Saturday, March 17, 2018


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 reviewForgetting 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-ClarkWhy 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." 

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 This post is undergoing changes and updates. To Be Revised  Excuse typos. 3-20-18

©2018 LT/ThinkAlgebra 

Friday, March 16, 2018

Eureka Math

Math should be made as simple as possible. 
After, all, kids are not little mathematicians; they are novices, so simplicity is best. But, over the years, reformers have made basic arithmetic much more complicated than it should be, cluttering it with "standards for mathematical practice (for experts)" and a host of "alternative, nonstandard algorithms." Kids are not experts; they are beginners.   

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 are considered old-fashioned and not good teaching. The reformers are wrong, of course. 

An example of reform math is Eureka Math. It is a version of Common Core and the same as EngageNY, but Eureka produced student and teacher materials. The 3rd-grade Teacher Editions consisting of the seven modules, if put together, measure 6 inches thick. For example, Module 7 Teacher Edition is over 500 pages long, Module 5 is over 400 pages long, and Module 1 and Module 2 are each over 300 pages long. It is reform math run amok. The reform math curriculum is overcrowded and often pushes standard arithmetic to the back burner. For example, in 1st-grade Eureka math, the memorization of addition facts is not an objective, and the standard algorithm is missing. In 3rd-grade Eureka math, the standard algorithms for both multiplication and long division are missing. Instead, students focus on alternative algorithms. Most students do not memorize the single-digit multiplication facts for long-term memory recall. In short, mastering essential content is not the aim of reform math.   

In contrast, we should articulate the specific content students need to learn (i.e., master), not a bunch of low-value stuff that clutters the curriculum. We should focus on the mastery of essential content, not test-based proficiency.  The problem has been that many elementary teachers don't know what the essentials are anymore. Many are, themselves, the products of reform math from K-12 schools to Schools of Education and PD (i.e., Professional Development).

Teachers are required to follow the curriculum handed to them, even if it is lousy. Unfortunately, in many reform math programs, teachers are bogged down with teaching tenets of "mathematical practice" and alternative algorithms. They tell me they have to teach x (e.g., a nonstandard, alternative algorithm) because x is on the state test. In the real world, one uses x to perform the algorithm. They use the standard algorithm, which not taught or marginalized. Therefore, many students never master the basics. 

Teaching to items-on-the-test is a fragmented and incomplete curriculum. We are fixated on metrics (test scores), transparency, and accountability. We punish schools, teachers, and students who do not meet state standards.   

I like some of the ideas in Eureka Math, but the curriculum is cluttered, and test-based, and does not get kids where they need to be. The basics are not valued.  

To make curriculum decisions, we should look at results, not a methodology. Performance matters!  Reform math is a messed up methodology that hasn't worked well for decades.  

© 2018 LT/ThinkAlgebra

Tuesday, March 13, 2018


Tom Loveless (Brown Center Chalkboard 2-23-18) wrote, "Scores from the 2017 National Assessment of Educational Progress (NAEP) will be released in the coming weeks. NAEP is often referred to as “the nation’s report card,” providing a reliably accurate estimate of national academic achievement every two years." 

Like Tom, I can't wait. 

Here are the NAEP results comparing the past two tests (2013 and 2015) for students who scored at or above Proficient. 


Some critics argue that the "Proficiency" cut scores are set too high, such as in mathematics, but I don't buy the argument. The NAEP scores show that the vast majority of students on average are not performing to the expected level in mathematics, reading, or science. NAEP is a national test given every two years. 

In math and reading, for example, we should look at classroom practice, policies, and programs that sound good but do not work, from the metrics of Common Core and state testing (Every Student Succeeds Act: ESSA) to minimal guidance instructional methods (e.g., discovery, etc.), tech/big data, and heterogeneous grouping that equalizes downward. 

Minimal Guidance = Minimal Learning. 

We are not teaching enough essential content and often substitute critical thinking for factual and procedural knowledge. The problem we have in the 4th grade and above starts in the 1st grade. Kids are not taught basic arithmetic that counts. Memorization of single-digit number facts and the standard algorithms, for example, are not taught for mastery. Instead, students are taught reform math with so-called "standards of mathematical practice" and a host of alternative algorithms. With ESSA and mandatory state testing, teachers teach the test items, which fragments the curriculum. 

Knowledge in long-term memory supports and enables thinking. 

© 2018 LT/ThinkAlgebra