Attacking the Global Mathematics Gap: "Knowledge for Teaching"

    In this post, I'm drawing from the materials of several outstanding thinkers in the field of elementary mathematics education who argue that elementary math should be taught by teachers with special skills and special knowledge, which might be called "mathematics for teaching."  This is the third in a series of posts in which I am arguing that we need to cure our global mathematics achievement gap -- the gap between American scores and scores in other industrialized countries -- by changing the way that we approach the teaching and learning of mathematics in the early grades.   I don't claim to have invented these arguments.   This fact:  that we must change our approach to the teaching of mathematics by improving the mathematics understanding of those who teach it, has been crying out for immediate attention for decades.  

    Before I get to the work of Dr. Loewenberg Ball and others, I want to begin with a parenthetical point.  Whenever one advances this idea--that we need to change our approach to elementary mathematics -- immediately someone steps forward to say that, yes, we need to use "manipulatives" and teach math to elementary students more concretely.  Using math manipulatives is often the centerpiece of reform, because installing manipulatives allows us to give the impression that we are doing something really different, and because children enjoy working with blocks, leggos and other concrete things.  But adding base-ten blocks, set cards, fish-counters, and cuisenaire rods, whatever their benefit in the classroom, completely misses the point.   When manipulatives are used by teachers who lack "mathematics for teaching", the manipulatives are not likely to solve the fundamental problem in American elementary mathematics....which is that we aren't teaching for understanding.

  The argument for fundamental reform begins with the this simple idea:  that in order to teach mathematics well, a teacher must know mathematics, but to know mathematics in a special way that makes teaching for understanding possible.     It is true that just about anyone can teach, and just about anyone can learn, tbasic math facts -- sums and times tables, how to subtract integers and do long division.   It is possible to learn these simple chores with a deck of flashcards and lots of repetition.  There was a time when a person who could do his sums, who could make change at the cash register, was considered to have mastered enough arithmetic to get by.    But these days, cash registers can do the arithmetic for you, and so being able to make change is barely enough to keep a minimum wage job.      More and more of us need to use mathematics for applications in the world of work that require understanding.   If we do not make the transition to concept based learning in mathematics, other countries are going to pass us by in engineering, computer sciences, telecommunications, architecture, banking, insurance, and a host of other areas that require application of mathematics to solve problems.  

    As I've been collecting information to write these posts, I ran across a great power point, online, that makes the points I've been advancing way better than I could ever do.   Its called "The Case for Elementary Mathematics Specialists", authored by Hung-Hsi Wu of the University of California at Berkeley (Cal) and Deborah Loewenberg Ball,  Dean of the University of Michigan school of education.  Dr. Hung-Hsi Wu, a research mathemetician and Ball, a teacher and now Dean, argue that its asking too  much of traditional elementary teachers to attempt to teach mathematics the way it needs to be taught.  The slideshow  summarizes ideas expressed in Ball's article in the American Educator, "Knowing Mathematics for Teaching."  (Fall 2005).  

Ball and Dr. Hung-Hsi Wu point out that a lot of people in the United States believe that it should be really easy to teach math --- that math is just a matter of learning to add, subtract, multiply and divide. If you're one of those people who believes that we just need to "get back to the basics", or even if you are not, I urge you to spend a few minutes reading through the power point, or the "Knowing Mathematics for Teaching."   Both are great supplements to the book by Liping Ma, that I discussed in my first post on elementary mathematics.   The point is, that"in order to really teach mathematics successfully, a teacher needs to explain the mathematical foundation of basic arithmetic, and "explaining mathematics...is not all that easy."    The point here is that you can do cash register math without understanding the mathematics behind elementary mathematics, but failure to understand elementary mathematics in a deeper way disables the teacher from teaching, and the a student from learning, enough math to serve as an appropriate foundation for high school and college coursework using math skills.  

Here is an example from the slideshow mentioned above.  

Many teachers can teach children to do multi-digit multiplication...that is the procedure commonly used to multiply with paper and pencil...but how many elementary teachers can explain why  his computation correct, and why it work?  

82 x 59 738 410   4838

In her book, Lipping Ma, discusses interviews of both American and Chinese teachers, in which she asks elementary teachers to explain multi-digit multiplication, and the answers are tremendously revealing.   When they were asked whether the answer would be different if a zero were added as a place holder after the 410 in this problem, many American teachers answered that adding a zero would be improper, because it would change the answer, or create mathematical confusion.  Some American teachers said that if a student wanted a place holder after 410, an X should be used, not a  zero, to make sure that students were not confused into thinking that 4100 was the same as 410.   In contrast, her interviews with Chinese teachers suggested that many had a much deeper understanding. They saw multi-digit multiplication as using the distributive law:   (82)*(59) =  (82) * (50+9)  = 9*82 + 50*82.  

"Clearly, being able to multiply correctly is essential knowledge for teaching multiplication to students.  But this is also insufficient for teaching.   Teachers do not merely do problems while students watch.  They must explain, listen, and examine students' work.  They must choose useful models or examples.  Doing these requires additional mathematical insight and understanding.

An elementary teacher should be able to:

• Analyze student work and understand what  misunderstanding has contributed to an error in the answer.
• Represent the meaning of mathematical algorithms or solutions that illustrate why they work.  
• Use the language of math in ways that are understandable to students, yet do not distort the fundamental mathematical principles. 
• Answer student questions like "why does this work," and "why can't we do it this way".  
• Find alternative ways to explain procedures, so that students have multiple windows on mathematical understanding. 

Ball and her co-authors in the 2005 article argue that there is a a body of mathematics knowledge that teachers need that they call "Mathematical Knowledge for Teaching."    Their research has demonstrated that students of teachers who possess this knowledge do better in elementary mathematics.    And in mathematics, students who do better this year, have a better chance to do better in subsequent years, because mathematics is a pyramid of skills and knowledge that must be built on a solid foundation.  

Those of us who want to "reform" education need to recognize that reforming education means reforming the way we deliver instruction.   Whether we organize the delivery education into charter schools, traditional public schools, or some non-unionized Walmart - style education factory, if math education is delivered by teachers who don't have "mathematical knowledge for teaching," the results are going to be the same. Too often, we believe that changing the organizational structure of education will magically transform what students learn.  But in mathematics, and in other fields, putting old wine in new bottles is not the solutoin.  When we discuss the reform of education, we need to focus, I believe, on the mechanism by which those reforms are going to actually translate into better math, better writing, stronger literacy skills, and better understanding of history, economics and civics.     

Blog Post Part II

House and Senate Bills Promise Compensation Freezes for Public Education

Some time ago, I wrote a post calling for a two year reprieve for public education. Two of those ideas coincide with bills introduced in the House and Senate. (See,"Eight Legislative Ideas to Provide a Two-Year Reprieve to Public Education"). Senate File 56 proposes one of those ideas, a two year compensation freeze for public and charter schools. It has been introduced in the Senate by Senators Thompson (Assistant Majority Leader); Olson (President Pro Tem); Hann (Assistant Majority Leader); Gazelka ; and Hall . The Bill would bar school districts from implementing compensation increased through the end of the next biennium.

The core text reads as follows;

From the effective date of this section through June 30, 2013, a school district or charter school must not increase the 2.24 rate of salary or wages for any employee. This section prohibits any increase including, but not limited to, across-the-board increases; cost-of-living adjustments; increases based on longevity; increases as a result of step and lane changes; increases in the form of lump-sum payments; increases in employer contributions to deferred compensation plans; or any other pay grade adjustments of any kind. For purposes of this section, salary or wages does not include employer contributions toward the cost of medical or dental insurance premiums, provided that employee contributions to the costs of medical or dental insurance premiums are not decreased.

The Bill contains a variety of complex provisions designed to accomplish the intended objective, which is to implement a comprehensive "hard freeze" for public school employees. You can see a full text of the Bill by clicking here.

A second proposal is contained in House Bill 92, with a lengthy list of co-sponsors, ( Downey ; Benson, M. ; Banaian ; Bills ; Beard ; Kiel ; Shimanski ; Petersen, B. ; Kieffer ; Kelly ; Anderson, S. ; Myhra ; Crawford ; Leidiger ; Fabian ; Kiffmeyer ; Barrett). House Bill 92, if passed, would repeal the bargaining penalty deadline. This penalty has cost our district a total of 1/2 million dollars in the last two bienniums, because the District has refused to agree to pay compensation which would require massive employee layoffs.

Together, these proposals would provide a major element of the relief needed for two years of breathing space on the cost side for school districts across the state, a reprieve from the persistent downward financial spiral that has gripped school districts, in which every two years, districts feel forced to agree to compensation increase that they cannot afford. The other form of relief that school districts will need to bring their budgets into balance, is for the legislature to fund the increased mandate costs that are coming from the State itself. The two most significant of these State imposed cost increases would be increases in special education mandated deficits and the substantial increases in the teachers' retirement contributions which the legislature imposed on school districts.


While the sponsors of these bills are currently in the majority party, I would hope that the minority party and the Dayton administration would join in the effort to bring fiscal stability to public education. Returning financial sustainability to public education requires adequate revenues, true, but it also requires the ability to keep costs and revenues in balance. Under the system as it exists today, that balance no longer exists, because of deep structural problems with the way in which schools are funded, the way in which mandates are distributed without compensating funds, and the way in which compensation is set . We need a time-out so that all parties concerned can step back and arrive at a new and sustainable system.

The cost relief provided by this proposed legislation will not completely solve the financial crisis confronting public education. But it would represent a giant first step. In a previous post, I explained the need for a two year reprieve by pointing to the experience of the Anoka School District. As I explained in that post

In 2007, Anoka-Hennipen took a referendum to the voters seeking $30-$50 million per year in new taxes. The voters approved question 1, providing $29 million a year for five years. It also passed question 2, providing for $15 million per year. According to campaign proponents, the two levies together would provide funds necessary to prevent precedent-setting cuts: School closings, hundreds of teachers laid off, and hundreds of dollars added to activities fees.

But in 2009, Anoka-Hennipen found itself $10 million short in funding just a year after receiving that levy support. It solved this problem by taking $3 million out of its budget reserves and making $7 million in budget reductions. To produce the 7 million dollars in “savings” last year, Anoka-Hennipen had to cut 73 licensed positions and 42 non-licensed positions. Taking money out of reserves is a temporary solution. When you take money out of reserves, you are maintaining programs at a level that are unsustainable. You must eventually make that up with $3 million in additional cuts. That means, even if the State were to “hold education harmless,” whatever that means, A-H has another $3 million in cuts to make, even if it provides no compensation increases whatsoever. Thus shortly after passing a levy to prevent crippling cuts, A-H started making the very crippling cuts it had hoped to avoid.

A-H’s fiscal problems can be traced to two major problems, an unaffordable settlement, and massive increases in its special education cross subsidy. Last year, although the State provided no formula increase for the biennium, and imposed major cuts in special education on Districts like A-H, Anoka-Hennipen settled its contracts with increases that were measured as having “total package cost” increase of 4 percent for the biennium. This 4 percent represents an understatement of the amount of the increase, because for several years now, the MSBA has intentionally understated the percentage increase in bargaining settlements, so as to make school boards appear more frugal than they really are.

But I also pointed out that the rising teacher compensation costs is not the only problem faced by school districts. State special education funding requirements are also blowing a hole in public school budgets. According to the MDE, in the last reporting period, Anoka Hennipen’s Adjusted Special Education Cross subsidy was $28 million, or $614 per pupil. Two years before, the Anoka Hennipen cross subsidy was $427 per pupil, or $22 million. There was a 50 percent increase in the deficit over a two year period, and the current state budgeting plans for further substantial increases in that deficit. Minneapolis' deficit has skyrocketed to $985 per student in the last reporting period. If the State had fully funded A-H’s Special education expenditures, A-H could have avoided all $7 million of the cuts it made, kept its $3 million in its fund balance, and added $18 million back to its fund balance. Of the $7 million in cuts that A-H made last year, all but one million could have been eliminated if the State had merely held A-H harmless from the increase in special education cross subsidy that has occurred in the last two years.

To fix the public school financial crisis, then, the legislature must do more than impose a two-year freeze. It must take action to close the funding gap in special education. I have argued that it should freeze special education spending statewide, again to give public education a two year reprieve, and at the same time provide revenues to reduce the current state mandated deficit. The two actions together, stopping the growth in special education costs, and funding the existing deficit with additional revenues, would bring the first period of financial stability to public education in a long time.

The changes proposed in this new legislation will not solve the financial problems of public education permanently. They do not address the fact that health expenditures are growing at an unsustainable rate. They do not address the fact that we must provide adequate compensation to our staff over the long run. They do not address the need for compensation reform that really works for teachers, nor do they address deficiencies in the way we evaluate. But they do give us a time to take a break from the compensation wars, and work on a permanent fix.

Lane Improvement a Cost Component in School Finance

I've been writing recently about the budget issues that our school board faces. The financial cost drivers that I am describing are not unique to our district, at all. But I've been using the particularized costs that we face over the next several years, in hopes that it will propagate a better understanding of the way school finance works under our current state system.

Today, our Board of Education will conduct a workshop to begin the budgeting process for budget year running from July 1, 2011 through June 30, 2012. In my first post, I explained how the governor and legislature added increased retirement costs, $250,000 per year, to the District's obligations in each of the next four years, without providing compensating revenue increases, so that unless the State legislature provides revenues to offset these increases, we will have to find a total of $1 million of offsets over the next four years. In the second post, I explained how the so-called excess cost aid pro-ration formula subtracts $350,000 in revenue from out district each year. As a result, unless and until the State corrects this problem, we will lose $350,000 in special education revenue each year, but will be prohibited from making compensating cost adjustments in special education to balance that budget.

Today, I'm going to talk about automatic lane compensation increases and how it drives up our compensation costs without regard to state funding. The average increased cost to the district resulting from lane improvement compensation is about $190,000 per year, so that it contributes to about $380,000 in increased costs for the biennium in our district.

I have several more cost and revenue items to explain in this series of posts. I want to emphasize again, that as I examine personnel costs I am not in any way disparaging the need for adequate compensation for public school employees. But we need to discuss these costs, if we are to understand them.

A lane is earned when a teacher adds, generally, 15 credits beyond a basic degree in university coursework. The cost of the automatic lane adjustments to the district depends upon the number of teachers in a given year who earn enough credits to move into a new lane. (They are called lanes because the salary grid for a school district is a chart with horizontal rows that are "steps," and vertical columns that are lanes. A teacher moves from one row to the next by increasing the number of years of service, and from one column to the next by earning sufficient lane credit.) As I have said, the total increased cost of lanes to the district would be about $190,000 in additional compensation cost a year. If reaffirmed in the new agreement, then, the total lane adjustment costs amount to total cost increases of about $380,000 for a two year collective bargaining agreement.

Teacher compensation in Minnesota has for decades been based on training and experience. Training has been recognized by lane movement. Experience has been recognized by step improvement. For many years, the State actually encouraged this system by providing equalizing revenue to provide additional compensation to districts with more highly experienced teachers and more highly trained teachers. As you can imagine, a district that is older, and has not been recently growing, has a larger percentage of its teachers with advanced degrees, and a larger percentage of teachers with longevity and earning step improvement pay. On the other hand, a district that was recently small, and is in a major growth spurt, has a very high percentage of its teachers on the lower level lanes and the lower level steps. As a result, those districts make a profit on the revenues provided in the general formula, and begin to build up reserves, whereas the districts with declining enrollment have lots of teachers bunched at the top steps and lanes, and start to run a deficit. So the training and experience revenue was designed to make up for that disparity. However, the State ultimately concluded that training and experience revenue should be discontinued, probably as an incentive to districts to change their system, or possibly as a raw cost saving measure. But all across the state, the system has persisted.

Now in many districts, including ours, lane improvement raises continue beyond the end of the collective bargaining agreement, while the new contract is being negotiated. Under Minnesota law, public employee contracts have a termination date at the end of two years. Our current contracts, for example, run from June 30, 2009 to July 1, 2011. Now under the current bargaining law, when a contract expires without a replacement contract, all permanent employees work under a so-called "continuing contract." The old collective bargaining agreement continues in force, beyond its expiration date. When that happens, all teachers and other employees work at the rate of compensation established by the old agreement.

The question of whether teachers get lane improvement compensation under the continuing contract is governed by the old collective bargaining agreement. And in most districts it is. It is a form of automatic increase that drives up compensation costs automatically before the new contract is signed. Why do we, and many other districts, agree to make these increases automatic during the continuing contract? The answer is that until recently, the legislature has generally provided us with enough funding so that we could cover that cost. In addition, the teachers who are moving up the ladder of lane improvement have put in a significant amount of work, so we have generally been able to cover the cost of lane improvements out of increased revenues from the State.

There is a significant body of research that suggests that the lane compensation system is not really all that good for school districts, nor all that good for the teaching profession. What I mean is, if you are going to spend the $190,000 more each year for teachers, there are ways to spend that money that would actually be better for everyone involved, and here I don't mean compensation based on test results. The lane credit is expensive for teachers. They have to spend a significant amount of money in tuition to earn 15 credits, and frankly, there is very little evidence that the education that they receive in the coursework they take has a significant professional benefit, let alone a benefit in the classroom. From time to time, we've suggested that it would be better to replace the lane credit system with something that was less costly to the teachers earning their lanes and of more benefit to children in the classroom. For example, teachers could earn lane credits by taking on new leadership responsibilities. But that idea has never garnered support on the other side, for some reason.

Incidentally, I would encourage you to become familiar with a new resources that allows you to view not only our agendas online, but also the attachments to those agendas that provide valuable background information on what the board of education will be discussing at its upcoming meeting. You can also review past meetings and look at the backup material in this way. You can get to the materials by going to the District 742 website and choosing the menu selection for the school board, and foll wing the links to the meetings and agenda sections. Or, you can get directly to the public board-book side by clicking this link: Click Here.

Reform agenda for divided government

School board members across the state are facing next year's legislative session with a great sense of anticipation and apprehension. Once again, we face divided government. The voters have handed the legislature to the Republicans and the Governor's mansion to Democrat Mark Dayton. (The chances that a recount can overturn a nearly 9000 vote margin are slim to none). Republicans have promised significant reforms in education, but what does that mean? Dayton arrives at the Governor's mansion independent of traditional constraints, because he was not the party-candidate. The sense of anticipation arises from the hope that Dayton and the Republicans might join forces and develop an agenda of proven reforms--reforms that have actually worked in the real world. The sense of apprehension is that partisan gridlock and partisan ideology might bring us gridlock, and worse, a few scatterbrained faddish reforms designed to help each party run against the other in the next elections.

There are all sorts of "reform" ideas that the Dayton crowd and the republican legislative crowd might advance that would be red-meat to their MSNBC and FOX watching zealots. Many of these reforms actually have very little basis in proven results. Few of them show up in the growing literature on what actually works in transforming schools. Many of them are veto bait that, whatever their merits, promise to head us down the path toward gridlock.

In the next few weeks, as time permits, I'm going to advance some reform ideas that should be passed and signed by the legislature in bipartisan fashion. I'm going to suggest that they are ideas that will make governance and operations of school districts vastly easier to operate, and which will translate into real benefits in the classroom for kids. And, from time to time, I'm going to attack some reform ideas that have been advanced as magic bullets, but which really have no sound research basis to support them, and I'm going to argue that its time for the Dayton people and the Republican legislative leadership to roll up their sleeves and work together to pass a courageous package of reforms that actually work for schools. Those of you who think that we should can tenure completely, for example, are going to find me skeptical that this would result in significant improvements that warrant deadlocking the legislature and Governor, which would most certainly be the result.

Here are some of the reforms that I think should pass, and can pass, that would make a significant positive difference for education:

• Eliminate the additional state mandates for special education that force local districts to spend significantly more on special education than required by federal law. In our District, that would reduce our expenses by about one million of unfunded dollars per year. Listen: the Republicans have campaigned on eliminating unfunded mandates. If they can't enact this reform early in the session, then they didn't mean what they said.
• Fully fund the balance of the unfunded special education mandate. It is fundamentally unfair, and I believe unconstitutional, for the legislature to force some school districts to spend vastly more to carry a statewide responsibility, than other school districts. Combine this initiative with a rigorous program of scrutiny to assure that local districts are spending their special education funds appropriately. Until now, the State Department has exerted supervision of local districts with largely push them to spend more, rather than spend efficiently.
• Eliminate the bargaining penalty.
• Prohibit collective bargaining contracts from containing automatic compensation increases that occur beyond the expiration of the duration of the contract. This feature in many school district contracts, places management in the position of having agreed, before bargaining, to increases significantly beyond the amount of funding increases coming from the legislature.
• Grant management the power to implement the Pawlenty quality compensation reforms in return for the additional funding. I've said in the past, and I'll say again, in my opinion, the compensation part of quality compensation is the least important of the reforms found in quality compensation.
• Prohibit strikes designed to force school boards to increase compensation at a rate faster than reimbursed by state funding. Public education will not be viable if school districts are forced to increase class size and cut programs in order to fund compensation increases. In tandem with this, develop a coherent legislative strategy to provide school districts with sufficient funds to attract and retain quality professional teachers.
• Develop financial incentives to promote strategies that are proven to work in closing the achievement gap.
• Transform professional staff development for teachers away from the existing program of university or quasi-university seminar education, toward internal professional staff development that is focused on implementing school and district improvement plans. Stop granting lane advancement for taking a course at the local community college, and grant lane credit for becoming a teacher leader who implements a great science program, a great math program with proven results in a local school.

Some of you may find these reforms too aggressive. Some may feel that they don't go far enough. In future posts, I'll discuss my rationale for these ideas, and discuss reform ideas that have been advanced by others.