If the vision of the Sustainable Development Goals (SDGs) is that Mother Earth is heading for trouble and we must collectively solve global problems, then the underfunding of global public goods (GPGs) must be addressed. As the world becomes increasingly globalized, the need for global public goods increases: from action on climate change, financial stability, limiting the spread of diseases, management of conflicts, responding to natural disasters, terrorism, and cyber-warfare. At some level even the eradication of extreme poverty and more inclusive and sustainable development could be considered a global public good because more poverty and unequal development breeds conflict, increases environmental stress, state failure, terrorism, and piracy, thereby increasing the need for the global public goods required to address these issues.

Missing in the recently agreed Addis Ababa Action Agenda (AAAA) and in the Paris Conference of Parties (COP21) are steps that should be taken at a global level that will positively impact many countries, such as:

• A global set of standards on migration to curb exploitation and human rights standards for the migrant population;
• Better coordination of monetary and fiscal policies so as to avoid huge volatility in financial markets, which have large costs on vulnerable countries;
• Strengthened global disaster response mechanisms to handle increasing climate volatility and natural disasters;
• No agreement on a global tax institution demanded by many developing countries and civil society groups; and,
• No progress on carbon taxation.

There is considerable underfinancing of GPGs as it is difficult to get countries to pay for activities outside their borders. Official Development Assistance (ODA) has fallen well short of the agreed target of 0.7 percent of GDP—and in fact is closer to just 0.2 percent. GPG funding from ODA is estimated at only about 10 percent of the total. This problem even afflicts other sources of financing. Multilateral development bank (MDB) financing also underfunds regional, multi-country projects for addressing regional public goods as countries are unwilling to use their country allocations for multi-country projects even if the return on them is higher than the marginal country project.

Global thematic funds to support specific development challenges—Global Alliance for Vaccination and Inoculation (GAVI), Global Fund to Fight AIDS, Tuberculosis and Malaria (GFATM), Global Environmental Fund (GEF) and earlier funds like the Consultative Group for International Agricultural Research (CGIAR)—have been successful in addressing specific development challenges through projects in specific countries, especially for agriculture, the environment, and health. They have also drawn in private philanthropic financing in addition to public resources. But global funding for global public goods has not had the same success, and systematic and sustained financing for disasters, biodiversity, desertification, and even for Ebola outbreaks has been difficult.

The Green Climate Fund, which will begin its work this year and will devote 50:50 share of funding for adaptation and mitigation has very limited funding so far – despite the commitment to provide $ 100 billion per year over and above ODA. But neither the AAAA, nor the SDG’s address many of the trade-offs involved between climate change and poverty eradication. COP 21 also did not provide greater guidance on these matters – despite high expectations that it would. Given the need for rapid economic growth to eradicate poverty for the LDC’s  as well as their need to deal with huge adaptation costs, it probably makes sense not to focus excessively on mitigation in these countries. These countries would increase their global carbon footprint by at best 2-3 percent of the total carbon emissions. The big tradeoffs will arise in the need for rapid growth in middle-income countries to address poverty and their increased emissions, which will accompany faster growth.

Protection of biodiversity is given specific mention in the AAAA, and the Global Strategic Plan for Biodiversity for 2011-20 is endorsed along with its 20 Aichi biodiversity targets. But progress in meeting these targets is slow and at current trends unlikely to be achieved. The AAAA does not address this slow progress or suggest ways to accelerate it. It does endorse the U.N. Convention to Combat Desertification and the African Union Green Wall Initiative; but again with no specificity on how progress on these commitments will be accelerated. The same is true of the attention on oceans and marine resources, where the U.N. Convention on the Law of the Sea is mentioned but with no concrete steps on how to finance, enforce, and protect vulnerable areas, especially the small island developing states (SIDS).

Private philanthropic foundations have played important catalytic roles, such as efforts by the Ford Foundation and the Rockefeller Foundation to help jump-start the Green Revolution in the 1960’s, and the eventual creation of the CGIAR. A somewhat similar role has been played by the Bill & Melinda Gates Foundation for global public health. But no such foundations exist for many underfunded issues, such as disaster relief, peacebuilding, and desertification. These types of activities can be much better funded by more globalized revenue sources. The AAAA does not even mention the need for any such revenue sources.

A key GPG is peacekeeping, international security, and the prevention of conflict. Surprisingly, military spending is also not touched upon in the AAAA but has increased sharply. It dropped in the late 1990s following the end of the Cold War, from $1.5 trillion to around $1 trillion globally, but has increased again to almost $ 2 trillion today. Cutting military expenditure—especially in many developing countries where it exceeds 4 percent of GDP—would be an important step and shifting some of those resources to peacekeeping and conflict prevention would improve public spending.

With the AAAA pushing for new modes of financing, its surprising that for GPG financing more global sources of finance are not considered. At least four such options exist and could go a long way towards financing the SDGs. The first is a carbon tax or auctioning of carbon emissions permits. This is an idea with huge appeal as it will also help dissuade use of fossil fuels and could lower emissions globally, but is opposed by all the major emitters. Carbon taxes have been used in several countries to reduce fossil fuel use without any damage to long-term growth. Emission permits have also been used in some countries to reduce emissions of some harmful chemicals. But they have not been used internationally.

The second is a so-called “Tobin tax,” a tax on all foreign exchange transactions, which might also discourage destabilizing short-term volatile capital movements. The third is to add a pollution tax on all shipping and air travel – whose pollutions costs are not fully captured by existing taxes and fees imposed on them. The fourth is to allow issuance of SDRs to finance GPG’s.

Unfortunately, all these proposals are currently opposed by the major G-20 countries for various reasons. While several European countries—and even some developing ones—have introduced carbon taxes, still more remain opposed to carbon taxation. The Tobin tax idea has been around now for several decades and is considered an anti-globalization proposal even if its revenues were to be used to finance GPGs.  At times in the past, some countries have imposed a tax on foreign exchange transactions, with the explicit purpose of slowing down volatility in capital markets.

Global taxation has the connotation of supra-nationality, which many rich country legislatures—especially in the U.S.—would oppose. One way around this might be to specify how these resources would be used or to use them through MDBs where the richer countries have a controlling vote. To some extent the Global programs—GAVI, GFATM, CGIAR, and now the Green Climate Fund—have done that, but their financing remains much too dependent on national budgets and not on automatic revenue-raising mechanisms. National lotteries have been used in some countries to raise resources for specific causes; global lotteries could be an option for financing some specific global goods. But the world must move to some global means of revenue-raising if it wants to address GPGs seriously. Private financing, innovative financing, and public-private partnerships touted in the AAAA and COP21 can be crowded in, but without more international public financing to address market failure, financing the SDG’s will be difficult.

The world needs to heed Ben Franklin advice in another context “We must hang together or surely we will hang separately.”

On November 7, millions of Americans will exercise their civic duty to vote. At stake will be control of the House and Senate, not to mention the success of individual candidates running for office. President Bush's "stay the course" agenda will either be enabled over the next two years by a Republican Congress or knocked off kilter by a Democratic one.

With so much at stake, it is not surprising that the Pew Research Center found that 51 percent of registered voters have given a lot of thought to this November's election. This is higher than any other recent midterm election, including 44 percent in 1994, the year Republicans took control of the House. If so, turnout should better the 1994 turnout rate among eligible voters of 41 percent.

There is good reason to suspect that despite the high interest, turnout will not exceed 1994. The problem is that a national poll is, well, a national poll, and does not measure attitudes of voters within states and districts.

People vote when there is a reason to do so. Republican and Democratic agendas are in stark contrast on important issues, but voters also need to believe that their vote will matter in deciding who will represent them. It is here that the American electoral system is broken for many voters.

Voters have little choice in most elections. In 1994, Congressional Quarterly called 98 House elections as competitive. Today, they list 51. To put it another way, we are already fairly confident of the winner in nearly 90 percent of House races. Although there is no similar tracking for state legislative offices, we know that the number of elections won by less than 60 percent of the vote has fallen since 1994.

The real damage to the national turnout rate is in the large states of California and New York, which together account for 17 percent of the country's eligible voters. Neither state has a competitive Senate or Governor's election, and few competitive House or state legislative races. Compare to 1994, when Californians participated in competitive Senate and governor races the state's turnout was 5 percentage points above the national rate. The same year New York's competitive governor's race helped boost turnout a point above the national rate.

Lacking stimulation from two of the largest states, turnout boosts will have to come from elsewhere. Texas has an interesting four-way governor's race that might draw from infrequent voters to the polls. Ohio's competitive Senate race and some House races might also draw voters. However, in other large states like Florida, Illinois, Michigan and Pennsylvania, turnout will suffer from largely uncompetitive statewide races.

The national turnout rate will likely be less than 1994 and fall shy of 40 percent. This is not to say that turnout will be poor everywhere. Energized voters in Connecticut get to vote in an interesting Senate race and three of five Connecticut House seats are up for grabs. The problem is that turnout will be localized in these few areas of competition.

The fault is not on the voters; people's lives are busy, and a rational person will abstain when their vote does not matter to the election outcome. The political parties also are sensitive to competition and focus their limited resources where elections are competitive. Television advertising and other mobilizing efforts by campaigns will only be found in competitive races.

The old adage of "build it and they will come" is relevant. All but hardcore sports fans tune out a blowout. Building competitive elections -- and giving voters real choices -- will do much to increase voter turnout in American politics. There are a number of reforms on the table: redistricting to create competitive districts, campaign financing to give candidates equal resources, and even altering the electoral system to fundamentally change how a vote elects representatives. If voters want choice and a government more responsive to their needs, they should consider how these seemingly arcane election procedures have real consequences on motivating them to do the most fundamental democratic action: vote.

The promise of geoengineering is placing average global temperature under human control, and is thus considered a powerful instrument for the international community to deal with global warming. While great energy has been devoted to learning more about the natural systems that it would affect, questions of political nature have received far less consideration. Taking as a given that regional effects will be asymmetric, the nations of the world will only give their consent to deploying this technology if they can be given assurances of a fair compensation mechanism, something like an insurance policy. The question of compensation reveals that the politics of geoengineering are far more difficult than the technical aspects.

What is Geoengineering?

In June 1991, Mount Pinatubo exploded, throwing a massive amount of volcanic sulfate aerosols into the high skies. The resulting cloud dispersed over weeks throughout the planet and cooled its temperature on average 0.5° Celsius over the next two years. If this kind of natural phenomenon could be replicated and controlled, the possibility of engineering the Earth’s climate is then within reach.

Spraying aerosols in the stratosphere is one method of solar radiation management (SRM), a class of climate engineering that focuses on increasing the albedo, i.e. reflectivity, of the planet’s atmosphere. Other SRM methods include brightening clouds by increasing their content of sea salt. A second class of geo-engineering efforts focuses on carbon removal from the atmosphere and includes carbon sequestration (burying it deep underground) and increasing land or marine vegetation. Of all these methods, SRM is appealing for its effectiveness and low costs; a recent study put the cost at about $5 to $8 billion per year.¹

Not only is SRM relatively inexpensive, but we already have the technological pieces that assembled properly would inject the skies with particles that reflect sunlight back into space. For instance, a fleet of modified Boeing 747s could deliver the necessary payload. Advocates of geoengineering are not too concerned about developing the technology to effect SRM, but about its likely consequences, not only in terms of slowing global warming but the effects on regional weather. And there lies the difficult question for geoengineering: the effects of SRM are likely to be unequally distributed across nations.

Here is one example of these asymmetries: Julia Pongratz and colleagues at the department of Global Ecology of the Carnegie Institution for Science estimated a net increase in yields of wheat, corn, and rice from SRM modified weather. However, the study also found a redistributive effect with equatorial countries experiencing lower yields.² We can then expect that equatorial countries will demand fair compensation to sign on the deployment of SRM, which leads to two problems: how to calculate compensation, and how to agree on a compensation mechanism.

The calculus of compensation

What should be the basis for fair compensation? One view of fairness could be that, every year, all economic gains derived from SRM are pooled together and distributed evenly among the regions or countries that experience economic losses.

If the system pools gains from SRM and distributes them in proportion to losses, questions about the balance will only be asked in years in which gains and losses are about the same. But if losses are far greater than the gains; then this would be a form of insurance that cannot underwrite some of the incidents it intends to cover. People will not buy such an insurance policy; which is to say, some countries will not authorize SRM deployment. In the reverse, if the pool has a large balance left after paying out compensations, then winners of SRM will demand lower compensation taxes.

Further complicating the problem is the question of how to separate gains or losses that can be attributed to SRM from regional weather fluctuations. Separating the SRM effect could easily become an intractable problem because regional weather patterns are themselves affected by SRM.  For instance, any year that El Niño is particularly strong, the uncertainty about the net effect of SRM will increase exponentially because it could affect the severity of the oceanic oscillation itself. Science can reduce uncertainty but only to a certain degree, because the better we understand nature, the more we understand the contingency of natural systems. We can expect better explanations of natural phenomena from science, but it would be unfair to ask science to reduce greater understanding to a hard figure that we can plug into our compensation equation.

Still, greater complexity arises when separating SRM effects from policy effects at the local and regional level. Some countries will surely organize better than others to manage this change, and preparation will be a factor in determining the magnitude of gains or losses. Inherent to the problem of estimating gains and losses from SRM is the inescapable subjective element of assessing preparation. 

The politics of compensation

Advocates of geoengineering tell us that their advocacy is not about deploying SRM; rather, it is about better understanding the scientific facts before we even consider deployment. It’s tempting to believe that the accumulating science on SRM effects would be helpful. But when we consider the factors I just described above, it is quite possible that more science will also crystalize the uncertainty about exact amounts of compensation. The calculus of gain or loss, or the difference between the reality and a counterfactual of what regions and countries will experience requires certainty, but science only yields irreducible uncertainty about nature.

The epistemic problems with estimating compensation are only to be compounded by the political contestation of those numbers. Even within the scientific community, different climate models will yield different results, and since economic compensation is derived from those models’ output, we can expect a serious contestation of the objectivity of the science of SRM impact estimation. Who should formulate the equation? Who should feed the numbers into it? A sure way to alienate scientists from the peoples of the world is to ask them to assert their cognitive authority over this calculus. 

What’s more, other parts of the compensation equation related to regional efforts to deal with SRM effect are inherently subjective. We should not forget the politics of asserting compensation commensurate to preparation effort; countries that experience low losses may also want compensation for their efforts preparing and coping with natural disasters.

Not only would a compensation equation be a sham, it would be unmanageable. Its legitimacy would always be in question. The calculus of compensation may seem a way to circumvent the impasses of politics and define fairness mathematically. Ironically, it is shot through with subjectivity; is truly a political exercise.

Can we do without compensation?

Technological innovations are similar to legislative acts, observed Langdon Winner.³ Technical choices of the earliest stage in technical design quickly “become strongly fixed in material equipment, economic investment, and social habit, [and] the original flexibility vanishes for all practical purposes once the initial commitments are made.” For that reason, he insisted, "the same careful attention one would give to the rules, roles, and relationships of politics must also be given to such things as the building of highways, the creation of television networks, and the tailoring of seeming insignificant features on new machines."

If technological change can be thought of as legislative change, we must consider how such a momentous technology as SRM can be deployed in a manner consonant with our democratic values. Engineering the planet’s weather is nothing short of passing an amendment to Planet Earth’s Constitution. One pesky clause in that constitutional amendment is a fair compensation scheme. It seems so small a clause in comparison to the extent of the intervention, the governance of deployment and consequences, and the international commitments to be made as a condition for deployment (such as emissions mitigation and adaptation to climate change). But in the short consideration afforded here, we get a glimpse of the intractable political problem of setting up a compensation scheme. And yet, if the clause were not approved by a majority of nations, a fair compensation scheme has little hope to be consonant with democratic aspirations.

¹McClellan, Justin, David W Keith, Jay Apt. 2012. Cost analysis of stratospheric albedo modification delivery systems. Environmental Research Letters 7(3): 1-8.

²Pongratz, Julia, D. B. Lobell, L. Cao, K. Caldeira. 2012. Nature Climate Change 2, 101–105.

³Winner, Langdon. 1980. Do artifacts have politics? Daedalus (109) 1: 121-136.

This is part two of my analysis of instruction and Common Core’s implementation.  I dubbed the three-part examination of instruction “The Good, The Bad, and the Ugly.”  Having discussed “the “good” in part one, I now turn to “the bad.”  One particular aspect of the Common Core math standards—the treatment of standard algorithms in whole number arithmetic—will lead some teachers to waste instructional time.

In 1963, psychologist John B. Carroll published a short essay, “A Model of School Learning” in Teachers College Record.  Carroll proposed a parsimonious model of learning that expressed the degree of learning (or what today is commonly called achievement) as a function of the ratio of time spent on learning to the time needed to learn.     

The numerator, time spent learning, has also been given the term opportunity to learn.  The denominator, time needed to learn, is synonymous with student aptitude.  By expressing aptitude as time needed to learn, Carroll refreshingly broke through his era’s debate about the origins of intelligence (nature vs. nurture) and the vocabulary that labels students as having more or less intelligence. He also spoke directly to a primary challenge of teaching: how to effectively produce learning in classrooms populated by students needing vastly different amounts of time to learn the exact same content.^[i] 

The source of that variation is largely irrelevant to the constraints placed on instructional decisions.  Teachers obviously have limited control over the denominator of the ratio (they must take kids as they are) and less than one might think over the numerator.  Teachers allot time to instruction only after educational authorities have decided the number of hours in the school day, the number of days in the school year, the number of minutes in class periods in middle and high schools, and the amount of time set aside for lunch, recess, passing periods, various pull-out programs, pep rallies, and the like.  There are also announcements over the PA system, stray dogs that may wander into the classroom, and other unscheduled encroachments on instructional time.

The model has had a profound influence on educational thought.  As of July 5, 2015, Google Scholar reported 2,931 citations of Carroll’s article.  Benjamin Bloom’s “mastery learning” was deeply influenced by Carroll.  It is predicated on the idea that optimal learning occurs when time spent on learning—rather than content—is allowed to vary, providing to each student the individual amount of time he or she needs to learn a common curriculum.  This is often referred to as “students working at their own pace,” and progress is measured by mastery of content rather than seat time. David C. Berliner’s 1990 discussion of time includes an analysis of mediating variables in the numerator of Carroll’s model, including the amount of time students are willing to spend on learning.  Carroll called this persistence, and Berliner links the construct to student engagement and time on task—topics of keen interest to researchers today.  Berliner notes that although both are typically described in terms of motivation, they can be measured empirically in increments of time.     

Most applications of Carroll’s model have been interested in what happens when insufficient time is provided for learning—in other words, when the numerator of the ratio is significantly less than the denominator.  When that happens, students don’t have an adequate opportunity to learn.  They need more time. 

As applied to Common Core and instruction, one should also be aware of problems that arise from the inefficient distribution of time.  Time is a limited resource that teachers deploy in the production of learning.  Below I discuss instances when the CCSS-M may lead to the numerator in Carroll’s model being significantly larger than the denominator—when teachers spend more time teaching a concept or skill than is necessary.  Because time is limited and fixed, wasted time on one topic will shorten the amount of time available to teach other topics.  Excessive instructional time may also negatively affect student engagement.  Students who have fully learned content that continues to be taught may become bored; they must endure instruction that they do not need.

Jason Zimba, one of the lead authors of the Common Core Math standards, and Barry Garelick, a critic of the standards, had a recent, interesting exchange about when standard algorithms are called for in the CCSS-M.  A standard algorithm is a series of steps designed to compute accurately and quickly.  In the U.S., students are typically taught the standard algorithms of addition, subtraction, multiplication, and division with whole numbers.  Most readers of this post will recognize the standard algorithm for addition.  It involves lining up two or more multi-digit numbers according to place-value, with one number written over the other, and adding the columns from right to left with “carrying” (or regrouping) as needed.

The standard algorithm is the only algorithm required for students to learn, although others are mentioned beginning with the first grade standards.  Curiously, though, CCSS-M doesn’t require students to know the standard algorithms for addition and subtraction until fourth grade.  This opens the door for a lot of wasted time.  Garelick questioned the wisdom of teaching several alternative strategies for addition.  He asked whether, under the Common Core, only the standard algorithm could be taught—or at least, could it be taught first. As he explains:

Delaying teaching of the standard algorithm until fourth grade and relying on place value “strategies” and drawings to add numbers is thought to provide students with the conceptual understanding of adding and subtracting multi-digit numbers. What happens, instead, is that the means to help learn, explain or memorize the procedure become a procedure unto itself and students are required to use inefficient cumbersome methods for two years. This is done in the belief that the alternative approaches confer understanding, so are superior to the standard algorithm. To teach the standard algorithm first would in reformers’ minds be rote learning. Reformers believe that by having students using strategies in lieu of the standard algorithm, students are still learning “skills” (albeit inefficient and confusing ones), and these skills support understanding of the standard algorithm. Students are left with a panoply of methods (praised as a good thing because students should have more than one way to solve problems), that confuse more than enlighten. 

Zimba responded that the standard algorithm could, indeed, be the only method taught because it meets a crucial test: reinforcing knowledge of place value and the properties of operations.  He goes on to say that other algorithms also may be taught that are consistent with the standards, but that the decision to do so is left in the hands of local educators and curriculum designers:

In short, the Common Core requires the standard algorithm; additional algorithms aren’t named, and they aren’t required…Standards can’t settle every disagreement—nor should they. As this discussion of just a single slice of the math curriculum illustrates, teachers and curriculum authors following the standards still may, and still must, make an enormous range of decisions.

Zimba defends delaying mastery of the standard algorithm until fourth grade, referring to it as a “culminating” standard that he would, if he were teaching, introduce in earlier grades.  Zimba illustrates the curricular progression he would employ in a table, showing that he would introduce the standard algorithm for addition late in first grade (with two-digit addends) and then extend the complexity of its use and provide practice towards fluency until reaching the culminating standard in fourth grade. Zimba would introduce the subtraction algorithm in second grade and similarly ramp up its complexity until fourth grade.

It is important to note that in CCSS-M the word “algorithm” appears for the first time (in plural form) in the third grade standards:

3.NBT.2  Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

The term “strategies and algorithms” is curious.  Zimba explains, “It is true that the word ‘algorithms’ here is plural, but that could be read as simply leaving more choice in the hands of the teacher about which algorithm(s) to teach—not as a requirement for each student to learn two or more general algorithms for each operation!” 

I have described before the “dog whistles” embedded in the Common Core, signals to educational progressives—in this case, math reformers—that  despite these being standards, the CCSS-M will allow them great latitude.  Using the plural “algorithms” in this third grade standard and not specifying the standard algorithm until fourth grade is a perfect example of such a dog whistle.

It appears that the Common Core authors wanted to reach a political compromise on standard algorithms. 

Standard algorithms were a key point of contention in the “Math Wars” of the 1990s.   The 1997 California Framework for Mathematics required that students know the standard algorithms for all four operations—addition, subtraction, multiplication, and division—by the end of fourth grade.^[ii]  The 2000 Massachusetts Mathematics Curriculum Framework called for learning the standard algorithms for addition and subtraction by the end of second grade and for multiplication and division by the end of fourth grade.  These two frameworks were heavily influenced by mathematicians (from Stanford in California and Harvard in Massachusetts) and quickly became favorites of math traditionalists.  In both states’ frameworks, the standard algorithm requirements were in direct opposition to the reform-oriented frameworks that preceded them—in which standard algorithms were barely mentioned and alternative algorithms or “strategies” were encouraged. 

Now that the CCSS-M has replaced these two frameworks, the requirement for knowing the standard algorithms in California and Massachusetts slips from third or fourth grade all the way to sixth grade.  That’s what reformers get in the compromise.  They are given a green light to continue teaching alternative algorithms, as long as the algorithms are consistent with teaching place value and properties of arithmetic.  But the standard algorithm is the only one students are required to learn.  And that exclusivity is intended to please the traditionalists.

I agree with Garelick that the compromise leads to problems.  In a 2013 Chalkboard post, I described a first grade math program in which parents were explicitly requested not to teach the standard algorithm for addition when helping their children at home.  The students were being taught how to represent addition with drawings that clustered objects into groups of ten.  The exercises were both time consuming and tedious.  When the parents met with the school principal to discuss the matter, the principal told them that the math program was following the Common Core by promoting deeper learning.  The parents withdrew their child from the school and enrolled him in private school.

The value of standard algorithms is that they are efficient and packed with mathematics.  Once students have mastered single-digit operations and the meaning of place value, the standard algorithms reveal to students that they can take procedures that they already know work well with one- and two-digit numbers, and by applying them over and over again, solve problems with large numbers.  Traditionalists and reformers have different goals.  Reformers believe exposure to several algorithms encourages flexible thinking and the ability to draw on multiple strategies for solving problems.  Traditionalists believe that a bigger problem than students learning too few algorithms is that too few students learn even one algorithm.

I have been a critic of the math reform movement since I taught in the 1980s.  But some of their complaints have merit.  All too often, instruction on standard algorithms has left out meaning.  As Karen C. Fuson and Sybilla Beckmann point out, “an unfortunate dichotomy” emerged in math instruction: teachers taught “strategies” that implied understanding and “algorithms” that implied procedural steps that were to be memorized.  Michael Battista’s research has provided many instances of students clinging to algorithms without understanding.  He gives an example of a student who has not quite mastered the standard algorithm for addition and makes numerous errors on a worksheet.  On one item, for example, the student forgets to carry and calculates that 19 + 6 = 15.  In a post-worksheet interview, the student counts 6 units from 19 and arrives at 25.  Despite the obvious discrepancy—(25 is not 15, the student agrees)—he declares that his answers on the worksheet must be correct because the algorithm he used “always works.”^[iii] 

Math reformers rightfully argue that blind faith in procedure has no place in a thinking mathematical classroom. Who can disagree with that?  Students should be able to evaluate the validity of answers, regardless of the procedures used, and propose alternative solutions.  Standard algorithms are tools to help them do that, but students must be able to apply them, not in a robotic way, but with understanding.

Let’s return to Carroll’s model of time and learning.  I conclude by making two points—one about curriculum and instruction, the other about implementation.

In the study of numbers, a coherent K-12 math curriculum, similar to that of the previous California and Massachusetts frameworks, can be sketched in a few short sentences.  Addition with whole numbers (including the standard algorithm) is taught in first grade, subtraction in second grade, multiplication in third grade, and division in fourth grade.  Thus, the study of whole number arithmetic is completed by the end of fourth grade.  Grades five through seven focus on rational numbers (fractions, decimals, percentages), and grades eight through twelve study advanced mathematics.  Proficiency is sought along three dimensions:  1) fluency with calculations, 2) conceptual understanding, 3) ability to solve problems.

Placing the CCSS-M standard for knowing the standard algorithms of addition and subtraction in fourth grade delays this progression by two years.  Placing the standard for the division algorithm in sixth grade continues the two-year delay.   For many fourth graders, time spent working on addition and subtraction will be wasted time.  They already have a firm understanding of addition and subtraction.  The same thing for many sixth graders—time devoted to the division algorithm will be wasted time that should be devoted to the study of rational numbers.  The numerator in Carroll’s instructional time model will be greater than the denominator, indicating the inefficient allocation of time to instruction.

As Jason Zimba points out, not everyone agrees on when the standard algorithms should be taught, the alternative algorithms that should be taught, the manner in which any algorithm should be taught, or the amount of instructional time that should be spent on computational procedures.  Such decisions are made by local educators.  Variation in these decisions will introduce variation in the implementation of the math standards.  It is true that standards, any standards, cannot control implementation, especially the twists and turns in how they are interpreted by educators and brought to life in classroom instruction.  But in this case, the standards themselves are responsible for the myriad approaches, many unproductive, that we are sure to see as schools teach various algorithms under the Common Core.