Stories & Grievances
The Train Wreck of Everyday Math
Everyday Math - or Connected Math - is a program that is bad for kids, teachers, and ultimately, Principals and their schools. It makes no sense.
One Step Ahead of the Train Wreck
Barry Garelick - May 15, 2009
The first math tutoring session with my daughter and her friend Laura had ended. I sat in the dining room, slumped in my chair. "You look sick," my wife said.
"I am," I said.
My daughter—subjected to the vagaries of Everyday Mathematics, a math program her school had selected and put in effect when she was in the third grade—was having difficulty with key concepts and computations. She was now in 6th grade, and with fractional division, percentages and decimals on the agenda, I wanted to make sure she mastered these things. I decided to start tutoring her using the textbooks used in Singapore’s schools. I was familiar with the books to know they are effective. To make the prospect more palatable, I suggested tutoring her friend at the same time, since Laura’s mother had mentioned to me that her daughter was also having problems in math.
I figured I would start with the fourth grade unit on fractions which was all about adding and subtracting fractions, which they had already done, and then move rapidly into fifth grade, and start on the rudiments of multiplication. "This'll be easy," I thought. "They've had all this before.”
We only made it into two pages of text in the fourth grade book. I came to find out that the concept of equivalent fractions (1/2 = 2/4 = 3/6 and so on) was new to them. This was the beginning of my attempt to teach my daughter what she needed to know about fractions while trying to stay one step ahead of the train wreck of Everyday Math (EM).
Train Wreck Defined
To understand why I refer to Everyday Math as a train wreck, I need to provide some context. First of all, some information about me: I majored in mathematics and have been working in the field of environmental protection for 36 years. I not only use mathematics myself, but I work with engineers and scientists which requires a fairly good proficiency in it.
As I mentioned, my daughter’s school in Fairfax County, Virginia started using the program when she was in third grade. By fourth grade, I was seeing some of the confusion caused by EM’s alternative algorithms. This aspect of EM has been written about extensively so I won’t dwell on it here except to say I wanted to make sure my daughter understood the standard algorithms for two-digit multiplication and for long division. Her teacher insisted they use the alternative algorithms offered by EM; she did not teach the standard algorithm for long division. Some of the teachers at her school offered tutoring services, so we hired one of them to teach her the standard algorithms.
The teacher/tutor did as we instructed and after four sessions, my daughter was excited to show me how she could do long division. She wrote out a long division problem but got stuck along the way when she didn’t know the answer to 28 divided by 7. Long division is predicated on students knowing their multiplication facts. My daughter was not alone in this; many of the students in her class did not know them. Perhaps her tutor had discussed what to do in such instances. It was apparent that whatever she told her was not to brush up on her facts, but rather go back to first principles, since my daughter was now drawing 28 little lines on the sheet of paper and grouping them by 7’s. I decided to inquire.
“WHAT ON EARTH ARE YOU DOING?” I asked. My daughter began to cry.
I felt bad about yelling. Later, my wife, daughter and I sat down and reached an agreement. It was too expensive to keep on having her tutored-- I had spent $200 so far on tutoring and really could not afford any more. We would therefore halt her tutoring and I would take over provided that I would not yell.
I helped her on an ad hoc basis. If she needed help, I would step in. The problem is that when she needed help, it was generally too late, and I would end up having to do damage control. One problem I was having was that EM does not use a textbook. Students do worksheets every day from their “math journal” a paperbound book that they bring home. Without a textbook, however, it is not always apparent what was taught—particularly when the student doesn’t remember. Any explanation that a student has received about how to solve such problems is done in class. The technique is contained in the Teacher’s Manual, but that is something neither students nor parents have. There is a student’s reference manual, a hardbound book containing topics in alphabetical order and which can provide some guidance, but does not necessarily cover what was said in class. Thus, there is no textbook a student (or parent) can refer to go over a worked example of the type of problem being worked. Worse, sometimes problems are given for which students have no prior knowledge or preparation. They appear to be reasonable problems—it is just not evident to the parent who steps in to help the struggling child that they have had little or no preparation for such problems. Then there is the issue of sequencing, or lack thereof—which I will discuss later.
By the time my daughter was in fifth grade, she would get a problem like 8÷0.3. They had not had fractional division, and limited work with decimals—certainly nothing like this problem before. A typical dialogue would then proceed as follows:
Me: What did the teacher say about how to solve this?
Daughter: I don’t know.
Me: Whattya mean you don’t know? You were there weren’t you?
Daughter: I don’t know what he said; he just said do the problems.
Me: Well, how do they expect you to do this? You’ve never had anything like this before. SO OF COURSE THEY GIVE YOU SOMETHING THAT YOU CAN’T DO AND YOU’RE SUPPOSED TO FIGURE IT OUT?
Wife: (offstage) what’s the yelling about?
Daughter: It’s OK, he’s not yelling at me.
Me: I’m not yelling at her.
Wife: (offstage) I heard yelling. Are you getting mad at her?
Daughter: He’s not getting mad at me; he’s mad at the book.
My daughter’s fifth grade teacher shared my disdain for EM and supplemented it heavily with photocopies of pages from an older textbook. I told him once in an email that I was not happy with EM and asked him his opinion. I’ve asked other teachers this question and they usually chose not to answer—perhaps out of fear for their jobs. I was surprised therefore when he responded: “I totally agree with you on everything you said about Everyday Math. It has been very difficult for me to use the book.”
Despite his knowledge and good teaching, there was still lack of a textbook and he was still consigned to the pacing and sequence of EM. I believe these factors contributed to the lack of knowledge about fractions exhibited by my daughter and Laura.
The Long March to Fractional Division
Knowing that in 6th grade, they would learn fractional division, as well as decimals and percents, I feared a train wreck if I didn’t get to my daughter first. Given how little they knew about fractions during the first lesson, I felt that my fears were justified.
Fortunately, things progressed nicely with the two girls after that first lesson. But I only had about four weeks before they hit fractional division—not a lot of time. Therefore, I decided to teach each chapter on fraction in the Singapore Math, from 4th grade to 6th grade textbooks in a concentrated burst. Although I really should have started all this back in 4th grade, doing it this way had an unexpected benefit: they saw almost immediately the connections between multiplication and division of fractions. This was no coincidence—the curriculum is very carefully sequenced. And while fractional division isn’t presented formally until the 6th grade, students are working on aspects of fraction division long before they reach the 6th grade. By the time students reach the 6th grade unit on fraction division, they have done hundreds of these problems leading to an understanding of the meaning of and connection between fraction multiplication and division.
The heavy lifting with Singapore worked well; when they got to EM, it was a review. It was almost anticlimactic. It was a one page worksheet asking questions such as “How many ¾ inch segments are there in 3 inches?” After four such questions, the text presented a formula in a box in the middle of the page, titled “Division of Fractions Algorithm”. The algorithm was stated as a/b÷ c/d = a/b * d/c. Unlike in Singapore Math, there was nothing to connect any invert and multiply relationships to previous material. In fact there was nothing that appeared to lead up to this—just a rule to be memorized despite EM’s pledge to teach “deep understanding”. As I and many other parents I’ve spoken with have found, EM lacks the sequencing to pull it off; and that is the crux of the train wrecks that were to come.
The Spiraling Train Wreck
Numbers with Points in Them
Despite the victory with fractional division, the following week’s tutoring session left me slouched in my chair with my hand over my eyes.
“You look sick,” my wife said.
“I am,” I said. “Just when you think everything is going great, it isn’t.”
I had planned to focus on word problems in fractional division to cement in the concept, but apparently the day’s math lesson at school had confused Laura, and before my lesson could begin, she asked me the following question:
"I'm confused about something," she said. "How do you get from a number on top and number on the bottom of a line into a number that has a point in it?"
I had her repeat the question a few times before I understood she was asking how you convert a fraction to a decimal. Now, Laura was bright and she knew what a numerator and denominator were, and what a fraction was, but apparently the EM lesson they were working on sprung this on them without warning
I wasn’t planning on teaching decimals that day, but seeing that the train wreck of conversion of fraction to decimal was upon us, I took this as a cue. Singapore presents conversions for the first time in the 4th grade text (Singapore Math 4A) showing 6 dimes divided into 3 groups yielding 2 dimes per group, which is expressed first as 6 “tenths” divided by 3 is 2 “tenths”. They then take it to the next step: 0.6÷3 = 0.2. After a few more similar problems, Singapore then introduces 2÷ 4 and shows a boy thinking "2 is 20 tenths."
At the end of the unit they are solving problems like 2.4÷ 6, 3 ÷ 5 and 4.2 ÷7 as well as non-terminating decimals such as 7 divided by 3. What is striking about this lesson is that while its focus is decimal division, the lesson implicitly teaches how to convert fractions into decimal form by virtue of students having learned earlier that fractions are the same as division. That is, they have learned earlier that 1÷ 4 is the same as ¼. The lesson on dividing decimals was situated in the context of fractions—and treating fractions (i.e., tenths) as units—a unifying theme that extends throughout the Singapore series.
I’ve thought about why Laura could not understand the lesson at school, to the extent she could no longer recognize what a fraction was. I believe it is because while Singapore situates decimals in the context of fractions, EM situates decimals in the context of the unfamiliar. The EM program is predicated on the theory known as the “spiral approach”:
“The Everyday Mathematics curriculum incorporates the belief that people rarely learn new concepts or skills the first time they experience them, but fully understand them only after repeated exposures. Students in the program study important concepts over consecutive years; each grade level builds on and extends conceptual understanding.” (Everyday math; Education Development Center; Newton MA; 2001)
This does in fact make sense considering that for most people a particular concept or task starts to make more sense after they have moved on to the next level. But this phenomenon occurs when there is mastery at each previous level. For example, I became fairly good at arithmetic and developed a deeper understanding of it after I took algebra; I fully understood analytic geometry after calculus and so on. Each previous bit of learning seems that much more apparent at the next level of understanding.
In EM, however, students are exposed to topics repeatedly, but mastery does not necessarily occur. Topics jump around from day to day. Singapore Math’s very strong and effective sequencing of topics is missing in Everyday Math. While Singapore develops decimals by building on previous knowledge of fractions, in Everyday Math, students are presented with fractions and decimals at the same time. The topic of conversion of fractions to decimals occurs in the fourth grade in the context of equivalent fractions, and is called “renaming a fraction as a decimal”. The “Student Reference Manual presents fractions that can easily be expressed as an equivalent fraction with a denominator of a power of 10 such as ½, or ¾. For fractions that cannot be directly expressed with power of 10 in the denominator, the Student Reference Manual provides the following instruction: “Another way to rename a fraction as a decimal is to divide the numerator by the denominator. You can use a calculator for this division. … For 5/8 key in: 5 ÷ 8; “enter”; Answer: 0.625.” (University of Chicago School Mathematics Project; 2004. Everyday mathematics. Student reference book. 2002. SRA/McGraw-Hill; Chicago (p. 59)
It is not surprising then that Laura would fail to see what was going on. Without knowing what the connection was between fractions and decimals, the fraction ceased being a fraction in her mind and was just a number on top and a number on the bottom with a line in between. And somehow that strange looking number got transformed into a number with a point in it.
What the Casual Observer Doesn’t Know
A casual glance at Everyday Math’s workbook pages does not reveal that there is anything amiss. The problems seem reasonable, and in some cases they are exactly the same type given in Singapore Math. What the casual observer doesn’t know is what sequencing has preceded that particular lesson, nor how that lesson is conducted in class. What is supposed to happen is that students are given a series of problems to work (in small groups). The Teacher’s Manual advises teachers to monitor students as they work through the worksheet and look to see if students can answer certain key questions. If a student cannot, it is an indication that the student needs more help. This means “reteaching”. Reteaching amounts to having students read about the particular topic of concern in the Student Reference Manual.
If the lack of proper sequencing, lack of direct instruction, lack of textbook and lack of mastery of foundational material prevents a student from making the necessary discoveries, he or she can be “pulled aside” and given material to read. So teachers are left with a three ring circus of kids getting it, kids not getting it, and are expected to “adjust the activity” as needed.
By the time EM gets to 6th grade, the workbooks are loaded with Math Boxes—the term for worksheet review sessions that come in the midst of a particular unit and consist of a mixture of problems from past years in the hope that the kids will finally master the material. Students get ever increasing amounts of Math Boxes. The expectation is that the nth time through the spiral is the charm. With EM, every day is a new train wreck of repeated partial learning.
Connecting Home with School
The danger of an “after schooling” program such as I was conducting is a tendency for the students to think of the math learned at home to be different or unconnected with the math learned at school. My goal of staying one step ahead of train wrecks worked to get to the topics first, so that by the time they got to it in school, they had seen it before. This was difficult since I was held hostage to EM’s topsy turvy sequencing and occasionally was forced to tackle things like geometry that came out of nowhere. All in all, the crash course that I cobbled together on fractions provided the proper framework to then work with Singapore Math’s lessons on percents, ratios, proportions and rates. The rest of the semester came without undue problems and both girls got A’s in the class I’m happy to say.
I’ve told this story to many people since it happened—mostly people who have asked me what to do when their school has a problematic math program. My last retelling was to my wife; it’s a recurrent theme in our house. We were reminiscing about when I had our daughter’s toy blackboard set up in the dining room, and I was teaching her and Laura the math they weren’t learning at school.
There was no need for me to finish the conversation, because the conclusion is always the same: Poorly structured math programs are not fair to students, parents or teachers. It is unfair to students because they are essentially attending another class after a fully day in addition to finishing their homework for school. It is unfair to parents who have to either teach their kids or hire tutors—and are held hostage to the school’s math program whether they like it or not. And it is not fair to teachers who are expected to teach students based on an ineffective and ill-structured program. Through no fault of the teachers, math taught via EM is math taught poorly. It is by no means easy to teach math correctly. But it is even harder to undo the damage by math taught poorly.
Many teachers do not realize that they have been given an unenviable and impossible task. In fact, I have spoken with new teachers who speak of EM and other poorly conceived programs in glowing terms, speaking of them as leading to “deeper understandings of math.” Some have said “I never understood math until I had this program.” But it is their adult insight and experience that is talking and creating the illusion that the math is deep. Children cannot make the connections the adults are making who already have the experience and knowledge of mathematics.
Through my experience teaching my daughter and her friend, I have come to believe that an essential requirement of textbooks is that they teach the teachers. This may happen to some degree with EM, but based on my experience with the program, not much gets transferred to the students. With Singapore Math or any well structured and authentic mathematics program, both teachers and students greatly benefit.
Shortly after this experience, I began taking evening classes at a local university to obtain certification to teach math when I retire. I have no illusions—I’m told that it isn’t easy. I’m not out to save the world—just to educate one child at a time. That said, I will remain forever grateful to my daughter and Laura for having taught me so much about fractions.
School math controversy
Posted by Letters editor, The Seattle Times
Kids will suffer consequences of decision
Last night, for the first time all school year, my second-grader uttered the words, "I like math!" She surprised herself, I think.
For the first time all year, my husband and I --no math slouches ourselves -- were thrilled to understand what her homework was asking for and be able to help as needed. This is why: Having completed the Everyday Math curriculum she is required to teach, my daughter's teacher had sent home a math homework packet from another curriculum -- what a difference!
Everyday Math is the elementary math curriculum adopted by Seattle Public Schools two years ago over the impassioned objections of teachers and parents. Now the district has done it again and selected Discovery Math for Seattle's high schools.
I don't know what the political machinations are behind the district's decisions. There must be some, though, because why else would they opt for what is clearly the lesser curriculum in both cases over the strenuous objections of the community they are supposed to serve? The opinions of the teachers, who devote their lives to helping our kids learn, and the parents, whose primary concern is obviously their children's well-being, carry no weight.
I feel so powerless and frustrated by the district's lack of accountability to us. I feel terrible that all our kids will be suffering the consequences of the district's misguided decision for years to come.
-- Robin Kelson, Seattle
Concept-driven math leads to achievement
Real Math as been tossed around as the bad guy for years. Real Math is math for understanding. It is concept-driven. The math Bruce Ramsey speaks of is regressive memorization ("School Board fails math test," Opinion, column, May 13).
Math watered down for parents to understand because they were not taught concepts when they went to school will not put our students on the road to high achievement. I have taught in an international school with many Asian students whose parents often have tutors for them. Perhaps that's what parents need to do if they cannot help with homework.
Thanks to the School Board, which took the advice of the math committee.
I know many teachers and students will thank you, too. I am a retired teacher after 43 years in the classroom. I learned to teach math for understanding late in my career, and it was life-changing for my students.
-- Patricia Martin, Issaquah
A misleading endorsement
Imagine if a real-estate agent were selling a house. If she listed all the features of the house but failed to point out that two independent contractors had declared the home to be unsound, she would be in huge trouble and liable.
Yet that is what the Office of Superintendent of Public Instruction (OSPI) representative, Greta Borneman, did at the Seattle School Board meeting with respect to math curriculum. She touted the quantitative review, the results of curriculums mapping to our new math standards, but did not include the fact that Prentice-Hall and Glencoe were ranked third and fourth, respectively. She did show Core-Plus, which was ranked fifth, as though it were ranked third.
Borneman then downplayed the qualitative review, not mentioning the results: two mathematicians independent of OSPI and the publishers found Discovering AGA to be unsound. It was a dishonest presentation.
OSPI gave an implicit endorsement to the district to adopt Discovering AGA. Prentice-Hall was the district's second-choice curriculum. OSPI withheld the information that would have allowed the board to make a truly informed decision.
Some 80 percent of the people who showed up to speak about math to the board begged them not to adopt Discovering AGA. Included in that group were at least four district teachers and one UW science professor.
How can we change course in this state to ensure our students learn real math if we are fighting OSPI?
-- Laura Brandt, Sammamish
May 16, 2009 at 11:06 PM
Speaking as a student, the discovery method is garbage. Teachers answer questions with questions and you never actually learn anything. I know a person who is good at math but is failing because she simply isn't taught how to do it. When she asks me for help she always catches on really fast with only the most basic explanation of how to do it. It's a GPA killer. For the last two years of high school I have learned so much math really well, but next year the school is switching to this method and I had to put up with it in middle school so I am dreading it, but I can't fight it because the school district doesn't care what students think.
Everyday Mathematics was developed at the University of Chicago through grant from the Education and Human Resources Division of the National Science Foundation in the early 90’s. It has been implemented in many public schools in the U.S. Parents have often protested its adoption and in some cases have prevented it from being used, or succeeded in getting the program halted. In other cases (such as in Palo Alto, California most recently), it has been adopted despite protests from parents.
The Singapore math texts are part of the Primary Mathematics curriculum, developed in 1981 by Curriculum Planning & Development Institute of Singapore. Singapore’s math texts have been distributed in the U.S. by a private venture in Oregon, singaporemath.com, formed after the results of the international test TIMSS spurred the curiosity of homeschoolers and prominent mathematicians alike.
Said this on 5-15-2009 At 08:06 am
I think it really helps to have parent's give their perspective on the destruction from these fuzzy math programs. Sometimes you think something is wrong with your child and their ability to understand. Many parents do not realize it's the program itself. This leads to kids feeling something is wrong with them. How do kids with neglectful parents deal with this program? Or parents who lack math skills themselves? They are doomed.
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Said this on 5-15-2009 At 09:01 amTry the Seattle results after one year of Everyday Math:
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Said this on 5-15-2009 At 06:40 pm
This should be required reading for every school administrator and math coordinator that have inflicted this program on their students. I have 3 kids, now in college, who were poorly prepared in math thanks to Everyday Math, Connected Math, and Integrated Math. If Doctors used experimental fads such as this on their patients, they'd be sued for malpractice!
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Said this on 5-16-2009 At 05:23 pm
For 20 years, I have had the pleasure of building support for Project SEED, a supplemental program that brings math specialists with degrees in math, science, technology and engineering, into 3rd through 7th grade classes in low-income elementary and middle schools. They teach abstract math in an energetic, interactive manner that keeps students intensely involved and learning. Doubters ask, "How does that help students learn the Standard math?" "And why bother students with abstract math before they've mastered the Standard topics?"
Project SEED students outperform matched-comparison students after one year (14 weeks of instruction). After three years they outperform their matches by up to a full year.
There is compelling research that shows why:
University of Illinois at Chicago Professor David Page studied the similarities between learning the language of advanced mathematics and learning a foreign language, and found that both are best achieved at an early age.
Page, David, University of Illinois, Department of Mathematics, personal communication, 1996.
A study reported in the scientific journal, Nature, showed that when children learn second languages by age eleven, they use a different part of the brain than that which is used after age eleven. Apparently, language learning is more effective at this location, “the original language learning center,” than at other parts of the brain. This may be why young children, in comparison to older children and adults, learn new languages readily, and it underscores the importance of teaching the “language of (advanced) mathematics” to children at an early age.
There is also strong research evidence to show why our professional development for the classroom teachers is effective.
Please see Project SEED
Click on Video Room, then on the top box to view a 12-minute video.
Joanne L. Blum
Director of Development
From Betsy Combier: My daughter protested when she was 9 years old, and a 4th grader at Manhattan's PS 6. I wrote the article below.
WHY TERC? Asks a 9 year old, Who Questions the Value of 'Fuzzy Math' For Her Future Academic Goals
Many years ago several very wealthy and powerful individuals got together and discussed what to do with the problem of not having enough math and science teachers to teach rigorous, higher-level math courses in our nation's high schools and colleges. Also under discussion was the issue of the high fees 'expert' mathematicians could expect if hired from scientific think-tanks, and their non-union status (as in UFT, NEA, etc).
A brilliant solution was decided upon: a political-public relations-textbook industry-teaching college partnership that would design a curriculum supporting the notion that true learning came from children who felt comfortable working out problems and finding information without being told by a teacher what the 'facts' were. Children would, this partnership decided, be graded on how well they thought up their own answers and how well they 'understood' math processes, rather than just getting the right answer. In fact, this new 'constructivist' doctrine declared in the quickly published textbooks for teachers, the 'right' answer was not something that should be considered important at all. In other words, if a child understood that multiplying 16 "times" 19 meant the same as 16 "added" 19 times, then he or she was right, and must be given credit for constructing the reality around the process. It didn't matter whether the child arrived at the "real" answer, 304, or not, as long as they provided an understanding of the process, that
16 X 19 actually meant:
16+16+16+16+16+16+16+16 = 304, or 305, or 310, or 204, or.....
Baloney like this has swept the US for years. In New York City, the District 2 miracle"started by Tony Alvarado funded the dumbing down of elementary school children. Teachers and Principals who did not want to go along were forced to, and still are today. Stuyvesant High School, a premier High School for very smart kids, two years ago started a new math course to accommodate those 9th and 10th graders who could not do the traditionally challenging math curriculum.
Several years ago my 9-year old daughter Marielle, a busy professional opera singer and very strong math student at a premier elementary school that was the leading proponent of the TERC constructivist nonsense, got upset with the time needed by her math homework and decided to write a letter to the NY SUN newspaper. Her letter was published on August 29, 2002. Below is the unedited version:
by Marielle Combier-Kapel
4th Grade, PS 6
Parents are making tutors crazy calling them all the time because of TERC math. Kids don't have time to do anything because all they do after school is get tutored in math. There is no one to have playdates with anymore!
Citywide math scores are falling, but Board Of Education officials say that the District 2 math scores on the Standardized tests are high, therefore the TERC math curriculum is a good thing. Many District 2 parents are spending lots of money on tutoring, which brings up the scores, giving the impression that TERC is good for us kids.
Tutoring is great if your parents have money to spend on this. TERC math shouldn't be the only kind of math schools teach to their students. Just because some students aren't that smart, the schools are sending flyers home to parents saying that they should not teach their child traditional math which includes long division and algorithms. I like long division!
My mom says:
"Fuzzy math condemns our kids by not allowing them to establish an understanding of base computations which will empower them as they reach higher levels of problem-solving. The Board of Education policy to implement TERC math and ONLY this curricula is assuring our kids an immediate future of confusion, or worse, boredom, and a long-term disability in math achievement and academic performance in non-math
subjects as well. Learning traditional math as a reference is similar to having a Spanish dictionary when you are trying to write something in Spanish."
Parents are now calling other parents to find out if they tutor their children in math or not, and are signing up my friends. One of my sisters' teachers at Stuyvesant told my mom that the math at the Freshman level may have to be changed to a lower achievement level, as kids from District 2 who are getting in
are having trouble with the traditionally rigorous math program. A teacher at my other sister's honors program told my mom that she has never seen children in 7th grade who are not able to do long division.
What may happen is that I may be unable to compete for college places because the math teaching I have received is not teaching me what I should know. Is that fair?"
We received this admonition from her teacher about the 4th Grade math homework:
In mathematics, our class is starting a new unit called Arrays and Shares. This unit focuses on multiplication and division. Students begin the unit by looking at things that are arranged in rows, for example, juice packs, egg cartons, and rows of chairs. Through examining these rectangular arrangements (or arrays), they begin to visualize important aspects of multiplication, for example, that the solution to 7 X 6 is the same as the solution to 6 X 7.
As students go on to work on two-digit multiplication and related division problems, it is critical that they visualize how to pull apart the numbers they are working with. To solve these harder problems, students learn to use related problems they already know how to solve. For example, the problem 7 X 23 can be solved by breaking the problem into more familiar parts: 7 X 10, 7 X 10, and 7 X 3.
While our class is studying multiplication and division you can help in the following ways:
Look for items around your house or at the grocery store that are packaged or arranged in rectangular arrays. Tiles on the floor, egg cartons, window panes, and six-packs of juice cans are examples of rectangular arrays. Talk with your child about the dimensions (rows and columns) and discuss ways to figure out the total number.
Play the Array Games that your child brings home for homework.
Help your child practice skip counting by 3's, 4's, 5's, and so forth.
When your child brings home problems, encourage your child to explain his or her strategies to you. Ask questions, such as "How did you figure that out?" and "Tell me your thinking about this problem", but don't provide answers or methods. Show that you are interested in how your child is thinking and reasoning about these problems.
Please don't teach your child step-by-step procedures for computing multiplication and division. Too often we find that children at this age memorize the multiplication and division procedures but cannot recognize situations in which multiplication and division are useful. We will gradually support students this year in developing several strategies for carrying out multiplication and division problems, but we would prefer they not memorize procedures at this time.
Thank you for your interest in your child's study of mathematics. We are looking forward to an exciting few weeks of work on multiplication and division.
Marielle was subsequently told that she could not do math, and in 5th grade she was not put back onto the Math Team. She agreed to try for the Johns Hopkins Center For Talented Youth, and was a high scorer in Math, becoming one of the top 2% of 5th Graders in the United States. Her self-esteem was pushed back up, despite the comments from her teacher.
...then there is the textbook industrial complex:
Imagine the textbook industry's happiness with a new market for their publications (we have not seen them, but they must be out there) such as " The Math Classroom: Designing Desk Groups To Optimize Constructing Comfortable Math Thinking" or "Desk Arranging To Assist Teachers in Student Conversations During Math Class". Chapter after chapter must provide colorful pictures of where rugs could be placed, what size rocking chair to use, and how to arrange desks in the most creative way to encourage student discussion, in case the teacher has to go to another room, to the bathroom, a retreat in the country, etc.
In conclusion, we congratulate Rocky Mountain News reporter Linda Seebach, and NYU Professor Alan Siegel, in their exposure of the false claims of constructivism!
Seebach: An illusory math reform; let's go to the videotape
by Linda Seebach, Rocky Mountain News, August 7, 2004
American children come off badly in international comparisons of mathematics performance, and they do worse the longer they're in school.
One such comparison, the Third International Mathematics and Science Study, tested more than 500,000 children in 41 countries, starting in 1995. As part of the study, researchers videotaped more than 200 eighth-grade math lessons.
These lessons have been studied intensively in an effort to figure out why Japanese students do so well in math while American students do so badly. Alan Siegel, a professor of computer science at New York University, has reviewed the videos and calls the teaching "masterful."
He also believes that many of the TIMSS studies contain "serious errors and misunderstandings." If you have doubts, he says on his Web site, "go review the tapes and check out the references. After all, that's what I did" (www.cs.nyu.edu/faculty/siegel/). His paper also appears in a recent volume of essays on testing published by the Hoover Institution, Testing Student Learning, Evaluating Teacher Effectiveness.
The eighth-grade geometry lesson Siegel discusses is based on the theorem that two triangles with the same base and the same altitude have the same area, and it is framed in nominally "real world" terms as a problem in figuring out how to straighten the boundary fence between two farmers' fields so that neither farmer loses any land.
This is of course highly relevant to urban Japanese youngsters, who are likely to be called upon frequently to accomplish this task.
The teacher first primes the class by reminding them of the theorem, which they had studied the previous day. Then he playfully suggests with a pointer some ways to draw a new boundary, most of them amusingly wrong but a couple that are in fact the lines students will have to draw to solve the problem (though they aren't identified as such).
Then he gives the students a brief time, three minutes, to wrestle with the problem by themselves, and another few minutes for those who have figured out a solution based on his broad hints to present it. Then he explains the solution, and then he extends the explanation to a slightly more complex problem, and finally assigns yet another extension for homework.
As Siegel describes it, "The teacher-led study of all possible solutions masked direct instruction and repetitive practice in an interesting and enlightening problem space.
"Evidently, no student ever developed a new mathematical method or principle that differed from the technique introduced at the beginning of the lesson. In all, the teacher showed 10 times how to apply the method."
But that's not the way the lesson has been described in the literature. A 2000 commission report from the U.S. Department of Education, Before It's Too Late, gushes that in Japan, "teachers begin by presenting students with a mathematics problem employing principles they have not yet learned. They then work alone or in small groups to devise a solution. After a few minutes, students are called on to present their answers; the whole class works through the problems and solutions, uncovering the related mathematical concepts and reasoning."
How could Japanese children solve problems based on "principles they have not yet learned"? Why, in the same way that Meno's slave solved a mathematical problem on the exact same day that Socrates happened to be asking him questions.
As to how this confusion might arise, Siegel notes that a report by J.W. Stigler and others for the National Center for Education Statistics, The TIMSS Videotape Classroom Study, uses this very lesson as an example of how their data analysts were trained to identify solutions discovered by students.
"Altogether, this lesson is counted as having 10 student-generated alternative solution methods, even though it contains no student-discovered methods whatsoever," Siegel says.
Furthermore, the mathematicians who wrote about the study subsequently didn't see the original tapes; they relied on the misleading coding done by the data analysts.
Why does it matter? Because so-called "discovery learning" is the promised land of mathematics reform, and if only we follow the prophecies of the National Council of Teachers of Mathematics across the River Jordan, all our failings and failures as a nation will vanish away. And we know the prophecies are true, because the Japanese have gone before us.
Only they haven't. This is teaching in the traditional mode, beautifully designed and superbly executed, but nothing like the parody of instruction that goes by the term "discovery learning" in math-reform circles in the United States.
The videotape shows, Siegel says, that "a master teacher can present every step of a solution without divulging the answer, and can, by so doing, help students learn to think deeply. In such circumstances, the notion that students might have discovered the ideas on their own becomes an enticing mix of illusion intertwined with threads of truth."
Illusion prevails in far too many American classrooms.
The New York Sun • August 6, 2004
The Answer Goes Back To Basics
By ANDREW WOLF
The National Endowment for the Arts recently released a report on our nation's literacy. Americans are not reading as much as they used to. This trend has been accelerating in the past two decades. There has been a lot of anguish over this news, which has drawn much comment on both the left and right, and no shortage of theories attempting to explain how we came to this. Everything gets blamed, from television to videogames to the Internet. I have my own theory.
It is curious is that the movement of our young people away from the joy of reading for pleasure appears to coincide with the rise of the whole language (often labeled "balanced literacy") method of teaching reading, and the associated content-poor "progressive" teaching model that encourages students to "construct" their own knowledge.
This pedagogy, now the predominant way American (and as we shall see, British) children are taught, is supposed to instill a love of reading and literature. "Libraries" are placed in every classroom, part of an effort to create a "literature rich" environment. Small groups of students are organized into "book clubs."
Meanwhile, the role of the teacher as a conduit of knowledge has been subverted. Here in New York City, teachers have been directed to arrange classroom desks in clusters, in which groups of children face each other to facilitate the group projects that have replaced direct instruction by teachers. "Authentic literature" has supplanted textbooks as the tools of learning subject matter.
All this is done to promote the love of independent learning, made possible by promoting the love of reading. This is at the center of the ideology promoted by literacy gurus such as Lucy Calkins of Columbia University Teachers College, Diane Snowball of the Australian United States Services in Education, or AUSSIE, and Lauren Resnick of the University of Pittsburgh Institute for Learning. Our Department of Education is funneling tens of millions of dollars to these theorists, spending perhaps a quarter of a billion dollars in the past year to enforce compliance with this approach. This is wasted money.
As more American children are taught by these methods, the love of reading is apparently not increasing, but diminishing at an alarming rate, as the NEA study demonstrates. Can the way children are taught in school actually be the cause of this? I believe it is.
The idea that children learn to read and develop a love of reading from merely immersing them in a "literature rich" environment is akin to teaching the children of Gary, Ind. how to play musical instruments using the "think" system. Prof. Harold Hill and the storyline of "The Music Man" is, I believe, fiction. Knowledge is not acquired by osmosis. It is a process of building, one fact upon another. Those children who develop a love of reading don't do so for its own sake. It is because these children thirst for more knowledge.
That quest is not triggered by being surrounded by books such as the ones in the classroom libraries mandated by the Department of Education. Most of these books are works of fiction, mostly carefully scrubbed and filtered, with content largely designed to build self-esteem rather than impart knowledge. I submit that this kind of literature is the least likely to encourage children to become voracious readers.
Are textbooks obsolete? They are now much maligned and in danger of becoming extinct. But in the real world, I have found that they often were motivators for further reading. They provided overview and context, and in my distant youth often lured me into further study.This is not to say that the textbooks of today are without problems. Censors from the left and the right have attempted, and in many cases succeeded, to remake textbooks to advance their political and even religious agendas, undermining their value.
America is not alone in exhibiting a rising concern over the anti-intellectual proclivity of the teaching methodologies that have become so widespread. In Britain, the same debate is raging. Even the Prince of Wales has recently weighed in.
In a speech to teachers of English and history in state-run secondary schools in late June, Charles lamented that the "faddish" curriculum is resulting in students who are becoming "culturally disinherited." The prince suggested that the content-poor British curriculum could be a "potentially expensive and disastrous experiment with people's lives."
The traditional instruction apparently favored by the prince is more in line with the thinking of Professor E.D. Hirsch Jr., whose Core Knowledge curriculum is based on the idea that knowledge is like Velcro – its acquisition is facilitated by the previous knowledge already accumulated.
This suggests that if we believe that reading is good for society and we want our education system to result in a literate and knowledgeable people, we will not find the answers in whole language, balanced literacy, or constructivist ideology. The answers are to be found in a true "back-to-basics" movement. The kind of instruction that candidate Bloomberg promised us, but Mayor Bloomberg has thus far failed to deliver.
The havoc wrought by today's "modern" math
By Dr. Charles Orms/ Guest Columnist, North Andover Citizen
Friday, September 24, 2004
If you are a parent of elementary school children you've probably seen it: elaborate make-work homework assignments, cutting and pasting extravaganzas, overly complex and roundabout procedures to add or multiply numbers, estimation exercises that won't quit, and the use of calculators in place of traditional arithmetic methods.
You thought: "Of course, the educational professionals must know what they are doing. Once my children catch on to these clever techniques, they will develop into mathematics geniuses!" Unfortunately, what you discover is that they never learn the core facts and methods, their confusion grows, they lose their self-confidence, they decide they just can't do math, and you are stuck paying for tutoring. Even worse, children who might have become exceptional mathematicians, engineers, or scientists are denied their rightful future.
What went wrong? Years ago the educational establishment decided that teaching mathematics had to either consist of rote memorization (without real understanding) or students had to discover mathematics through trial and error because it was assumed that only 'if they discovered it themselves' would they truly understand it. While this view presented a false choice (there are much better alternatives), the educational community was sold on the second alternative because it had an intellectual cache that was lacking in "rote memorization." What resulted are the various "modern" math curricula for our children that under emphasize learning math facts, that bend over backward to avoid teaching standard methods for addition, subtraction, multiplication, and division, and that refuse to teach traditional processes for manipulating fractions.
Almost all our public schools today use one of the "modern" math curricula and millions of promising math-based technology careers are being ruined every year.
Permit me a short excursion. I have a technical/engineering background (PhD MIT '74) that rests largely on advanced mathematics. One reason I followed this path is that, as a student, I always had poor memorization skills and, therefore, subjects such as biology, chemistry, and foreign languages were very difficult for me. I loved math and physics because there was very little memorization; I could derive any formula I needed during a test. "Understanding" was a much more powerful asset for me than memorization. If anyone would be inclined to favor understanding over memorization it would be me. But the modern math is a disaster. I'm convinced that if I had been "taught" math with it 50 years ago, I would probably have become a poet (my English teacher is rolling over in her grave).
So do I favor just rote memorization? Of course not. Successful math education requires that students learn the techniques that true geniuses have developed over the last 3000 years. You may think the standard technique used to add numbers is trivial (place value concepts, carrying, etc.), but it was not obvious to the ancient Greek philosophers. Multiplication, long division, and fractions are even more complex. Teaching the techniques first and then exploring the underlying concepts and why these techniques work is the most efficient way to achieve true understanding. If Socrates and Aristotle couldn't invent our modern arithmetic system, why do we think the typical third or fourth grader can?
The impact of "modern" math on students in the US has been devastating. Just look at how the US stands up against other countries.
Place an equal emphasis on method mastery AND conceptual understanding, and you have the makings of a powerful elementary math curriculum. A curriculum that leads to real learning, that builds self-esteem and, rarest of all, a child that comes home and says, "Hey mom, I really love math!"
This approach to math education is not new. It is what a well-taught traditional mathematics course always emphasized. In some cases, poor teaching may have led to over-emphasis on rote memorization drills, but that is no reason to stop teaching the critically important mathematical methods.
How are we doing with early mathematics education in this area? As the table shows, not very well. While the differences in scores between towns/cities may be accounted for by socio-economic factors, the percent of students who are not proficient (meaning they scored as "Needs Improvement" or "Warning/Failing") in fourth grade (90 percent in Lawrence, 60 percent in Methuen, 50 percent in North Andover, and 34 percent in Andover) cannot be excused. Even worse is the lack of progress after four more years of what passes for math "education".
2003 MCAS SCORES(Percent ADVANCED OR PROFICIENT)
Town/City - 4th Gr - 6th Gr - 8th Gr
Andover -- 66% - 76% - 66%
Lawrence - 10% - 9% - 9%
Methuen -- 40% - 41% - 34%
North Andover - 50% - 51% - 60%
AVERAGE -- 42% - 44% - 42%
The trend towards "modern" math may finally be slowing. Parents are upset with the lack of a rigorous math curriculum and the need to hire tutors or enroll their children in remedial after school programs. A nationwide movement is growing to expose the failures of "modern" math and restore an academically sound curriculum. For information, visit mathematically Correct