Wednesday, December 30, 2015

Adding It Up

Adapting Thinking: Adding It Up 

Why is a 2nd-3rd-grade question given to 8th graders? Expectations are low. 

Only 61% of 13-year olds selected the correct answer. In my view, it is a 2nd-grade question, not a middle school question, and clearly indicates the fundamental relationship between addition and subtraction that all 13-year olds should know, but, apparently, many don't. If arithmetic were taught well, then most 2nd graders would have selected the correct answer without calculating. The example from Adding It Up (2001) [1] shows how poorly arithmetic has been taught under NCTM reform math. However, in my opinion, Adding It Up seems to confirm many reform math practices while ignoring the science of learning. For decades, the use of calculators and so on, which are typical NCTM math reforms, have pushed aside standard arithmetic in K-8 schools, an error in judgment. 

Adding It Up: "Only 61% of 13-year-olds chose the right answer, which again is considerably lower than the percentage of students who can compute the result." What percentage might that be? 90%? 100%? Adding It Up erroneously assumes or suggests that students who practice standard algorithms for mastery have little understanding of number relationships. The reason that only 61% of the 8th graders selected the correct equation, rather than a higher percentage, is that the fundamentals of arithmetic (via NCTM reform math) have not been taught well. The relationship between addition and subtraction is basic arithmetic, but so is competence in calculating via the standard algorithms (paper-pencil). Also, G. Polya (How To Solve It) states that understanding in mathematics is in the doing of arithmetic, i.e., applying it. 

Note. The Adding It Up report (PreK-8) of 2001 from the National Research Council is hardly the final word, of course. The Adding It Up theory of proficiency in mathematics is a fabricated on five intertwined strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. The far-reaching theory of proficiency is based more on judgment than on science and has never been tested. In fact, there are countries that clobber US students in math, yet the Adding It Up report asserts that "no country--not even those performing highest on international surveys of mathematics achievement do all students display mathematical proficiency as we have defined it in this report." Put simply, the five intertwined strands of proficiency are not practical and almost impossible for typical kids to achieve, even the best kids.   

Adding It Up provides cover for NCTM reform math programs, such as Investigations (TERC), a program that is still used in many schools and embodies the math reform movement that focuses more on understanding than on learning standard arithmetic. The Investigations curriculum uses "minimal guidance during instruction" methods, that is, child-centered discovery activities. After examining the 5th-grade materials, mathematician W. Stephen Wilson (Johns Hopkins University), wrote that Investigations was not standard arithmetic. He called it pre-arithmetic. Professor Wilson writes, "Arithmetic is the foundation. Arithmetic has to be a priority, and it has to be done right." Starting in 1st grade, Singapore math does it right most of the time [2]; however, Investigations and other reform math programs do not.

Adding It Up has had a profound influence in math education, and, often, not in a good way. Its central premise is that proficiency is too narrowly defined. The report states, "Mathematical proficiency, as we see it, has five (intertwined) strands." Really? More judgment, less research. Also, the report states, "Many educational questions, however, cannot be answered by research." Education depends on "judgments" that "often fall outside the domain of research," especially in curriculum and instruction. Really?

I disagree. Math is hierarchical: one idea builds another and everything fits together logically. A good math curriculum starts with standard arithmetic. We know the essential content and skills (the curriculum) needed to get kids off to a world-class start starting in 1st grade. Indeed, content and its associated skills are hierarchical, along with intellectual skills (Gagne: instructional design and prerequisites) [3]. Because math builds in long-term memory, the proper sequencing that creates coherence (a learning hierarchy) in a math curriculum is paramount [4].

Moreover, Gagne writes that "intellectual skills are arranged in a hierarchical order so that successful instruction begins with teaching lower-order skills and progresses upwards." Furthermore, in addition to arithmetic, the elementary school math curriculum should include parts of algebra, geometry, and measurement to prepare for a full course in algebra by middle school. Also, we know from cognitive science that direct instruction is strikingly more efficient than the favored minimal guidance methods of teaching, group work, nonstandard algorithms, manipulatives, and multiple representations, which are among the least effective. In short, the diverse group of Adding It Up writers ignores the cognitive science of learning.

Despite what you may have heard from reform math apostles, there is nothing intrinsically wrong with standard arithmetic. Indeed, it is the keystone for higher-level math. Therefore, very young students should practice standard algorithms for mastery, grasp the rules of arithmetic that govern the behavior of numbers, memorize math facts for auto recall in problem-solving, and apply math concepts to everyday problems.

Beginners need lots of factual and procedural knowledge to do the math, says Daniel Willingham, a cognitive scientist. Indeed, knowing and doing math well requires factual and procedural knowledge in long-term memory. The modern reform math methods of minimal guidance during instruction (e.g., discovery, inquiry, problem-based, etc.) don't work, say, Kirschner, Sweller, & Clark. They point out, "Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture [working and long-term memories], expert-novice differences, and cognitive load." 

Children are not pint-sized mathematicians or experts; they are novices. Contrary to Adding It Up, beginners don't need to explain their reasoning, make drawings, or engage in group work to learn arithmetic well. Students need straightforward instruction via carefully thought out, coherent, hierarchically organized worked examples. The standard algorithms are the most efficient ways to do arithmetic, but for years, they have been under brutal attack. Reformists say the standard algorithms are too hard for some students to learn, threaten a student's growth in independent thinking, and are obsoleted by calculators. These are bogus arguments. The National Mathematics Advisory Panel (2008) explicitly stated that students must master standard arithmetic to prepare for algebra, not something that looks like arithmetic or something that has no long-term value. 

According to Adding It Up, "Nearly all second graders might be expected to make a useful drawing of the situation portrayed in an arithmetic word problem as a step toward solving it." In short, making a picture is the first step needed to solve a word problem. Nonsense! The idea of "making a drawing" as a necessary step for problem-solving has emerged as a best practice in reform math programs via NCTM and now Common Core state standards, etc. The idea is misguided.

Add It Up Box 5.15 Never teach these strategies. Teach the standard algorithm.

Adding It Up sharply criticizes the standard algorithm for division. I won't go into details, but the report is wrong.

Instead, the report offers two alternative versions. One is Box 5-15. The other is the "partial quotients" model shown in Box 5-16 below. 

No one uses the partial quotient method to do long division (Box 5-16) or the area model to do multiplication, etc., much less the methods shown in Box 6-15. Who would calculate this way? They are a waste of classroom time.

Add It Up Box 5-16 Teach the standard algorithm, not this.

Still, in modern reform math classrooms, a disproportionate amount of classroom time is spent on these and other similar calculation strategies leaving efficient standard algorithms, which are vital, on the back burner. In my opinion, nonstandard, complicated, multiple strategies to do simple arithmetic are usually a waste of valuable instructional time. Kids need to know the standard algorithms (Box 5-14)

The long division standard algorithm should start no later than 3rd grade with up to 4 digits divided by one digit, sometimes two digits.
This is the standard algorithm. Teach it first.

[4] Break a problem into smaller problems. 

This is a fundamental idea taught in mathematics, and it can carry over to everyday life. Also, the idea that new knowledge builds on old knowledge is central to learning math. Because math builds in long-term memory, the proper sequencing that creates coherence (a learning hierarchy) in a math curriculum is paramount. Here is a sequencing example from Science--A Process Approach (SAPA), which uses Gagne’s hierarchical approach. It is not hit and miss. The sequencing (learning hierarchy) must work in the classroom, which is the reason SAPA was tested extensively and rewritten several times before it was released to the public.

Part C is 2nd Grade - 1967
FYI: Integers were introduced in Part B, 1st Grade. 

[1] Adding It Up is a product of the National Research Council, specifically the Mathematics Learning Study Committee, Division of Behavioral and Social Sciences and Education (2001). According to the report, Adding It Up was written by a committee composed of "diverse backgrounds." Its theory of proficiency in mathematics is based on five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. The theory is based more on judgment than on science and never tested. What, no mathematicians?

[2] Singapore 1st-grade students learn much more standard arithmetic than American 1st-grade students and so on up the grades. The curriculum is better than in most countries. Still, the Singaporean 1st-grade math curriculum isn't perfect. In my view, it lacks some essential content, especially algebra and integers, topics I typically teach to 1st-grade students. Moreover, the overemphasis on bar models (drawings) to solve arithmetic problems in Singaporean math can be distracting. A few kids might benefit from drawing bar models, but, for many kids, making a drawing slows up and disrupts cognitive activity. 

[3] Robert Gagne greatly influenced the hierarchy of Science A Process Approach (SAPA) by identifying the processes and prerequisites: observing, classifying, using numbers, measuring, predicting, inferring, formulating hypotheses, and interpreting data. But these processes are actually skills, which are essential to inquiry, analytic thinking, and problem-solving, explains Henry P. Cole (Process Education, 1972). The processes are actually skills of doing something, so they are measurable. The thinking is hidden in the doing.  

To Be Continued. 

©2016 LT/ThinkAlgebra