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. 

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©2018 LT/ThinkAlgebra