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NWEA Testing and Individualized Instruction

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NWEA Testing and Individualized Instruction

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On a hot August day, students packed into a nondescript storefront on Snelling Avenue to do something atypical for the middle of summer - work math problems.

When any of the more than a dozen students - who ranged in age from elementary to high school - got stuck, they raised their hand and a Mathnasium tutor, such as Sarah Koehler, came over.

"It's great to help them get past their frustration," the Macalester College student said. "I know how these kids feel."

It's becoming increasingly clear that personalized interventions, such as those provided by Mathnasium and a dozen other private tutoring companies in the Twin Cities, are key to improving student achievement in math. The challenge for educators and parents alike has been getting help to all students who need it.

Now, thanks to a new adaptive online testing system that gives immediate results, teachers can quickly identify and address individual student deficiencies for the first time on a broad scale.

"It is a more precise measure," said Brenda Cassellius, commissioner of the Minnesota Department of Education, of new online mathematics assessments. "The immediacy of the test results allows teachers to better prepare students through instruction. When students can see their results, they are going to try harder and get more information out of the test."

But not every student has that opportunity. The new assessments are only for elementary and middle-school students and are available only online. Paper-test takers don't get the feedback that online test takers do.

Most Minnesota elementary and middle-school students took online math assessments this year, and scores in those grade levels climbed an average of 7 percentage points from the year before. Only paper tests were given to high school students, who historically have struggled with the 11th-grade proficiency test, and scores dropped by 6 percentage points.

Statewide, 66 percent of districts posted gains on elementary and middle-school tests but saw scores fall at the high school level, a Pioneer Press analysis of testing data shows.

Previously, all students took the math test once, and teachers received the results during summer break.

But when students take the online test, teachers get the results immediately and can work with students to help them pass. Students can retake the test twice.

Schools with some of the largest gains in the east metro - from Stillwater and Anoka-Hennepin to Lakeville and West St. Paul - all gave the online test.

"It is about learning, not about a snapshot of any given point in time," said Lisa Snyder, superintendent of Lakeville schools, where students took the online test and scores rose 12 percentage points at the elementary and middle schools, but fell 16 points at the two high schools. "That is the ultimate goal. Not to just give the test on one day and say 'gotcha.' "

'UNEVEN PLAYING FIELD'

Districts without the resources to give the computerized test or that decided to give the paper-and-pencil version missed out on the chance to bring into focus areas where students were struggling.

Among the state's four largest school districts, Anoka-Hennepin and Minneapolis schools gave the online tests and posted gains of 6 percentage points and 4 points, respectively. St. Paul and Rosemount-Apple Valley-Eagan gave students paper tests, and gains were about half as large.

Eric Vanden Berk, interim director of research for Minneapolis schools, said while district students took the online tests, most took it only once and only students at a handful of schools had an opportunity to receive coaching and retake the test.

"We did see pretty good growth in those schools," Vanden Berk said. "I'm convinced if you give someone three opportunities to take the test, they'll do better."

Vanden Berk added that Minneapolis school officials were hopeful that taking the online test would not give students an advantage. Giving the electronic test was a logistical challenge that was more than made up for by the ease of administering the test, he said.

Online tests may put large metropolitan school districts at a disadvantage, said Scott Croonquist, executive director of the Association for Metropolitan School Districts.

"They often don't have the quantity of computers to offer the adaptive test on a districtwide basis. When you use your computer labs like that, you are putting them off limits for instructive purposes," Croonquist said. "I think it is fair to say that we have an uneven playing field when it comes to access to technology."

The Pioneer Press requested a list of schools that administered the tests electronically, but state Education Department officials said the list wouldn't be available until later this month. Statewide, 88 percent of elementary and middle-school students took the online tests, state officials said.

Mike Redmond, a former middle-school principal in the North St. Paul-Maplewood-Oakdale district who is now curriculum director for Stillwater Area Schools, said he believed the online tests and teachers' ability to coach helped raise test scores.

"Doing something early in the year puts real results in the hands of teachers, who can target instruction for students," Redmond said. "You're getting real feedback in real time."

Charlene Briner, Cassellius' chief of staff, said it was unclear to the Education Department how strongly the online adaptive tests influenced results.

"I think it is hard to say how much improvement is attributable to that," Briner said. "There is no question that having multiple opportunities to take an exam will help improve student performance on a test."

State officials hope to increase the number of districts using the online test next year, but it is unknown how they will do so.

"I think we are always concerned about equity," Briner said. "That is a lens through which we view every decision we make, quite frankly."

'PERSONALIZED TUTORING'

Ensuring equal access to digital tests is tough to accomplish nationwide, said Robert Schaeffer, spokesman for the National Center for Fair and Open Testing. A move toward a national "Common Core" curriculum has states from Maine to Washington looking for ways to offer uniform, digital assessments.

States "have failed disastrously" to implement uniform online assessments, Schaeffer said. Building the infrastructure for every student to have the same access is expensive and time-consuming.

"It's certainly isn't a level playing field when one group of kids has an opportunity that others do not," he said. "You're basically giving some students personalized tutoring. Kids who get the extra tutoring have a huge leg up. They have gotten specialized training."

Schaeffer added that individual instruction could greatly improve learning when done properly. However, he said it shouldn't be tied too closely to standardized tests.

"The more customized the teaching, the better the learning, no question," Schaeffer said. "The problem is, the customization is around one thing, test taking. ... The curriculum becomes test prep. It doesn't necessarily improve learning."

Next year, the state will tighten the rules around the online test in hopes of leveling the playing field. Students will still get a chance to take two online practice tests earlier in the school year, but only the test given in the spring will count.

Last year, schools could count any of the three scores, a provision some criticized as too generous.

The change is designed to give the state more accurate information, Briner said.

"In order to have the best and most reliable information, we have to have the same window," she said. "We want to compare apples to apples."

Because of security and other logistical concerns, there are no plans to allow students repeated attempts at the paper version of the exam, Briner said.

For that, districts will have to switch to the online exams. Rosemount-Apple Valley-Eagan school officials have already decided to make the change.

Christopher Magan can be reached at 651-228-5557. Follow him at twitter.com/cmaganPiPress. Read our blog Ahead of the Class at http://blogs.twincities.com/education/.

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Online math tests may help improve scores, but aren't available to all

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On a hot August day, students packed into a nondescript storefront on Snelling Avenue to do something atypical for the middle of summer - work math problems. When any of the more than a dozen students - who ranged in age from elementary to high school - got stuck, they raised their hand and a Mathnasium tutor, such as Sarah Koehler, came over.

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Home > Administrator's Desk Channel > Administrator's Desk Archive > Administrator's Desk Columnists > Larry Bell Archive > Larry Bell Article

Somebody Needs You

Are math word problems killing your state test scores? Or maybe fractions are? Or basic math facts? Whatever it is, you need to take action to do something about it. I have a suggestion that I have seen work very effectively.

At each workshop I present, I challenge teachers to become members of my 100 Percenters Club. In order to qualify as a 100 Percenter, 100 percent of a teacher's students must pass the grade-level state tests. That includes at-promise kids, kids of color, special ed inclusion kids -- all kids.

While members of the 100 Percenters Club might have picked up a few valuable tips from me, I have learned far more from them. The members of the Club have helped me to compile a list of 20 simple things that excellent teachers do to raise scores.

Among the 20 tips, those that are most near and dear to my heart are the ones that raise math achievement test scores. Perhaps those are most dear to me because I was not a good math student. I see immediately how some of these tips might have helped me to be a better student.

One of the simplest tips is this: when teachers disaggregate school-wide test data, they should look for the test questions that lowered their school's scores the most. What skills are associated with those questions? Those are the skills that it will pay to focus on in the year ahead.

Are word problems killing your scores? Or maybe it is fractions? Or basic math facts? Whatever it is, you need to take action to do something about it.

In many cases that I have examined, I have found that word problems are killing the scores of young students of color, particularly young black males. If that is the case in your school, I have a suggestion that I have seen work very effectively.

First, email every teacher on the staff a list of 30 word problems that are the same kind of word problem used on the standardized tests your students take. Those questions might be created by your school's math teachers or, if your state allows, they might be sample problems taken directly from the previous year's test. Email the questions and the answers to all teachers in your school. On the answer page, don't simply list the answers; provide the detailed steps that led to each answer.

Then, at the start of every class, every teacher in every single class must randomly select one math word problem, use an overhead projector to display that problem, and ask students to solve it. Give students a few minutes to do the math, and then take another few minutes to go through the steps to solve the problem.

This idea -- like all 20 of the ideas I have collected -- can be done in 7 to 10 minutes, max, of class time. These ideas can be done at the start of the class, the end of the class, during a moment of transition between activities, or any other time. This might not sound like an effort that will result in higher test scores but, take it from me and the 100 Percenters, it works!

The beauty of the approach is that if you get every teacher in the building -- whether he or she teaches science or language arts or industrial arts -- to work at solving a math word problem during the first 6 or 7 or 10 minutes of every class four times, five times, or six times a day, think of the cumulative effect of having every student do this all year long. This is a tremendous way to help kids get over the fear of these types of problems.

If you teach in an elementary school where students stay in the same classroom all day, every teacher might do this at regular intervals two or three or more times a day. They might do it as a bell-ringer activity at the start of every day, immediately after recess, and immediately after lunch.

If you don't rope the entire school into doing this, why not just the math department? Or the math and science departments? The key is the repetition. Students will see these problems over and over. (You might switch to a new set of 30 problems at semester break. Or give them another skill on which to work.) They almost can't help but absorb the process for solving these problems. It will become second nature to them. Most important, when they see a math word problem on your state's standardized test, they won't get frustrated and freeze up like they did last year. They will be so comfortable with math word problems that your scores can't help but climb.

If you know you have students, especially at-promise students, who are intimidated by and don't do well with a certain type of math, why not start on Day One in September and have every teacher join together to conquer this weakness?

In sports, we make sure kids practice over and over and over those things that intimidate them. Practice is a great tool to use with any skill that can greatly increase the test scores of your must vulnerable students.

My friends, somebody needs you.

Larry Bell

Education World®
Copyright © Education World

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A Simple Idea for Improving Math Achievement

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Home > Administrator's Desk Channel > Administrator's Desk Archive > Administrator's Desk Columnists > Larry Bell Archive > Larry Bell Article Are math word problems killing your state test scores? Or maybe fractions are? Or basic math facts? Whatever it is, you need to take action to do something about it.

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Educators vs. Republican Politicians ' Educational Answers

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Educators vs. Republican Politicians ' Educational Answers

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Official Tickets and Your Source for Live Entertainment | AXS.com

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AXS.com brings you inside access to tickets, artist news, and exclusive stories on concerts, tours, sports teams, family events, arts, theater, and festivals - nationally and in your town.

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More on differences in opinions on how to improve math scores, these have no research to back them up

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More on differences in opinions on how to improve math scores, these have no research to back them up

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Richard Perry/The New York Times

Only 39 percent of fourth graders and 34 percent of eighth graders scored at or above the proficient level on the nationwide math test given this spring. With little improvement over the past six years, it's seems unlikely that all children will reach grade-level proficiency by 2014, a central goal of the No Child Left Behind law, which imposed federal testing rules on schools nationwide.

Do the poor results suggest that testing requirements under "No Child Left Behind" have been ineffective and should be abandoned? What's lacking in math education that makes progress so hard to achieve?

Rethink Federal Oversight
Bruce Fuller, professor of education and public policy at University of California, Berkeley, is author of "Standardized Childhood: The Political and Cultural Struggle Over Early Education."
This week's dismal test-score results are bad news for President Obama, just as his education secretary, Arne Duncan, begins to rethink Washington's role in lifting the schools, searching for a less punitive, more robust blend of policies that energize students and teachers alike.

In New York, just two-fifths of the state's fourth graders were found to be proficient in math. This, despite Mayor Michael Bloomberg's claims of astonishing progress.

Many governors advanced tough-love policies, starting in the 1980s. This state-led accountability strategy - blending clear curriculum standards, atomizing knowledge to fit standardized tests and punishing lifeless schools - worked well for almost two decades. Racial achievement gaps closed significantly during the 1990s. (The school attainment of African-American mothers, a far stronger driver of children's development, had climbed as well, thanks to earlier affirmative action.)

Don't Shoot the Messenger
Lance T. Izumi is the senior director of education studies at the Pacific Research Institute.
Disappointing test scores, like the recent National Assessment of Educational Progress math scores, don't mean that testing systems need to be abandoned. The key question is what have schools, school districts and states done with those results to improve teaching in the classroom? Often times the answer is not much.

There are certainly problems with No Child Left Behind, but the law's encouragement of testing is not one of them. Oakland Charter Academy is a hugely successful middle school in one of the toughest inner-city neighborhoods in Oakland, Calif. One hundred percent of the student body is socio-economically disadvantaged. Yet, on the state's rigorous algebra 1 exam in 2009, 100 percent of the school's eighth graders scored at or above the proficient level. Oakland Charter Academy's principal, Jorge Lopez, says that No Child Left Behind's testing and progress requirements makes it the "best law that ever happened to minority kids."

It's the Curriculum, Not the Test
Holly Tsakiris Horrigan, an economist, is the mother of three children who attend public school in Massachusetts.
Not meeting a goal is never justification for deeming the goal unworthy. Standardized tests in can and do help students. Data support this. Standardized tests are essential in identifying what we as adults are doing right for our children and where we are falling short.

William C. Lopez for The New York Times

If we want to improve mathematics education, we should banish nonsensical curricula like Trail Blazers, Everyday Math and Investigations and make sure that our teachers are properly educated and proficient in math content. These curricula substitute writing, drawing and calculator usage for solid math content, leaving children unprepared for more advanced math topics. The precious few children who go on to succeed in higher math after being subjected to one of these programs are those that had the benefit of substantial reteaching outside of the classroom.
In wealthy towns around Massachusetts where these curricula are widely used and state standards are well defined, tutoring shops and independent instructional programs thrive. Parents in my community with extra time and resources make sure their children get what they need to meet or exceed the standards. It is not surprising that children lacking access to these outside resources aren't succeeding with the content-poor, misguided curricula.

Focus on Mastery of Concepts
Richard Bisk is professor and chair of the Mathematics Department at Worcester State College. He was an adviser to the Massachusetts Department of Education in the development of the "Guidelines for the Mathematical Preparation of Elementary Teachers."
To give students a firm foundation in math, we must start in the elementary grades by providing three things: a substantial improvement in elementary teachers' knowledge of mathematics; a more focused curriculum that emphasize core concepts and skills; and more challenging textbooks that teach for mastery and not just exposure.

In Massachusetts, we've begun to address this issue through new certification requirements. Prospective elementary teachers will have to pass a math test that assesses their understanding of the mathematics that is taught in the elementary grades. Institutions of higher education in Massachusetts are being asked to provide prospective elementary teachers with a sequence of three math courses to develop their math skills and knowledge. Since these changes do not affect teachers who have already been certified, they need professional development to gain the background we now expect of new teachers.

Content Over Process
Barry Garelick, an analyst, is the co-founder of the U.S. Coalition for World Class Math, a parent advocacy group.
The recently released national math test scores for 2009 are part of a growing body of lackluster results nationwide since the inception of the No Child Left Behind law. The exam, which has been deemed the "gold standard" of testing is hardly what one would call challenging with respect to math, and yet even so, the scores are barely indicative of a pulse.

What is needed is not another test, but sound mathematics instruction that stresses content over process. The education establishment needs to understand that even process is based on skills and exercises, and a logical sequence of topics whose mastery builds upon itself. We need solid math curricula and textbooks that are based on the premise that procedural fluency leads to conceptual understanding. Problem solving does not exist independently of exercises. The belief that content-based math teaches only definitions and procedures needs to be abandoned.

Students don't need skills-free math and "real world" problems that are nothing more than data mongering and statistical approaches. Students need to learn the skills and concepts necessary to solve challenging problems.

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Richard Perry/The New York Times Only 39 percent of fourth graders and 34 percent of eighth graders scored at or above the proficient level on the nationwide math test given this spring.

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Differences in opinion on how to improve math test scores

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Differences in opinion on how to improve math test scores

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Parents signing petitions. Governments forced to defend their positions. A declaration that Canada's problem is on the scale of a national emergency.

Welcome to the Math Wars, a battle that's been brewing for years but heated up last month when this country dropped out of the top 10 in international math education standings.

The battleground is fractured and the sides aren't clearly drawn, but at the centre of the debate over so-called discovery learning is this question: Should the teacher be a sage on the stage, a guide at the side, or both?

Parents in provinces such as Alberta, Saskatchewan and Ontario look at their children's math homework and see little, if anything, of the fundamentals they were taught just decades ago. Gone are the days, in much of the country, of long division, mad-minute multiplication, addition with a carry and subtraction with a borrow.

Today, children in provinces that have introduced the Western and Northern Canadian Protocol (WNCP) curriculum - a vast swath of the country - learn instead by investigating ideas through problem-solving, pattern discovery and open-ended exploration.

"If you look at what's been happening, predominantly over the last decade, there's been an unprecedented emphasis on discovery learning," said Donna Kotsopoulos, an associate professor in Wilfrid Laurier University's education faculty and former teacher.

Robert Craigen, a University of Manitoba mathematics professor who advocates basic math skills and algorithms, said Canada's downward progression in the international rankings - slipping from sixth to 13th among participating countries since 2000 - coincides with the adoption of discovery learning.

"The word 'emergency' suggests a suddenness, but I don't think there's anything particularly sudden about this," said Prof. Craigen, referring to comments made in December by John Manley, the head of the Canadian Council of Chief Executives, that the results of the OECD's 2012 Programme for International Student Assessment (PISA) were "on the scale of a national emergency."

"What we're seeing is the final demonstration that things have been going downhill," Prof. Craigen said.

And with Canada's slip on the global stage comes an anxiety at home, one that's palpably sweeping across the country as governments tout different solutions to an almost universal problem. Parents in Alberta, Ontario and British Columbia, for example, launched petitions over the Christmas holidays, calling on their governments to revamp curriculums with a greater emphasis on basic math skills.

But the governments themselves are taking different approaches, free to do so since provinces have jurisdiction over education and there are no national standards or strategies. Ontario has no plans to change its curriculum and is instead banking on ramped-up teacher training. Manitoba is watering down discovery learning with more "back to basics" fundamentals, while British Columbia appears to be heading in the opposite direction as it revises its own curriculum.

Meantime, Quebec, with its intensive training and teachers who apparently refuse to shirk algorithms despite reforms, enjoys the best scores in Canada and is now at the centre of math-education research.

And then somewhere in the middle of conventional math and discovery learning is JUMP Math, a Toronto-based non-profit whose curriculum reaches 100,000 Canadian students and counting.

"In all the debates that are going on, we think it's a bad idea to throw [discovery learning] out, but the other side is right also," said JUMP founder John Mighton, a mathematician and author whose vision has caught the attention of The New York Times. "The parents and those who want to go back to basic math are right in some sense: You need to combine the discovery with guidance."

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There is no national strategy for improving math scores because provinces have jurisdiction over education. Here's what the provinces are doing - and whether they plan to make curriculum changes.
BRITISH COLUMBIA
Landscape: The province withdrew from the Western and Northern Canadian Protocol (WNCP) for financial reasons in 2011 and is currently revising its curriculum. The ministry of education dictates what must be taught, but not how to teach it. Here's an example: A Grade 4 student is required to learn mental math strategies for adding two-digit numbers, but the teacher chooses whether to teach the strategies by adding from left to right, or top-to-bottom with a carry. Some critics argue the current math curriculum is too conceptually based, with parents in the province launching a petition calling for greater emphasis on basic math skills. JUMP Math, which is being touted as a "third way" that emphasizes rehearsing the basics and breaking problems into small parts, reaches roughly 10 per cent of B.C. students, mathematician and JUMP founder John Mighton said.

PISA 2012 ranking, when provinces are included: 12

PISA paper-based math score change between 2003 and 2012: -16

Outlook: As the province plans for a major curriculum overhaul that will put a greater emphasis on "personalized learning" across all subjects - a focus on concepts and processes rather than factual content - the ministry says math will likely change the least out of all subjects. The ministry, however, maintains that students still must be strong at elements such as mathematical algorithms and memorizing times tables. While the new curriculum is still in draft stage, expected changes include a greater emphasis on financial literacy and interdisciplinary thinking. The province is also reviewing teacher training.
ALBERTA
Landscape: Alberta's current math curriculum, based on the WNCP and discovery learning, is under revision. There are mandatory learning expectations for Grades 2 to 6 "that address the understanding of addition, subtraction, multiplication and division with an emphasis on mental mathematics," an education ministry spokesperson said. That means Grade 3 students are being asked to add and subtract two-digit numbers and multiply to 5 x 5. By the end of Grade 5, students are expected to know their multiplication tables.

PISA 2012 ranking, when provinces are included: 17

PISA paper-based math score change between 2003 and 2012: -32

Outlook: Based on the PISA scores, Alberta's share of low performers has doubled over the past decade. "It tells me we've got some work to do," said Jeff Johnson, the province's minister of education. Parents and teachers are pushing for a better balance between the WNCP learning strategies and students knowing basic math facts. Despite a petition calling for Alberta to focus on fundamental skills, the ministry contends the curriculum already places emphasis on addition, subtraction, multiplication and division. Changes to the math curriculum won't materialize until 2016.
SASKATCHEWAN
Landscape: Saskatchewan's ministry of education follows the WNCP curriculum. In 2012, the government responded to concerns that basic math was being overlooked in favour of new concepts. It was announced that work would be done to help parents understand new math, while there would also be improvements in teacher training.

PISA 2012 ranking, when provinces are included: 22

PISA paper-based math score change between 2003 and 2012: -10

Outlook: Following consultations, the ministry ultimately said its curriculum was "in line with what is taught in jurisdictions across North America and will continue."
MANITOBA
Landscape: Though the province is still part of the WNCP, it has started moving toward conventional learning. This past fall, the government responded to a push from parents and announced curriculum revisions for students in kindergarten to Grade 8. The government wanted to "ensure that there's a strong emphasis on giving kids basic math skills," Education and Advanced Learning Minister James Allum said in an e-mail. The province set clearer expectations around counting and memorizing math facts. Students are also now taught all four standard methods for arithmetic - addition with a carry, subtraction with a borrow, long multiplication and long division. The curriculum also stresses that children in kindergarten through Grade 3 not rely on calculators. The Manitoba government has also added JUMP Math to its recommended resource list, Mr. Mighton said.

PISA 2012 ranking, when provinces are included: 35

PISA paper-based math score change between 2003 and 2012: -36

Outlook: Besides the curriculum changes, the province has reduced class sizes in kindergarten to Grade 3, and is piloting a new program at the University of Winnipeg to strengthen teacher education in math.
ONTARIO
Landscape: In the late 1990s, research into discovery learning drove changes to the math curriculum, which was last revised in 2005. Students are required to be able to multiply by 9 in Grade 4, but there is no requirement that they memorize multiplication tables. They aren't required to learn basic algorithms such as long division or adding a column of figures, either. Instead, they are encouraged to break problems down into smaller parts to work through them, or use physical materials to help them understand.

PISA 2012 ranking, when provinces are included: 19

PISA paper-based math score change between 2003 and 2012: -16

Outlook: The PISA results have got Ontario worried, with parents launching a petition and Education Minister Liz Sandals vowing to tackle the problem. The province is currently in the early stages of consulting experts on math education, said Lauren Ramey, spokeswoman for Ms. Sandals. But asked this week whether the province would revamp its curriculum, Ms. Sandals said "no." On Wednesday, the government announced it will spend $4-million this year to help pay for more current teachers to take advanced qualifications courses in math and upgrade their skills. In 2015, the province will also move from a one-year teachers' college program to a two-year program.
QUEBEC
Landscape: The province is the Canadian leader in math, scoring with the likes of Japan and Macau. While Quebec adopted the "discovery learning" method during major educational reforms in the early 2000s, the approach was already in the works as early as the 1980s. That may have given the province the benefit of a longer lead-time in implementing it, said Stéphane Cyr, a University of Quebec in Montreal math-education professor. But some educators say Quebec is succeeding in spite of its curriculum. Nathalie Morel, vice-president of the Fédération autonome de l'enseignement, a major teachers' union, said educators have insisted on teaching and testing children on math fundamentals. Still, the province hasn't abandoned rote memorization - the curriculum, for example, calls for children to learn their multiplication and division tables by heart, starting in Grade 3.

PISA 2012 ranking, when provinces are included: 8

PISA paper-based math score change between 2003 and 2012: -1

Outlook: Few studies compare why students in Canada have such different math scores, but researchers have started focusing on Quebec's intensive teacher training and its curriculum, which balances traditional math drills with problem-solving approaches. Montreal's McGill University is taking part in a study on math teaching across Canada, and early findings suggest Quebec's four-year math-teacher course may be a model to emulate. Grade school teachers, for example, must take as many as 225 hours of university courses in math education, whereas in some Canadian jurisdictions, the number can be as low as 39, according to Annie Savard, a McGill professor of math education who is part of the national research team.
THE ATLANTIC
Landscape: The Atlantic provinces have adopted or adapted the WNCP math curriculum, said Robert Craigen, a math professor at University of Manitoba. Newfoundland and Labrador, for example, several years ago decided to launch a new curriculum, adopted from the WNCP. In 2012, the Nova Scotia government announced the province would adopt Alberta's math curriculum, which is the WNCP's, explained Michael Zwaagstra, a Frontier Centre for Public Policy research fellow who specializes in education policy.

PISA 2012 ranking, when provinces are included: New Brunswick, 25; Nova Scotia, 31; Newfoundland and Labrador, 37; PEI, 46

PISA paper-based math score change between 2003 and 2012: New Brunswick, -10; Nova Scotia, -18; Newfoundland and Labrador, -27; PEI, -21

Outlook: PEI Education Minister Alan McIsaac said his government is focusing on interventions in the early years; for example, investing in a play-based curriculum for preschool. After Nova Scotia ranked in the bottom half of the provinces in PISA 2012 scores, the government issued a press release entitled "Nova Scotia Students Perform Well In International Assessments." The release noted the math results "are a concern," but the education minister noted "math is a priority area" and that a "new curriculum is being introduced."
Sixty-five countries participated in PISA 2012. No data were collected in the three territories, which are part of the WNCP. With reports from Kathryn Blaze Carlson, Adrian Morrow and Caroline Alphonso in Toronto; Andrea Woo in Vancouver; Allan Maki in Calgary; Ingrid Peritz in Montreal.

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Math wars: The division over how to improve test scores

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Parents signing petitions. Governments forced to defend their positions. A declaration that Canada's problem is on the scale of a national emergency. Welcome to the Math Wars, a battle that's been brewing for years but heated up last month when this country dropped out of the top 10 in international math education standings.

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Project MUSE - Assessing the Effects of the

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As teachers of all grade levels know very well, it is extremely difficult to teach students higher order math algorithms when they are not fluent with their basic facts (addition, subtraction, multiplication and division). If students are not automatic in responding to math facts, their attention is necessarily taken away from the multiple steps necessary to solve more complex problems.

Think about long division. Students must be able to come up with the answers to multiplication, subtraction and division facts all in the context of the higher order algorithm. Less than automatic facility with math facts often results in either errors in the algorithm or fact errors. Both kinds of errors appear to be the result of carelessness when in fact it is the result of being distracted by having to figure out the answers to facts. Below are 12 principles necessary to successfully teaching math facts.

There is a big difference between figuring out the answer and memorizing facts. If students have too many facts to learn at one time, they necessarily fall back on figuring out the answer. Instead, we want students to REMEMBER the answers without having to figure them out.

Only add more facts to learn as the previous set has been mastered

Once students have learned a set a set of facts to mastery, it is now possible to add two or three more facts to be learned. Student success is greatest when they have only two or three things to learn in a sea of material they have already mastered.

Practice must be structured in a way that facts which have been previously deemed mastered continue to appear along with the two or three new facts that are being learned.

Students should memorize facts in a way that forms a verbal chain

Students should always practice by saying the whole problem and the answer aloud. In this way, students memorize a verbal chain. As a result of this kind of practice, students hear/see 8x7 and can't stop from saying/writing 56.

Automaticity is the ability to say the answer to a problem immediately after reading the fact. There should be no hesitation.

Students write at different speeds. The speed at which students can write the answer to facts is limited by their writing skill. Automaticity means that students should write answers to math facts exactly as fast they can write.

Practice is most effective when it is distributed throughout the year. Learning math facts is just one part of a math program therefore efficient use of limited time is critical. A standard daily routine is necessary in order to accomplish this goal.

Whenever students hesitate or give an incorrect answer, a corrective procedure should give them the answer, ensure they know it and provide them with a delayed test. It is very important that the delayed test occur before the student has a chance to forget the correct response.

Students cannot maintain focus on drill for more than 2-4 minutes at a time. Practice sessions may occur more than one time during the day, but should remain short.

If students are really learning math facts, the number of facts they can answer within a set time period should gradually increase. Periodically, students should be given a timed test of all the facts in the operations they are learning to see if their fluency is improving.

Because fractions demand instantaneous recognition of multiplication facts, these must be mastered before fractions can be successfully learned. Even those students who need their fingers to add and subtract in Grade 4 and above, should learn multiplication facts to mastery or their continued progress in math is in jeopardy.

Students will be motivated to learn math facts if teachers act like it is important. Finding ways to celebrate students' success is the best way to demonstrate the importance of learning math facts.
Share your tips for teaching math facts in the comments section!

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As teachers of all grade levels know very well, it is extremely difficult to teach students higher order math algorithms when they are not fluent with their basic facts (addition, subtraction , multiplication and division ).

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November 2007 | Volume 65 | Number 3
Making Math Count Pages 16-21

Marilyn Burns

Paul, a 4th grader, was struggling to learn multiplication. Paul's teacher was concerned that he typically worked very slowly in math and "didn't get much done." I agreed to see whether I could figure out the nature of Paul's difficulty. Here's how our conversation began:

Marilyn: Can you tell me something you know about multiplication?

Paul: [Thinks, then responds] 6 × 8 is 48.

Marilyn: Do you know how much 6 × 9 is?

Paul: I don't know that one. I didn't learn it yet.

Marilyn: Can you figure it out some way?

Paul: [Sits silently for a moment and then shakes his head.]

Marilyn: How did you learn 6 × 8?

Paul: [Brightens and grins] It's easy-goin' fishing, got no bait, 6 × 8 is 48.

As I talked with Paul, I found out that multiplication was a mystery to him. Because of his weak foundation of understanding, he was falling behind his classmates, who were multiplying problems like 683 × 4. Before he could begin to tackle such problems, Paul needed to understand the concept of multiplication and how it connects to addition.

Paul wasn't the only student in this class who was floundering. Through talking with teachers and drawing on my own teaching experience, I've realized that in every class a handful of students are at serious risk of failure in mathematics and aren't being adequately served by the instruction offered. What should we do for such students?

Grappling with Interventions

My exchange with Paul reminded me of three issues that are essential to teaching mathematics:

It's important to help students make connections among mathematical ideas so they do not see these ideas as disconnected facts. (Paul saw each multiplication fact as a separate piece of information to memorize.)

It's important to build students' new understandings on the foundation of their prior learning. (Paul did not make use of what he knew about addition to figure products.)

It's important to remember that students' correct answers, without accompanying explanations of how they reason, are not sufficient for judging mathematical understanding. (Paul's initial correct answer about the product of 6 × 8 masked his lack of deeper understanding.)

For many years, my professional focus has been on finding ways to more effectively teach arithmetic, the cornerstone of elementary mathematics. Along with teaching students basic numerical concepts and skills, instruction in number and operations prepares them for algebra. I've developed lessons that help students make sense of number and operations with attention to three important elements-computation, number sense, and problem solving. My intent has been to avoid the "yours is not to question why, just invert and multiply" approach and to create lessons that are accessible to all students and that teach skills in the context of deeper understanding. Of course, even well-planned lessons will require differentiated instruction, and much of the differentiation needed can happen within regular classroom instruction.

But students like Paul present a greater challenge. Many are already at least a year behind and lack the foundation of mathematical understanding on which to build new learning. They may have multiple misconceptions that hamper progress. They have experienced failure and lack confidence.

Such students not only demand more time and attention, but they also need supplemental instruction that differs from the regular program and is designed specifically for their success. I've recently shifted my professional focus to thinking about the kind of instruction we need to serve students like Paul. My colleagues and I have developed lessons that provide effective interventions for teaching number and operations to those far behind. We've grappled with how to provide instruction that is engaging, offers scaffolded instruction in bite-sized learning experiences, is paced for students' success, provides the practice students need to cement fragile understanding and skills, and bolsters students' mathematical foundations along with their confidence.

In developing intervention instruction, I have reaffirmed my longtime commitment to helping students learn facts and skills-the basics of arithmetic. But I've also reaffirmed that "the basics" of number and operations for all students, including those who struggle, must address all three aspects of numerical proficiency-computation, number sense, and problem solving. Only when the basics include understanding as well as skill proficiency will all students learn what they need for their continued success.

Essential Strategies

I have found the following nine strategies to be essential to successful intervention instruction for struggling math learners. Most of these strategies will need to be applied in a supplementary setting, but teachers can use some of them in large-group instruction.

1. Determine and Scaffold the Essential Mathematics Content

Determining the essential mathematics content is like peeling an onion-we must identify those concepts and skills we want students to learn and discard what is extraneous. Only then can teachers scaffold this content, organizing it into manageable chunks and sequencing these chunks for learning.

For Paul to multiply 683 × 4, for example, he needs a collection of certain skills. He must know the basic multiplication facts. He needs an understanding of place value that allows him to think about 683 as 600 + 80 + 3. He needs to be able to apply the distributive property to figure and then combine partial products. For this particular problem, he needs to be able to multiply 4 by 3 (one of the basic facts); 4 by 80 (or 8 × 10, a multiple of 10); 4 and by 600 (or 6 × 100, a multiple of a power of 10). To master multidigit multiplication, Paul must be able to combine these skills with ease. Thus, lesson planning must ensure that each skill is explicitly taught and practiced.

2. Pace Lessons Carefully

We've all seen the look in students' eyes when they get lost in math class. When it appears, ideally teachers should stop, deal with the confusion, and move on only when all students are ready. Yet curriculum demands keep teachers pressing forward, even when some students lag behind. Students who struggle typically need more time to grapple with new ideas and practice new skills in order to internalize them. Many of these students need to unlearn before they relearn.

3. Build in a Routine of Support

Students are quick to reveal when a lesson hasn't been scaffolded sufficiently or paced slowly enough: As soon as you give an assignment, hands shoot up for help. Avoid this scenario by building in a routine of support to reinforce concepts and skills before students are expected to complete independent work. I have found a four-stage process helpful for supporting students.

In the first stage, the teacher models what students are expected to learn and records the appropriate mathematical representation on the board. For example, to simultaneously give students practice multiplying and experience applying the associative and commutative properties, we present them with problems that involve multiplying three one-digit factors. An appropriate first problem is 2 × 3 × 4. The teacher thinks aloud to demonstrate three ways of working this problem. He or she might say,

I could start by multiplying 2 × 3 to get 6, and then multiply 6 × 4 to get 24. Or I could first multiply 2 × 4, and then multiply 8 × 3, which gives 24 again. Or I could do 3 × 4, and then 12 × 2. All three ways produce the same product of 24.

As the teacher describes these operations, he or she could write on the board:

It's important to point out that solving a problem in more than one way is a good strategy for checking your answer.

In the second stage, the teacher models again with a similar problem-such as 2 × 4 × 5-but this time elicits responses from students. For example, the teacher might ask, "Which two factors might you multiply first? What is the product of those two factors? What should we multiply next? What is another way to start?" Asking such questions allows the teacher to reinforce correct mathematical vocabulary. As students respond, the teacher again records different ways to solve the problem on the board.

During the third stage, the teacher presents a similar problem-for example, 2 × 3 × 5. After taking a moment to think on their own, students work in pairs to solve the problem in three different ways, recording their work. As students report back to the class, the teacher writes on the board and discusses their problem-solving choices with the group.

In the fourth stage, students work independently, referring to the work recorded on the board if needed. This routine both sets an expectation for student involvement and gives learners the direction and support they need to be successful.

4. Foster Student Interaction

We know something best once we've taught it. Teaching entails communicating ideas coherently, which requires the one teaching to formulate, reflect on, and clarify those ideas-all processes that support learning. Giving students opportunities to voice their ideas and explain them to others helps extend and cement their learning.

Thus, to strengthen the math understandings of students who lag behind, make student interaction an integral part of instruction. You might implement the think-pair-share strategy, also called turn and talk. Students are first asked to collect their thoughts on their own, and then talk with a partner; finally, students share their ideas with the whole group. Maximizing students' opportunities to express their math knowledge verbally is particularly valuable for students who are developing English language skills.

5. Make Connections Explicit

Students who need intervention instruction typically fail to look for relationships or make connections among mathematical ideas on their own. They need help building new learning on what they already know. For example, Paul needed explicit instruction to understand how thinking about 6 × 8 could give him access to the solution for 6 × 9. He needed to connect the meaning of multiplication to what he already knew about addition (that 6 × 8 can be thought of as combining 6 groups of 8). He needed time and practice to cement this understanding for all multiplication problems. He would benefit from investigating six groups of other numbers-6 × 2, 6 × 3, and so on-and looking at the numerical pattern of these products. Teachers need to provide many experiences like these, carefully sequenced and paced, to prepare students like Paul to grasp ideas like how 6 × 9 connects to 6 × 8.

6. Encourage Mental Calculations

Calculating mentally builds students' ability to reason and fosters their number sense. Once students have a foundational understanding of multiplication, it's key for them to learn the basic multiplication facts-but their experience with multiplying mentally should expand beyond these basics. For example, students should investigate patterns that help them mentally multiply any number by a power of 10. I am concerned when I see a student multiply 18 × 10, for example, by reaching for a pencil and writing:

Revisiting students' prior work with multiplying three factors can help develop their skills with multiplying mentally. Helping students judge which way is most efficient to multiply three factors, depending on the numbers at hand, deepens their understanding. For example, to multiply 2 × 9 × 5, students have the following options:

Guiding students to check for factors that produce a product of 10 helps build the tools they need to reason mathematically.

When students calculate mentally, they can estimate before they solve problems so that they can judge whether the answer they arrive at makes sense. For example, to estimate the product of 683 × 4, students could figure out the answer to 700 × 4. You can help students multiply 700 × 4 mentally by building on their prior experience changing three-factor problems to two-factor problems: Now they can change a two-factor problem-700 × 4-into a three-factor problem that includes a power of 10-7 × 100 × 4. Encourage students to multiply by the power of 10 last for easiest computing.

7. Help Students Use Written Calculations to Track Thinking

Students should be able to multiply 700 × 4 in their heads, but they'll need pencil and paper to multiply 683 × 4. As students learn and practice procedures for calculating, their calculating with paper and pencil should be clearly rooted in an understanding of math concepts. Help students see paper and pencil as a tool for keeping track of how they think. For example, to multiply 14 × 6 in their heads, students can first multiply 10 × 6 to get 60, then 4 × 6 to get 24, and then combine the two partial products, 60 and 24. To keep track of the partial products, they might write:

14 × 6

10 × 6 = 60

4 × 6 = 24

60 + 24 = 84

They can also reason and calculate this way for problems that involve multiplying by three-digit numbers, like 683 × 4.

8. Provide Practice

Struggling math students typically need a great deal of practice. It's essential that practice be directly connected to students' immediate learning experiences. Choose practice problems that support the elements of your scaffolded instruction, always promoting understanding as well as skills. I recommend giving assignments through the four-stage support routine, allowing for a gradual release to independent work.

Games can be another effective way to stimulate student practice. For example, a game like Pathways (see Figure 1 for a sample game board and instructions) gives students practice with multiplication. Students hone multiplication skills by marking boxes on the board that share a common side and that each contain a product of two designated factors.

Figure 1. Pathways Multiplication Game

9. Build In Vocabulary Instruction

The meanings of words in math-for example, even, odd, product, and factor -often differ from their use in common language. Many students needing math intervention have weak mathematical vocabularies. It's key that students develop a firm understanding of mathematical concepts before learning new vocabulary, so that they can anchor terminology in their understanding. We should explicitly teach vocabulary in the context of a learning activity and then use it consistently. A math vocabulary chart can help keep both teacher and students focused on the importance of accurately using math terms.

When Should We Offer Intervention?

There is no one answer to when teachers should provide intervention instruction on a topic a particular student is struggling with. Three different timing scenarios suggest themselves, each with pluses and caveats.

While the Class Is Studying the Topic

Extra help for struggling learners must be more than additional practice on the topic the class is working on. We must also provide comprehensive instruction geared to repairing the student's shaky foundation of understanding.

The plus: Intervening at this time may give students the support they need to keep up with the class.

The caveat: Students may have a serious lack of background that requires reaching back to mathematical concepts taught in previous years. The focus should be on the underlying math, not on class assignments. For example, while others are learning multidigit multiplication, floundering students may need experiences to help them learn basic underlying concepts, such as that 5 × 9 can be interpreted as five groups of nine.

Before the Class Studies the Topic

Suppose the class is studying multiplication but will begin a unit on fractions within a month, first by cutting out individual fraction kits. It would be extremely effective for at-risk students to have the fraction kit experience before the others, and then to experience it again with the class.

The plus: We prepare students so they can learn with their classmates.

The caveat: With this approach, struggling students are studying two different and unrelated mathematics topics at the same time.

After the Class Has Studied the Topic.

This approach offers learners a repeat experience, such as during summer school, with a math area that initially challenged them.

The plus: Students get a fresh start in a new situation.

The caveat: Waiting until after the rest of the class has studied a topic to intervene can compound a student's confusion and failure during regular class instruction.

How My Teaching Has Changed

Developing intervention lessons for at-risk students has not only been an all-consuming professional focus for me in recent years, but has also reinforced my belief that instruction-for all students and especially for at-risk students-must emphasize understanding, sense making, and skills.

Thinking about how to serve students like Paul has contributed to changing my instructional practice. I am now much more intentional about creating and teaching lessons that help intervention students catch up and keep up, particularly scaffolding the mathematical content to introduce concepts and skills through a routine of support. Such careful scaffolding may not be necessary for students who learn mathematics easily, who know to look for connections, and who have mathematical intuition. But it is crucial for students at risk of failure who can't repair their math foundations on their own.

Marilyn Burns is Founder of Math Solutions Professional Development, Sausalito, California; 800-868-9092; mburns@mathsolutions.com.

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November 2007 | Volume 65 | Number 3 Making Math Count Pages 16-21 Marilyn Burns Paul, a 4th grader, was struggling to learn multiplication. Paul's teacher was concerned that he typically worked very slowly in math and "didn't get much done." I agreed to see whether I could figure out the nature of Paul's difficulty.

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Cross-post of an article I wrote for The Apple, you can read the original article here.

For those of you who aren't familiar with it, The Apple is a social network for educators that is also a great resource for inspiration, lesson ideas, educational news, videos, and job postings.

Kelly Tenkely | TheApple.com

Differentiating instruction can be challenging. Student's educational strengths and weaknesses can be widely varied, making it a difficult task to meet each student's needs in any given lesson. Math is one such subject area where student skill levels can be very different. Some students have their basic math facts so well mastered that completing more complex tasks isn't hindered. Other students have a logical mathematical mind and math just seems to click for them. Then there are the students who really struggle with math facts; they just can't seem to get them down. Every other area of math feels painful because they don't have the building blocks mastered. Still other students have no trouble solving equations but when a story problem is introduced, they are stopped dead in their tracks.

For most students, math takes a lot of practice. Unfortunately, the students who need the most practice are the most reluctant to do so because they haven't been successful in the past. Many of these students have convinced themselves, through negative self-talk, that "I'm just not good at math." What is a teacher to do with such a mix of skill and comfort levels in the math classroom? Technology can be a great equalizer in the classroom. It provides struggling students with extra support and attention at exactly their skill level, and makes it possible for the student to pace their own learning. Students who have mastered the basics and need to be challenged, can attend to tasks that require them to think in new ways using the skills they already have. The Internet abounds with outstanding free math resources that can be used to encourage struggling students and challenge students who are excelling in math. These websites can be used as a center in the math classroom that is limited to a few computers, for individual learning and practice in the computer lab (1 to 1 setting), or used with the whole class and an interactive whiteboard or projector. The sites are well designed to meet the various learning needs in your classroom, and motivating enough to keep all of your students engaged in learning and practice.

Offline Math Practice:

1. If you are lacking student computers for math practice, consider visiting iPlay Math Games. http://iplaymathgames.com has a collection of printable math games for students in kindergarten through twelfth grade. The games are easily searched by grade or skill level. These math games can be played using common items like dice, cards, and other manipulatives. The games have been created to build skills such as addition, subtraction, multiplication, division, fractions, decimals, percents, algebra, long division, measurement, graphing, exponents, problem solving, puzzles, geometry, probability, logic, reasoning, numeration, patterns, and counting. Search for games that will meet your different learning needs. Next, group students together to play the games. Grouping students with similar learning needs helps students practice the math skill with others who are in need of the same practice. Create a math learning center where students can easily access the games and manipulatives needed for the games. Hold a math mania day once a week where students can practice the skills they have been learning in math class by playing these fun games.

2. 3D Vinci ( http://www.3dvinci.net/ccp0-display/freestuff.html) has downloadable pdf lesson plans to teach math concepts using Google's Sketchup. The site also has an impressive collection of printable puzzles and mosaics for students in kindergarten through twelfth grade to solve. This site helps bring 3-D concepts to life in a way that textbooks can't. 3D Vinci also has a YouTube channel where students can watch several ModelMetricks projects.

Math Fact Practice:

Nothing is more motivating than a little friendly competition. Several sites offer competition that helps students practice their math facts, some are seasonal competitions, while others are available year-round. These sites make it a breeze to differentiate math practice. Each student can play games at their own ability level. This keeps the lower math students engaged and feeling successful in their math skills while the higher level students are challenged.

3. Math Mania and World Math Day- Vmath Live hosts these online competitions each year for students around the world to compete in. Math Mania ( http://www.voyagerlearning.com/mathmania.do) is a back to school challenge where students compete against other students in real time using the Vmath Live math program. Students face off with other students from around the world while practicing their math facts. As students answer questions, they can see, in real time, how other students from around the world are doing answering the same questions at the same time. The goal is to answer all of the questions as quickly as possible (correctly) and finish before the other students. The live competition is highly motivating and has students asking, "can I play this at home?" World Math Day ( http://worldmathday.com/) is a competition that is held each year in March. Just like Math Mania, World Math Day is a competition between students from around the world. Each math fact game lasts for 60 seconds and improves student's mental math. The World Math Day competition is for students in kindergarten through twelfth grade. Visit both competition sites for more details about the competition and dates.

4. Make 5 ( http://www.fi.uu.nl/toepassingen/00091/toepassing_wisweb.en.html) Make 5 is a fun way for students to practice addition, subtraction or multiplication. Students can play this game individually or against another student. This game provides a great platform for students to practice math facts and to start recognizing relationships between numbers. Students are given a target number and a goal to choose an equation on the grid that equals that target number. There are multiple correct answers on the grid. The goal of the game is to get 5 answers in a row on the grid (like Tic-tac-toe or connect 4). If you don't have access to classroom computers to play this game, split your class into two groups and play with the whole class with an interactive whiteboard or projector. Each student should be given the chance to represent their team and match an equation with a target number. Give teams 30 seconds to plan their move before sending up the representative.

5. Arcademic Skill Builders ( http://www.arcademicskillbuilders.com/) is a collection of fun arcade-like games that help students practice their math facts. Students can play games for addition, subtraction, multiplication, division, fraction, and ratios. The games are research based and standard aligned. It blends the fun of arcade games and key math skills into fun online games that engage, motivate, and teach students. All games have the option of single or multi-player mode.

6. Tut Pup ( http://tutpup.com) is a free math fact practice website. Students are matched up with other students from around the world where they play fact games and compete in real time. This site does not collect personal information about students and there are several games to choose from, each with multiple levels. Students can practice addition, subtraction, multiplication, division, algebra, and a mixture of those skills.

7. Free Rice ( http://freerice.com) is an amazing website that helps students practice math facts, it has the added bonus of donating rice for every correct answer. Students can choose to practice basic math facts or multiplication. For each answer that is correct, grains of rice are donated to hungry people around the world. Add some additional math practice by figuring out how many grains of rice are needed to create a bowl of rice, and keep tally of how many total grains that your students have earned.

Math Puzzles:

Math requires students to look at the world in different ways than they may be used to. It is about seeing relationships and patterns. Students may not be used to thinking in this way. Math puzzles can be a great way to help students start to think about relationship and pattern. This is a fun way to get your students in the math mind set. Your reluctant math students won't realize that they are practicing math skills as they complete these puzzles.

8. Tangram House ( http://games.ztor.com/tang/) This Chinese puzzle challenges students to create a shape using only 7 tans (shapes) to complete a puzzle. The tans can be rotated, flipped, and dragged into place. Tangram House can be used as a math center on classroom computers, completed individually in the computer lab setting, or used as a whole class activity with an interactive whiteboard. To play as a whole class activity, split students into teams that will take turns at the board forming the tangrams. Students who are not at the board can help their teammate using good directions and clues for the student working on the puzzle. This is a great way for students to practice giving and receiving quality instructions and descriptions.

Better Than a Text Book:

Math textbooks can be very dry and hard for students to read for understanding. So much of math involves relationships, patterns, and 3-D objects that a static textbook just can't do justice to. Online supplements can give students a better understanding of relationship, pattern, and 3-D objects with video and animation.

9. Number Nut ( http://www.numbernut.com/index.html) is an interactive math textbook, there are multiple pages for each math concept and each page is followed by two interactive practice areas. Topics on Number Nut include shapes, color, numbers, counting, addition, subtraction, multiplication, division, order of operations, date, time, fractions, decimals, percent values, estimation, rounding, ratios, and money math. The site is standard aligned and has an exceptional math glossary. Number nut is an excellent alternative to math textbooks because students get immediate feed back as they practice a skill they have just learned. With traditional math textbooks, a student doesn't know if they understand a concept until they turn in their math worksheet or test and get it back a few days later full of red marks. Often they will not go back and re-practice unless they are required to. This site is ideal in a computer lab setting where each student can work at his or her own pace.

10. Learning Clip ( http://www.learningclip.co.uk/index1.aspx) is a collection of interactive activities for primary math. Resources and activities include topics such as using and applying math, understanding numbers, number facts, calculating, understanding shape, measuring, and handling data. Each activity includes a brief description, a pdf of notes that accompany the activity, the age appropriateness, and a corresponding worksheet that can be printed out. When an activity is loaded, a video clip explaining the math concept plays. Each activity includes a game or interactive space for students to practice what they are learning. This site is ideal for struggling students, they can replay the video portion of the lesson as many times as they need to so that they understand the concept. The practice area is a great place for students to try what they have learned.

11. Harcourt Multimedia Math Glossary ( http://www.harcourtschool.com/glossary/math2/index3.html) will help students in kindergarten through sixth grade understand math vocabulary. The glossary is easily searched by grade and alphabetic order. This glossary is not your typical textbook glossary, it doesn't just define math terms, it actually shows them what it looks like. Each word and definition can be read aloud to students by clicking on the speaker icon next to the word.

12. Math TV ( http://www.mathplayground.com/mathtv.html) is a series of video story problems for math. Each math problem comes with a step by step video solution, followup problems, an online calculator, and a sketch pad. Word problems include topics such as fractions, percentages, ratios, probability, geometry, averages, and algebra. Story problems are often a source of anxiety and frustration, these videos are perfect for visual, auditory, and kinesthetic learners. You can't go wrong!

Real World Math Practice:

"When will I use this?" is a common phrase muttered in math classrooms around the country. As students advance in math and learn more complex skills and equations, they are often puzzled by how it relates to real life. Students want to know that what they are learning is relevant to their lives. Bringing real world math practice into the classroom helps students relate what they are learning with how it is applied in life.

13. The Stock Market Game ( http://stockmarketgame.com/) gives students a chance to practice hands on real-world math. Each student gets the chance to invest a hypothetical $100,000 in an online portfolio. While playing the game, students learn economic and financial concepts that they will use the rest of their lives. The game is intended for students in fourth through twelfth grades.

14. Kids and Cookies ( http://www.teacherlink.org/content/math/interactive/flash/kidsandcookies/kidcookie.php) is a flash game that teaches students about fractions in a real-world situation. Students choose characters to be their friends and then choose how many cookies they have. Students must evenly share their cookies with their friends and can use different cookie cutters to divide them up. This is a great introduction to how rational number fractions are used in life.

Interactive Math:

15. Illuminations Resources for Teaching Math ( http://illuminations.nctm.org/Activities.aspx?grade=1&grade=2&grade=3&grade=4) is a treasure trove of activities for the math classroom. The site has more than 100 interactive math activities for teaching and practicing math in the kindergarten through twelfth grade classroom. The activities help students understand concepts that can be difficult by inviting them to interact with the concept. These activities are ideal for an interactive whiteboard but can also be a wonderful math center activity.

16. Math Arcade ( http://www.funbrain.com/brain/MathBrain/MathBrain.html) is a website where students in first through eighth grades can practice math facts and math skills through fun interactive games and activities. Math Arcade plays like a board game, each space on the game board is another activity that must be completed before students can move on. Students receive a special password that they can write down to return to their saved game. There are 25 games at each grade level to play. Students can play games that build skills at the specific level they are at.

17. Math Playground ( http://www.mathplayground.com/index.html) is packed full of games, videos, and puzzles all centered around math for elementary and middle school students. The games are interactive and a great place to practice the skills they are learning. Students get immediate feedback as they play the games and go through the activities. If they are correct, they can move on to the next game or problem. If they are incorrect, they will try again. The word problem section is divided up by grade level, ensuring that students are challenged at an appropriate level. The Math video section contains videos that introduce students to math concepts.

There are oodles of other ways to meet the needs of your individual learners in math, check out these posts for some more ideas!

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17 Ways to Meet Individual Learning Needs in the Math Classroom

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Cross-post of an article I wrote for The Apple, you can read the original article here. For those of you who aren't familiar with it, The Apple is a social network for educators that is also a great resource for inspiration, lesson ideas, educational news, videos, and job postings.

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Is it possible to eliminate the bell curve in math class?

Imagine if someone at a dinner party casually announced, "I'm illiterate." It would never happen, of course; the shame would be too great. But it's not unusual to hear a successful adult say, "I can't do math." That's because we think of math ability as something we're born with, as if there's a "math gene" that you either inherit or you don't.

School experiences appear to bear this out. In every math class I've taken, there have been slow kids, average kids and whiz kids. It never occurred to me that this hierarchy might be avoidable. No doubt, math comes more easily to some people than to others. But the question is: Can we improve the methods we use to teach math in schools - so that everyone develops proficiency?

Looking at current math achievement levels in the United States, this goal might seem out of reach. But the experience of some educators in Canada and England, using a curriculum called Jump Math, suggests that we seriously underestimate the potential of most students and teachers.

Peter Bregg

"Almost every kid - and I mean virtually every kid - can learn math at a very high level, to the point where they could do university level math courses," explains John Mighton, the founder of Jump Math, a nonprofit organization whose curriculum is in use in classrooms serving 65,000 children from grades one through eight, and by 20,000 children at home. "If you ask why that's not happening, it's because very early in school many kids get the idea that they're not in the smart group, especially in math. We kind of force a choice on them: to decide that either they're dumb or math is dumb."

Children come into school with differences in background knowledge, confidence, ability to stay on task and, in the case of math, quickness. In school, those advantages can get multiplied rather than evened out. One reason, says Mighton, is that teaching methods are not aligned with what cognitive science tells us about the brain and how learning happens.

In particular, math teachers often fail to make sufficient allowances for the limitations of working memory and the fact that we all need extensive practice to gain mastery in just about anything. Children who struggle in math usually have difficulty remembering math facts, handling word problems and doing multi-step arithmetic ( pdf). Despite the widespread support for "problem-based" or "discovery-based" learning, studies indicate that current teaching approaches underestimate the amount of explicit guidance, "scaffolding" and practice children need to consolidate new concepts. Asking children to make their own discoveries before they solidify the basics is like asking them to compose songs on guitar before they can form a C chord.

Mighton, who is also an award-winning playwright and author of a fascinating book called "The Myth of Ability," developed Jump over more than a decade while working as a math tutor in Toronto, where he gained a reputation as a kind of math miracle worker. Many students were sent to him because they had severe learning disabilities (a number have gone on to do university-level math). Mighton found that to be effective he often had to break things down into minute steps and assess each student's understanding at each micro-level before moving on.

Take the example of positive and negative integers, which confuse many kids. Given a seemingly straightforward question like, "What is -7 + 5?", many will end up guessing. One way to break it down, explains Mighton, would be to say: "Imagine you're playing a game for money and you lost seven dollars and gained five. Don't give me a number. Just tell me: Is that a good day or a bad day?"

Courtesy of Mary Jane Moreau

Separating this step from the calculation makes it easier for kids to understand what the numbers mean. Teachers tell me that when they begin using Jump they are surprised to discover that what they were teaching as one step may contain as many as seven micro steps. Breaking things down this finely allows a teacher to identify the specific point at which a student may need help. "No step is too small to ignore," Mighton says. "Math is like a ladder. If you miss a step, sometimes you can't go on. And then you start losing your confidence and then the hierarchies develop. It's all interconnected."

Mighton saw that if he approached teaching this way, he could virtually guarantee that every student would experience success. In turn, the children's math anxiety diminished. As they grew more confident, they grew excited, and they began requesting harder challenges. "More than anything, kids love success," he says, "and they love getting to higher levels, like in a video game."

As the children experienced repeated success, it seemed to Mighton that their brains actually began to work more efficiently. Sometimes adding one more drop of knowledge led to a leap in understanding. One day, a child would be struggling; the next day she would solve a problem that was harder than anything she'd previously handled. Mighton saw that if you provided painstaking guidance, children would make their own discoveries. That's why he calls his approach "guided discovery."

Courtesy of Mary Jane Moreau

The foundation of the process is building confidence, which Mighton believes should be the first goal of a math teacher. Confidence begets attention, which begets rich learning. "I've never met a teacher who will tell you that a student doesn't need to be confident to excel in school," explains Mighton. "But I've never seen a math curriculum that follows the implications of that idea rigorously." Math is well-suited to build confidence. Teachers can reduce things to tiny steps, gauge the size of each step to the student and raise the bar incrementally.

When math is taught this way, surprising things happen.

Consider some of Jump's results. It's been used for four years in the public schools in Lambeth, one of the most economically depressed boroughs of London, England. Teachers placed into Jump the students who were struggling most in math. Among the 353 students who entered the program in fifth grade, only 12 percent began at grade level. Most were at least two grade levels behind and the vast majority were not expected to pass England's grade six (KS2) national tests. But 60 percent did.

In rural Ontario, Jump was recently evaluated in a randomized controlled study involving 29 teachers and about 300 fifth-grade students (controlled studies of math programs are rare). Researchers from Toronto's Hospital for Sick Children and the Ontario Institute for Studies in Education arranged for a control group of teachers to use their district's standard curriculum while another group used Jump. Each set of teachers was given two days of training relevant to the materials they would be using.

In five months, researchers found substantial differences in learning. The Jump group achieved more than double the academic growth in core mathematical competencies evaluated using a well known set of standardized tests. (The study has not yet been published.) "Kids have to make pretty substantial gains in order to see this kind of difference," explained Tracy Solomon, a developmental psychologist in the Research Institute at the Hospital for Sick Children who is the study's lead author. "It's impressive over a five-month period."

Solomon believes that the key to Jump's effectiveness is the way it "breaks math down to its component parts and builds it back up." And she notes that this "flies in the face of the way math is typically taught."

Interviewing teachers and principals, I have heard numerous stories of results like these. At times, they seem hard to reconcile with our assumptions about math. Isabel Grant, principal of the General Wolfe Elementary School, in Vancouver, British Columbia, has seen Jump produce impressive results in two schools where it has been used by a variety of teachers. Schools in British Columbia evaluate students based on whether they meet expectations for learning outcomes. "Teachers who used Jump were suddenly finding that they had all of their kids in the 'fully meeting expectations' category," Grant told me. "It was such a foreign experience. It doesn't typically happen when we're teaching science or language arts. And they were kind of at a loss. 'What do we do about this?'"

Another example is Mary Jane Moreau, who teaches at the Mabin School, an independent school in Toronto that does not screen students based on academic ability. Moreau, an experienced educator, dabbled with Jump for a year and started to see progress among her students, so she decided to immerse herself in the philosophy. "I was used to getting a bell curve in the past," she told me, "but what I started seeing was all the kids getting between 90 and 100 percent on tests, and within months, they were all getting between 95 and 100 percent."

She decided to see if the results would transfer to the standardized Test of Mathematical Abilities. Moreau teaches the same cohort of kids in fifth and sixth grades. Each September, for four years, the students wrote the test. From 2006 to 2007, the class percentile average jumped from 66 percent to 92 percent. From 2008 to 2009, with a new cohort, it increased from 54 percent to 98 percent.

Notably, the bell curve of the students' scores shifted to the right and narrowed - which is to say that the performance differences between the "slow" kids and the "whiz" kids began to fade away. Moreau encouraged her sixth-grade students to enroll in the Mathematica Pythagoras contest, which attracts only five percent of Canadian students, most of whom would be deemed "gifted" in math. All but one did. For each group, 14 out of 17 students beat the contest average.

Moreau is a dedicated teacher - and she has the benefit of small classes - but, even so, she hadn't seen results like this before. And it troubled her to think of students she had taught who didn't have the opportunity to learn math this way. "When I think about what we've been doing for years when we could have been doing something else," she told me, "I feel like I have to run so hard on this because I'm coming to the end of my career. But if I don't help to change attitudes, I'll feel like a criminal."

Jump is a modest outfit. Mighton has a staff of 10 to create materials and conduct teacher trainings. With decisions about math curriculum highly politicized, it's difficult for a small group to influence the debate. Big textbook companies and paid math consultants have a big say - and big investments - in what gets used.

It will take independent-minded educators to use Jump and see if its results can be replicated in more classrooms and schools. It's hard to imagine what society might look like if we could undermine the math hierarchies that get solidified in grade school. These patterns tend to play out across society, often in negative ways. Wasn't it the whiz kids who invented financial derivatives and subprime mortgages? And how many adults got themselves into hot water with their mortgages because, at bottom, they didn't really understand the risks?

Even deeper, for children, math looms large; there's something about doing well in math that makes kids feel they are smart in everything. In that sense, math can be a powerful tool to promote social justice. "When you have all the kids in a class succeeding in a subject, you see that they're competing against the problem, not one another," says Mighton. "It's like they're climbing a mountain together. You see a very healthy kind of competition. And it makes kids more generous to one another. Math can save us."

On Friday, I'll reply to comments, explain how Jump has helped one teacher to conquer her own math fears, and I'll get into some more details about how the program works - including the vital role of bonus questions.

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David Bornstein is the author of "How to Change the World," which has been published in 20 languages, and "The Price of a Dream: The Story of the Grameen Bank," and is co-author of "Social Entrepreneurship: What Everyone Needs to Know." He is the founder of dowser.org, a media site that reports on social innovation.

# data[].attributes.attachment_link.description
Is it possible to eliminate the bell curve in math class? Imagine if someone at a dinner party casually announced, "I'm illiterate." It would never happen, of course; the shame would be too great. But it's not unusual to hear a successful adult say, "I can't do math."

# data[].attributes.body
Who is affected?

Students appear to not have mastered math facts, making high order

math more difficult and demanding on them than it should be.

Who or what is suspected of causing the problem?

Teachers seem to focus on teaching from the textbook, which does not

place any value on mastery.

What kind of problem is it?

This is a skill problem that the students have.

What is the goal for improvement?

I want all of my students to have mastered the addition, subtraction, and

multiplication math facts from 0-10, so that they can use the skills that they

have mastered to apply to other math problems.

What do I propose to do about it?

To accomplish this...

# data[].attributes.body
Factors:

Parental support, grades, teacher support,

competition among peers, self-esteem.

# data[].attributes.body
How will I keep my variable constant?

# data[].attributes.body
How will I define growth and/or

decide how much knowledge my

students have obtained.

# data[].attributes.body
Possible issues that might arise...

# data[].attributes.body
Topic: Students struggle with

mastery of math facts. What

method of incorporating math

fact skills will be most beneficial

to use in the classroom?

# data[].attributes.body
How can this research project

be completely done within a timely

manner?