they would be about 6 inches thick. I measured them. One 3rd-grade Teacher Edition module is over 500 pages long. Do the Eureka Math people believe that teachers have the time to read through all that stuff? Moreover, there is no evidence that it works. Eureka Math stresses complicated, alternative algorithms and pushes questionable "standards for mathematical practice" over necessary factual and standard procedural knowledge. In short, Eureka math clutters the curriculum. (Common Core: K-12 "standards for mathematical practice" include, construct viable arguments and critique the reasoning of others, attend to precision, use appropriate tools strategically, persevere is solving problems, reason abstractly and quantitatively, model with mathematics, etc." Really?)

Reform math is popular among educators and impacts instruction in most of our schools. Critical thinking is stressed over facts. Memorization of facts and drills to build arithmetic skills for mastery are considered old-fashioned and not good teaching. For decades, the standard algorithms have been pushed aside. The reformers are wrong, of course.

   

Note: This post is incomplete. I want to cite more sources. Some of the quotes and ideas on this page come from David Didau (The Learning Spy), Kirschner-Sweller-Clark, Anders Ericsson, Johns Hopkins Center for Talented Youth, Genius (National Geographic), Daniel Willingham, Investigations (Grade 5), Larry Cuban, Richard Rusczyk, and others. 

Model Credit1: GabbyB

Gabby, giggling with her Mom!

Model Credit2: ChloeM

March 19, 2018

For comments, please email me at ThinkAlgebra@cox.net. This post is undergoing changes and updates. To Be Revised  Excuse typos. 3-20-18

©2018 LT/ThinkAlgebra