Dr. Ball:

We have never met but I write articles concerning math education and am committed to better math instruction and curricula in the U.S. You recently wrote an op-ed in the Huffington Post calling for effective teachers and better teacher training. You have given a brief description of what you feel constitutes effective teaching. In it you said:

We need to agree on a common core of the most essential capabilities for skilled teaching. Examples include being able to diagnose common student difficulties, explain core ideas and procedures understandably, lead a productive discussion, manage a classroom for learning, and work effectively with students with special needs. On this central core, we need to build assessments, using the best technology and tools available, that enable us to determine candidates' readiness for practice, and to monitor teachers' capability and continued development over time.

Given the importance you place on diagnosing student difficulties and explaining core ideas and procedures understandably I would like you to view the video that is posted here. You'll notice that the little girl is illustrating how to add two large numbers using a method from Investigations in Number, Data and Space, a program for which you are listed as an advisor/consultant for the second edition. Given your interest in better teaching and teachers, and your connection with this program, I would greatly appreciate it if you would answer the following questions:

What is your opinion of the method the girl has been taught for adding large numbers?

Investigations teaches the standard algorithm for adding large numbers in the fourth grade. Do you agree that it is beneficial for the girl not to be allowed to use "stacking" until then?

Do you see any harm in the approach that the third grade Singapore "Primary Math" textbook uses to teach this same procedure? (See illustration)

Do you think the girl's mother has done the girl harm by teaching her the standard ("stacking") method?

Do you think the process the girl is using is more important than her getting the

right answer? If so, why?

If two teachers were to teach this method, and one used the Investigations method illustrated in the video and the other used the method in Singapore's textbook, how would you judge the effectiveness of each teacher?

Do you believe that a teacher who opposes the Investigations method and uses instead the method Singapore's textbook employs is an effective teacher? Do you believe such judgment on the teacher's part indicates a lack of "readiness for practice"?

Lastly, tell us where you stand on Investigations, Everyday Math, and Connected Math Program. Do you support them and believe they are effective? Why? Many of us await your answer.

Illustration of Singapore Math 3rd grade textbook lesson on adding large numbers:

## 61 comments:

Okay, setting aside for the moment the tone of the letter to Dr. Ball...

Here is one video of one student doing one problem. She does several things that are very solid mathematically: representation, mental arithmetic, unitizing, etc. These conceptual developments will serve her well as she progresses through math. They do not develop from learning the algorithm of the stacking method.

If the point of addition was to teach get answer in smallest space, the traditional algorithm is pretty efficient. You know what's more efficient? A calculator. If efficiency is your principal goal, why bother with an algorithm at all? I do want proficiency and automaticity for my students in computation, but it is one goal among several.

How many students has any math teacher had who makes ridiculous mistakes with that algorithm because of lack of conceptual understanding and number sense? Research shows that invented algorithms result in fewer and smaller mistakes on average.

What I am happy about for the girl in the video is that she is reflective, she exhibits metacognition, and she is making connections between different areas of mathematics. That's important.

Investigations provides opportunities for problem solving, for student discussion, for multiple representations and more. A good teacher using it will result in leaners with a high degree of automaticity built on a foundation of conceptual understanding.

From my point of view, it's not a defect to help learners grasp underlying concepts by using graphic or concrete represenations of the processes. But it sure is a defect to use those methods any longer than is necessary, because there is an opportunity cost, and too many children using Everyday Math and its cousins are not reaching automaticity (OR understanding). Others are bored and frustrated by the constant instruction to invent new methods when the already-learned ones work well.

John said, "Research shows that invented algorithms result in fewer and smaller mistakes on average."

Really? Is that a valid study? Want to reference it?

Here's a study titled "ACHIEVEMENT EFFECTS OF FOUR EARLY ELEMENTARY SCHOOL MATH CURRICULA: FINDINGS FROM FIRST GRADERS IN 39 SCHOOLS"

http://ies.ed.gov/ncee/pubs/20094052/pdf/20094053.pdf

Here's part of the concluding section:

“Student math achievement was significantly higher in schools assigned to Math Expressions and Saxon, than in schools assigned to Investigations and SFAW. Average HLM-adjusted spring math achievement of Math Expressions and Saxon students was 0.30 standard deviations higher than Investigations students, and 0.24 standard deviations higher than SFAW students. For a student at the 50th percentile in math achievement, these effects mean that the student’s percentile rank would be 9 to 12 points higher if the school used Math Expressions or Saxon, instead of Investigations or SFAW.”

(SFAW: Scott Foresman Addison-Wesley)

Seems to me if we are interested in achievement, Investigations produces sub-par results.

Investigations math even spawned a comic strip.

www.weaponsofmathdestruction.com

John, I'm not sure why you took your post down, maybe you're editing it, but the study you referenced in it at the TERC website url below, actually shows that Pearson paid for the study. That's sort of like Phillip Morris running a study that determines smoking is safe. :)

http://investigations.terc.edu/impact/2ndEd/2ndEdFindings.cfm/

I've heard people, like John, who support programs like Investigations repeatedly state that children "do not develop [conceptual understanding] from learning the algorithm of the stacking method." Know what I've never heard or seen? Anything to support this allegation. No research. No studies. Not even anecdotal evidence. Just that same blanket statement with nothing to back it up.

Know what we do know? The standard algorithms have been taught to children for hundreds of years. Based on that instruction we were able to build skyscrapers, develop computers, create microscopic robotic arms that can assist surgeons in performing open heart surgery, and go to the moon. Just a few moments ago the Space Shuttle took off for what is supposed to be the last time. All of that, and more, was accomplished by scientists and engineers who were taught the standard algorithms.

So much for the standard algorithms providing inadequate conceptual understanding!

There is nothing wrong with teaching children the standard algorithms. If taught properly, like the example provided in the exert from the Singapore text, children can actually develop a greater understanding of place value, how numbers relate to one another, and how the basic operations work together and build on one another.

Why I took down my post is because it ate three of my references and all my text. Then off to a meeting.

We've had traditional instruction on the standard algorithm since the industrial revolution, not for hundreds of years. Are the people who built the computers and skyscrapers typical or exceptional? What is the typical American's view on mathematics? Where did they learn to hate it?

I have taught problem solving centered mathematics and seen dramatic results with all age learners. Age 3 to 40. Changes in ability and attitude. I have children. Last year my daughter could divide fluently and accurately. This year, the only acceptable method is the traditional algorithm and she hates it. Se can do it, she is bright and has adjusted.

I am a Ph D mathematician, and was educated traditionally. Success story right? But I hated math until high school, where I had greater freedom to work on my own. I made sense of the material on my own, and watched as my classmates were classified as unable to do the material, or turned off of it, or just bored. The math was interesting to me, but classwork was agonizing.

The video is another anecdote. Yes the girl got the right answer quickly with the standard algorithm. But she was interested in the other way, and exhibited a lot of deep understanding. Did that come from being taught to carry? I do want students to compute efficiently, as do 99% of the math teachers I know. But it is not my only or most important goal.

I'm a mathematician, and I never have to simplify rational polynomial functions. Never. Do you? It is now required for all Michigan students. It will help future STEM students with future math classes, which is important. But what about everyone else? If it is an opportunity for problem solving or reasoning, that can make it worthwhile to all students, at no detriment to future engineers.

I'll post the references later, if people are still interested.

I do want to be clear: I love that people are passionate about this, and want high quality education for their kids. I may disagree with the particulars, but feel people who are deeply interested in education have more in common than what separates us.

The girl in the video is very fortunate to have someone at home who is teaching her how to do math efficiently. We had to teach our daughter math at home since it wasn’t being taught at school using the TERC Investigations program. There was a lot of work to develop conceptual understanding but no work to develop basic facts or standard algorithms.

Students tend to use what they learn first. The conceptual development will not serve well if the transition is not made to more efficient methods of working problems. As a teacher, I have worked with many students who were taught inefficient algorithms in prior grades with no transition. Some students never would make the shift, always dropping back on the familiar.

A calculator may be more efficient but it is a major crutch and is no substitute for learning basic math facts. I have seen seventh grade students who give all indication of understanding multiplication but can’t tell what 3 times 8 is without a calculator and are unable to work with fractions. These students will be required to take algebra in grade eight. Investigations is the program used with these students when they were in elementary school.

John said in his comments, “Research shows that invented algorithms result in fewer and smaller mistakes on average.” I often hear “research shows” to be followed with a claim for which no reference is provided. If valid peer reviewed research exists showing this, I would like to see it. My observations of students who invent and use their own algorithms do not support the claim made above.

I want to clarify a statement in my comment. I said, “We had to teach our daughter math at home since it wasn’t being taught at school using the TERC Investigations program.” To some this may not be clear. My daughter’s school used the TERC Investigations program. We had to teach our daughter math at home. As a math program, TERC was inadequate in helping her develop the skill, concepts, and understanding she needed.

For those who like programs like TERC Investigations, I hope your child thrives and is successful with the program. I hope you have the choice to have your child in such a program while at the same time, I would like to have a choice for my child and others to have a different program. We never had the choice.

Dan Dempsey, a school board member in the Seattle area, sent me a letter a while ago that contained this information:

**********

"Good teachers plan classroom instructional design not just on past experiences but also on relevant data. One of the best places to look for empirical evidence likely to increase achievement is the book "Visible Learning", a synthesis of over 800 meta-analyses relating to achievement that collectively looked at 83 million students. It reports the following effect sizes:

Problem based teaching = 0.15

Inquiry based teaching = 0.31

Direct Instruction = 0.59

An effect size of 0.30 or lower is ineffective. Thus direct instruction is far superior to discovery teaching. The following should be a non-issue: Example Based "Explicit/Direct Instruction" vs. "Discovery/Inquiry".

***************

I don't have the book but it sounds interesting.

Oak

http://investigations.terc.edu/impact/2ndEd/2ndEdFindings.cfm/ has specific results about the 2nd edition of TERC. It is true that Pearson helped fund it, but if you read the articles, they are reliably done.

Elementary School Students Mental Computation Proficiencies

Filiz Varol1,2 and Dale Farran1 Early Childhood Education Journal, Vol. 35, No. 1, August 2007 ( 2007)

This article has a nice section on the connection between computation and conceptual understanding.

Journal for Research in Mathematics Education

2001, Vol. 32, No. 4, 368-398

The Impact of Two Standards-Based Mathematics Curricula on Student Achievement in Massachusetts

Julie E. Riordan and Pendred E. Noyce, The Noyce Foundation

This article is an older very focused study comparing reform curricula to traditional on standardized tests.

Journal for Research in Mathematics Education

1999, Vol. 30, No. 1, 3-19

Relationships Between Research and the NCTM Standards

James Hiebert, University of Delaware

This article is a masterful overview to me. The references for this article provide plenty of future reading on which to follow up.

John, nobody relies on self-funded studies. There is a huge risk of compromise because even if the study showed poorly for you, you would find the silver lining and twist something into a positive. Overall, TERC is a disaster.

For example, there was a COMAP study a few years ago that reviewed TERC and this was their conclusion:

"...The principal finding of the study is that the students in the NSF-funded reform curricula consistently outperformed the comparison students: All significant differences favored the reform students; no significant difference favored the comparison students. This result held across all tests, all grade levels, and all strands, regardless of SES and racial/ethnic identity. The data from this study show that these curricula improve student performance in all areas of elementary mathematics, including both basic skills and higher-level processes. Use of these curricula results in higher test scores."

Holy smokes Batman! This is a silver bullet. It's the best under every circumstance at every level. Just one problem. TERC funded the study which completely invalidates the results.

How else do I know? Sandra Stotsky was the assistant commissioner of education in Massachusetts in 2000 when TERC was piloted there. TERC showed favorably in the results, but this was what Ms. Stotsky wrote to me several years ago concerning that study.

"I am aware of several major problems with the MA part of the study. (1) As the Executive Summary admits, mostly high--income "white" schools were using the "reform" programs in the MA grade 4 sample, (2) no information is given on the supplemental tutoring that exists in these suburban communities (a hard factor to get information on without labor-intensive exploration at each school), (3) no information is given about supplemental curriculum materials the teachers themselves may have used--all we are told is that the schools that were contacted said they fully used the reform program. I know that many teachers in these high-income schools use supplemental materials to make up for the "reform" programs deficiencies, (4) no information is given on the amount of professional development the "reform" teachers had (a huge amount in all probability) in comparison to the teachers in the comparison group (if no new math program, no professional development), (5) no information is given on the amount of time spent on math in the reform schools compared to the comparison group (the "reform" programs require a lot more time per week than most schools had been allotting math for many years. For example, I discovered that one Newton elementary school with top scores was considered a model because it taught math one hour each day!), and probably most important and relevant (6) the MCAS grade 4 math test was originally designed with a great deal of advice from TERC. TERC also shaped the math standards in the 1995 standards document that were being assessed by this test in 2000 (it is acknowledged in the intro to this document). TERC's supporters (and EM supporters) were on the assessment advisory committees that made judgments about the test items and their weights for the math tests. It is well-known that the grade 4 test reflects "constructivist" teaching of math. In other words, the grade 4 test in MA in 2000 favored students using a "reform" program."

I also wouldn't quote Noyce anywhere since they are afraid to tell anyone where they performed their study. What's also troubling is their lack of revealing they've previously financed reform math programs so they have an incentive to not look bad by showing they don't work. They are compromised in my view.

http://www.csun.edu/~vcmth00m/noyce.htm

John, I have nothing against using constructivism in the classroom when it's appropriate (which I view as 10-20% of the time). But on the whole, it's been shown an utter failure. Project Follow Through proved that decades ago. As a culture, we rely on transmitting knowledge from one generation to another. The faster we can transmit knowledge with conceptual understanding, the better. Real research shows that that happens best with direct instruction methods.

Another thought from Anonymous (the first one in this thread): math can be taught as if it were a series of puzzles (and this approach seems pretty strong in discover/inquiry-based curricula). On the other hand, it can be taught as a set of tools for solving puzzles and other tasks. Traditional math instruction leans in this direction, and it is entirely possible for the mechanical application of tools to be stressed to the exclusion of understanding. However, for most people, it's more important to master the tool aspect (with underlying conceptual understanding) than to teach the whole discipline through the puzzle aspect.

“Research shows that invented algorithms result in fewer and smaller mistakes on average.”

Does one of the four papers/articles show empirical research findings supporting the claim that invented algorithms result in fewer and smaller mistakes on average? If so which one?

Interested parties may want to check the findings of the National Math Advisory Panel.

Studies need funding. But there is a difference between self-study and self-publication and peer-reviewed research. Peer review is far from perfect, but the best validity check we have. JRME has the highest math ed standards for peer review.

What your measure is obviously is also crucial. If your dominant goal is quick accurate computation, that might be able to be most affected by direct instruction for some or even most learners. Effect size is a tough measure to understand; the criteria about which effect sizes matter is nebulous. The best study I could find on this reported that curriculum choice mattered less than instructional methodology.

"Cooperative learning methods, in which students work in pairs or small

teams and are rewarded based on the learning of all team members, were found to

be effective in 9 well-designed studies, 8 of which used random assignment, with

a median effect size of +0.29. ... Another well-supported approach included programs that focus on improving teachers’ skills in classroom management, motivation, and effective use of time, ... Studies supported programs

focusing on helping teachers introduce mathematics concepts effectively... Supplementing classroom instruction with well-targeted supplementary instruction is another strategy with strong evidence of effectiveness." From Robert E. Slavin and Cynthia Lake; Effective Programs in Elementary Mathematics: A Best-Evidence Synthesis;REVIEW OF EDUCATIONAL RESEARCH;2008 78: 427

The goals about which I care most are not addressed by direct instruction. To learn the mathematical processes, students need time and task to explore, communicate and consolidate.

In my 25 years as a teacher, I see direct instruction helping insufficiently those who need help most. It is most effective for those for whom need help least. Is lecture sometimes appropriate? Yes. The majority of the time? Not for me.

In instructing addition, I care about computation, place value understanding, understanding of the operation, connections to multiplication, connections to algebra, connections to number, and other factors. I do not just care about this one class of problems, but how they connect to previous and future topics.

I am interested in the affective piece that none of you have addressed yet. Math, as it has been taught, generates loathing, and a majority of the population that is anti-math. What do you think about that?

Have any of you read or watched any Jo Boaler?

The classic computation article is Carroll, W. M. (1996), Mental Computation of Students in a Reform-Based Mathematics Curriculum. School Science and Mathematics, 96: 305–311.

Jong, C., Pedulla, J. J., Reagan, E. M., Salomon-Fernandez, Y. and Cochran-Smith, M. (2010), Exploring the Link between Reformed Teaching Practices and Pupil Learning in Elementary School Mathematics. School Science and Mathematics, 110: 309–326. This article demonstrates some effect of reform methods on student learning as measured by regular district assessments. Their lit review also covers a lot of relevant ground.

The Hiebert article I cited before is nice for helping us form expectations about what can be demonstrated by research.

Math, as it has been taught, generates loathing, and a majority of the population that is anti-math. What do you think about that?Any cites for that rather sweeping and somewhat personal statement? "Math as it has been taught" refers to what era? Always?

As far as your statement that direct instruction helps those who need it least, I think the tutoring and learning center industry (and it is a substantial industry in this country) would tend to disagree with you.

I don't think anyone is saying that math should be taught by pure lecture and I certainly wasn't taught that way (I grew up in the 50's and 60's). Teachers asked us questions and there was a back and forth. Traditional math is frequently mischaracterized as rote memorization and procedures taught in isolation. A glance at any number of books from that era proves such supposition wrong, and this particular blog offers many examples of that.

Iowa test scores in the State of Iowa from the 40's through mid 60's show a steady increase in math, in 4th, 6th, 8th, and high school grades. The decline in scores starting in the mid-60's mirrors similar declines seen in the US as a whole, and there have been many theories and reports offered as to why. But the increase in ITBS scores in Iowa prior to that offers some evidence that the traditional math being offered at that time certainly wasn't failing.

Have you read the National Math Advisory Panel's report? Are you aware that much of the research you cite was reviewed and rejected by that panel as not meeting the research standards they established?

A couple years ago I spoke with representatives of Benchmark charter school in AZ and at the time they were the top school in the state and on a school wide survey 94% of children listed math as their favorite class. Their curriculum was Singapore's Primary math series, a direct instruction method, though if it was constructivist and produced a love of math and the top school in the state, I'd have to reconsider my loathing for the program.

I need a citation for antipathy towards mathematics? Really?

It is true that the panel rejected much research as evidence of efficacy. Not that it was poorly conducted or written, but that it was being misapplied in this debate. The "effective programs..." article is using the panel approved methodology for the studies it reviews.

I apologize if you feel I am mischaracterizing traditional instruction. Do you feel reform instruction is always being portrayed accurately in this debate?

Fair point about the tutoring industry. What is the difference between the traditional instruction there and the traditional instruction in the school that is failing the student? (Most likely, according to TIMSS and any number of other teaching characterization studies.)

It feels as if you are playing two games. My claims need to be research validated in broad studies, your claims can be made anecdotally. What about a study like the one this month at What Works Clearinghouse:

"One study of Saxon Math that falls within the scope of the High School Math review protocol meets What Works Clearinghouse (WWC) evidence standards with reservations. The one study included 278 high school students in two districts in Colorado.3

Based on the one study, the WWC considers the extent of evidence for Saxon Math on high school students to be small for math achievement.

Effectiveness

Saxon Math was found to have no discernible effects on math achievement for high school students."

http://ies.ed.gov/ncee/wwc/reports/hs_math/saxon/index.asp

Can we discuss the ideas here? How do you see problem-solving being taught and evaluated in traditional instruction? What are your goals other than computational efficiency? Do you want achievement for all students or just enough students (somehow)?

First, Saxon Math high school texts did indeed earn a dubious result according to WWC. But Saxon elementary materials are among the most highly rated by WWC. If we want to get good results, we at least use Saxon in grades K-8. (I did, on an Indian reservation and then in an all-white Seattle elementary school.) Second, Anyone, ANYONE, who references Joanne Boaler as an authentic math researcher has been snookered beyond belief. Third, Dr. Heibert's article about how to use research findings effectively is very clear in its basic premise: It's the VALUES of those who want to affect math curriculum that impact how the curriculum is researched and ultimately written (or, in the case of reform math, how research articles were originally cherry-picked for results wanted). His report, having been requested by NCTM, ends up supporting many of the reform programs' ideas. Fourth, I quote from Dr. Hiebert's study with his approval in my biography published last year titled, John Saxon's Story, a genius of common sense in mathematics. I encourage folks on both sides of the traditional vs. reform math education issues to read this book, available from Amazon.com or http://saxonmathwarrior.com. (John's CEO, Frank Wang, earned his Ph.D. in pure mathematics from MIT, so his perspective is also included in the book.) It has reams of data proving the validity and reliability of Saxon Math, compared to the non-existent research prior to the NCTM's pushing its Standards on to schools, beginning in 1989; Saxon also doesn't have parent revolts against a politically-supported, failed constructivist program of the past 20 years. Fifth, "traditional" math, therefore, isn't the one that has produced a 70% enrollment rate in community college remedial math classes (and up to 40% in 4-year universities).

Last, the one thing at which John Saxon was also very good was studying his opponents' history and success rate according to three criteria: How many of their students were relegated to remedial/basic classes? How many were enrolled in advanced math and science courses? What were their students' college board scores? His opponents never bothered to do the same for his record.

My issue with the drawing pictures method is that it isn't "conceptual" at all. It's simply another procedure, with pictures.

The child may very well have explained the exact same thing standing in front of the traditional algorithm.

One clarification regarding the fact that Saxon Math's high school texts received a poor rating from WWC: I want to know when the students in the study first began using the Saxon high school books. If the students have not had the K-8 Saxon series (or some equally good traditional curriculum, which is probably Singapore Math), they will have a dismal time with Saxon books--unless their teachers are absolutely super in remediating the missing basic skills found in a majority of our high school students. (Look at state and national test scores.) I tried to introduce my Seattle high school students to Saxon's Alg 1 and Alg 2 texts and they absolutely rebelled against the books. "They're too hard!We're used to making A's and B's!" they declared. They demanded a return to the integrated, effort-based series that had given them soft grades for simply trying, not for mathematically-accepted results. I ultimately had to switch back to those textbooks that I knew were NOT preparing my students for college work.

In addition, Houghton Mifflin Harcourt, the new owner of Saxon Publishers, has changed some of the high school textbooks, which were written by John Saxon himself. (They can do that now that he's deceased.) I therefore need to know which editions of the books were used in the WWC report. (The K-8 series can't be messed with because those authors' rights are binding with HMH.)

What is the typical American's view on mathematics? Where did they learn to hate it?Today, children are learning to hate math in classrooms using constructivist math curricula.

I interviewed my cousin about her child's experience with Everyday Math.

The words my cousin used to characterize her daughter's experience were:

frustrating

demoralizing

boring

One family here in town told me their daughter hated Trailblazers so much she independently took to calling it "Failblazers."

How do you see problem-solving being taught and evaluated in traditional instruction? What are your goals other than computational efficiency? Do you want achievement for all students or just enough students (somehow)?There may be some traditional texts that emphasize only computational efficiency. The traditional method can be done poorly it's true. But there are also traditional types of texts and classrooms that are done well. Such texts/teachers emphasize both procedure and problem solving and in fact procedural fluency is part and parcel to conceptual understanding and problem solving. Katharine has posted many examples showing the types of problems that Singapore Math uses, and other types of textbooks, and compares the problems with those in Investigations, CMP, etc. In fact, the post above this one has such a comparison. Here's a problem in the 4th grade Singapore textbook: "A worker mixed 13.45 lb of cement with sand. The weight of sand used was 3 times the weiht of the cement. How many pounds of sand did he use?"

This problem can be solved using arithmetic methods rather than algebra. Do you think this is a bad problem? The textbook has many multistep type problems. The advantage of multi-step problems is that the order of the steps can be made to vary in many ways, so it isn't just a matter of "memorizing a procedure" and applying it to a problem. There is a degree of understanding that is required that goes beyond just procedures.

I don't understand your second question. Do you feel that texts such as Singapore only reach some or "just enough" students?

Per John

"I need a citation for antipathy towards mathematics? Really?"Yes.

You see, I've heard that statement repeated frequently by supporters of constructivist / reform / standards based math, but I've never seen it quantified. From what I can tell, many of the supporters of constructivist / reform / standards based math hated math as children and therefore assume that everyone else did. They also appear to assume that their hatred of the subject somehow prevented them from excelling at math, yet, at the same time they expect us to consider them experts in math and how math ought to be taught.

Really? You stunk at math and hate it, so you created this nifty new math program that removed the actual math from math, and we're supposed to believe that this program you created will teach our kids math? Hugh?

Yes we have been teaching Mathematics, and the standard algorithms, for hundreds of years. Thomas Jefferson professed that one of his favorite subjects as a child was math, and, as a young child, he learned, dare I say it, the standard algorithms. I don't think they held him back, do you?

What is wrong with teaching the standard algorithms, anyway? Contrary to what is expressed by people like John, learning them proerly depends on solid number sense and understanding of place value and how numbers relate to one another.

As a society, in the US and around teh qworld, we've taught children teh standard algoritms for many many years with tremendous results. Sure the Architects, Engineers, Scientists, Researchers, and Doctors of the past were exceptional people. The Architects, Engineers, Scientists, Researchers, and Doctors of today are just as exceptional, but by and large they are no longer American. Why? Because we've dumbed down math and science instruction to make it "fun" for all rather than concerning ourselves with actually teaching our kids what they need to know.

I don't care that John or anyone esle disagress with me. And honestly, for every study that shows programs like TERC are the bestest things ever, there's a study which shows it's the worst thin ever.

The probelm, as I see it, isn't that I'm right and John's wrong, or that John's right and I'm wrong. The problem is that school districts choose one approach and force our kids to follow them, even when we think the programs stink. Parents ought to be given a choice in how their children are taught math (or reading or science or sex ed). If I think Investigations, or something like it, stinks, then I ought to be able to enroll my child in a program that follows an approach I do support. And if I thik Investigations is the greatest thing since sliced bread, then I ought to be able to enroll my child in an Investigations or similar program.

Having nearly completed the requirements for an undergraduate math major and physics major (at a relatively well-respected institution), I definitely agree with one of the above comments that "longer" methods should be used no longer than necessary. After all, that's exactly the philosophy behind virtually all middle- to higher-level math and physics pedagogy: first take an elementary conceptual approach (like the drawing method), then use that approach to build more powerful tools (like the stacking algorithm, which is really the same thing), then use the more powerful tools to develop even more powerful tools, etc. Stacking, like the drawing method, *is* grouping large numbers of small quantities into smaller numbers of large quantities. The conceptual approach and the fast algorithmic approach need not be mutually exclusive.

Why is elementary math suddenly exempt from the same logic used in high school and beyond? You use Euclidean postulates (or the definitions of algebraic structures like groups, rings, and fields) to prove theorems, and then you use the theorems to solve more complex problems. You don't solve complex problems using elementary postulates. You use Newtonian principles to solve elementary problems in mechanics, and then you use those principles to derive Lagrangian mechanics, from which you solve (a certain class of) more complex problems. You don't solve that class of complex problems directly with Newton's laws. You learn point-set topology and then use it to make claims about infinite series, then continuity, and then derivatives; you don't cite compactness when taking the derivative of a polynomial over the reals.

The same philosophy dictates that students should learn about grouping numbers using an intuitive pictorial method and then you move on to the more powerful stacking algorithm. Adding huge numbers pictorially wouldn't make any more sense than solving higher-level problems with highly elementary tools-- it's tedious, frustrating, and totally unenlightening.

PS I'm not the same Barry as Barry Garelick.

Another point about the use of pictures and manipulates, as illustrated in the video: extended use of this method is tilted in favor of highly compliant and patient children. Children who grasp the concept early on, and would be happy to use an efficient proxy for the pictures/manipulatives process (i.e. would be happy to use the standard algorithm) are frustrated by the time it takes to use pictures. Mature, patient children like the girl in the video may be willing to slog their way through the pictures; hyper, impatient children (more likely to be boys) will tune out.

John said:

"I am a Ph D mathematician, and was educated traditionally. ... But I hated math until high school, where I had greater freedom to work on my own. I made sense of the material on my own, and watched as my classmates were classified as unable to do the material, or turned off of it, or just bored."John, you worked on your own AFTER you gained math skills, which students of today are not able to do. How are you expecting them to work on their own in high school, as you did, when they don't know how to work with fractions, don't know long division, have no number sense, can't subtract a negative, don't know how to multiply, and when any number that pops out of their calculator could be correct, whether it's 10 or 10,000?

How will they ever become PhD mathematicians, as you did?

I just poked through some

Everyday Mathbooks andInvestigations in Number, Data, and Spacebooks andConnected Mathematicsbooks, so that I can show them to people in the community math forums I'm holding.Wow. Have you seen these materials? No wonder our children are confused. No wonder they begin to hate math in about the 3rd or 4th grade. No wonder they grow up to be math illiterate. If you think reform math is producing a generation of children who understand math, or who even LIKE math, you are dreaming.

Wake up. Look around you. See the absolute devastation that reform math and excessive constructivism have brought us. If you are indeed a mathematician, then you know the value of math for nearly every citizen of this country. You also know that what matters are efficiency, proficiency, sufficiency, and correct answers.

Look up and see where we are. If reform worked, John, it WOULD have worked by now.

What good is reform math when it clearly doesn't produce

what matters?Laurie H. Rogers

author of "Betrayed: How the Education Establishment Has Betrayed America and What You Can Do about it:

http://betrayed-whyeducationisfailing.blogspot.com/

wlroge@comcast.net

Re: citation for antipathy. I was imagining that many commenters were math teachers, sorry. When you are a math teacher, people go out of their way to tell you about all their negative math experiences. Runs from 10:1 negative:positive at a college to > 100:1 negative:positive irl.

The goal of these curricula is not to have students use blocks their whole life. It is to get to invented algorithms. Often a variety of them. Given a computation, powerful math learners have 4 or more choices for how to solve. Students that have only been given one way (typically, in my experience, a way that does not make sense to them) are literally less fluent.

An example 21003-8899. Deadly to many students with traditional algorithm. If I had to do this, I solve 21104-9000 instead, which probably I do as -10000+1000. I have choices, which results in increased efficiency, number sense and accuracy. I want my students to have choices also.

If a teacher is helping students to make sense of what they are doing, and using the traditional algorithm, I am okay with that. But that is not how I most often see it used or taught. So @J.D. Fisher - I'd be happy to see an understanding explanation of the traditional algorithm from a student. I also think that nontraditional methods can be taught rotely without understanding, and I'm against that, too.

@Barry (not G.) I want to see cchildren doing mathematics. Not doing what they are told. Mathematicians (I am one, I will repeat here, with a Ph D in math) make sense of novel situations, make conjectures and construct proofs. As a profession, we are not interested in repeating something someone else did. Hopefully in your program you are getting a chance to do math as opposed to just repeat it.

@Mom22 I did not stink at math. The authors of these programs are excellent mathematicians. They LOVE the subject. They are geeks about it. A good problem or new idea will consume groups of them for days. To think I or they dislike the subject is just a misconception. And unfair.

@BarryG Thank you for being conversational. The Singapore curriculum does have some good problems. Given that one example, what I want to know is how is it used. If the students have a chance to explore it, figure it out, and share their understanding - great. If students are taught a method, and then the problem is practice for the method, and the teacher tells them if they are right or wrong - that's icky to me. And a missed opportunity for a rich context with some realism.

The curriculum issue for me comes down to which one offers the most support for teachers that want to teach math as a sense-making, problem-solving discipline. Many curricula just do not have rich problems, or exile them to one or two after a ream of practice. That is not math as mathematicians do it. Problems come first, data is gathered, then generalizations and abstractions follow.

@Laurie I am very familiar with these curricula. The idea that they are the majority textbooks used in the US is far from true.

Places with high implementation of reform show remarkable achievement, like the Michigan representative group that outperformed Singapore in the TIMSS-repeat.

If reform had been implemented, we may have been able to judge if it worked. Traditional mathods got us where we are today as a nation.

Cf. TIMSS-R data in

Mullis, I. V., Martin, M. O., Gonzalez, E. J., O’Connor, K. M., Chrostowski, S. J., Gregory, K. D., Garden, R. A., & Smith, T. A. (2001). Mathematics benchmarking report TIMSS 1999 – eighth grade.

International Study Center, Lynch School of Education, Boston College.

“Research shows that invented algorithms result in fewer and smaller mistakes on average.”

What research shows this? Please provide a reference to the specific research findings that support this claim.

John listed some papers but didn’t indicate which, if any, are research findings supporting the claim. The old Riordan and Noyce paper was listed. The controversy surrounding this work pales in comparison to some of the controversy surrounding Jo Boaler and her work.

An Email Exchange about the Quality of Mathematics Education Research and "Everyday Mathematics"

http://www.csun.edu/~vcmth00m/noyce.htm

A Close Examination of Jo Boaler’s Railside Report

by Wayne Bishop , Paul Clopton , and R. James Milgram

ftp://math.stanford.edu/pub/papers/milgram/combined-evaluations-version3.pdf

As for a constructivist, discovery, or problem based approach, there is a place---once students have a solid foundation in the skills and knowledge to do more than flounder. Minimal guidance at one time sounded good and promising. Over time, has it really worked?

Why Minimal Guidance During Instruction Does Not

Work: An Analysis of the Failure of Constructivist,

Discovery, Problem-Based, Experiential, and

Inquiry-Based Teaching

Paul A. Kirschner , John Sweller, Richard E. Clark

EDUCATIONAL PSYCHOLOGIST,41(2),75–86

http://www.cogtech.usc.edu/publications/kirschner_Sweller_Clark.pdf

http://www.cogtech.usc.edu/publications/

kirschner_Sweller_Clark.pdf

John, I appreciate your attempt to reply, but yours wasn't a response to anything I actually said. My point is that using elementary tools to build more complex tools both enlightens students about why complex tools work and enables them to solve more complex programs on their own-- or, in your words, to do their own math. At least, that's the pedagogical theory employed by every math and physics Ph.D. professor I've had in the past four years. It's also how they suggest we learn to do *real* math. Maybe in your particular Ph.D. program you enjoyed using highly elementary methods in place of more powerful tools, but that doesn't seem to be the mainstream pedagogical theory, and it probably isn't going to help the next generation of quants, engineers, chemists, and physicists do what they need to do. The drawing method is an example of something that should be done as a way to gain intuition about the stacking method of addition, and the stacking method should quickly replace the drawing method en route to more advanced mathematics. Disallowing the stacking method, as suggested in the video, is totally misguided.

I apologize if I am repeating something someone has posted earlier but I got tired of reading comments that asked for research and then summarily dismissed it. Picking sides and attacking sources is not helpful to addressing important educational issues. It only serves to make a point - well your point is made. (full disclosure - I am a colleague of John's)

Continuing to focus on skills and procedures prepares students for the 18th century not the 21st. Adding "with conceptual understanding" doesn't mean it happens. In fact, this focus makes it harder because we jump to a product without understanding how historically we got there. You want research? Try googling ontogeny recapitulates phylogeny.

Let's set aside the research though and acknowledge that reformed-based textbooks are being used effectively in some districts and not in others. The same could be said about traditional texts A lot depends on the teachers (the level of mathematical understanding and PD offered) and the children (prior experiences and home support). So how are we supporting teachers and children to be successful? And what does that look like?

From the post I read, it seems that we can agree that we want learners to have the capacity and agency to do mathematics with understanding. For me, a reform-based text would support my efforts because of my philosophy and experience. Someone else might pick Saxon because of his or her strengths. As long as the learners can do the math WITH understanding and wants to do the math - that's what matters to me.

“Research shows that invented algorithms result in fewer and smaller mistakes on average.”

delta_dc: I have asked for the research on this. I do not feel I have summarily dismissed any of the research mentioned, I have simply asked which, if any, of the mentioned papers have research findings that show invented algorithms result in fewer and smaller mistakes on average. It still has not been answered.

Anonymous (not sure which one).

As I said, I didn't read everyones' post - no offense intended to you by my comment.

If I were interested in "Research shows that invented algorithms result in fewer and smaller mistakes on average" then I'd google it. I did and I found the following that looked promising:

Carroll, W. M. (1996). Use of invented algorithms by second graders in a reform mathematics curriculum. Journal of Mathematical Behavior, 15(2), 137-150.

Disclaimer, I haven't read it. Just sharing how I might find the information. Good luck and keep searching.

Thanks. I hope to look at it, unfortunately it will be later today. I do appreciate you posting it. I do know how to search for things on the computer, but when someone says "research shows", I often like to know specifically which research they are referring to... and not be left to have to try to find such research on my own---and maybe not find the same source.

Again, thanks. You have at least attempted to answer a question---what research---that should have been provided with the "research shows" claim.

While we may not see things the same way, it is okay to have differences of opinion. It is helpful and often nice to know what information has helped formed the opinions. Even at that, some may not agree as to the validity or quality of the research. And that is okay as well. It would sure be boring if everyone agreed about everything.

@Anonymus, I posted that reference above along with another in response to your request the first time.

Also, more broadly:

Adding it up: Helping children learn mathematics; Kilpatrick, J.; Swafford, J.; Findell, B.; 2001

The 1998 NCTM Yearbook:The teaching and learning of algorithms in school mathematics

Making Sense: Teaching and Learning Mathematics with Understanding Hiebert, Fennema, Fuson, Murray, Carpenter

John,

Thanks. You posted four references with no mention as to whether any of these supported the "research shows". Do all four have research findings show that invented algorithms result in fewer and smaller mistakes on average?

Sorry, but I do hear "research shows' a lot with no references provided supporting the claim. As a college student, making such a statement without providing the reference would have resulted in an unacceptable grade. The professors were not expected to go off and search the literature with no reference. It is a good practice to provide an information source---it was required when I was in school. That way everyone can access the same source. That does not mean everyone will agree or interpret it in the same way.

This reminds me of a student I had one year. He turned in a science report. He knew he was required to provide references for his information sources. Maybe he forgot to keep track of them as he gathered his information. He put down peasandcorn.com as the reference for everything. He did provide a reference, never mind it was not the source of his information. For anyone interested in learning more about what he had written and wanted to go read his source of information for themselves, his references were not helpful.

John,

Thanks. I did see the four references you posted. There was no indication as to whether one, all, or none of the references had research showing that invented algorithms result in fewer and smaller mistakes on average.

As a student in college, if I made a statement in a paper it was my responsibility to be clear about referencing my source of information. To not do so would result in an unsatisfactory grade. Maybe that has changed and accounts for why I often hear and read “research shows” with no reference provided.

I laughingly call this peasandcorn.com research. One student I had knew he was supposed to provide references for his source information. Maybe he forgot to keep track of his sources. He turned in a science paper and used peasandcorn.com as his source of information for everything. While he did provide references, they were not to his sources of information. For anyone interested in learning more about what he had written about, his references did not help at all.

I still am not certain if this is a peasandcorn.com research paper, where I am interested in reading the research showing that invented algorithms result in fewer and smaller mistakes on average. I don’t think it is an unreasonable request to ask since a reference was not initially provided. I do have an interest in seeing the paper that supports this claim but am not going to go searching, only to wonder if what I find is or isn’t the same source of information. So, I will ask again. It is true that references have been provided but it has not been indicated as to whether these references are the ones on which the “research shows” claim is made. It is reasonable to ask specifically which one(s) show empirical research findings supporting the claim that invented algorithms result in fewer and smaller mistakes on average?

For those interested in more research, I would suggest reading Dr. James Zull's "The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning." I happened to be reading it on the plane today for an upcoming book discussion and came across this - "Calculating a right answer is important, but it does not generate understanding." (p. 162). This seemed germane to our conversation.

Turns out that the biology says that in this case it is not about being left-brained or right-brained but whether we are using the front of the brain or the back. I'm sure I'm not doing his writing justice - check out the book.

To the first commenter -- why in god's name would any teacher or parent want their child to "learn" math by using a calculator? I want my kids to know how to do math in their head or on paper without a calculator. Calculators hide the fact that teachers cannot teach math because they do not know math. Those teachers who allow the use of calculators in their classroom, on test and on the annual state tests are doing their kids (and themselves) a huge disservice. How in the world do you decide if the kids don't how to use the calculator, don't understand the actual math concept, or you don't know how to teach? Go back to the basics...save the calculator for the office...please!

What is more important?: Understanding or fluency? Would you want a person who "understands" math but can never compute a correct answer or a person who doesn't understand math but can calculate any answer to any problem 100% of the time?

I find it interesting that educators tend to emphasize understanding, when 99% of people in the world do not understand math. How fair is it to demand of a 3rd grader what most adults cannot do?

From the post I read, it seems that we can agree that we want learners to have the capacity and agency to do mathematics with understanding.Why?

In Elementary, Middle and high school, all I want is for students to be able to

domath. If they need/want to know the whys and wherefores, they can learn them in college. The vast majority of people neither need to have an "understanding", or care.Here's one of the comments that didn't appear earlier:

John,

Thanks. I did see the four references you posted. There was no indication as to whether one, all, or none of the references had research showing that invented algorithms result in fewer and smaller mistakes on average.

As a student in college, if I made a statement in a paper it was my responsibility to be clear about referencing my source of information. To not do so would result in an unsatisfactory grade. Maybe that has changed and accounts for why I often hear and read “research shows” with no reference provided.

I laughingly call this peasandcorn.com research. One student I had knew he was supposed to provide references for his source information. Maybe he forgot to keep track of his sources. He turned in a science paper and used peasandcorn.com as his source of information for everything. While he did provide references, they were not to his sources of information. For anyone interested in learning more about what he had written about, his references did not help at all.

I still am not certain if this is a peasandcorn.com research paper, where I am interested in reading the research showing that invented algorithms result in fewer and smaller mistakes on average. I don’t think it is an unreasonable request to ask since a reference was not initially provided. I do have an interest in seeing the paper that supports this claim but am not going to go searching, only to wonder if what I find is or isn’t the same source of information. So, I will ask again. It is true that references have been provided but it has not been indicated as to whether these references are the ones on which the “research shows” claim is made. It is reasonable to ask specifically which one(s) show empirical research findings supporting the claim that invented algorithms result in fewer and smaller mistakes on average?

I am an elementary teacher and also a math tutor Kindergarten-Geometry. I am in my young 50's so was educated in the traditional way. I am in no way an expert, however I know I have a great understanding of the math required K-9. Here is my observation from the trenches.

90% of my young colleagues (under 30) are not proficient in elementary math. Most were educated using constructivist programs and swear by the effectiveness of them.

Our district mandates Everyday Math, and these teachers are the ones that insist children may not use the traditional methods. These teachers do not understand either method.

Our district in undertaking district wide professional development to increase teacher effectiveness in math instruction. Within our district 23% of 10th graders were proficient in math on the state test. Every participating teacher was given an arithmetic test that was at a 7th grade level. 80% of the elementary teachers failed. This is the problem in my district. The teachers do not understand what they are being asked to teach.

Is it time to have math specialists that can actually understand math teaching only math in elementary schools?

In Elementary, Middle and high school, all I want is for students to be able to do math. If they need/want to know the whys and wherefores, they can learn them in college. The vast majority of people neither need to have an "understanding", or care.The push for understanding comes from the mischaracterization of traditional math as teaching procedures in isolation, so that students may know 5 x 4 = 20 but may not know what the multiplication represents, or that problem solving is taught in a "rote" procedural fashion that does not engender problem solving strategies. In addition, there is a belief that if a student cannot explain how an algorithm works, then all they are doing is "mere computation". I address these issues in an article found here.

My children are taught in a district that mandates TERC Investigations, despite the fact that the program failed to meet a sufficient number of state standards of learning to be recommended by the Dept of Ed for use in public schools. Since TERC was mandated the percentage of students passing state exams is down, but only slightly, while the percentage of students achieving and advanced score (88% or more correct) has dropped more than 20%, on average.

Our district used to rank in the top 25% of schools in the state based on the percentage of students passing the state math exams. Since TERC was mandated, because our pass rates were unchanged or down slightly while the rest of the districts in the state improved their performance, our rank in the state has dropped from the top quarter to the bottom half.

Officials in our district were challenged by parents to compare correct responses on the state exams by standard content area from before TERC to after TERC, and against district mandated thresholds to identify areas where our students might be struggling and instruction might need tweaking. District officials refused to examine the test data because they felt the program provided by TERC was sufficient.

One of the parents in our district looked at the data and discovered areas where our students were struggling (namely subtraction, division, multiplication outside of fact families, fractions, decimals, estimating, and likelihood of outcome). District officials scoffed at her work and refused to consider it. Worse, they refused to admit that students were struggling with certain topics or that instruction needed to be tweaked in any way, shape, or form.

In my experience that head in the sand mentality is what you get with a TERC district.

John:

You said: "@Laurie I am very familiar with these curricula. The idea that they are the majority textbooks used in the US is far from true."What is your reference for that? I have several references from the NSF, and others, that brag about how many districts implemented various reform curricula.

Regardless, in Spokane Public Schools, we've had various reform curricula - K-12 - for more than a decade. Our teachers and students have been beat to death with reform. Administrators dismiss all complaints as "isolated" -- and they continue to buy more reform -- while blaming all problems on teachers, parents, poverty, money, legislation, standards, testing, and hormones (yes, I've actually heard that).

Our students now have low levels of math skills, high remediation rates in math in college -- most testing into elementary math or below -- and low pass rates on our state math tests. Last year, the 10th-grade math test pass rate was 38.9% on a test that required just 56.9% to pass.

Like you, our administrators continue to blame "traditional" math, even though our students haven't had it in years.

You said: "Places with high implementation of reform show remarkable achievement, like the Michigan representative group that outperformed Singapore in the TIMSS-repeat."Your reference is from 2001. I don't know how those curricula were implemented. My references are from 2009-2011. Our children -- after 15 years of reform math and excessive constructivism -- have almost no skills in mathematics.

You said: "If reform had been implemented, we may have been able to judge if it worked. Traditional mathods got us where we are today as a nation."You have pulled those statements out of the air. They ignore all of the evidence around you. To me, it's as if you see a chair in front of you, and you're insisting on calling it a table.

You don't work for Spokane Public Schools, do you?

Laurie H. Rogers

author of "Betrayed: How the Education Establishment Has Betrayed America and What You Can Do about it:

http://betrayed-whyeducationisfailing.blogspot.com/

wlroge@comcast.net

So @J.D. Fisher - I'd be happy to see an understanding explanation of the traditional algorithm from a student. I also think that nontraditional methods can be taught rotely without understanding, and I'm against that, too.I'm not sure why the explanation has to come from a student in order to be effective (or perhaps that's not what you meant).

I've never had a problem with rote learning (whatever that is). I'll often spend three or four sessions with tutoring students going over the long division algorithm until they have it down pat. Once that's done, I hit 'em with this, and suddenly a lot of things become clearer.

But nothing about the rote learning has destroyed their brain. In fact, it has better prepared them to understand the conceptual part of it.

I'm one of those who have been through the "new math" and the old math. I've done it all. I've also taken advanced math (through complex-variable calculus) and graduated with a technical degree requiring it. I have experience on the practical end of this subject.

The so-called "researchers" who claim that students using "invented algorithms" have a lower error rate than those using traditional methods are WRONG. WRONG.

WRONG WRONG WRONG!I can look at these 4-5 digit addition problems and do them in my head... not despite knowing the traditional methods backwards, forwards and sideways, but BECAUSE of them. The associative and commutative properties are NOT invented. Realizing that 8899 = 9000 - 101 is NOT an invention, it's an insight. These insights are impossible without intimate knowledge of the traditional basis of arithmetic. And that's all it is, arithmetic. Mathematics such as algebra requires knowledge of arithmetic so that fractions (and related concepts, such as proportions) are familiar and easy. Any student who has to draw three-dimensonal shapes to represent the thousands place in an integer will be so hopelessly bogged down in hypersimplified details to ever grasp broader principles; they lose the forest in the trees. Any instructional method which requires such crutches deliberately cripples those who are ready to move ahead.I, as a software engineer, am PAID to do things like hyper-simplifying these processes so that silicon-based morons manipulating 4-bit integers can construct useful arithmetic operations from them. Humans are supposed to be smarter than that, and usually are. That is, unless they are politically-indoctrinated "teachers" practicing an art they call "social justice" instead of pedagogy. In that case, all bets are off.

Jonh said: I am a Ph D mathematician, and was educated traditionally...I'm a mathematician, and I never have to simplify rational polynomial functions. Never. Do you?" I am an mechanical engineer, and I have. Always.I wonder how John got his PhD in math?

I'm fully versed in ring theory. I understand polynomial division on a deep level. I have to teach it regularly. IN REAL LIFE, despite all this equipping to find them, I have never seen a need for polynomial division. I thought that was understood. If Anonymous means they do know real applications of polynomial division or rational functions, dish!

Wow, John's got a bee in his bonnet here.

John, you say you're a PhD in math and teach in a mathematics department. So am/do I. I found these things pretty hard to reconcile with your statement:

"Research shows that invented algorithms result in fewer and smaller mistakes on average."

Either you teach in some sort of educational bubble and are completely insulated from real students or, like me and the many thousands of math professors across this continent you realize that the main problem we have is students using their own approaches, having learned something completely wrong and internalizing it. We spend a good deal of our time UNteaching things, which is the least effective mode of teaching.

As the reform/fuzzy math you appear to be advocating works its way through school and now into students entering mathematics, this problem is rapidly getting worse. What has changed is that "inventing your own way of doing things" is now being taught as a virtue.

It's great when students have UNDERSTANDING and can develop valid alternative approaches. I see this all the time with students who are WELL GROUNDED in basic core materials such as the four standard algorithms.

As Daniel Ansari (Cognitive Developental Science, University of Western Ontario) said at a recent conference we held here, understanding and skills are not antithetical -- as "reformists" often argue -- but mutually necessary because they form mutual scaffolding. You cannot have one without the other.

Further, he goes on to say, there is no chicken-and-egg problem as this might appear to imply, because is is clear that skill often must come before understanding -- particularly in early years. Why? Because skill can be developed in a vacuum, but understanding cannot. You can't have understanding of something without a background of skills within which to frame it.

Anyone who has watched a child learn to count knows this. The ALGORITHM comes first. Understanding comes later, through REPETITION of the algorithm until it has created well-worn pathways through the mind. I get into this discussion with education professors all the time and I use this example. Usually I'm met with a gaping jaw and bald-faced assertions that I'm wrong -- usually they say I clearly don't know anything about learning processes. But what they don't ever do is explain why this isn't an instance in which skill must precede understanding. Perhaps you'd like to try, John?

As for long division, your argument goes around and around. Actually in my second-year calculus course (multivariable calculus, sequences, series etc.) I counted 6 times we required long division in the first month. Only in a couple of cases did we "use" long division to calculate something. MOST of the time it was needed for UNDERSTANDING something. And there's the point John: We require tools like this to be taught early, and for students to be well-versed in them because they are important scaffolds for understanding. Any advocate of understanding in mathematics would understand why they are important in this regard.

You say you've never used long division in "real life". Hmmm. That's a road that leads to an educational trash heap. Try it with trigonometry. Where have you ever used Arctan in "real life"? What about even sin, cos, etc.? Maybe we should just scrap trig courses all together -- after all, if we ever need to calculate those functions, we've got dime-store calculators that'll do it for us.

The same is true for almost everything in mathematics, except you can probably make a case for learning how to count, and how to read statistical charts. Won't math be fun then, John.

In fact, how about base 10 blocks? When has anyone ever needed them in ''real life"? Get rid of them. And the pictorial represenations used in reform math that represent base 10 blocks. Scrap these -- they're never used in "real life".

But why stop with math? Why not sift through all of the sciences and arts, discarding anything rarely used in "real life". Solar system? Gone. Periodic table of elements? Who needs it? Shakespeare? Early American lit? Really, who'll ever use that stuff?

You get the picture.

We've always had a problem with students thinking they've got a better way to do something than they've been taught -- but they're wrong. But now we're seeing students who have been taught according to an assumption that inventing your own stuff is a REPLACEMENT for core topics and skills. This is utter Malarky.

Your statement is ludicrous, John. Please cite an actual study here and we'll discuss it. I don't mean to be insulting, but when people base their entire argument for their claims upon an unsupported assertion that "Research shows" something in this discussion, the first thing that jumps into my mind is that they must be in Teacher Education, because that seems to be a standard proof technique over there. I'm not a "teacher ed" mathematician, John. I'm a real mathematician; appeal to authority does not hold water with me, especially if you don't even cite the authority. If you want to appeal to research, show us the money and then let us engage in a real discussion about what is actually shown.

I'm expecting, by the way, that you are likely to cite Carpenter, or Kamii/Dominick. If you really want to be taken seriously, I suggest you find something better, but if you like we can discuss these pieces.

I am a math coach from a K-8 school using Investigations in Number, Data, and Space and CMP2. We have been using Investigations and CMP2 for 7 years now and have seen a tremendous gain in scores. We have made AYP every year in a Title I school with 76% low-SES students and 20% students with IEPs. For the last 3 years, our African American students have performed as well or better than the white students (closing of the achievement gap). The scores of students with IEPs have also steadily gone up each year.

A few things that I have noticed in comparison to other places that I have worked in the past that I think are worth noting:

1) Our 5th graders are able to do division mentally, faster than their parents are able to do the traditional algorithm with paper and pencil (we've raced!) because they understand what division is and have figured out more effective ways to get to the answer.

2) Our 2nd and 3rd graders are able to solve multi-step multiplication and division problems long before ever having been taught an algorithm because they have developed skills in problem solving and perseverance. When we give our benchmark assessments which include these types of problems that they have not yet encountered, we are always surprised at how many students are able to figure them out. In both of my previous schools, when giving the same benchmark assessment, students shut down, apply incorrect procedures, or don't even try to solve these problems.

3) Our 4th and 5th graders do not make the same mistakes with fractions as students who are taught traditionally (mistakes such as thinking that 1/4 is less than 1/8 because 4 is less than 8, or thinking that 1/2 + 1/2 is 2/4 because 1 + 1 = 2 and 2 + 2 = 4).

4)Our students transition well into Algebra because they have been taught all along to find unknowns, to study patterns, and to make sense of situations.

We have put a ton of time into hiring the right teachers and giving them the support and professional development needed to be successful. We've also done extensive work in teaching parents the math so that they know how to help (and how not to help) their kids at home. When you have an entire school community focused on the same goals, you can do incredible things. This approach may not work for everyone, but it is working for us.

Hmmm, I think the sense of accomplishment in the fact that they're doing division faster without using the traditional algorithm is a little short-sighted. The traditional algorithm might be slower for doing 3200 divided by 20 or some other such easy-to-do-in-one's-head problem, but it's the clearest, most efficient way to go when manipulating messy polynomials for use in higher maths (not just arithmetic) such as calculus. Mental math tricks aren't enough to help deal with tough problems like 6.0221415x^3 divided by 3.25x^1.5 or other problems that tax the working memory.

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