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Fixing Fuzzy Math


What constitutes quality mathematics instruction? And, if we don't have it in schools now, how do we get it? In this Education Week Commentary, Jere Confrey warns that setting standards will not be enough to raise mathematics achievement. Testing must accompany standards and, even then, the task of fixing American mathematics education will not be complete.

The missing piece is improvement in how the subject is taught. This improvement of the "instructional core" requires long-term, strategic federal investment and action, as well as national recognition, writes Confrey. Lacking federal funding and a comprehensive plan to improve instruction, math achievement will continue to flounder.

What do you think? Could federal policies aimed at strengthening the instructional core improve mathematics achievement?


I agree that instruction is key to solving the math concerns students are facing nationwide. Much of the responsibilitiy falls on the colleges and universities as they prepare the preprofessionals for the mathematics classrooms. The syllabi for the math courses taught to preparing educators need to address a variety of instructional strategies, differentiated instruction, the needs of the ELL, and the inclusiion of students with disabilities. Beginning teachers are on a huge learning curve and need to be well equipped to face the high standards they face in the schools.

I believe that mathematics is the content area that shows the strongest correlation between the level of the teacher's education and the achievement of students. Even those who are teaching at the elementary level need to have a solid mathematics background in college--not just pedagogy.

And special education teachers need to be working collaboratively with regular education teachers in the area of mathematics--recognizing that students with special needs require both a teacher with the academic background in mathematics and a teacher who is able to adapt, accommodate and differentiate. We have to stop assuming that those who know enough math to be able to DO what is required of their students, also know enough to be able to TEACH their students HOW to DO.

Jere Confrey has a number of good points. Among them is DIAGNOSTICS. Find what kids are specifically having problems with. I saw kids doing so much more work and harder work than necessary on problems that were designed to be easy - if you knew how to use the properties to simplify them. But they had learned the properties as FACTS, disconnected from the problem solving process, rather than as tools to make complicated problems simple. Another important point is to get to kids in Elementary school. It is critical to fix the problems BEFORE they lose confidence, BEFORE they 'give up'!

I agree with almost everything Jere Confrey says however there are a couple of things that I have found in my work with teachers and parents alike. 1. There is a real failure among many to see that mathematics as a subject is a content area like any other. Instead many see mathematics as only a series of skills to be acquired. The current testing and commercial curricula (including some of the reform curricula) do little to counter these beliefs. 2. The yearly high stakes testing does more than narrow the curriculum it also fosters teaching methods focused on speed and short term memory development instead of long term understanding. In my work with both high income and low income school districts there is a stark contrast in institutional fear and action as a result. These two foci create situations where, in the name of leaving no child behind, even more students are being left further behind.

One thing that appears to be missing in mathematics instruction is a clear transition from basic skills to more complex ones. Math is a subject that requires basic knowledge begore more complicated concepts can be introduced. Students need to understand addition, subtraction, multiplication and division of whole numbers before they are exposed to the use of these same processes with abstract concepts. The basics prepare the way by showing what the functions are doing. It is fairly easy to explain that 2+2=4, but it gets to be more difficult when the student must add x and y, or any other letter to gain an answer. For instance, if x+y=4, are x and y equal? Do both letters stand for the number two/ Does the equation x+y=4 indicate that one or the other letters stand for 3 and the other, therefore 1?
If the elementary steps are not understood, the advanced steps will be even more difficult.

As a middle school math teacher, it has been very difficult to teach the 6th, 7th and 8th grade curriclum due to the fact the majority of my students do not know how to multiply. They are very good at scoffling but have not memorized their math facts. Secondary educators can not go back and teach the basics. Once in secondary school environment, without the basic skills students "shut down" when the content becomes more intense. Please elementary teachers make sure students have memorized their basic math facts before moving them on - you are not doing them a favor or the parents.

Teresa, you are so funny if you think that it is the elementary teacher that is at fault for promoting students that have not memorized their multiplication facts. Elementary teachers are responsible for the entire curriculum plus mandated ELL minutes, PE minutes, music, health, character education..... Perhaps the parents need to be held accountable for student learning. Fine the parents for low scores....

Teresa and Rebecca, while I understand how challenging it is to teach all subjects and I admire those who do so, I want to agree with Teresa's comments about multiplication in particular. Too many curricula and those who teach it permit students to go on without really understanding multiplication and division and "working in multiplicative space". Just learning the basic facts, independent from treating those facts as repeated addition, is needed, but so is understanding how to move around with multiplicative reasoning-- using factors and primes etc. Ask your students how to go from 15 to 24 using only multiplication and division and you will see the kinds of skills and thinking I mean. Try a few of these "daisy chains" and see what happens. Thanks to all for reading the article and adding your comments.

Speaking as an elementary teacher who has spent the last 6 years working as a math coordinator - the quality of instruction is the key. It is about knowing the content and knowing the pedagogy, but it is more about the content pedagogical knowledge. Teachers know how to do it and know how to teach what they know how to do. Usually they teach it the way they were taught. The question is - What do they do when the students don't learn what was taught? Can they peel back the layers of the concept to see what the next step should be?Example - I was working with students who had not been successful at showing the different ways to make $.27. Did they know money? When I probed their knowledge, I found they knew the values of coins. Some knew the value of 3 nickels, but could not tell me the valus of 4. Or they could tell me how much 4 dimes was but could not tell me the value of 4 dimes and a nickel. What they could not do was count by 5s or 10s in a long enough series or they could not start with with 10s and move to 5s. We needed to go back and practice their counting skills. Instead of covering a topic we need to know how to uncover it. The focus of this work needs to be both preservice and in service teachers. It must be ongoing.

Marty Solomon raised two interesting issues in http://www.edweek.org/chat/transcript_02_08_2006.html, but I currently refer the reader to the second. He suggests that many elementary school teachers carry their own math-phobia into the classroom. I know from personal experience (see http://www.edweek.org/chat/transcript_10_25_2006.html) that most people who are good at math have job opportunities that pay significantly better than teaching available to choose from. While I sincerely believe that ALL teachers are sorely underpaid, I believe that one way to "fix" the fuzzy math problem is to pay people who are good at both math and teaching a salary that can compete with industry. And while I am aware that being good at math does not automatically imply that one can teach it, I work with MANY people who can do both and would choose to do so IF they could earn a decent wage. But we can't. (Instead we lurk on web sites like this).
Fix the problem by getting those of us who love math into the classrooms to teach it. I'm watching to see what will happen with the "Adjunct Teacher Corps".

As Prof. Confrey say, "the mising piece is improvement in how the subject is taught" and to address that she makes a proposal many mathematicians have already endorsed, and that is to train and hire math-science specialists. But not only in the upper elementary grades, but at every grade level. Early last summer, in fact, I circulated a letter among friends to be sent eventually to foundations that support education. In that letter I fleshed out this proposal as follows.
"Enclosed is a plan, already widely endorsed by the mathematics community, for reversing the disastrous underperformance of our students in the area most critical to America's future economic viability. It calls for training and giring math-science specialists in the elementary and middle grades. Such specialists would meet daily with small groups of 6-10 students in small "math-science" labs equipped with coputers, film strips and other learning aids. By allowing the lengths of these lessons to vary, we provide the flexibility needed to challenge the most ambitious and creative students with hard problems and enrichment projects while offering slower learners the individual attention and extra support they may need to master the material."

I go on to say:

"Needed, too, are more stimulating and challenging currricula... All too often our school math and science programs, mired in dull procedural 'stuff', damp down the natural curiosity children bring to school. The kinds of curricula we have in mind, chock full of mathematical vitamins and scientific nutrients, can double as college level texts for the training of these specialists. For what better than to teach them the materials they will actually be usuing in their classrooms? Several such curricula authored by mathematicians already exist and others are on the way."

Thus, I see no need to involve the federal gov't in the design of school curricula. Needed, instead, are controlled experiments comparing the effectiveness of different curricula and teaching styles. My letter continues:

"No realistic turnaround plan can iognore the need to upgrade the knowledge and sklls of the teachers already in the system. My proposal is that they be given the opportunity too become math-science specialists-- and salary inducements to do so-- by attending series of weekend workshops or summer institutes to be held by well-qualified math and science Ph. D.s, including college faculty and retired college teachers. Upon passing a final exam, the participating teachers would be accredited to teach the relevant grades of the chosen curriculum."

In addition, teachers most successful in achieving the announced learning goals of their preferred curriculum would be eligible for bonuses or raises.

The tuition for the workshops could be paid by local school districts, with additional aid coming from foundations and state, local or federal gov't sources.

I am in process of completing one such curriculum. It is organized around four main strands: (A) arithmetic, algebra and algorithms, (B) geometry and visual representation, (C) modeling and measurement, and (D) reasoning and problem-solving. It is designed to support inquiry-driven interactive learning (such as the small-group sessions I envisage lend themselves to). But, again, I agree with Prof. Confrey's view that differences in how the material is taught have distracted us from the main task, which is to develop the kinds of rich, challenigng and stimulating curricula that will more adequately prepare students for college level study of math and reflect the excitement we mathematicians feel about our subject.

Kudus, then, to Prof. Confrey for emphasizing these points.

Roger Rosenkrantz (email: [email protected])

I am a parent of three and I have worked professionally in the technological (Network Engineer) and educational (9-12 Math and Programming Teacher) fields now for 15 years (and non professionally for 20 years). 80% of K-12 education is to train the mind (and body in respect to PE), not to teach what students are "going to use" in life. Math is a subject that trains the mind to think technically. Thinking technically requires building blocks. In agreement with Bob Frangione, we need to get back to basics early on. Students leaving Elementary School need to know the basics...basic operations and how to perform those operations with long numbers (ON PAPER), fractions and operations on them, ratios, proportions, and percents. THATS IT!! And its not the teachers' fault for this....they are just doing what they are told to do. All these "fuzzy, spiraling Math" programs (ie "Everyday Math", "Integrated Math", IMP) are just money makers for publishers and TI. A few may think they have good intentions, but unfortunately it is counterproductive to young minds. Many of those in the field of Mathematical Education have become blinded with all their research in how to teach and assess Math. Research does not need to be done. It should be taught the way we learned it (you know what I mean). Going back to basics first means TAKING CALCULATORS AWAY FROM THE STUDENTS!! I am very technologically oriented and skilled. I love computers and graphing calculators....BUT THEY ARE NOT TO BE USED TO LEARN BASIC OR INTERMEDIATE MATH. There is nothing that abstract or difficult that would require a graphing calculator. In India, students are forbidden to use calculators. Calculators are great tools to be use later in education (maybe Trig on up). PENCIL AND PAPER MATH IS KEY TO DEVELOPING THE MATHIMATICAL MIND. Some say that the calculator allows students to concentrate on “more in depth mathematical concepts” and computational time is freed up. COMPUTATIONAL TIME IS EXACTLY WHAT ELEMENTARY (and some middle school) STUDENTS NEED. When the basics are solidified, the conceptual and abstract understanding will follow. This leads in to my second point, kids need to memorize and do rote practice. Math is mechanical and memorization helps solidify their previous knowledge and development. I don’t need to do "research" to support this....it’s been proven with past generations and we have been doing just fine with it. Have good background in basics and an understanding of the conceptual and even abstract stuff will follow. Lets stop trying to reinvent mathematics with "new assessments" and new "methods" of teaching. The applications of math will follow indirectly in Science and problem solving skills later in life. ITS THAT SIMPLE. And lets face it, NCLB, not all students are going to get it no matter how it is taught, but the REAL way (back to basics) will not only target many, many more. If NCLB is that concerned, they will try to change attitudes of parents and students....it cannot be done through education.

So please, stop trying so hard to “fix” Mathematics. It was never broken. We know that “fuzzy Math” has been popping its head up for 50 or so years now. We also know that it has been widely used with the reduction of prices in graphing calculators. And we (the teachers) also know that it needs to go away for the good of our students.

Wow...Dan thanks for the great post. I have a similar background to yours except I don't teach. Drilling is taboo in education today. Number sense is better taught with indirect instruction than direct instruction introduced with fuzzy math. You get number sense by working with numbers doing real, exact math problems by hand over and over again. Once a child has a firm grasp of exactness, you don't need to teach him/her how to compute in fuzzy ways. It's a by product of working with numbers. The emphasis on working with inexact numbers is disturbing because in real world financial and engineering applications precision matters a great deal. The government isn't going to accept that the reason you didn't pay $20,000 in income taxes is because you rounded down. The overemphasis on rounding beginning in the third grade up through the 8th grade is one major problem with the curriculum today.

The next major problem is that we are trying to stuff advanced math topics into the minds of elementary school kids. It's wrong headed and unnecessary. All the great minds in this country that have been creating all these high tech products over the past 20 years weren't introduced to algerbra until the middle school. Why are we try to introduce it into elementary school kids today? Of course they can't multiply by rote, they're working too hard to learn algerbra before they've mastered long division and fractions.

No doubt we are on a national slide in education. Lack of parental support has always been an element of student failure, but today that is amplified by a curriculum that makes parental support even more necessary to overcome the weakness in today's curriculum. In my day, my parents didn't have the background to pick up the missing pieces. It goes to show you that with good curriculum parents become less important to education success.

I guess it's been the constant emphasis on failing schools over the past 30 years that has led educators to change and consequently throwing the baby out with the bath water. The need to show that everyone is achieving has led educators to embrace ways to measure and show academic success where there is none. The death spiral in education today has been a deadly embrace between the educators(all levels), parents, and politicians. Dan, you are rare among educators, laying fault where it rightly is -- in the classroom. Keep up the good work. Kids need more champions like you.

Thanks Bill. Teachers, however (and ironically) are virtually powerless as to what they are required to teach in the classroom. We are concerned about losing our jobs if we voice our opinion to upper channels. I don't know why teachers' opinions are never taken into consideration when decisions about what is to be taught are being made. I have a pretty good idea that the teachers (Math and Science) and parents who know how Math should be taught far, far outweight those who choose otherwise. It's really a matter of caring about the students. Again, I think some advocates of fuzzy Math truly think that it is helping, but they need to be re-enlightened.

I agree fully with the article in Ed. Week and especially regarding releasing test data. ("And too many states fail to release actual test items with results. Withholding these denies teachers and students valuable feedback about the test and is unethical.") Assessment should be used to inform instruction. Teachers must consider what student errors mean.

Regarding the post about students not knowing their multiplication facts, I believe that in this country we put too little pressure on the family. Elementary teachers have a much fuller curriculum than say 20 years ago. The day is only so long. By fourth grade, if a student is weak in basic skills, it is not inappropriate for parents to feel some pressure to help at home. In other cultures, not all responsibility rests with the public schools. The cultural expectations in Europe and certainly in Asia place responsibility on individual children and families. I have taught in Germany and in the Philippines and most recently in middle school in NJ. I am a parent of two, and always spent car time as valuable drill and practice time. Parents need to be encouraged to participate in the education of their children.
Additionally I teach pre-service teachers. The pre-service teachers have very weak math skills and with only one math methods class required, this is not likely to be a recipe for success in the elementary math classroom. Paid weekend workshops and summer institutes with follow up accountability for the teachers would be beneficial.

It's refreshing to read the common sense "Fuzzy Policy, Not 'Fuzzy Math,' Is the Problem." However, a major point has been missed.

"Any time, Any Place, Any Path, Any Pace" and technology should be used to combat the fuzzy policy. Many texts have been published and the knowldge required for high school graduation is known or else tests could not be prepared for assessment purposes.

So, why not prepare Interactive Automated Assisted Learning (IAAL) and use portable devices like video game platforms like PlayStation Portable and many other devices which can be used any where at any time? They can be connected to the Internet too. These devices have triggers: one to back up and replay a segment, another to skip forward, and a buttom to freeze frame a graphic or stop a video or movie from playing.

That's what learnng is about. That's the way most people learn. That's why there are more than 100 million video games platforms in homes around the country with more coming onto the market everyday. Pre-K kids and adults love them. So why not use them for learning purposes?

Software can be prepared for math from zero to differential calculus with off shoots to statistics and genomerty as a continuous course to eliminate arbitrary grade levels so people can progress at their own pace. An added advantage is that tutors, parents and teachers all could use them to ensure that what is being taught is correct. And, assessment is built-in.

Video game devices like PlayStation, Nintendo, Xbox and others should be tools for implementing the "Seven-Point Plan for Strengthenng the Instructiional Core" instead of books. They are inexpensive and are reaily available since they cost $130 to $200. More sophisticated platforms cost more. And, course software can be updated easily at little cost to meet different needs.

Given the success of Schoolhouse Rock, why not extend it? I recently contacted the writer/singer of Multiplication Rock, and he had great ideas for extending the series that could not be pursued at the time. If you go back and watch them again (as we are doing with our kids), he snuck all kinds of properties and other good stuff in there! It was successful before, and could be again...

I think one of the most overlooked problems with mathematics instruction is the notion of spiraling in the curriculum. While many think that all reform-based mathematics supports this idea, this is not the case. For example, Everyday Math does, while CMP does not.

I teach at a high school where we have several different traditional and non-traditional options (we use IMP for the non-traditional). In the traditional option pathway, so much time is spent spiraling through the same topics year after year after year. This is supposed to allow us to go deeper each time until mastery is attained, but the reality seems to be that we teach and then re-teach again and again at the same superficial levels with no time left to spend building meaning, conceptual understanding, or mastery. To many, spiraling looks good on paper, but I feel that too often it leaves us with curriculum that's an inch deep and a mile wide.

Mathematics lends itself to building from one concept to another. It lends itself to taking simple and applying it the complicated to provide a solution. It has its own language and notation.

I am against reform mathematics; especially TERC ("Investigations in Number, Data, & Space", Everyday Math, Connected Math, Integrated Math. I have a second grader and a first grader. I am against the material because of my first hand experience with TERC. "Joining and separating", "bits and pieces" ... whats wrong with teaching that its adding and subtracting and fractions.

I am against curriculum that "spiral", ever leaving the child unprepared for the next level until they finally hit the wall. They reach a level where they are assumed to have masterd material through all those spiralling years but yet never did. Thats what grade band (e.g., K-2) curricula gete you.

Blame the parents? In my town, parents are having their children do after schooling, tutoring, or attending math centers. These are kids with stellar report cards but yet their parents notice they cant add without a manipulative, dont recognize the standard mathematical notation, and more.

Blame the teachers? Heavens no. They hold one of the most difficult jobs. And it is taboo to even constructive criticism to a teacher.

Who to blame? I dont care whom is to blame at this point.

As a parent, I long for clear and concise listing of the mathematics curriculum. Accountability from the schools. A voice in this debate on mathematics.

As a parent, I am ultimately responsible for my child's learning and yet in our school systems, a parents voice is the last one heard.

Let's not forget about student to teacher-to-teacher ratio. How in the world am I supposed to engage 32 students who are all at different ability levels? Trust me, I've tried my hardest but they are not all interested in the same thing at the same time. I think 18 would be the most practical max teacher-to-student and that is only when the students are well-behaved. I try hands-on activities quite often but it is tiresome running around to 8 groups of 4. Plus, I only have 42 minutes per class period which, less the prep time for getting the project started and a few minutes to close, we are talking 22 minutes which is 2 minutes per group or 30 seconds per student. Of course, that is ONLY if there are not problems that pop up that aren't EVEN math related.

I am looking for a fresh start in math. Could someone point me to a lesson plan for those that have been defeated by elem. alg.

I really need help on doing subtraction in 1 min. I keep getting 59% F!I hate that! It's to compclatied!

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