In Standardized Test Preparation and Tips for Success, you'll find:
Don't miss CT4ME's Test Prep Help this School Year!
See CT4ME's Common Core Resources for high school learners. Use these all year long to address each of the domains within the Common Core standards.
Visit our extensive collection of resources for Preparing Your Students for the Ohio Graduation Test in Mathematics. Educators will appreciate our Six Steps to Success and resources to help your students to review concepts and practice questions correlated to grades 8-10 mathematics benchmarks. Each set of strand resources for the state high school exam is accompanied by a downloadable test prep booklet. Students, regardless of the state in which you live, can benefit.
Hot news of January 2012! CT4ME.net was named in the Schools.com list of 20 indispensible online test prep resources.
No Child Left Behind legislation required states to measure students' progress in reading and mathematics annually in Grades 3-8 and at least once in Grades 10-12 by 2005-2006.
The Every Student Succeeds Act (ESSA) maintains the requirement that each state implement "a set of high quality student academic assessments in mathematics, reading or language arts, and science" (114th Congress, 2015, p. S.1177-24) among its provisions. Further, mathematics and reading or language arts assessments will be administered in each of grades 3 through 8, and at least once in grades 9 through 12 (p. S.1177-25).
Beginning in 2014-2015 school year, learners faced a new testing challenge in that their assessments of learning will be via online testing of the Common Core standards. Assessments are being developed by organizations such as PARCC and SBAC. Tests may take learners from 8-10 hours to complete (Doorey, 2014; Gewertz, 2013). As a result, educators have become concerned about the nature of these tests and what appears to be an excessive focus on test preparation. Online testing has posed additional concerns about required technology, sufficient bandwidth, computerized test security, learners' technology skills, and new forms of test anxiety.
Become Familiar with Updated Policies for Computerized Testing
Computerized testing raises new issues that require updating of test security laws and policies, as policies written for standardized testing administered via paper-and-pencil are no-longer sufficient. ACT has a highly relevant report in this regard: The End of Erasures: Updating Test Security Laws and Policies for Computerized Testing by Michelle Croft (2014).
Croft (2014) outlined many concerns, noting that computerized testing does not eliminate cheating and test piracy. Such practices just take on different forms. Unique risks include such things as "educators logging in to tests to view questions or change student responses, computer hacking; keystroke logging; printing, emailing, or storing test information in a computer outside the test delivery system. ... there is a greater risk of students accessing the Internet and other programs during testing" (p. 1). There's concern about students using their own devices for testing and who has administrative privileges. How should workstations be positioned and secured so that students can't see what's on the monitors of others.
Croft (2014) recommended that states update their state statutes and regulations to reflect the shift to computer-administered assessments, concentrate efforts on controlling test access, and ensure that there is a single test security section within the updated manual that contains answers for any question that a test administrator has about test security. For example, policies should consider how student login information is secured. There should be rules on how tests are reactivated if disrupted:
The rules should emphasize having more than one proctor aid in the reactivation, and most importantly, proctors should maintain a log of all reactivations to provide documentation in the event of an investigation. Likewise, the technology should be secure and the testing window should be as short as possible to reduce the likelihood that items are compromised. Finally, states should actively monitor test access issues through data reports to determine if there have been excessive logins or logins at times when testing should not occur (e.g., on the weekends), and have clear policies in place detailing how violations will be handled. (p. 4)
The test security section should also include an itemized list of what materials are secure (e.g., work folders, student authorization tickets with IDs and passwords, session rosters, scratch paper, reference sheets). "Information about who can access the test should be clearly articulated. In addition, there should be information on how to report test security concerns and possible violations, which can be applicable regardless of the testing format" (Croft, 2014, p. 4).
Thus, educators should become familiar with any new policies regarding computerized test administration, including what they, test proctors, and students may and may-not do.
Use Effective Test Prep Strategies with Learners
In A Case Study of Key Effective Practices in Ohio's Improved School Districts, Research Associates Aaron Kercheval and Sharon Newbill (2002) reported the key effective test preparation strategies included:
According to Douglas Reeves (2004), "Even if the state test is dominated by lower-level thinking skills and questions are posed in a multiple-choice format, the best preparation for such tests is not mindless testing drills, but extensive student writing, accompanied by thinking, analysis, and reasoning" (p. 92). Silver, Strong, and Perini (2007) found that student success on standardized tests, regardless of grade level or content area, hinges on 12 core skills relating to those ideas. They grouped those skills into four categories in what they call "Hidden Skills of Academic Literacy." They said, "If we expect students to perform well on state tests, we must teach them how to apply these skills without cutting into content." Unfortunately, the skills that follow have been "radically undertaught and rarely benchmarked" (Part One: Introduction section):
Emphasis on literacy was one key effective practice in Ohio's improved school districts (Kercheval & Newbill, 2002). In other words, good instruction is the best test preparation!
Ensure Learners Acquire Technology Skills
What has changed for Common Core assessments, however, in relation to that good instruction is that teachers need to ensure their learners also have the technology skills to perform well on tests administered online. Per Kristine Gullen (2014), "If we want an online assessment to capture a student's level of learning, rather than that student's ability to navigate technology, teachers must integrate these skills into their instruction, giving students practice before administering high-stakes exams on a computer" (p. 69). Hence, among the best ways to prepare learners for new assessments is to integrate test preparation into every day lessons using CCSS-type questions linked to the curriculum and to use technology for assessments and as a content learning tool. It's more effective than a two-week or more crash effort for test prep prior to the actual high-stakes exam (Miller, 2014).
Learners need practice with the new testing formats, and new types of questions. For example, multiple choice questions might have more than one answer. They need practice with the same and must be able to enter test responses via a keyboard, sometimes placing those in boxes on the screen. Fluid keyboarding skills will help minimize frustration when answering constructed-response questions. They also need good "mousing" or "touchscreen" skills to enter or remove responses via click or drag and drop into particular places on a screen. And they need to build endurance for working long periods of time at the computer via gradual focused exposure to the new testing scenario.
Gullen (2014) noted that learners also need skills highlighting text, drawing lines and creating graphs on a screen, operating an online calculator, and using a scroll bar. The need to use a scroll bar might be increased if learners also need to increase font size, an accommodation feature. Above all learners need "opportunities to build a level of comfort with the actual keyboards, screens, eternal mouse or touch pads, and so on that they'll use during the assessment" (p. 71). Piloting online assessments from PARCC or SBAC will help in this endeavor, as will debriefing learners following those about their difficulties and recommendations for additional skills they need to develop.
Teach to the Standards, Not to the Test
Teaching to the Test?--An Answer to Consider
Jeff Weinstock (2008) of T.H.E. Journal provided food for thought for critics of standardized testing. "When the system works the way it should, teaching to the test is a misnomer. It's not the test that teachers are teaching to, but the state learning standards embedded in the test. Has the student learned this, that, and the other?...Count me among those who think introducing some accountability into math instruction is an idea whose time has come. I can't suffer another generation of supermarket cashiers who become disoriented when I hand over $8.07 for a $7.82 bill" (p. 8).
Read Dr. Patricia Deubel's commentary, Accountability, Yes. Teaching to the Test, No featured April 10, 2008, in T.H.E. Journal.
Test Prep and Math Realities
Read Dr. Patricia Deubel's commentary, "Test Prep and Math Realities," featured September 27, 2007, in T.H.E. Journal.
We would hope that teachers use a broad range of curricular materials and activities that address standards--what we have identified as important for students to know and be able to do. Teaching to the test is not a new practice brought about by NCLB, nor will it be any different for preparing learners for testing of the Common Core standards. Teachers have been doing it for as long as standardized tests have been used to make important educational decisions.
Years ago, William Mehrens (1989) stated, "Although teaching to the test is not a new concern, today's greater emphasis on teacher accountability can make this practice more likely to occur. Depending on how it is done, teaching to the test can be either productive or counterproductive" (para. 2, 3). Those words are still true. He and his colleague Kaminski (1989, cited in Mehrens, 1989) suggested the following seven points on the continuum along which practices range from ethical to unethical, or legitimate to illegitimate.
Cross-over point depends on inferences you wish to draw from the test and lies between:
Mehrens (1989) indicated, "The inferences you typically wish to draw from test scores are general in nature and will be inaccurate if you limit instruction to the actual objectives sampled in the test or, worse yet, to the actual questions on the test" (Summary section).
Educators will observe, however, that current test prep efforts do include using questions from old tests, which state departments of education release. Technically, these are not parallel forms of the same test.
Plan for Good Instruction
So how does one plan for good instruction? For many teachers, instruction has come to mean addressing as many standards as possible. But, state exams do not test every benchmark annually. To assist teachers with providing good instruction leading to improved student outcomes, O'Shea (2005) suggested that teachers have copies of standards and frameworks for each subject they teach, and use them along with related state documents to plan lessons in regularly scheduled grade-level or subject matter team meetings. Instruction should not include "unchallenging student desk work, including word searches, sentence completion exercises, puzzles, and other forms of response sheets not linked to standards" (p. 13). In addition to identifying a topic and rationale, a truly standards based lesson would include:
Further, O'Shea (2005) indicated that districts should:
Joan Herman and Eva Baker (2005) said that there should be a strong predictive relationship between students' performance on benchmark tests and their performance on state assessments. They cautioned, however, that aligning benchmark tests too closely with a state's tests may accelerate curriculum narrowing. Tests should "focus on the big ideas of a content area" and be designed to "allow students to apply their knowledge and skills in a variety of contexts and formats" (p. 49). For mathematics test items, this might include giving short answers, using multiple choice with extended explanations for why an option was selected, and drawing pictures to demonstrate a concept. They provided the following six criteria to help educators make benchmark testing effective:
Many are concerned that standards-based instruction neglects the diverse learning needs of students. However, Carol Ann Tomlinson (2000) indicated, "There is no contradiction between effective standards-based instruction and differentiation" (i.e., attending to the diverse needs of learners). "Curriculum tells us what to teach: Differentiation tells us how." For any standard, "Differentiation suggests that you can challenge all learners by providing materials and tasks on the standard at varied levels of difficulty, with varying degrees of scaffolding, through multiple instructional groups, and with time variations. Further, differentiation suggests that teachers can craft lessons in ways that tap into multiple student interests to promote heightened learner interest in the standard. Teachers can encourage student success by varying ways in which students work: alone or collaboratively, in auditory or visual modes, or through practical or creative means" (p. 9).
Consider the Big Picture--Everyone Counts
A major survey study conducted by Williams, Kirst, Haertel, et al. (2010) in the 2008-2009 school year in 303 California middle schools echoes many of the recommendations noted above. The study illustrates a culture change that must take place within districts to focus on student outcomes and illustrates that test preparation and standards-based instruction is not solely the role of individual teachers. Student responsibility and parent involvement are key, as are the roles of district, state, and federal leadership. Although the study yielded practices that have a potential to positively affect achievement on standardized tests in math and English language arts (ELA) at the middle school level, the major findings are worthy of consideration for implementation at all K-12 levels for ELA and math. Readers are cautioned that this was correlational research, not experimental research. Hence, cause and effect conclusions between practices and outcomes can not be drawn. Illustrative major findings in the Narrative Summary include:
You can read a more complete review of this study at our Education Research page for Standards, Raising Achievement, Assessment, and How People Learn.
Become a Smart Learner--Raise your Skills! According to an Education Week Quality Counts (2010) report, many states provide educators with benchmark assessments or item banks linked to their state standards (p. 41). Learn more about standardized tests in your state, and the National Assessment of Educational Progress (NAEP) using resources in this section.
Are your students ready for the Common Core math exams? You can find out by using Benchmark Now!, a free online tool from Naiku for Grades 3-High School. It is designed to help identify students' knowledge and skills with professionally developed assessments using questions from Houghton Mifflin Harcourt’s Assess2Know® Benchmark Item Bank. Questions in the item bank are aligned to Common Core State Standards. There are beginning-of-year assessments help identify students’ readiness by domain and multiple interim assessments for each grade. Domain specific assessments provide teachers insight into student performance by standard.
Learn more about the Common Core Standards and Standards in Your State, as provided by the Common Core State Standards Initiative.
Concerned about CCSS Math Tests?
Read Dr. Patricia Deubel's commentary, Are We Ready for Testing under Common Core State Standards?, featured September 15, 2010, in T.H.E. Journal. Learn about the rise of online testing and concerns for educators who will be preparing students for new Common Core State Standards assessments.
Readers might be interested in CCSSI Mathematics, a blog that "takes an independent look at the Common Core State Standards Initiative." Among concerns are those on question designs and learner potential problems in using technology to answer them.
Join the national conversation on Common Core tests.
Testing Talk "provides a space for you to share your observations of standardized tests your students are taking this year. What works? What doesn't?" That test might be PARCC, SBAC, state tests by Pearson or McGraw-Hill, or the ACT. Parents, teachers, administrators, students, researchers, testing experts, teacher-educators, and others are invited to share their experiences.
PARCC and SBAC Consortia Assessments
PARCC (2013) Common Core Mathematics Test Specification Documents are intended to help teachers prepare their learners for the online assessments that begin in school year 2014-2015. The following are also helpful:
SBAC Practice and Training Tests include "sets of assessment questions aligned to the Common Core for grades 3–8 and 11 in both English language arts/literacy and mathematics." (para. 1). SBAC provided an important limitation of these tests.
SBAC Digital Library "is an online collection of instructional and professional learning resources contributed by educators for educators. These resources are aligned with the intent of the Common Core State Standards and will help educators implement the formative assessment process to improve teaching and learning." The library is "available to educators in subscribing member states and territories." (K-12 Education, Teachers section)
Consortia Developing Alternative CCSS Assessments
For English Language Learners:
For Learners with Cognitive Disabilities:
For the latest developments on the six assessment consortia, see the Educational Testing Service: K-12 Center.
Brainchild Online Assessment: Subscription based by schools or individuals. But demo questions are available online for your state. Lessons include multimedia instruction, study mode with immediate feedback, test mode with review of mistakes, self-directed student learning plan.
National Assessment of Educational Progress has released numerous questions from past NAEP assessments, along with data about student performance on specific questions. An overview of NAEP and major findings from past assessments are included. NAEP reports, the tools featured in Explore NAEP Questions "can be used to supplement classroom instruction, provide additional insight into the content of the assessment, and show what students nationally or in your state or district know and can do." Readers should consider, however, that the NAEP is not considered a high stakes test. Gerald Bracey (2009) reported on characteristics that make it a poor accountability tool. For example, no student ever takes the entire test, nor do districts, schools, or individual students find out how they performed. Thus, students might not take NAEP as seriously as they would the ACT or SAT or their state high stakes tests (p. 33).
Programme for International Student Assessment (PISA) from the Organisation for Economic Co-Operation and Development surveys "15-year-olds in the principal industrialised countries. Every three years, it assesses how far students near the end of compulsory education have acquired some of the knowledge and skills essential for full participation in society." The U.S. is among participating countries. Sample questions are available. Gerald Bracey (2009) noted that PISA is not a high stakes test and points out flaws in using results as a measure of the quality of U.S. schools. Chief among those is comparing results of a nation with a diverse population of over 300-million people to results of small "homogeneous city-states like Hong Kong and Singapore." Formal schooling differs among nations as to when students start school, policies differ in relation to repeating grades, and schools might not be serving the entire population, particularly those from low-income families. The design of test items also fall into question when one considers difficulty in translating questions into several languages, and keeping those questions free of culture bias (p. 34).
Alabama: Alabama Department of Education: Student Assessment includes information about its state tests.
Alaska: Item samplers/practice tests for math, science, reading, and writing are available for grades 3-10 and the high school exam.
Arkansas: Arkansas Department of Education Testing and Student Assessment (ArkansasEd.gov) contains booklets with released test items.
Colorado: Colorado Student Assessment Program (CSAP) Released Items.
Connecticut: Connecticut State Department of Education: Bureau of Student Assessment. Among resources are released test questions for the Connecticut Academic Performance Test, and a set of skills checklists.
Florida: Florida K-12 Student Assessment at the Florida Department of Education. Get released test questions and Florida practice tests for math grades 3-8, algebra 1, geometry, algebra 2. Note: the Florida Department of Education, the National Science Foundation, and FCR-STEM have provided funding for the CPALMS (Collaborate, Plan, Align, Learn, Motivate, Share) initiative whose mission is to provide instructional resources vetted by peers and experts and professional development for implementation of the standards. You'll find curriculum, activity, and general resources and an extensive collection of lesson plans, and more.
Georgia: Georgia Department of Education Testing/Assessment Also see the Mathematics section at GeorgiaStandards.org for additional resources.
Idaho: Idaho Standards Achievement Tests (ISAT) includes practice and training tests. Math sample items are listed for grades 3-8 and high school in their portal: http://idaho.portal.airast.org/training-tests/
Illinois: Illinois State Board of Education: Assessment includes assessment resources.
Kansas: Kansas Department of Education Note: The Career, Standards, and Assessment Services section also has an Assessment Literacy Project available online with 21 modules appropriate for all educators. W. James Popham provided the introductions to these modules.
Kentucky: Kentucky Department of Education released test items for end of course and K-PREP.
Louisiana: The Louisiana Department of Education provides the Louisiana state content standards, lesson plans, web resources, and sample assessment items aligned to Louisiana content standards. See sections for assessment and academics.
Maine: Maine's Comprehensive Assessment System. Maine participates in the New England Common Assessment Program (NECAP) and there are released test items to view.
Maryland: Maryland State Department of Education Instruction and Assessment includes sample items for PARCC tests, high school assessments, and the Maryland School Assessment (MSA).
Michigan: Michigan Department of Education Student Assessment Program The Michigan Student Test of Educational Progress (M-STEP) is the new state assessment system beginning in Spring 2015. Math tests will be administered in grades 3-8 and 11. Read the M-STEP Mathematics Summary and get sample questions at the website.
Minnesota: Minnesota Department of Education has testing resources and item samplers.
Mississippi: Mississippi Department of Education Office of Student Assessment includes practice test items for grades 3-8 and high school (e.g., algebra 1).
Nebraska: Nebraska Statewide Assessment (NeSA) practice tests for math include paper/pencil forms for grades 3-8, 11.
Nevada: Nevada Department of Education Standards and Assessments; Note: The Southern Nevada Regional Professional Development Program includes math resources for elementary, middle, and high school. These latter contain content units with notes (many indicating alignments with Common Core math standards), worksheets, quizzes, practice tests.
New Hampshire: New Hampshire Department of Education Math Curriculum, practice test questions and information, and NECAP released test questions for math. Note: New Hampshire Department of Education, Rhode Island Department of Education, and Vermont Department of Education have developed a common set of Grade-Level Expectations, known as the New England Common Assessment Program Grade-Level Expectations (NECAP GLEs).
New Jersey: New Jersey Department of Education: Assessment. Released questions from the testing program are found under elementary, middle, and high school.
New Mexico: New Mexico Standards Based Assessments includes released items with student work, which are representative of the kinds of questions learners in grades 3-8 will encounter.
North Carolina: North Carolina Public Schools released test forms for grades 3-8 and high school.
North Dakota: North Dakota Department of Public Instruction Standards, including standards-based assessments.
For this School Year! Target your test prep with CT4ME resources. CT4ME developed Preparing Your Students for the Ohio Graduation Test in Mathematics. Help your students to review concepts and practice questions correlated to grades 8-10 mathematics benchmarks. These materials are also relevant for students in other states.
Oklahoma: Oklahoma Department of Education: Office of Assessments Assessment materials for grades 3-8 and "end of instruction" secondary tests include a set of representative released items. High school math includes algebra l, algebra II, and geometry items.
Oregon: Oregon Department of Education Common Core Assessment includes sample items and practice for students.
Pennsylvania: Pensylvania Department of Education Standards Aligned System (SAS) contains an Assessment section with options such as: Project-based assessment, the state's Keystone Exams with high school sample questions in algebra 1, algebra 2, and geometry, an Assessment Creator, Reference Materials (e.g., formative assessment), and more.
South Dakota: South Dakota Department of Education: Assessment has information on its state testing program.
Tennessee: Tennessee Comprehensive Assessment Program (TCAP) sampler items and practice tests for grades 3-8 and high school.
Vermont: Vermont Department of Education: New England Common Assessments Program (NECAP) with released test items, practice tests, and support resources. Note: Rhode Island and New Hampshire also participate in the NECAP.
Virginia: Virginia State Standards Practice tests in mathematics, science, and technology. Mathematics tests are available for grades 2-8 and algebra I, algebra II, and geometry. Practice concepts in multiple choice form with answers. Exams can also be customized by number of problems presented and strands.
West Virginia: Teach 21 and Learn 21, the latter of which is an interactive website with educational games, online video instruction, virtual field trips and other internet resources for preK-2, 3-5, 6-8, and 9-12. Math is a focus area, and there are other subjects.
Wisconsin: Wisconsin Department of Public Instruction assessment resources include sample items from various tests given by the state, including for the Wisconsin Forward Exam.
According to the National Mathematics Advisory Panel (2008) in its Foundations for Success:
"Anxiety about mathematics performance is related to low mathematics grades, failure to enroll in advanced mathematics courses, and poor scores on standardized tests of mathematics achievement. It also may be related to failure to graduate from high school. At present, however, little is known about its onset or the factors responsible for it. Potential risk factors for mathematics anxiety include low mathematics aptitude, low working memory capacity, vulnerability to public embarrassment, and negative teacher and parent attitudes." (p. 31)
Most likely, everyone has experienced math anxiety at one time or another. It is an emotional response that often comes from negative experiences working with teachers, tutors, classmates, or family members. Symptoms include panic (feeling helpless about an ability to do better and putting pressure on yourself, which affects your ability to concentrate), paranoia (feeling that everyone but you knows the answer), passivity (feeling that regardless of what action you might take, you were just not born with math ability; hence you do nothing to overcome the problem), no confidence (you continually question yourself and approach math by memorizing rules and procedures, rather than through understanding concepts). Identifying the source of your problem may be a first step in overcoming it.
Do you have math anxiety? Take a self-test from Dr. Ellen Freeman of Mathpower.com.
Anxiety's Role in Math Performance
Does math anxiety lead to poor performance or does poor math performance lead to math anxiety? Research is mixed as to which comes first, the emotion or poor performance. Carey, Hill, Devine, and Szücs (2016) discussed theories:
In addition to outward behavior manifestations of math anxiety, there is biological evidence of its existence. According to Dr. Venod Menon of the Stanford University School of Medicine, "Math anxiety is an under-studied phenomenon, which still lacks formally established diagnostic criteria ... it is possible for someone to be good at math, but still suffer from math anxiety" (Digitale, 2012, para. 5). Menon and his team of researchers used brain scans in their study of 46 second- and third-grade students with low and high math anxiety, and found that "Children with high math anxiety were less accurate and significantly slower at solving math problems than children with low math anxiety. The results suggest[ed] that, in math anxiety, math-specific fear interferes with the brain’s information-processing capacity and its ability to reason through a math problem" (para. 12-13). "His team’s observations show[ed] that math anxiety is neurobiologically similar to other kinds of anxiety or phobias" (para. 7). Results may lead to new strategies for treatment of it, as in ways suggested for treatment of other anxieties and phobias.
What teaching strategies help minimize math anxiety?
Findings from research are particularly relevant for pedagogical methods that have been successful with learners, particularly in terms of reducing math anxiety. However, the potential for a bidirectional relationship between math anxiety and math performance per Carey et al. (2016, Reciprocal Theory section) would suggest that any interventions should address both math anxiety and math performance.
Blazer (2011) reviewed strategies that teachers can use to help reduce students' anxiety, thus leading to better achievement for all:
Researchers at Stanford University School of Medicine (2015) found that one-on-one tutoring can be used to relieve math anxiety. Grade 3 learners with high math anxiety participated in the study. The tutoring "fixed abnormal responses in the brain's fear circuits," as noted by functional MRI brain scans.
Can learners reduce negative affects of anxiety?
Write about your anxiety.
Educators must take math anxiety seriously. One way to potentially help learners reduce the negative affects of their math anxiety is writing about it prior to taking a math test, as suggested in Neuroscience and Education: A Review of Educational Interventions and Approaches by Neuroscience (Howard-Jones, 2014) commissioned by the Education Endowment Foundation in London. Howard-Jones noted:
An intervention focused on controlling negative emotional response has reported improved achievement. Writing about anxieties may be one way to rehearse such control. Following on from laboratory-based studies, a school-based intervention was carried out among students aged 14-15 years (N=106). These students first rated their own anxiety and were then randomly allocated into two groups. For 10 minutes, immediately before a maths test, one group wrote about mathematics-related anxieties and the other about a topic not in the test. Amongst students who had had rated themselves as more anxious, those who wrote about their anxieties significantly outperformed maths anxious students in the other group, performing similarly to less maths anxious students. (p. 16)
This aforementioned study was conducted by Gerardo Ramirez and Sian L. Beilock in 2011. Read the details in Writing About Testing Worries Boosts Exam Performance in the Classroom.
Learn How to Learn Math.
Another potential strategy to minimize anxiety is for students to learn how to learn math. Dr. Jo Boaler of Stanford University has a short, free course at youcubed.org called "How to Learn Math" for any student in all levels of mathematics. There are six short lessons about 10 to 20 minutes each. You'll get some key information on the brain and learning, and effective strategies for learning math. Concepts include overcoming myths about math, math and mindset, mistakes and speed; number flexibility, math reasoning, and connections; number patterns and representations; and math in life, nature, and work. The course also features videos of math in action.
Learn more about what math anxiety is, how to take possession of your math anxiety, and get some strategies for how to study math and take tests.
Read Coping with Math Anxiety at Platonic Realms.
Deconstructing Maths Anxiety: Helping Students to Develop a Positive Attitude Towards Learning Maths by Dr. Sarah Buckley (2013) of the Australian Council for Educational Research.
Cuesta College provides short and long term relaxation techniques and how to overcome negative self-talk in How to Reduce Text Anxiety, referencing Winning at Math, a 1997 work by Paul D. Nolting, Ph.D.
HOT! Strategies for Reducing Math Anxiety, Volume 1102, September 2011, is an Information Capsule from the Research Services of Miami-Dade County Public Schools. It contains specific strategies for teachers, parents, and students to use to reduce math anxiety. The full list of Information Capsules are available at http://drs.dadeschools.net/InformationCapsules/IC.asp Per its description: "An Information Capsule is a review of recent publications in education. These reviews include summaries of educational research articles, reports, books - publications of any kind. The publications are relevant to educational administrators, teachers, and staff for the purpose of policy and management decision-making. These documents are considered condensed versions of what is commonly thought of as literature reviews."
HOT! Quick tips for standardized test preparation: Read Duke and Ritchhart's article No Pain, High Gain at Scholastic. They discuss strategies for reading comprehension, mathematics, reducing test-taking stress, and teaching format fundamentals. In mathematics, for example:
ACT provides online test prep for this exam and tips for success.
HOT! Formative assessments for standardized test preparation: ASSISTments is a free web-based tutoring program for grades 3 to high school, principally with content for mathematics aligned to the Common Core standards. The program was developed by Professor Neil Heffernan at Worcester Polytechnic Institute (WPI) in Massachusetts and numerous colleagues (e.g., at Carnegie Mellon and WPI students, faculty and staff as well as many cooperating teachers) with funding and support from a variety of sources, including grants from the US Department of Education and the National Science Foundation. The word “ASSISTments” blends tutoring assistance with assessment reporting to teachers. This gives teachers fine grained reporting on skills that the system tracks. The developers identified skills using standards from many states, and their observations of many state assessments. As a formative assessment tool, math teachers assign problem sets to their students to do on the computer, and students are tutored on the items they get wrong. Teachers log on to the system and study detailed reports about their students’ difficulties and strengths. Teachers can use content developed at WPI or write their own content. Numerous schools in the US are taking advantage of ASSISTments for their students.
Adapster SAT Math is an iPhone or iPod app available for a small price at iTunes. After taking a diagnostic test, it differentiates and individualizes learning via a customized study plan based on the results. It has over 1000 questions and 150 lessons covering major math strands. See a review of this product.
Chegg.com provides ACT and SAT test prep, including tips for success and a free online practice test.
CollegeAtlas.org: Study Skills Guides for College Students. While meant for college, this site has tips beneficial for all students, regardless of level. You'll find sections devoted to general study skills, reading and writing, test taking and preparation for a variety of test types, time management, memory techniques, and subject specific study skills that also include for math.
College Board offers test preparation materials, tips for success, and other information related to its tests: SAT, PSAT/NMSQT, the Advanced Placement program (AP Central), and College Level Examination Program (CLEP). Visit the SAT Resource Center for Educators. Note: Effective Spring 2016, the College Board will implement a revised SAT. The College Board and Khan Academy have a SAT Practice site designed to help learners prepare for the revised test. There are Key Changes to the SAT Redesign. Major math changes include the following:
"The exam will focus in depth on three essential areas of math: Problem Solving and Data Analysis, the Heart of Algebra, and Passport to Advanced Math. Problem Solving and Data Analysis is about being quantitatively literate. It includes using ratios, percentages, and proportional reasoning to solve problems in science, social science, and career contexts. The Heart of Algebra focuses on the mastery of linear equations and systems, which helps students develop key powers of abstraction. Passport to Advanced Math focuses on the student’s familiarity with more complex equations and the manipulation they require." (About section: Key Content Changes)
College Entrance Exam Math Prep: "EduCAD Learning Solutions has taken 50 of the most complex type of math problems that a student would see when taking the SAT or ACT and put them into a free application that can be used by anyone preparing for these college entrance exams. ... This is not a drill system ... the system presents problems in a way that allows you to work through them with help offered when you need it." Steps in solutions are provided.
Cuesta College: Math Study Skills includes multiple pages of academic support devoted to math study skills and test taking skills, referenced from Winning at Math, a 1997 work by Paul D. Nolting, Ph.D. Of particular value are the 10 steps to better test-taking.
DOME SAT Review is a free comprehensive SAT Test Prep program online with a catalog of material including over 100 study guides, a database of 5,000+ SAT vocabulary building flashcards, practice quizzes, and simulated practice tests.
Family Education Network: Standardized Tests: Preparation and Advice. See some sample questions by grade level (elementary, middle, high school), and get more tips for success. A section for SAT and ACT test advice and practice questions is included.
Formative Assessment Item Bank was originally developed by the Educational Testing Service, acquired by Northwest Evaluation Association, and is now offered by Certica Solutions, which acquired the item bank from NWEA in February 2015. It includes assessment coverage for K-12 Math, English Language Arts, Science, and Social Studies. Assessments are aligned to state, Next Generation Science Standards, and Common Core standards. Item types include multiple-choice, constructed response with rubrics, writing prompts with rubrics, and technology-enhanced items.
Gojimo is a free app for standardized test prep, including the SAT, ACT, and AP exams (e.g., AP Calculus). It's available in iTunes (designed for iPad and iPhone) and on Google Play. There are also test prep options for those in the UK.
HSTutorials.net has animated and step-by-step audio-visual tutorials in pre-algebra, algebra, and geometry and tutorials and resources related to test prep for California's High School Exit Exam for math.
HOT for CCSS: Illustrative Mathematics Project is a work in progress to produce illustrative tasks that students would be expected to do related to each of the Common Core State Standards for Mathematics. Sections address illustrations for the K-8 and High School standards. The project is an initiative of the Institute for Mathematics & Education and the Bill & Melinda Gates Foundation.
Internet4Classrooms: Access activities on specific concepts within mathematics strands for grades 1-8 and an extensive list of standardized testing practice sites.
Intervention Central provides intervention ideas in the areas of general academic strategies, reading, writing, math, behavior modification, studying and organization, classroom management, and making rewards work. This site is brought to you by J. Wright, a school psychologist in Syracuse, New York.
IXL Math from IXL Learning is a math practice site, which has problem sets for preK-8, algebra 1, algebra 2, geometry and precalculus. The site provides a colorful, engaging environment for mastering skills. Guests can access "20 problems per day" for free with feedback on answers to help with understanding; however, the service is subscription based. Full benefits (e.g., student progress tracking and reports; and an awards system for learners who reach their goals) are gained with membership. You can identify specific activities for learners corresponding to state standards.
Jefferson Lab (VA), although primarily for science education, has some good puzzles and games suitable for use with elementary students to help them master basic math facts using addition, subtraction, multiplication, and division; also practice use of < = >, place value, and coordinates. Speed to complete exercises in noted as a motivation element.
Khan Academy: SAT Test Prep: The College Board has teamed up with Khan Academy for free SAT Test Prep materials. Students will find hundreds of questions and a set of videos with step-by-step solutions to help prepare for this exam.
Kidtest.com helps students from kindergarten through college to do better on achievement tests. The site contains several practice U.S. state and Canadian province achievement tests online and features near immediate grading and feedback reports. An online educational supplies store and online flashcards are offered.
Kids Place Mathematics from Houghton Mifflin Company contains online quizzes and tests and brainteasers correlated to their mathematics textbooks for grades 1-6, and additional test taking skills.
MathDrills by Elias Saab of the University of Missouri will help students to prepare for Mathcounts, SAT and ACT math problems. In addition, the basic skills sections can be used by students in upper elementary through high school settings. Answers and hints are provided. Sections include problems on distance, speed, and time; problems on job completion, roots of polynomials, factoring polynomials, percentage word problems, arithmetic and fraction attack (+, -, x, /), bases, linear equation drills, prime factorization, and LCM and GCD. Elias Saab also maintains the Online Test Page.
HOT for CCSS: Mathematics Common Core Toolbox has sample task items for elementary, middle, and high school from the PARCC Prototyping Project, posted by The Charles A. Dana Center at the University of Texas at Austin.
Math 10 Pure: Lance Burns at the Argyll Center in Edmonton, Canada, has developed materials for this course. He includes notes, video explanations, online interactive quizzes with explanations for answers to problems, and unit exams for the following topics: polynomials, rational expressions, relations and functions, real numbers, numbers patterns, coordinate geometry, measurement, and statistics. This site provides great reinforcement and review on many of the topics included on U.S. high school assessment exams.
McGraw-Hill Education Online Learning Center: Glencoe Mathematics contains self-check quizzes, chapter tests, standardized test prep questions, and vocabulary questions. Multiple-choice is included. Select your state, then textbook.
Number2.com has free online test prep for the SAT, ACT, and GRE exams.
HOT for CCSS: orgLib.com includes test-prep resources aimed to prepare K-12 students for SBAC and PARCC computer-based tests. You'll find technology-enhanced items, activities and worksheets that are aligned with common core state standards. You can search for assessments by standard, questions by topic, and worksheets by standard. There is also an assessment management system.
HOT! Open-Ended Math Problems posted at the Franklin Institute web site is specifically designed for middle school students to prepare for solving open-ended math problems on standardized tests. Although these are geared to Philadelphia (PA) standards, all middle school learners and even some high school learners will benefit. There are problem sets for each month September through April with three levels of difficulty. Problems are included for each major math strand: number theory, measurement, geometry; patterns, functions, algebra; and data, statistics, and probability. Possible answers and rubrics for assessment are also provided.
Saxon Publisher's Online Activities include over 125 practice activities to help students master content presented in their K-12 math texts. As activities are clearly titled, these will benefit learners regardless of text used.
Shmoop offers fee-based test prep for multiple tests (e.g., SAT, ACT, AP exams, Common Core). Shmoop Common Core test prep in math include PARCC testing for Algebra 1, Algebra 2, and Geometry; and SBAC for grade 8 and grade 11. Some learning resources are free, including study guides and videos for math--well worth examining.
SparkNotes: Math Study Guides include review explanations and problem sets for pre-algebra, algebra 1 and 2; geometry 1,2, and 3; trigonometry, pre-calculus, and advanced placement calculus levels AB, BC 1, and BC2. There's also test prep for SAT, ACT, GRE, and AP courses. Other subjects are also included at this site.
Study.com has a series of video lessons to help prepare learners for standardized tests. Among those are GED Math (93 video lessons on algebra, geometry, and arithmetic); AP Calculus Exam Prep (125 video lessons, a full course); PSAT: Practice & Study Guide (113 video lessons, including reading and math); SAT: Practice & Study Guide (110 video lessons, including reading and math); and ACT: Practice & Study Guide (176 video lessons on multiple subjects, including math).
Study Guides and Strategies contains several sections: study skills, preparing for tests, taking tests; improving research, project management, reading, writing, science, and math skills.
Study Island is a commercial product for standardized test prep in your state for elementary, middle, and high school grade levels and exit exams or end of course exams--whatever your state requires. The developers link their multiple choice questions to specific state standards. The program is web-based offering diagnostics and instruction and will generate various reports to help monitor mastery. Study Island also has a Common Core Benchmarking Program to help identify areas of student proficiency in relation to the Common Core State Standards. Study Island is a product of Edmentum, which stated "Students can work through questions using a standard test format, an interactive game format, printable worksheets, or a classroom response system." Pricing is available for the home, and schools/districts.
Test Preparation Study Center has a blog with tips and information for students preparing for exams. You'll also find help for specific exams, including state high school tests and college entrance and teacher certification. Among popular category pages are:
That Quiz is a real find. K-12 students can select practice tests (customized for their needs) with varying degrees of difficulty using integers, fractions, concepts (time, money, measurement, place value, graphs), geometry, algebra, calculus, probability, and more. Some are interactive and offer manipulatives (e.g., ruler, protractor). Select to view in Spanish, if needed.
Trends in International Mathematics and Science Study Test your mathematics and science knowledge by completing TIMSS items in the Dare to Compare challenge! TIMSS provides reliable and timely data on the mathematics and science achievement of U.S. students compared to that of students in other countries. See how well your students stack up. Answers are provided as feedback.
University of Houston System has two online test prep courses for high school learners: Preparing for the AP Calculus AB Exam and Preparing for the AP Statistics Exam. Courses are provided for free and are offered via Coursera. Each runs for six weeks, and includes a practice test at the end of the course.
USATestprep is a subscription based online product to help prepare high school, middle school, and elementary students for standardized testing. Materials include diagnostic assessments, performance tracking, practice questions (e.g., multiple-choice, free response), games, interactive skill work, performance tasks, video content, instant feedback, and more. Free trials are available to qualified educators. Review products are aligned to individual state standards, including the Common Core (Take A Tour section, Engaging Content).
Varsity Tutors: Practice Tests is a free section of the Varsity Tutors website where you will find practice test questions in multiple choice format and flashcards in multiple subject areas. When viewing math tests, for example, learners can select the concept to practice. Explanations for answers are included. Among K-12 math categories are basic arithmetic, basic geometry, algebra 1, algebra 2, trigonometry, precalculus, calculus (including AP), and math related to the ACT, GED, CLEP, GMAT, GRE, HSPT, ISEE, and SAT exams.
Know the Purpose of the Test You Take!
According to W. James Popham (2016), we need to acknowledge that "validity depends on the purpose for which a test is to be used" (p. 46). There are three primary purposes of tests:
There are two phrases describing tests of student achievement that are discussed in the literature: norm-referenced and criterion-referenced. Mark O'Shea (2005) provided a difference between the two:
Although O'Shea (2005) noted two kinds of standardized tests, Popham (2014) indicated: "Although test developers may build tests they believe will provide accurate norm-referenced or criterion-referenced inferences, a test itself should never be characterized as norm-referenced or criterion referenced" (p. 64). It's a common misconception. "What's criterion referenced or norm-referenced is the inference about, or the interpretation of a test taker's score" (p. 64). This clarification is important if one is using precise language. To emphasize, "it's score-based inferences--not tests--that are criterion-referenced or norm-referenced" (p. 64). Thus, educators should know how test results will be interpreted. According to Popham, "To support actionable instructional decisions about how best to teach students, norm referenced inferences simply don't cut it" (p. 64).
Per Popham (2016), "the primary purpose of a particular educational test ... should dominate the decision making of those who are building the test as well as those who are evaluating it. Currently, emphasis on purpose is absent from U.S. educational testing" (p. 49).
Become Familiar with Standardized Testing Terms
The following will help you to better understand terms associated with standardized testing: Glossary of Standardized Testing Terms from the Educational Testing Service.
Understand Test Accommodations for Students with Special Needs
Students with special needs such as those with disabilities, limited English language and English language learners also are subject to taking large-scale assessments, including standardized tests. CTB/McGraw-Hill (2005) developed Guidelines for Inclusive Test Administration to help educators use appropriate test accommodations and then make valid and useful interpretations for both criterion- and norm-referenced test scores. Guidelines fall within three categories:
Category 1. "Category 1 accommodations are not expected to influence student performance in a way that alters the standard interpretation of either criterion- or norm-referenced test scores. Individual student scores obtained using Category 1 accommodations should be interpreted in the same way as the scores of other students who take the test under default conditions. These students’ scores should be included in summaries of results without notation of accommodation(s)" (p. 8). Examples: Students take the test alone or in a study carrel, or have directions read aloud or recorded. ELL might need bilingual directions. Some students might need to give responses to a scribe or use sign language.
Category 2. "Category 2 accommodations may have an effect on student performance that should be considered when interpreting individual criterion- and norm-referenced test scores" (p. 9). Examples: Students are given extra time to complete a timed test. ELL are given audiotaped test items provided in native language version or a side-by-side bilingual test or translated version provided for content other than Reading and Writing.
Category 3. "Category 3 accommodations are likely to change what is being measured and have an effect that alters the interpretation of individual criterion- and norm-referenced scores. This occurs when the accommodation is strongly related to the knowledge, skill, or ability being measured (e.g., the use of a Braille test where not all items in the non-Braille version are administered in Braille)" (p. 9). Example: Students are permitted to use calculators or tables on a math computation test when the intention is to measure computation skills without calculator use.
Have you made appropriate accommodations for students with disabilities?
The National Center for Educational Outcomes provides a list of State Websites for Accommodations Information.
The Partnership for Assessment of Readiness for College and Careers (PARCC) includes a series of common policies related to assessment administration, scoring and the reporting of Common Core test results. There is a manual that addresses accessibility and test accommodations for learners with disabilities and English language learners.
The Smarter Balanced Assessment Consortium (SBAC) assessments will include summative assessments for accountability purposes and optional interim assessments for instructional use and will use computer adaptive testing to the greatest extent possible. Assessments will go beyond multiple-choice questions to include extended response and technology enhanced items, as well as performance tasks that allow students to demonstrate critical-thinking and problem-solving skills.
Read the SBAC Usability, Accessibility, and Accommodations Guidelines that outline the kinds of testing supports and tools that will be made available to all students, and particularly those with disabilities and English-language learners for the Common Core assessments.
The U.S. Department of Education provides SES information among its SES resources. Studies and reports on SES are also among resources. Contact your state for a list of providers in your area.
The Every Student Succeeds Act of 2015 does not mandate tutoring. Within Section 1004: Direct Student Services, a local education agency applying for funding "may include high quality academic tutoring" among components for a "personalized learning approach" (114th Congress, 2015, p. S.1177-18). In determining who can provide "high quality academic tutoring" the local education agency may select from "a variety of providers of such tutoring that are selected and approved by the State and appear on the State’s list of such providers" (pp. S. 1177-18 - S. 1177-19).
See NCLB Supplemental Educational Services Non-Regulatory Guidance (Jan 14, 2009) at http://www.ed.gov/policy/elsec/guid/suppsvcsguid.doc. This 55-page document from the U.S. Department of Education provided information about state and local education association responsibilities, monitoring requirements and services, arranging for such services, the role of parents, provider responsibilities and funding. Answers to frequently asked questions on school choice and SES are at http://www.ed.gov/parents/schools/choice/choice.html.
Just as in the classroom, tutors need to be qualified. They need subject-matter expertise. Certification and prior teaching experience is a plus. A tutor needs to know if the student has a learning disability, and if so, the tutor should have skills in working with the specific disability. If not, then the tutor and/or program might not be appropriate for that student. Edward Gordon (2006) provided the following suggestions on what to look for in a good tutoring program.
Gordon and his colleagues Ronald Morgan, Judith Ponticell, and Charles O'Malley (2004) provided the same and additional advice in Tutoring Solutions for No Child Left Behind: Research, Practice, and Policy Implications.
Note: A teacher (shorelineschools.org) recommended TeachingTextbooks.com as a source for texts (grades 5-7, pre-algebra, algebra 1 and 2, geometry, and precalculus) that have the potential to minimize a need for personal tutors (personal communication, July 31, 2008). According to the developers, "Using a Teaching Textbook is like having a friendly tutor available at the push of a button, but for only a fraction of the cost." Texts are designed to be used by independent learners and home-schooled students. They provide maximum explanation and are accompanied by CDs with "audiovisual step-by-step explanations for every single one of the almost 3,500 problems in the book" (FAQ section).
114th Congress of the United States. (2015). Every Student Succeeds Act. Retrieved from http://www.ed.gov/esea
Blazer, C. (2011). Strategies for reducing math anxiety. Information Capsule IC1102. Miami, FL: Miami Dade County Public Schools, Research Services, Office of Assessment, Research, and Data Analysis. Retrieved from http://drs.dadeschools.net/InformationCapsules/IC1102.pdf
Bracey, G. (2009). The big tests: What ends do they serve? Educational Leadership, 67(3), 32-37.
Carey, E., Hill, F., Devine, A., & Szücs, D.(2016, January) The chicken or the egg? The direction of the relationship between mathematics anxiety and mathematics performance. Frontiers in Psychology, 6, article 1987. doi: 10.3389/fpsyg.2015.01987. Retrieved from http://journal.frontiersin.org/article/10.3389/fpsyg.2015.01987/full
Croft, M. (2014, October). The end of erasures: Updating test security laws and policies for computerized testing. Retrieved from http://www.act.org/research/policymakers/reports/endoferasures.html
CTB/McGraw-Hill. (2005). Guidelines for inclusive test administration. Monterey, CA: Author. Retrieved from http://www.ctb.com/ctb.com/control/researchArticleMainAction?articleId=475&p=ctbResearch
Digitale, E. (2012, March 21). Imaging study reveals differences in brain function for children with math anxiety. Stanford, CA: Stanford University School of Medicine. Retrieved from https://med.stanford.edu/news/all-news/2012/03/imaging-study-reveals-differences-in-brain-function-for-children-with-math-anxiety.html
Doorey, N. (2014). The common core assessments: What you need to know. Educational Leadership, 71(6), 57-60.
Education Week. (2010). Standards, Assessments, Accountability, pp. 39-41. In Quality Counts 2010. Retrieved from http://www.edweek.org/media/ew/qc/2010/17sos.h29.saa.pdf
Gewertz, C. (2013, March 5). Assessment consortium releases testing time estimates [Web log post]. Retrieved from http://blogs.edweek.org/edweek/curriculum/2013/03/assessment_consortium_releases_2.html
Gordon, E. (2006, November 29). America needs to wise up about need for quality tutoring. Chicago Sun Times.
Gordon, E., Morgan, R., Ponticell, J., & O'Malley, C. (2004, March). Tutoring solutions for No Child Left Behind: research, practice, and policy implications. NASSP Bulletin, 88(638), 59-68.
Gullen, K. (2014). Are our kids ready for computerized tests? Educational Leadership, 71(6), 68-71.
Herman, J. L., & Baker, E. L. (2005). Making benchmark testing work. Educational Leadership, 63(3), 48-54.
Howard-Jones, P. (2014, January). Neuroscience and education: A review of educational interventions and approaches by neuroscience. London, UK: Education Endowment Foundation. Retrieved from http://educationendowmentfoundation.org.uk/library/neuroscience-and-education-literature-review
Kercheval, A., & Newbill, S. (2002). A case study of key effective practices in Ohio's improved school districts. Bloomington, IN: Indiana Center for Evaluation and Education Policy. Available from https://web.archive.org/web/20111119041445/http://ceep.indiana.edu/projects/PDF/200202_Key_Effec_Prac_Final_Report.pdf
Mehrens, W. A. (1989). Preparing students to take standardized achievement tests. Practical Assessment, Research & Evaluation, 1(11). Retrieved from http://pareonline.net/getvn.asp?v=1&n=11
Miller, A. (2014, May 1). It's bigger than the backbone: 4 steps to prepare teachers for CCSS assessments. School CIO. Retrieved from http://www.schoolcio.com/cio-feature-articles/0109/its-bigger-than-the-backbone-4-steps-to-prepare-teachers-for-ccss-assessments/54940
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Retrieved from http://www.ed.gov/about/bdscomm/list/mathpanel/index.html
O'Shea, M. (2005). From standards to success. Alexandria, VA: ASCD.
Popham, W. J. (2016). Standardized tests: Purpose is the point. Educational Leadership, 73(7), 44-49.
Popham, W. J. (2014). Criterion-referenced measurement: Half a century wasted? Educational Leadership, 71(6), 62-66.
Ramirez, G., & Beilock, S. L. (2011). Writing about testing worries boosts exam performance in the classroom. Science, 331, 211-213. Retrieved from http://education.uci.edu/brownbags/RamirezBeilock_paper.pdf
Reeves, D. B. (2004). Accountability for learning: How teachers and school leaders can take charge. Alexandria, VA: ASCD. ISBN: 0-87120-833-4.
Reforming Ohio's Education System for the 21st Century. (n.d.). Retrieved from http://www.conversationoneducation.org/
Silver, H., Strong, R., & Perini, M. (2007). The strategic teacher: Selecting the right research-based strategy for every lesson. Alexandria, VA: ASCD. Retrieved from http://www.ascd.org
Stanford University Medical Center. (2015, September 8). Tutoring relieves math anxiety, changes fear circuits in children. ScienceDaily. Retrieved from www.sciencedaily.com/releases/2015/09/150908180441.htm
Tomlinson, C. A. (2000). Reconcilable differences? Standards-based teaching and differentiation. Educational Leadership, 58(1), 6-11. Available in ASCD archived issues: http://www.ascd.org/portal/site/ascd/menuitem.a4dbd0f2c4f9b94cdeb3ffdb62108a0c/
U.S. Department of Education. (2003, August). NCLB Supplemental Educational Services Non-Regulatory Guidance. Retrieved from http://www.ed.gov/policy/elsec/guid/suppsvcsguid.doc
Weinstock, J. (2008, February). Make it a test worth teaching to. T.H.E. Journal, 35(2), 8.
Williams, T., Kirst, M., Haertel, E., et al. (2010). Gaining ground in the middle grades: Why some schools do better. Mountain View, CA: EdSource. Available from http://hub.mspnet.org/index.cfm/22637