The post 75% of Oregon Community College Students Need Remedial Classes appeared first on Education News.

]]>A recent study performed for the national Institute of Education Sciences has found that 75% of the 100,000 Oregon high school graduates to continue on to community college needed to participate in remedial classes upon arrival.

Portland-based researcher Michelle Hodara, who conducted the study, said that the graduates were ill-prepared for college life. She added that race and income levels did not play a role in the need to take the high-school or middle-school level courses.

The most common remedial course taken was in math, with almost 75% of all graduates who attended community college taking a remedial math course. The majority of those students tested into an introductory algebra class. Less than one-third of the students who needed to take remedial math were found to have ended up taking and passing even one college-level math class. This could be attributed to three terms of remedial math classes needing to be taken and paid for prior to entrance into college-level courses, writes Betsy Hammond for *The Oregonian.*

Of the high school graduates in the state who enter community college ready to participate in a college-level math course, the study found that 54% will earn a two-year degree or a career-related certificate. However, less than one-third of students who enter community college needing to take remedial math will earn any type of credential within five years, according to the study.

In addition, it was found that more high school graduates of the state will attend community college than will go to a four-year university.

Hodara recommended that schools work harder to academically prepare their students for the work they will need to do in college. According to the study, the best way to do this is through the passing of the state’s reading and math tests.

Higher education leaders in the state have been pushing for additional funding, arguing that due to declining state funding they have had to rely more and more on extra funding coming in from rising tuition rates, writes Jonathan Cooper for *The Fresno Bee.*

The presidents of all 7 universities and 17 community colleges in the state wrote a letter to legislative leaders, saying that too many people in Oregon cannot afford the rising cost of higher education.

“Like never before, Oregon’s public universities and community colleges are aligned to advocate the imperative of improved funding for higher education,” the presidents wrote. “Beginning to restore funding after a decade of disinvestment is the right thing to do for students and, ultimately, it is the right thing to do for Oregon’s future.”

However, according to the new state education budget, little extra money will be available for new or expanded programs.

“We are going to be largely addressing base needs, and there’s only going to be a modest amount of funding out there for quote-unquote new initiatives,” said Sen. Richard Devlin, D-Tualatin. “In fact, I think very little, to be frank.”

The post 75% of Oregon Community College Students Need Remedial Classes appeared first on Education News.

]]>The post Math Problems: Knowing, Doing, and Explaining Your Answer appeared first on Education News.

]]>by Barry Garelick and Katharine Beals

At a middle school in California, the state testing in math was underway via the Smarter Balanced Assessment Consortium (SBAC) exam. A girl pointed to the problem on the computer screen and asked “What do I do?” The proctor read the instructions for the problem and told the student: “You need to explain how you got your answer.”

The girl threw her arms up in frustration and said, “Why can’t I just do the problem, enter the answer and be done with it?”

The answer to her question comes down to what the education establishment believes “understanding” to be, and how to measure it. K-12 mathematics instruction involves equal parts procedural skills and understanding. What “understanding” in mathematics means, however, has long been a topic of debate. One distinction popular with today’s math reform advocates is between “knowing” and “doing.” A student, reformers argue, might be able to “do” a problem (i.e., solve it mathematically), without understanding the concepts behind the problem solving procedure. Perhaps he has simply memorized the method without understanding it.

The Common Core math standards, adopted in 45 states and reflected in Common Core-aligned tests like the SBAC and the PARCC, take understanding to a whole new level. “Students who lack understanding of a topic may rely on procedures too heavily,” states the Common Core website. “… But what does mathematical understanding look like?” And how can teachers assess it?

“One way … is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.” (http://www.corestandards.org/Math/).

The underlying assumption here is that if a student understands something, she can explain it—and that deficient explanation signals deficient understanding. But this raises yet another question: what constitutes a satisfactory explanation?

While the Common Core leaves this unspecified, current practices are suggestive: Consider a problem that asks how many total pencils are there if 5 people have 3 pencils each. In the eyes of some educators, explaining why the answer is 15 by stating, simply, that 5 x 3 = 15 is not satisfactory. To show they truly understand why 5 x 3 is 15, and why this computation provides the answer to the given word problem, students must do more. For example, they might draw a picture illustrating 5 groups of 3 pencils.

Consider now a problem given in a pre-algebra course that involves percentages: A coat has been reduced by 20% to sell for $160. What was the original price of the coat?”

A student may show their solution as follows:

x = original cost of coat in dollars

100% – 20% = 80%

0.8x = $160

x = $200

Clearly, the student knows the mathematical procedure necessary to solve the problem. In fact, for years students were told not to explain their answers, but to show their work, and if presented in a clear and organized manner, the math contained in this work was considered to be its own explanation. But the above demonstration might, through the prism of the Common Core standards, be considered an inadequate explanation. That is, inspired by what the standards say about understanding, one could ask “Does the student know why the subtraction operation is done to obtain the 80% used in the equation or is he doing it as a mechanical procedure – i.e., without understanding?”

**Providing instruction for explanations—the road to “rote understanding”**

In a middle school observed by one of us, the school’s goal was to increase student proficiency in solving math problems by requiring students to explain how they solved them. This was not required for all problems given; rather, they were expected to do this for two or three problems per week, which took up to 10 percent of total weekly class time. They were instructed on how to write explanations for their math solutions using a model called “Need, Know, Do.” In the problem example given above, the “Need” would be “What was the original price of the coat?” The “Know” would be the information provided in the problem statement, here the price of the discounted coat and the discount rate. The “Do” is the process of solving the problem.

Students were instructed to use “flow maps” and diagrams to describe the thinking and steps used to solve the problem, after which they were to write a narrative summary of what was described in the flow maps and elsewhere. They were told that the “Do” (as well as the flow maps) explains what they did to solve the problem and that the narrative summary provides the why. Many students, though, had difficulty differentiating the “Do” section from the final narrative. But in order for their explanation to qualify as “high level,” they couldn’t simply state “100% – 20% = 80%”; they had to explain what that means. For example, they might say, “The discount rate subtracted from 100% gives the amount that I pay.”

An example of a student’s written explanation for this problem is shown in Figure 1.

For problems at this level, the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. While drawing diagrams or pictures may help some students learn how to solve problems, for others it is unnecessary and tedious. As the above example shows, the explanations may not offer the “why” of a particular procedure.

Under the rubric used at the middle school where this problem was given, explanations are ranked as “high”, “middle” or “low.” This particular explanation would probably fall in the “middle” category since it is unlikely that the statement “You need to subtract 100 -20 to get 80” would be deemed a “purposeful, mathematically-grounded written explanation.”

The “Need” and “Know” steps in the above process are not new and were advocated by Polya (1957) in his classic book “How to Solve It”. The “Need” and “Know” aspect of the explanatory technique at the middle school observed is a sensible one. But Polya’s book was about solving problems, not explaining or justifying how they were done. At the middle school, however, problem solving and explanation were intertwined, in the belief that the process of explanation leads to the solving of the problem. This conflation of problem solving and explanation is based on a popular educational theory that being aware of one’s thinking process — called “metacognition” – is part and parcel to problem solving (see Mayer, 1998).

Despite the goal of solving a problem and explaining it in one fell swoop, in many cases observed at the middle school, students solved the problem first and then added the explanation in the required format and rubric. It was not evident that the process of explanation enhanced problem solving ability. In fact, in talking with students at the school, many found the process tedious and said they would rather just “do the math” without having to write about it.

In general, there is no more evidence of “understanding” in the explained solution, even with pictures, than there would be in mathematical solutions presented in a clear and organized way. How do we know, for example, that a student isn’t simply repeating an explanation provided by the teacher or the textbook, thus exhibiting mere “rote learning” rather than “true understanding” of a problem-solving procedure?

**Requiring explanations undoes the conciseness of math**

Math learning is a progression from concrete to increasingly abstract. The advantage to the abstract is that the various mathematical operations can be performed without the cumbersome attachments of concrete entities – entities like dollars, percentages, groupings of pencils. Once a particular word problem has been translated into a mathematical representation, the entirety of its mathematically-relevant content is condensed onto abstract symbols, freeing working memory and unleashing the power of pure mathematics. That is, information and procedures handled by available schemas frees up working memory. With working memory less burdened, the student can focus on solving the problem at hand (Aditomo, 2009). Thus, requiring explanations beyond the mathematics itself distracts and diverts students away from the convenience and power of abstraction. Mandatory demonstrations of “mathematical understanding,” in other words, impede the “doing” of actual mathematics.

**“I can’t do this orally, only headily”**

The idea that students who do not demonstrate their strategies in words and pictures must not understand the underlying concepts assumes away a significant subpopulation of students whose verbal skills lag far behind their mathematical skills, such as non-native English speakers or students with specific language delays or language disorders. These groups include children who can easily do math in their heads and solve complex problems, but often will be unable to explain – whether orally or in written words – how they arrived at their answers.

Most exemplary are children on the autistic spectrum. As autism researcher Tony Attwood has observed, mathematics has special appeal to individuals with autism: it is, often, the school subject that best matches their cognitive strengths. Indeed, writing about Asperger’s Syndrome (a high functioning subtype of autism), Attwood notes that “the personalities of some of the great mathematicians include many of the characteristics of Asperger’s syndrome.” (Attwood, 2007)

And yet, Attwood adds (ibid.), many children on the autistic spectrum, even those who are mathematically gifted, struggle when asked to explain their answers. “The child can provide the correct answer to a mathematical problem,” he observes, “but not easily translate into speech the mental processes used to solve the problem.” Back in 1944, Hans Asperger, the Austrian pediatrician who first studied the condition that now bears his name, famously cited one of his patients as saying that, “I can’t do this orally, only headily” (Asperger, H., 1991 [1944]).

Writing from Australia decades later, a few years before the Common Core took hold in America, Attwood adds that it can “mystify teachers and lead to problems with tests when the person with Asperger’s syndrome is unable to explain his or her methods on the test or exam paper” (Attwood, 2007, p. 241). Here in post-Common Core America, this inability has morphed into an unprecedented liability.

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers – from multi-digit arithmetic through to multi-variable calculus – doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?

Or is it possible that the ability to explain one’s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one?

**What is really being measured?**

Measuring understanding, or learning in general, isn’t easy. What testing does is measure “markers” or byproducts of learning and understanding. Explaining answers is but one possible marker.

Another, quite simply, are the answers themselves. If a student can consistently solve a variety of problems, that student likely has some level of mathematical understanding. Teachers can assess this more deeply by looking at the solutions and any work shown and asking some spontaneous follow-up questions tailored to the child’s verbal abilities. But it’s far from clear whether a general requirement to accompany all solutions with verbal explanations provides a more accurate measurement of mathematical understanding than the answers themselves and any work the student has produced along the way. At best, verbal explanations beyond “showing the work” may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.

As Alfred North Whitehead famously put it about a century before the Common Core standards took hold:

It is a profoundly erroneous truism … that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.

———-

**Katharine Beals** is a lecturer at the University of Pennsylvania Graduate School of Education and an adjunct professor at the Drexel University School of Education. She is the author of Raising a Left-Brain Child in a Right-Brain World: Strategies for Helping Bright, Quirky, Socially Awkward Children to Thrive at Home and at School.

**Barry Garelick** has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long term substitute in a high school and middle school in California: “Teaching Math in the 21st Century”.

———-

**References**

Anindito Aditomo. Cognitive load theory and mathematics learning: A systematic review, Anima, Indonesian Psychological Journal, 2009, Vol. 24, No. 3, 207-217

Hans Asperger. “Problems of infantile autism,” Communication: Journal of the National Autistic Society, London 13, 45-32), 1991 [1944]).

Tony Attwood. The Complete Guide to Asperger’s Syndrome, Jessica Kingsley Publishers, Philadelphia, PA, 2007. (p. 240)

G. Pólya, How to Solve It: A New Aspect of Mathematical Method, Doubleday, Garden City, NY, 1957

Richard Mayer. Cognitive, metacognitive, and motivational aspects of problem solving, Instructional Science, 1998, March, Vol. 26, Issue 1-2, pp 49-63

The post Math Problems: Knowing, Doing, and Explaining Your Answer appeared first on Education News.

]]>The post Book Review: Teaching Math in the 21st Century appeared first on Education News.

]]>“Teaching Math in the 21st Century”

By Barry Garelick; Introduction by Ze’ev Wurman; 2015, $9.95; 164 pages

As reviewed by Matthew Tabor, Editor, Education News

I spend my days working, and primarily reading, in education media. When it comes to Common Core, everyone is an expert and everyone has a solution (and a not-so-short list of villains). Hardly anyone bothers to write about what it’s like in the classroom — and to be fair, those who could usually can’t because of the professional consequences. It leaves us with a dearth of what is arguably the most important testimony.

In ‘Teaching Math in the 21st Century,’ Garelick, who has written on math education for The Atlantic, Education Next, Education News and others, takes advantage of a very unique set of circumstances. He’s an experienced insider as a math teacher; he’s an outsider as a school’s long-term substitute. He knows mathematics, having studied the actual discipline (i.e., not a watered-down math education degree). Since he’s a second-career teacher after having retired from the EPA, he understands how math is, and is not, applied in the workplace.

In short, Barry Garelick is the guy you want teaching your kid algebra.

Garelick has an impressive resume which, thankfully, he doesn’t rely on to offer another series of pronouncements about Common Core in a terribly-saturated intellectual marketplace. Instead, he lets his experience teaching math in the current climate of a largely confused, disorganized transition to Common Core in a California school district speak for itself.

Interactions with Sally, a district official armed with Common Core sound bytes and little substance, show just how nebulous Common Core implementation is in the average school district. Garelick’s accounts of how he matched proven, effective math instruction with muddy, CC-inspired district requirements shed light on how good teachers approach an uncertain landscape with equal parts reluctant-but-genuine compliance and cunning.

More importantly, Garelick shows how Common Core’s implementation affects students. The school is intent on reserving 8th grade Algebra for the “truly gifted,” a designation the district does not define. They admit to making up the selection process as they go along, which is comprised of a mini-gauntlet of testing the district says they can’t evaluate until they see the results. It’s a paradoxical mindset reminiscent of Nancy Pelosi advising that ‘we have to pass the bill so that you can find out what is in it.’

Garelick’s students and their parents are confused by the whole regime. Vignettes of several students, from struggling to solid, show just how difficult it is for the average 13 year old to get a sense of what’s going on in their math sequence — a point demonstrated best by a student Garelick describes as ‘bright’ ripping in half a placement test of ambiguous purpose. Garelick does a capable job of leading them through the swamp despite encountering several hurdles, from classroom management to interactions with individual students, that bring a refreshing honesty to the book (I’ve read enough self-congratulatory, humble-bragging teachers-as-superheroes accounts, and since I’m familiar with the realities of the classroom, I’m not interested in reading another.)

That Garelick’s account is based on teaching Algebra and pre-Algebra courses — the foundation for one’s future study of math — gives weight to his testimony.

Teaching in the 21st Century does not lay out every problem with Common Core math or the effects it may have on public education. It doesn’t address how to influence legislators and policymakers, or even how one might approach a local school board. It doesn’t advise you on the best math curriculum for your algebra-saddled student.

Instead, the book offers a brief glimpse into the eye of the storm that matters to kids, parents and teachers: the classroom as it functions under changing curricula and mindsets and how stakeholders deal with it. The book shows how great teachers are desperate to deliver a solid education in spite of proclamations from disconnected, poorly-grounded leaders; it shows how students just want to learn math and parents want to feel confident and informed about the education their kids are receiving.

That honest, unvarnished perspective — what educators really do in the face of a curricular sea change after having been given faulty maps, crude navigation devices and high expectations — is what we need more of right now in the education debate. In Teaching Math in the 21st Century, Garelick delivers it.

The post Book Review: Teaching Math in the 21st Century appeared first on Education News.

]]>The post Tories Tout Parent Math Classes Ahead of UK Election appeared first on Education News.

]]>The UK’s Conservative Party has plans to give math lessons to parents in schools in order for the parents to help their children with homework. The Daily Telegraph has been told that a Conservative government, after the general election, will be looking at an American-style idea which offers lessons for parents because so many adults are unfamiliar with the modern math curricula.

Peter Dominiczak of The Daily Telegraph explains that Tory advisers believe that education policy can be a plus for the party as the election nears. Inside sources are saying that currently the Conservative Party is neck-and-neck with Labour as far as education issues are concerned.

During former Conservative Education Secretary Michael Gove’s administration, the Tories were behind Labour in the polls on education matters. Gove’s time in office was dominated by significant changes to the education curriculum and frequent conflicts with teachers unions. David Cameron has stated that at a minimum of 500 extra free schools will be constructed if Conservatives win the election.

If the Prime Minister is re-elected, the Tories will continue to expand free schools and would forge ahead with his education policy. A new report has found that troubled schools were improving as much as twice as fast when a free school opened nearby, compared to the nationwide average.

In another article in The Daily Telegraph written by Dominiczak, the Conservative plans to promote “zero-tolerance of failure and mediocrity” are addressed. Students who fail their primary school leaving exams in English and math will be required to retake the tests in their first year of secondary school. Cameron says the Conservatives have become the “union for parents” and continued that a Tory government would not let students who fail their Sats reduce classroom standards for more dedicated pupils.

Students who fail the tests will have two more opportunities to pass the exams in the first year of secondary school in order for them to be in step with their fellow students. The plan will begin to be enforced in 2016.

The £500 “catch-up premium” for children who have not met the required standard in Year 6 will continue, but schools will have to show that at least 80% of students who failed the tests in primary school are passing the tests the second time around. Those who fail the exams again could face possible government intervention. Cameron said:

“There is no job that doesn’t require English and Maths and this is about making sure every child gets the best start in life and that our country can compete in the world.” Mrs Morgan added: “We know that the biggest predictor of success at GCSE is whether young people have mastered the basics at age 11. That means if we fail to get it right for young people at the start of secondary school they’ll struggle for the rest of their time in education.”

“More discipline, more rigor, zero-tolerance of failure and mediocrity.

He added that on the watch of the Labour Party, one in three students left primary school unable to read, write, and add. He explains that the Conservative Party has put reforms into place and has relied on teachers’ hard work to lower that number to one in five.

Math has changed over the last generation or so. Now, there is more importance put on a practical approach to numbers and how math applies to real life. In the UK, says Javier Espinoza, writing for The Daily Telegraph, children in Year 1 are doing complicated fraction and decimal work which was previously studied by secondary school students.

The post Tories Tout Parent Math Classes Ahead of UK Election appeared first on Education News.

]]>The post Matific Math Program Grows, Raises Investment Funds appeared first on Education News.

]]>New York-based Slate Science, the developer of Matific, has raised $12 million in its Series A round of funding despite being less than three years old.

One source said the funding came from “existing angel investors.” Previously, investors included Benny Schnaider, Roni Einav and Leon Kamevev.

Matific is a program that offers math activities and games for students in grades K-6 that developers say offers that “aha!” moment to learning math. The program is designed to be used in both in the classroom and at home and may be accessed through a browser or downloaded as an app for iPhone, Android, iPad and tablet use.

As of the last count in January 2015, over 15,000 teachers were using the program. Over 10,000 of those were reported to be in the US.

Guy Vardi, CEO of Slate Science, explained, “Matific appeals to children’s love of playing games. By making math interactive and hands-on, children learn the important fundamentals and enjoy the process of learning more. We’re proud of the product we’ve made and even more proud of the children who are learning because of it.”

According to the Matific website, the program creates its interactive mini-games and worksheets based on standard math curriculum and information from popular textbooks using a blended learning approach.

“At Matific we understand the importance of transparent tracking and analytics capabilities. The system features an intuitive reporting system that monitors progress and provides real-time and periodical status reports at both the class and student level.”

The program is free for teachers to use in their classrooms. There are two premium options for students who wish to access the program at home: schools can either purchase an extended use program, which allows the program to work off the school’s network, which would cost the school $10 per student, or parents can individually purchase the program for their child at a cost of $36 per year.

Vardi said the funding will be used to expand on the program on an international level. Currently the program may be accessed in 20 countries and seven languages. The company has plans to expand further into South America and Asia, writes Charley Locke for edSurge.

Vardi would also like to see more offerings within the program as it expands into more schools and countries. “We have a great product, but would like to integrate better with existing curriculum in schools,” he explained. He went on to say that doing so will become even more important as the program continues to expand to include more areas.

The convergence of math and educational video games is the result of two trends: more parents wanting to see their children succeed in STEM subjects, and the increasingly popular idea of gamification, which makes use of educational entertainment and incentive structures.

The post Matific Math Program Grows, Raises Investment Funds appeared first on Education News.

]]>The post UK EdSec Morgan Embarrassed by Simple Math Problem appeared first on Education News.

]]>UK Education Secretary Nicky Morgan is in the spotlight after refusing to answer a math question posed to her by a 10-year-old boy.

During a children’s newspaper interview for Sky News, the boy, Leon Remphry, asked Morgan what the cube root of 125 was. Morgan refused to answer his question.

Leon continued to request an answer from the secretary as she continued to dodge the question by joking that she would “not do maths on air.”

Leon responded to her avoidance by saying, “I’m afraid I’ve got to press that question actually, do you know what the answer is?” Morgan continued to deny to give a response, forcing Leon to provide her with the answer: five.

Leon then asked her the cube root of 1,728 minus 11. Morgan replied by saying the question was “one that I might just have to go away and work out.”

“I think politicians who answer maths questions or spelling questions on air normally come a cropper.”

He then discussed a number of pressing issues with the politician, including literacy and instilling a love of reading within children.

He raised concerns that this could not happen if libraries continued to be shut down, and did not appear happy with her response to the issue, saying: “I don’t think you’ve actually answered the question. Are the government going to take steps or are they not?”

Morgan had replied to the issue by saying the Government’s response was to remind local councils “that it is their duty to provide libraries which are, obviously, where people can borrow books for free which is the critical thing and, as education secretary, I want there to be libraries in schools.” She added that an independent report would be published on the importance of libraries.

The situation has left the nation wondering if she had in fact known the answer all along, or if she was denying to answer because she truly did not know.

Earlier this year, Chancellor George Osborne had been unable to answer a math question asked of him by a seven-year-old boy from the same newspaper. The boy had asked him for the answer to 7 x 8. In a radio interview, Former Labour schools minister Stephen Byers had answered “54″ to the same question.

According to the BBC, square and cube roots are learned by children in the country early on in their secondary school education.

Sir Anthony Seldon, headmaster of one of Britain’s leading public schools, Wellington College, said Morgan needs to become more radical in order to better help the country’s poor students. He said of Morgan, “She came to the job knowing little about schools and with no great interest in education.”

The piece was written in response to Morgan’s plan to spend almost $8 million on soldiers visiting schools to teach children about “grit” and “determination.”

Seldon believes the money would be better spent “providing the means for all independent schools to sponsor academies.”

The post UK EdSec Morgan Embarrassed by Simple Math Problem appeared first on Education News.

]]>The post Protecting Students from Learning appeared first on Education News.

]]>**by Barry Garelick**

I attended Mumford High School in Detroit, from the fall of 1964 through June of 1967, the end of a period known to some as the golden age of education, and to others as an utter failure. For the record I am in the former camp, a product of an era which in my opinion well-prepared me to major in mathematics. I am soon retiring from a career in environmental protection and will be entering the teaching profession where I will teach math in a manner that has served many others well over many years and which I hope will be tolerated by the people who hire me.

I was in 10th grade, taking Algebra 2. In the study hall period that followed my algebra class I worked the 20 or so homework problems at a double desk which I shared with Raymond, a black student. He would watch me do the day’s homework problems which I worked with the ease and alacrity of an expert pinball player.

While I worked, he would ask questions about what I was doing, and I would explain as best I could, after which he would always say “Pretty good, pretty good”—which served both as an expression of appreciation and a signal that he didn’t really know much about algebra but wanted to find out more. He said he had taken a class in it. In one assignment the page of my book was open to a diagram entitled “Four ways to express a function”. The first was a box with a statement: “To find average blood pressure, add 10 to your age and divide by 2.” The second was an equation P = (A+10)/2. The third was a table of values, and the last was a graph. Raymond asked me why you needed different ways to say what was in the box. I wasn’t entirely sure myself, but explained that the different ways enabled you to see the how things like blood pressure changed with respect to age. Sometimes a graph was better than a table to see this; sometimes it wasn’t. Not a very good explanation, I realized, and over the years I would come back to that question—and Raymond’s curiosity about it—as I would analyze equations, graphs, and tables of values.

The study hall was presided over by a high school counselor whose office was in the corner of the great room. The day came when we were to sign up for next semester’s courses, and she called out the names of the students in her custody. Raymond went in for his appointment. There was some discussion going on inside and suddenly the counselor, not given to sensitivity nor controlling the volume of her speech, blared out in unrelenting anger for all the study hall to hear: “You want to take algebra 2? You didn’t finish Algebra 1, and you got a “D” in the part you did finish! You will take ‘General Arithmetic’, young man!”

Raymond walked back and took his seat next to me. I said I was sorry about what happened, but he didn’t look in my direction. I was 15 and didn’t quite know what to do, so I didn’t do or say anything else to him. He didn’t speak to me for the remainder of the semester, and when I worked my algebra problems he found something else to do.

I never saw him again after that semester though I do know he graduated when I did. I suspect that he didn’t take any more math classes, arithmetic or otherwise. I don’t know whether his interest in math was based on my making it look easy, or whether he would have made a serious effort to get up to speed. I also don’t know whether his poor performance in the algebra course he took was because of poor teaching, lack of ability, or because he was a victim of neglect who had been passed on and promoted to the next grade as many students had been. I doubt his counselor knew either. Whether his counselor would have reacted differently had Raymond been white is also something I don’t know. What I do know is that his interest was strong enough to want to enroll in an algebra course, and the question “Why do you need to learn that stuff?” did not seem to be on his mind.

**The Exchange of One Inequity for Another**

I have written previously and extensively about math as it was “traditionally taught” because I feel strongly about it and it is what I know best. Two of the main criticisms about traditional math – which have also been levied against education in general for the era — are that 1) it relied on memorization and rote problem solving, and 2) it failed thousands of students.

This last criticism refers to the low numbers of students taking algebra and other math classes in the 50’s and 60’s and is taken as evidence that the techniques of traditional math—drills, memorization and word problems that were not necessarily related to the “real world”—worked only for bright students who learned math no matter how it was taught. Another side to this argument, however, is that the low numbers of students who took algebra and other math classes during this period was because of the tracking practices that were in force at the time.

The history of tracking students in public education goes back to the early part of the 1900′s. By the 20′s and 30′s, curricula in high schools had evolved into four different types: college-preparatory, vocational (e.g., plumbing, metal work, electrical, auto), trade-oriented (e.g., accounting, secretarial), and general. Students were tracked into the various curricula based on IQ and other standardized test scores as well as other criteria. By the mid-60’s, Mirel (1993) documents that most of the predominantly black high schools in Detroit had become “general track” institutions that consisted of watered down curricula and “needs based” courses that catered to student interests and life relevance. Social promotion had become the norm within the general track, in which the philosophy was to demand as little as possible of the students. The educational system in the U.S. pitted many groups against each other— skin color was not the only determinant. Children from farms rather than from cities, and children of immigrants, for example, were often assumed to be inferior in cognitive ability and treated accordingly.

During the 60’s and 70’s, radical critics of schools such as Jonathan Kozol, brought accusations of sadistic and racist teachers, said to be hostile to children and who lacked innovation in pedagogy. “Traditional” schooling was seen as an instrument of oppression and schools were recast in a new, “hipper” interpretation of what progressivism was supposed to be about. In moving away from the way things were, the education establishment’s goal was to restore equity to students rather than maintaining the tracking that created dividing lines between social class and race. The end product however was a merging of general track with college prep with the result that college prep was becoming student-centered and needs-based with lower standards, and less homework assigned. Classes such as Film Making and Cooking for Singles were offered, and requirements for English and History courses were reduced if not dropped. Social class and race was no longer a barrier for such classes as evidenced by the increasing numbers of white students began taking them.

By the early 80’s, the “Back to Basics” movement formed to turn back the educational fads and extremes of the late 60’s and the 70’s and reinstitute traditional subjects and curricula. The underlying ideas of the progressives did not go away, however, and the watchword has continued to be equal education for all. While such a goal is laudable, the attempt to bring equity to education by eliminating tracking had the unintended consequence of replacing it with another form of inequity: the elimination of grouping of students according to ability. Thus, students who were poor at reading were placed in classes with students who were advanced readers; students who were not proficient in basic arithmetic were placed in algebra classes. Ability grouping was viewed as a vestige of tracking and many in the education establishment consider the two concepts to be synonymous.

The elimination of ability grouping occurs mostly in the lower grades but also extends to early courses in high school. The practice of such full inclusion is now so commonplace that theories have emerged to justify its practice and to address the problems it brings. “Learning styles” and “multiple intelligences” are now commonplace terms that are taught in schools of education, along with the technique known as “differentiated instruction” to address how to teach students with diverse backgrounds and ability in the subject matter. Teachers are expected to “differentiate instruction” to each student, and to keep whole-group instruction to a minimum. To do this, the teacher gives a “mini-lesson” that lasts 10 to 15 minutes; then students work in small groups and told to work together. The prevailing belief is that by forcing students to solve problems in groups, to rely on each other rather than the teacher, the techniques and concepts needed to solve the problem will emerge through discovery, and students will be forced to learn what is needed in a “just in time” basis This amounts to giving students easy problems, but with hard and sometimes impossible approaches since they have been given little to no effective instruction to the mathematics that results in effective mathematics problem solvers.

The limitations of differentiated instruction work hand-in-hand with other aspects of the educational beliefs that shun “traditional” modes of instruction. (Beals (2009) describes the current trends in very accurate detail.) It is not unusual to hear parents concerned over art-based projects in English classes that call for students writing book reports in the form of a book jacket or poster—in which the artistic merits of the poster or book jacket may count as much as the actual composition. Exercises in grammar have declined to the point that they are almost extinct. Essays now are “student-centered” which is to say that students write about how they feel about certain events that occur in a story, relating it to themselves—this extends to history classes as well. They may be asked how Hester Prynne would write a profile about herself on Facebook, or George Washington on the eve of battle. Objective analysis, along with grammatical drill, sentence and paragraph structure and other tenets of a basic education are considered passé and not in keeping with the current watchword of 21st century education.

Brighter students are seated with students of lower ability in the belief that the brighter students will teach the slower ones what is needed. And frequently this occurs, though the fact that the brighter students are often obtaining their knowledge via parents, tutors or learning centers is an inconvenient truth that is rarely if ever acknowledged. The result is that brighter students are bored, and slower students are either lost, or seek explanations from those students in the know. Another inconvenient truth is that in lower income communities, there are unlikely to be students who have obtained their knowledge through outside sources; they are entirely dependent on their schools.

Students forced to endure this form of education do not progress as rapidly and do not master the essentials necessary to be successful in high school math courses. Even many of the classes for gifted and talented students are conducted in this manner. For example, one gifted class for seventh graders designed a new playground to a budget. The parent who told me this was quite proud to add that the playground was built with minor changes. The project took the bulk of the semester and there wasn’t much else in the way of geometry, proportions, rates, and pre-algebra concepts such as negative numbers, exponents and radicals—unless such concepts were visited on a “just in time” basis as discussed above.

In lower grade math classes, teaching mathematical procedures and algorithms has given way to more pictorial explanations, using alternative methods of adding, subtracting, multiplying and dividing in the name of providing students with “deep understanding”. Process trumps content. The results are that such students are passed on into algebra courses in high school with little to no mastery of the arithmetic procedures that are essential to move on to more abstract versions of the same. As such, they do not qualify for the honors track courses, nor—ultimately—AP calculus.

Many who make it to the honors have received the instruction and knowledge they need through tutoring/learning centers or their parents. In some cases, there are schools whose gifted and talented program consists of a traditional approach for math and other subjects. Thus, students who qualify for such programs are exempted from the one-size-fits-all, student-centered classes. In either case, students entering high school have been unintentionally split into groups of students, some of whom will qualify for honors classes and those who will not. Depending on the high school, the non-honors courses may be watered down versions often by necessity. These students are passed on through the system in some schools; in others they receive failing grades. Students, through circumstances beyond their control, may end up “tracked” in sub-standard courses and will be ill-prepared to take math courses in college, thus shutting out possibilities of a career in the sciences or engineering.

A recent study by William Schmidt of Michigan State University (Schmidt, et al., 2011) also observes the differences in learning opportunities and concludes that the differences are a function of the education system structure. Thus, there are differences in content depending on the area of the school district and that there is a fundamental relationship between content coverage and achievement. Schmidt states that with respect to mathematics, if the districts examined in his study were to hold generally for the U.S. then “any student can be disadvantaged simply due to differences in the rigor of the mathematics taught in the district in which they happen to attend school.” While a variety of factors contribute to the disadvantaging of students as discussed above, eliminating ability grouping is a big one. Through the efforts and philosophies of otherwise well-meaning individuals, full inclusion and equality for all has served as a form of tracking.

**Not Good Enough for Traditional**

Critics of the traditional model of education–particularly math–argue that traditional methods worked only for the gifted kids (for whom it is assumed they will learn what they need to know no matter how it is taught). And the corollary to such thinking is that students not gifted are not good enough for the traditional method. The move to homogenize skill levels in the classrooms has been entrenched now for several decades. It has come to the point now that students who have been forced through circumstances into non-honors tracks, and judged to not be able to handle the “traditional mode” of education and are thus “protected” from it. And in being protected from learning they are therefore not presented with the choice to work hard—and many happily comply in a system that caters to it.

Which raises the question of whether higher expectations and more teacher-centered instruction yield better results. Vern Williams is a middle school math teacher in Virginia, who teaches gifted students and served as a member of the President’s National Math Advisory Panel. He relates a story about how he was recently assigned a tutorial class made up of students who had failed and barely passed Virginia’s sixth grade math exams. When he first started teaching the class they wanted to play games, but Williams challenged them and included material that he was teaching to his seventh grade (gifted) algebra classes. He reports “Many of the students wanted to ditch their regular ‘baby’ classes and just attend mine. They viewed my class as not only interesting but serious.”

But students who have been put on the protection-from-learning track fulfill the low expectations that have been conferred upon them. The education establishment’s view of this situation is a shrug, and—despite their justifications for the inquiry-based and student-centered approach that brings out all children’s’ “innate” knowledge of math—respond with “Maybe your child just isn’t good in math”. The admonition carries to subjects beyond math and is extended to “Maybe your child isn’t college material.” And while it is true that a “college for all” goal is unrealistic, the view that so many students somehow are lacking in cognitive ability raises serious questions. Simply put, you no longer have to be a minority to be told you may not have cognitive ability. As Schmidt (2011) states in his paper: “To attribute achievement differences solely to differences in student efforts and abilities is grossly unfair and simpleminded and ignores the fundamental relationship between content coverage and achievement.”

There is now an in-bred resistance against ability grouping using explicit instruction. That such approaches may result in higher achievement, with more students qualifying for gifted and honors programs, is something that the education establishment has come to deny by default. What they have chosen instead is an inherent and insidious tracking system that leaves many students behind. They have eliminated the achievement gap by eliminating achievement. And many of those left behind disdain and despise education and the people who managed to achieve what they could not—just as I imagine Raymond must have many years ago.

—————

References

Beals, Katharine. 2009. “Raising a Left-Brain Child in a Right-Brain World”. Trumpeter. (Chapter 3).

Mirel, Jeffrey; David L. Angus. Equality, Curriculum, and the Decline of the Academic Ideal: Detroit, 1930-68; History of Education Quarterly, Vol. 33, No. 2 (Summer, 1993), pp. 177-207

Schmidt, W., et al. 2011. Content Coverage Differences across Districts/States: A Persisting Challenge for U.S. Education Policy, American Journal of Education, Vol. 117, No. 3; University of Chicago Press; (May 2011), p. 422

The post Protecting Students from Learning appeared first on Education News.

]]>The post Survey: UK Parents Struggling to Help with Basic Math Homework appeared first on Education News.

]]>Of 1,000 parents of United Kingdom primary school children surveyed, 46% said that they felt ill-equipped to help their children with their math homework – especially in the long division execution.

Other areas that are proving to be difficult for parents are conversion of decimals, fractions, and percentages. Math was considered the hardest subject to help their children with by 34%, while 8% said that English was the most difficult.

“Often, parents struggle to help with children’s maths homework because the method of teaching – for long division, subtraction etc – has changed. The Maths Factor sets out to specifically help parents, so parents can preview the next lesson for their child. In fact, many parents have found that they have actually got quite good at maths by watching the videos themselves.”

Research published this week, commissioned by Pearson, reveals that half of the parents surveyed are not aware of the changes in the math national curriculum in the UK for primary math, introduced this September, nor are they aware that the curriculum is designed to be more challenging.

Carol Vorderman, a former television game show host, also created The Maths Factor, an online math school for primary-age children. She says that the primary years of math study are crucial to the success of students as they make their way up to the higher levels of math education. Vorderman also says that if students are struggling with math at age 11, they, for the most part, will not pass their GCSE in the subject. She wants to make sure that parents have the support and guidance they need.

The *Press Association* reports that about half of the parents who participated in the survey could not do math problems designed for ten-year-olds. 19% of parents said they did not feel confident about helping their children with long multiplication and 6% said helping with multiplication tables would be difficult. Only 9% said they did not find math useful in their everyday life, and 82% believe that math at the primary school can help children solve more complex problems in later life.

Vorderman agreed with the majority:

Ms Vorderman said: “Maths skills are essential in everyday life and it’s perhaps concerning to see a divide opening up between those who are aware of the new curriculum and those who aren’t, and between those who have the confidence to help their children and those who don’t. As a parent myself, I know how busy life gets, but with a bit of support we can all easily become confident with numbers.”

Despite parents admitting that they did not feel they were up to the task of helping their children with their math homework, 82% of the survey subjects said they find math useful in working life, which was second only to English at 85%, reports Aled Blake, writing for *Wales News*.

The post Survey: UK Parents Struggling to Help with Basic Math Homework appeared first on Education News.

]]>The post LEGO Education’s MoreToMath Brings Bricks to Math Education appeared first on Education News.

]]>LEGOs, the long time favorite building blocks during playtime, are now also the building blocks of elementary mathematical education as LEGO Education announces the presale of their newest program: Lego Education MoreToMath 1-2.

The innovative classroom resource uses LEGO bricks to make abstract math tangible for first and second grade students, according to *Brick by Brick*, LEGO Education’s blog.

The hands-on tools include a LEGO brick set specifically designed for classrooms, training videos for teachers, curriculum, interactive whiteboard software, and worksheets for teachers and students that access the students grasp of the eight practices of mathematical problem solving that the Common Core Math standards have outlined.

Leshia Hoot, LEGO Education’s senior segment manager for preschool and elementary education, explained that educators were voicing their struggles about teaching the new Common Core math practices. MoreToMath is designed to help teach real world problem solving that supports the Common Core expectations, reports *Education World.*

“One of the primary things that we lack in our classrooms is the motivation piece and engagement in mathematics, and a lot of that has to do with reluctance of the teachers to know how to use hands on materials in the classroom. So, this particular product will offer teachers an opportunity to do something that is easily transferable from the textbook knowledge that they need to teach the kids and is easily blended into their daily mathematical teaching,” says Dr. Shirley Disseler, a co-developer of Lego Education MoreToMath 1-2.

Students are expected to learn 5 key learning values, as outlined by LEGO Education on Lego.com. The first is problem solving skills, understanding the basics of mathematical problem solving, comprehending problems, perseverance, modeling, representation, reasoning, and precision. Second, comprehension skills gained from hands on activities that reinforce algebraic thinking, areas of numeracy, operations in base 10, measurement, geometry, data, and spatial awareness. LEGO says that collaboration skills, communication skills, and technology skills are all developed with this product.

Students learn these lessons with LEGO figures Max and Mia and they work in pairs to complete each activity. While children have the actual Logo bricks to work with as they solve problems, they can easily share answers with each other using the smart white board software.

Dr. Disseler adds, “The other thing that is important with this product, that you don’t see in other products, is that it gives the kids visual, tactical and kinesthetic ways to show and do their math. [It] gives them an opportunity to converse about mathematical vocabulary, what their learning, learn to ask question of one another and actually be their own thinkers.”

The post LEGO Education’s MoreToMath Brings Bricks to Math Education appeared first on Education News.

]]>The post One Step Ahead of the Everyday Math Train Wreck appeared first on Education News.

]]>**by Barry Garelick**

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 (1), 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 6^{th} grade, and with fractional division, percentages and decimals on the agenda, I wanted to make sure she mastered these things. So, near the beginning of 6th grade, 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 (2). 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 in 4th and 5th grades.”

We only made it into two pages of text in the fourth grade book. I came to find out that despite their being in 6th grade, 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.

Everyday Mathematics was developed at the University of Chicago through a 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. For example, after a local parent group put pressure on the Bridgewater-Raritan Schools in New Jersey, a very comprehensive program evaluation was conducted (http://www.brrsd.k12.nj.us/files/filesystem/Math%20Evaluation%20Report.pdf) which resulted in a 9-0 school Board vote to replace Everyday Mathematics with a more balanced and traditional program, HSP Math by Harcourt School Publishers. 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.

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 [*i, ii, iii*] 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 6^{th} 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 4^{th} grade to 6^{th} grade textbooks in a concentrated burst. Although I really should have started all this back in 4^{th} 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 6^{th} grade, students are working on aspects of fraction division long before they reach the 6^{th} grade. By the time students reach the 6^{th} 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 4^{th} grade text [*iv*] 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.” [*v*]

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.” [*vi*]

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 after retirement. 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.

**References:**

[*i*] Braams, B. (2003). The many ways of arithmetic in UCSMP Everyday Mathematics. *NYC HOLD website.* February. http://www.nychold.com/em-arith.html

[*ii*] Braams, B. (2003). Spiraling through UCSMP Everyday Mathematics. *NYC HOLD website*, March. http://www.nychold.com/em-spiral.html

[*iii*] Clavel, M. (2003). How not to teach math. *City Journal,* March 7. http://www.city-journal.org/html/eon_3_7_03mc.html

[*iv*] Singapore Math 4A

[*v*] Everyday math; Education Development Center; Newton MA; 2001. Available at http://www2.edc.org/mcc/PDF/perspeverydaymath.pdf

[*vi*] University of Chicago School Mathematics Project; 2004. *Everyday mathematics. Student reference book. 2002. *SRA/McGraw-Hill; Chicago (p. 59)

The post One Step Ahead of the Everyday Math Train Wreck appeared first on Education News.

]]>