StrongGuidance

The topics are in no particular order.
Explicit Instruction: Children need strong, teacher-guided instruction.
Minimal-guided instruction has been an epic flop.
Progressive school reforms haven't worked.
Standard Algorithms: We make them up to fit the real world.
"Understanding" is a slippery slope because it is hard to measure.

Look at what reformers have done to simple arithmetic! It is bizarre! Common Core math has been interpreted and/or implemented as reform math by those in power over education in an effort to weaken "tried and true" traditional teaching of standard math. Below is a 5th grade Common Core math quiz from Kaplan. It represents a typical misinterpretation of Common Core as reform math.

Kaplan's answer to #3: Area Model

1. Find 15.7 + 9.72 by decomposing the numbers by place value. Show your work.
2. Find 9.53 - 4.6 using a place value chart. Show your work.
3. Find 5.3 x 2.4 using an area model. Show your work.
4. Find 4.8 / 0.8 using a number line model. Show your work.
5. Find 3.6  / 12 using a bar model. Show your work.

Who multiplies whole numbers, fractions, or decimals using an area model? Frankly, I have trouble believing that teachers would actually teach this junk and diminish the importance of standard algorithms, yet this seems to be the case in many classrooms. Standard algorithms get scant coverage, if they are taught at all. They are seldom  practiced for mastery. Without memorizing single-digit number facts for instant recall (key factual knowledge) and gaining proficiency in standard algorithms (efficient procedural knowledge), "students are severely handicapped as [they] attempt to pursue the next levels of mathematics," warns Professor W. Stephen Wilson.

Dr. W. Stephen Wilson, a mathematician at Johns Hopkins University, critiqued a popular reform math program (Pearson's Investigations, 5th Grade) and said it was actually "pre-arithmetic." Students never get to arithmetic, implying that students do not focus on memorizing single-digit number facts or on practicing the standard algorithms for mastery. In short, instead of  focusing on standard algorithms, students are often taught many inferior or weak methods or strategies that are not practical or useful. By pre-arithmetic, Professor Wilson means that kids learn something that looks like arithmetic, but it isn't the arithmetic that students need to know to advance to algebra by 8th grade. Is it any wonder that most students struggle with basic arithmetic and math in general? 

Traditional arithmetic works well when taught well. Students become better at mental math because they have memorized basic number facts. Furthermore, the standard algorithm always works. We keep forgetting that little kids are novices. They don't think like adults. Children need to memorize and practice to put mathematical knowledge, both factual and procedural, into long-term memory for instant use in problem solving. Students cannot do mathematics without knowing some mathematics. Also, understanding is a slow process. It does not produce competency, practice does. Do not expect instant understanding or hold kids back because their understanding is partial or incomplete. Furthermore, Jason Zimba, one of the two major writers of CC math standards, clarifies, "The standards also allow for approaches in which the standard algorithm is instructed in grade 1, and in which only a single algorithm is taught for each operation."

Note. Isaac Newton invented a fast way to calculate answers to physics problems, called calculus. It always worked (i.e., it was consistent with experimental data), but he didn't understand why the calculus, itself, worked; it just did. The "why" would take another 200 years. Indeed, maybe "understanding" is overrated and a slippery slope, something to stay away from. Newton was a true polymath, but, under Common Core reform math, he would not have been able to write a paragraph that explains why his procedures work; however, Newton might say through inductive reasoning, "It works in all the cases I have tired, therefore it must be correct."

Algorithms--we make them up to fit our experience in the real world. If we take 10 oranges and split each orange in half, then how many halves would we have? {20} That is, 10 ÷ 1/2 = 20 gives the same answer as 10 x 2 = 20. From this fact and others, together with the idea of reciprocals (1/2 and 2 are reciprocals because their product is ONE), we could make up an efficient algorithm for division of fractions that always works in the real world, and we have: To divide by a fraction, multiply by its reciprocal, which is the standard algorithm. It fits our experience. The proof, however, is left to mathematicians, not to novice kids. Kids should not be expected to reinvent arithmetic or be required to write explanations, but they do need to apply rules procedures and perform steps correctly. Furthermore, writing a "why" paragraph does not imply "deep understanding," whatever that is. Indeed, understanding is vague and very difficult to measure or pin down. In math, understanding varies widely, is always imperfect, and grows slowly over time with practice. It is a slippery slope best left alone.

Explicit teaching, which uses a carefully-planned sequence of worked examples in math works well for almost all students. Students learn concepts through examples, lots of practice, and repetition, says Zig Engelmann. However, since the 60s, teacher-led instruction has been called "old school" or the opposite of “good” teaching. Explicit, teacher-led instruction (using examples, practice, and repetition) “contradicts much of what educators are taught to believe about good teaching,” writes J. E. Stone (Clear Teaching).

Stone says that explicit teaching [the teacher is the academic leader that leads instruction by explaining examples on the board, etc.] has not been popular in K-8 schools, not because it didn't work but because it goes against progressive reform ideology taught in schools of education. The Progressive Era revolution of the 60s affected education by attacking teacher-led exercises, scripted lessons, skill grouping, choral responding, repetition, etc., says Stone. “Thus, education professors and theorists denigrate teacher-led practice as ‘drill and kill,’ its high expectations as ‘developmentally inappropriate,’ and its emphasis on building a solid foundation of skills as ‘rote learning’,” writes Stone. In short, kids have not been taught a solid foundation of arithmetic for multiple decades.

Today we have teachers as facilitators, not academic leaders; mainstreaming (inclusion); a weak, incoherent, narrowed curriculum; low expectations for students; popular reform methods of instruction (i.e., minimal guided, not teacher-led) that are inferior; reforms such as Common Core, intrinsically linked to standardized testing; etc. Education is no longer a "work hard and achieve" narrative; it is a political, test-centered, money-driven narrative.  

Note. I have quoted this study (Kirchner-Sweller-Clark: Why Minimal Guidance During Instruction Does Not Work...) since it first appeared in Educational Psychologist in 2006. The instructional methods in classrooms across the US--mostly group work activities with minimal teacher guidance or no teacher guidance--have failed our students for decades. The minimal guidance instructional methods, which come in different names or favors over the years (e.g., discovery, constructivist, problem-based, inquiry-based, etc.) and have been championed by schools of education, extend to Common Core. They are part of the progressive movement in education, starting with Dewey. Kids do a lot of group work, use manipulatives, etc. Their desks are in groups of 3 or 4, so kids face each other. Instead of being the academic leader in the classroom, the teacher's role has diminished to a "facilitator" of learning. In short, the teacher no longer teaches.

Kirchner-Sweller-Clark (Why Minimal Guidance During Instruction Does Not Work...) write [long quote], "Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although unguided or minimally guided instructional approaches are very popular and intuitively appealing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that consistently indicate that minimally guided instruction is less effective and less efficient than instructional approaches that place a strong emphasis on guidance of the student learning process. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide “internal” guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described."

Kirchner-Sweller-Clark write, "Cognitive load. Sweller and others (Mayer, 2001; Paas, Renkl, & Sweller, 2003, 2004; Sweller, 1999, 2004; Winn, 2003) noted that despite the alleged advantages of unguided environments to help students to derive meaning from learning materials, cognitive load theory suggests that the free exploration of a highly complex environment may generate a heavy working memory load that is detrimental to learning. This suggestion is particularly important in the case of novice learners, who lack proper schemas to integrate the new information with their prior knowledge."

Understanding is overrated and a slippery slope. If students can apply the math they have learned, then this cognitive outcome likely implies that they have some understanding of it, but I don’t know how much because I cannot measure it. I also think that understanding varies widely as does academic ability and that the two are correlated. Often, I hear reformers claim that novices need "deep understanding." What is that? How is it measured?

Reprinted [with additions and changes] from Strong Teacher Guidance, February 25, 2015, Math Notes by ThinkAlgebra

© 2015 LT, ThinkAlgebra,org