Exploring the College Math Placement Testing Process

Information Age Education (IAE) is an Oregon non-profit corporation created by David Moursund in July, 2007. It works to improve the informal and formal education of people of all ages throughout the world. A number of people have contributed their time and expertise in developing the materials that are made available free in the various IAE publications. Click here to learn how you can help develop new IAE materials.



Here are some quotes to get your mind moving in the  direction of thinking about math education and math tests. '''Click here to access Moursund's collection of Math Education Quotations. '''


 * “Each problem that I solved became a rule which served afterwards to solve other problems.” (René Descartes; French philosopher, mathematician, scientist, and writer; 1596-1650.)


 * "The most important single element in problem solving is the individual working on the problem. The secret of real success is the confidence and desire to succeed. One must try and try again, vary the methods and procedures, have brains and good luck. There are no infallible rules for solving problems." (John N. Fujii; American mathematician, math educator, and author.)

You have probably heard the statement, "Beauty is in the eye of the beholder." The correctness of a solution to a math word problem can also be in the eye/viewpoint of the problem solver. A variation of this joke is available in http://economicscience.net/content/JokEc.


 * Someone asked an accountant, a mathematician, an engineer, a statistician, and an actuary to answer "2 + 2 = ?"


 * The accountant used a hand-held calculator and said, "4."
 * The mathematician said, "It all depends on your number base."
 * The engineer used a slide rule and said, "Approximately 3.99."
 * The statistician consulted math tables and said, "I am 95% confident that it lies between 3.95 and 4.05."
 * The actuary said, "What do you want it to add up to?"

Introduction
The focus of this IAE-pedia document is the general topic of math placement tests used in higher education and how these tests align with precollege math education. The entry has two major goals:


 * Goal 1. To help college-bound secondary school students learn to self-assess their progress in learning math relative to the standards set in higher education. A student may well be getting good grades in secondary school math courses and still be making poor progress toward being prepared for the rigors and much higher demands of college math courses.


 * Goal 2. To help secondary school math teachers gain increased understanding of how well they are preparing their students to deal with college-level math courses.

The next section discusses the first of these goals. The second goal is discussed later in the Rigorous Math Courses section.

The main intended audience of this document is U.S. preservice and inservice teachers of math at the secondary school level. Other audiences include secondary school students and parents of these students, teachers of math teachers, and academic counselors/advisers of secondary school students. Similar audiences from other countries may want to compare their educational systems with the U.S. system.

We will explore a number of different free sample math tests: GED (General Educational Development); College Entrance Tests such as ACT and SAT; and College Math Placement Tests from a number of colleges and universities. While these are designed for and often used for different purposes, they overlap considerably. Indeed, a math test used to gain college entrance might also be used as a College Math Placement Test. However, the "cut scores" may differ for their different uses.

The SAT and ACT are the two major college placement tests. Math is a part of each of these tests. Here is 2014 information about how well high school students did on the ACT test.


 * Tyson, C. (8/202014). Still a Losing Game. Inside Higher Education. Retrieved 8/22/2014 from https://www.insidehighered.com/news/2014/08/20/acts-annual-score-report-shows-languishing-racial-gaps-mediocre-scores. Quoting from this article:


 * More students than ever are taking the ACT, says the ACT’s annual score report, released today. A record 1.84 million high school students who graduated in 2014 took the college readiness test – suggesting that more young people have college in their sights. But for many test-takers, succeeding in post secondary education might be an empty hope. Average scores remain stagnant. Only 39 percent of test-takers met three or more of the ACT’s college readiness benchmarks in English, math, reading and science – a percentage that’s unchanged from last year. And striking racial gaps persist. As in previous years, African Americans and Latinos scored much lower, on average, than their Asian-American and white peers. [Bold added for emphasis.]

The Self-assessment Math Education Goal
Notice that Goal 1 is different from a goal of helping students to prepare for a specific high-stakes math test such as a final exam that helps determine whether a student passes a particular course, a math exam that one must pass to graduate from high school, a GED math test, or college entrance exams such as the ACT and the SAT. Usually, the foremost goal in preparing for such high-stakes tests is to pass the test or to score as high as one possibly can on the test. A goal of gaining long-term math knowledge, skills, and understanding is typically given a much lower priority in studying for a specific high-stakes test.

The GED math exam mentioned above is a good example of a high-stakes test. GED exams are designed to measure whether a student has gained knowledge and skills roughly equivalent to a basic high school diploma. Quoting from the Wikipedia:


 * Although the "GED" initialism is frequently mistaken as meaning "general education degree" or "general education diploma", the American Council on Education, which owns the GED trademark, coined the initialism to identify "tests of general educational development" that measure proficiency in science, mathematics, social studies, reading, and writing. Passing the GED test gives those who did not complete high school the opportunity to earn their high school equivalency credential, in the majority of the United States, Canada, or internationally. [Bold added for emphasis.]

A GED is widely accepted as being "sort of" equivalent to a high school diploma, and so opens many employment and higher education doors for students who have not completed the work for a traditional high school diploma.

Currently several companies and states have developed such comprehensive exams. A variety of sample exams and programs of study are available on the Web. Students can use these in self-assessment and also in preparing for a GED or equivalent test.

Teachers and others can analyze such tests to see how well the traditional secondary school math curriculum is preparing students. They can also ask if the tests are fair, reliable, and valid. See the Smarter Balanced website that discusses fair, reliable, and valid tests being developed for Common Core mathematics.

Sample GED Math Questions
Click here for a website that provides sample GED math questions.

Here are the directions and the first two questions from the sample exam cited above:


 * Formulas you may need are listed under the Formulas link in the sidebar on the left-hand side of your screen. Only some of the questions will require you to use a formula. Not all the formulas given will be needed.


 * The use of calculators is allowed for questions 1–13. If you need help using your calculator, click on the Calculator Directions link in the sidebar on the left-hand side of your screen.

These directions make it clear that students need not have memorized all of the formulas they might need on the test. However, they will need to recognize when a formula is appropriate to a particular test question and how to use the formula. The types of calculators that can be used are restricted. Typically, test makers design their questions so that a calculator is not needed.


 * Sample question #1.
 * A 15-foot ladder is leaning against a 30-foot wall. The bottom end of the ladder is 9 feet from the wall. How many feet above the ground does the ladder touch the wall?

A)   1.7    B)    6 C)   12    D)    17.5 E)   144
 * Sample question #2.
 * What is the probability that, by guessing at random, one can correctly guess the month of birth of a stranger?

A)   1/12    B)    1/9 C)   1/6    D)    1/4 E)   1/3

Discussion. The first question is a geometry problem. It is designed to measure whether a student knows how to use the Pythagorean Theorem. A student needs to know that there is a formula useful in "solving" right triangle problems. The student can recall it from memory or find it in the list of formulas that are provided.

The remainder of the challenge in this problem is to figure out that the right triangle under consideration has a hypotenuse of 15 feet and one side of length 9 feet. The goal is to find the length of the other side. To complete the calculation, the student needs to calculate the (positive) square root of 144. This is accomplished from memory, by a little trial and error, or by use of a calculator.

Like many "word" or "story" problems one finds on tests, this one is flawed. It assumes that the student will assume the ground makes an exact 90-degree angle with the wall. This is not given in the statement of the problem, and it could well be a wrong assumption. For example, the wall might be the outside wall of a geodome, an igloo, a tepee tent, or the learning tower of Pisa.

The second question is a probability problem. I think it is a somewhat tricky problem. Since the selection of a month is to be done by a random process, it doesn't make any difference whether the person is a stranger or not. (It is common for word problems to include extraneous information.) The key idea is that in a random selection process, each month is equally likely. The student only needs to know that there are 12 months in a year, so the probability of a particular month being selected is 1/12.

I think that the list of possible answers is not as good as to might be. It is not immediately clear to me what type of thinking would lead students to the incorrect answers B, C, D, and E. Knowing that there are 12 months in a year might lead a student to guess the answer is A (which contains a 12) without having any insight into how to solve the problem.

Suppose that the question is sightly modified, so that the choice of a month is not random. The person selecting a month gets to specify a particular month. This changes the complexity of the problem in two ways. First, some months have more days than others. But second, more births occur on some days than others. A student is apt to know the number of days in each month, but is unlikely to have memorized which month or months have the most births per day in the month.

It turns out that it is quite difficult to write word problems that are designed to test specific math competencies and that are unambiguous. See Math Word Problems Divorced from Reality.

An Example of a Low-stakes Math Placement Test
While this document focuses mainly on U.S. College Math Placement Tests, the example given here is from the University of British Columbia, Canada.

UBC makes use of a self-assessment instrument to help a student determine whether to enroll in Math 001 or Math 002. These are both considered to be remedial courses.

Using a self-assessment math placement test is an interesting idea. The stated prerequisite for course Math 002 is course Math 001. However, a student can decide to go directly to course Math 002. To help a student decide whether to do this, a self-assessment exam is available. Quoting from the website:


 * Test Instructions


 * Open the exam below in the format you prefer (same test, different formats). [This is a 80-question (not multiple choice) test. The test begins with a number of ninth-grade questions and ends with two trigonometry 12-grade questions.]
 * Adobe format pdf (.pdf) (146 kb) Click here for the PDF version.
 * Microsoft Word format (.doc) (246 kb) Click here for the Microsoft Word version.
 * Write the exam within 2 hours.


 * Scoring Your Test


 * Mark your test using the answer key pdf (each question is worth 1 mark).
 * Total your marks (test is out of 89 questions).
 * 90% is equal to 80 questions answered correctly out of 89.
 * Goal is 80 questions answered correctly or better.
 * If you score lower than 80 out of 89, Math 002 may be too difficult for you; consider taking Math 001 first.

Notice the openness of this process. A student is trying to decide whether to enroll in Math 001 or Math 002. The test is taken by and scored by the student. The student uses the test results to help make a decision on whether to enroll in Math 001 or Math 002. The student is assumed to have a level of maturity and knowledge of self to make a decision. A teacher, advisor, or counselor does not intervene in the process.

Rigor as a Goal in Math Courses
The second goal of this document is:


 * Goal 2: To help secondary school math teachers gain increased understanding of how well their students are prepared to deal with college-level math courses.

The term rigor is often used in discussing the quality of a math course or a curriculum of study. The quality of math teachers and the rigor of the math courses they teach both make a significant difference in the math education of students. This section focuses on the rigor of courses.

A good example of a discussion of rigor is provided in Is High School Tough Enough?, a report from the Center for Public Education. Quoting from this report:


 * Is the high school curriculum tough enough? Education initiatives as broad as the Common Core State Standards, grant organizations such as the Bill & Melinda Gates Foundation, and legislative actions like ESEA reauthorization all raise the issue in some way. All of these initiatives agree that high schools should produce “college and career-ready graduates,” and that “a rigorous curriculum” is the way to do so.


 * Beyond that, the picture gets murky. What do those two phrases mean? How do we determine what makes a prepared high school graduate, and how do we know if the current high school curriculum produces them?


 * The following facts should raise some concerns:


 * Almost two-fifths of high school graduates “are not adequately prepared” by their high school education for entry-level jobs or college-level courses, according to a survey of college instructors and employers (Peter D. Hart Research Associates, 2005).


 * Many low-income schools lack access to a rigorous high school curriculum by any definition. The U.S. Department of Education’s Office for Civil Rights Data recently reported that 3,000 high schools serving nearly 500,000 students offer no classes in Algebra II, a key subject encompassed by the SAT and other indicators of college readiness (OCR, 2011).


 * See more at: http://www.centerforpubliceducation.org/Main-Menu/Instruction/Is-high-school-tough-enough-At-a-glance#sthash.FIgfFuEM.dpuf.

Thus, we can talk about the rigor of a whole program of study (such as a high school diploma), or a program of study in math, or a specific math course. The following article provides a very gloomy picture.


 * Brown, E., & Bui, L. (8/21/2013). Just 26 Percent of ACT Test-takers Are Prepared for College. The Washington Post. Retrieved 6/2/2014 from http://www.washingtonpost.com/local/education/just-26-percent-of-act-test-takers-are-prepared-for-college/2013/08/21/a99fba0e-0a81-11e3-8974-f97ab3b3c677_story.html. Quoting from the article:


 * Just more than one-quarter of students who took the ACT college entrance exam this year scored high enough in math, reading, English and science to be considered ready for college or a career, data released Wednesday showed.


 * That figure masks large gaps between student groups — with 43 percent of Asians, but only 5 percent of African Americans — demonstrating college readiness in all four subjects.

Notice the "toughness" of the criteria used in this discussion—demonstrating college readiness in all four subjects.

Here are some additional quotes from the Center for Public Education.


 * Using AP courses as a proxy for a rigorous curriculum continues to be a popular strategy. Nationally, AP enrollments are five times what they were twenty years ago. Minnesota, for example, reported that the number of high school students who took AP or IB tests rose from 2,114 in 2009 to 6,401 in 2010 – a dramatic 203 percent increase – compared with an increase of only 17 percent in the number of seniors.


 * Taking an AP course results in better performance in the subject. U.S. students who “failed” the AP Calculus exam still outperformed students from all other industrialized countries on the Trends in Mathematics and Science Study (TIMSS).


 * AP courses do show results in college, though study results are mixed. One study found that students who took AP courses were at least twice as likely to graduate from college in five years compared with those who did not take an AP course. The gains were particularly noteworthy for some minority and low-income students. For African-Americans, only 10 percent of those who did not take an AP course graduated in five years, compared with 37 percent of those who took an AP course but did not pass the exam and 53 percent of those who took an AP course and passed the exam. Comparable results exist for Hispanic and low-income students.


 * An AP curriculum is not a silver bullet, nor is it the only strategy. Klopfenstein and Thomas found that students who took AP courses performed the same as students with a non-AP curriculum [that was] strong in math and science.

Math Maturity
I have long been interested in the topic of Math Maturity. Quoting from this website:


 * It is well recognized that some rote memory learning is quite important in math education. However, most of this rote learning suffers from a lack of long-term retention, and from the learner’s inability to transfer this learning to new, challenging problem situations both within the discipline of math as well as to math-related problem situations outside the discipline of math.


 * Thus, math education has moved in the direction of placing much more emphasis on learning for understanding and for solving novel (non-routine) problems. There is substantial emphasis on learning some “big ideas” and gaining math-related "habits of mind and thinking skills" that will last a lifetime.


 * There is still more to achieving a useful level of math maturity. A student needs to learn how to learn math, how to self-assess his or her level of math content knowledge, skills, and math maturity, how to relearn math that has been forgotten or partially forgotten, how to make effective of online sources of math information and instruction, and so on.

Now, when I think about the goals of a math course, I think in terms of a student learning math content and the student gaining a level of math maturity appropriate to the level of math the student has studied. This thinking is reflected in the section that follows.

Transfer of Learning
Click here to access the IAE-pedia document, Transfer of Learning. Click here to access a free IAE book, Introduction to Using Games in Education: A Guide for Teachers and Parents, that includes a major emphasis on transfer of learning.

One of the goals of math education is to prepare students to make a transfer of the math knowledge and skills taught in math courses to other courses as well as to "real-world" problems. In addressing a math-related problem outside the context of a math course, the problem solver often has knowledge of the problem situation and can take advantage of this knowledge. For example, the "outside" knowledge may help the problem solver to make a reasonably good estimate of an answer, and that estimate can be used to help "catch" major errors in reasoning and/or in calculation in setting up and solving a math problem.

However, in developing word problems for a test to be used by a wide range of students, it is "unfair" to either require "real-world" knowledge or discipline-specific knowledge from any discipline other than math; all students cannot be expected to have such knowledge. So, math word problems in textbooks and on widely used tests tend to be rather inane, often unrelated to problems that a student is apt to encounter outside of a math class.

The math teacher is faced by the challenge of teaching for transfer and the math student is faced by the challenge  of learning for transfer. The traditional math book approach to this challenge is through providing word problems. The tests examined in the remainder of this IAE-pedia document contain a large proportion of word problems.

Levels of Math Placement
This section lists some general areas and levels that might be covered in a College Math Placement Test.


 * 1) Basic arithmetic knowledge and skills typically covered in a K-7 or K-8 pre-algebra curriculum. Math maturity (including problem solving and math understanding) covered in a reasonably rigorous K-7 or K-8 pre-algebra curriculum.
 * 2) Basic skills covered in a secondary school Algebra 1 course. Math maturity (including problem solving and math understanding) covered in a reasonably rigorous secondary school Algebra 1 course.
 * 3) Basic skills covered in a secondary school Geometry course. Math maturity (including problem solving and math understanding) covered in a reasonably rigorous secondary school Geometry course. Note that secondary school geometry courses vary considerably in the nature/level of "proofs" integrated into the content. Thus, the geometry covered in College Math Placement Tests may or may not include a focus on proofs. A Geometry course with little emphasis on proofs is considered to be less rigorous than one with substantial emphasis on proofs.
 * 4) Basic skills covered in a secondary school Algebra 2 course. Math maturity (including problem solving and math understanding) covered in a reasonably rigorous secondary school Algebra 2 course.
 * 5) Basic skills covered in a secondary school Pre-calculus course. Math maturity (including problem solving and math understanding) covered in a reasonably rigorous secondary school Pre-calculus course. In both cases, many of these courses include trigonometry.

There might also be a test to determine whether a student has the calculus knowledge, skills, and math maturity to begin in a second-term or third-term college calculus course. Such a test may determine whether a student has the calculus knowledge and skills to begin in a course that has calculus as a prerequisite.

Some placement tests provide students with a page of frequently used math formulas. A particular institution's test may or may not allow students to use a calculator. If calculators are allowed, typically there are careful specifications regarding the capabilities of the calculator. A "powerful, modern handheld calculator" may well be able to do all of the "straight" calculation problems found on College Math Placement Tests.

Math placement tests tend to make considerable use of word problems (story problems) that are stated in a combination of natural language and the language of mathematics. Such word problems are typically solved by extracting a "pure" math problem from the text, solving the pure math problem, and then checking to see if the results provide a solution to the original word problem. Of course, there is much more to math maturity than just being able to solve math word problems.

The rigor of college math placement tests varies. However, there appears to be quite a lot of similarity in these tests across the country. Indeed, many colleges make use of commercially developed and widely used tests. Companies marketing these tests typically also provide (sell) study guides and test-prep courses or programs of study.

In the sample tests I have examined, there typically is little information about the actual score a student needs to achieve in order to place for a certain math course. That is, a student taking a sample test cannot readily tell from performance on this test what college math courses he or she will place for.

From the point of view of an institution of higher education, it is desirable (usually considered necessary) for the math placement test to be relatively short (perhaps an hour or less) and machine scoreable. Such a test may be adaptive, i.e., a computer adjusts the level of questions presented to quickly hone in on a good measure of a student's performance.

Alternatively, it may draw on a very large databank of questions, so that students do not all receive the same questions and students retaking the test at a later date may receive different questions. Or a test may make extensive use of questions that are based on templates, where a computer generates specific numbers that are inserted in a template to make a test question. In some sense this is like drawing questions from a large databank.

A math placement test might well contain some open response questions in which students are expected to show and explain their work. Typically, the expectation is that these will be scored by a well-prepared human and partial credit is often given for answers that are not completely correct.

What Constitutes a "Good" College Math Placement Test?
Here are some of my thoughts about desirable features of a "good" college math placement test.


 * 1) The test is openly available and free to anyone—not just students who have been admitted to the specific institution of higher education offering the test. Thus, for example, secondary school students could take that institution's test to determine how well they are doing in their math learning. Students thinking of applying to a particular institution could take that institution's test to see how well they might do on it.
 * 2) The test is comprehensive and well balanced. It includes a significant emphasis on the various aspects of math maturity.
 * 3) The test provides analysis and feedback to the student. This feedback points out relative strengths and weaknesses.
 * 4) The institution and/or test system makes available comprehensive information about resources that are available free to the student and can provide good quality online instruction on the various areas covered in the test.

The fourth criterion is partially illustrated by Portland Community College, in Portland, Oregon. Quoting from their Preparing for Your Placement Tests:


 * IMPORTANT: Take your Placement Test seriously! Preparing well for your testing session can save you both time and money. Remember, you are only allowed one retest in three years. Placing into the right courses will help you achieve your educational goals faster. Come to your session well rested and fed. Allow plenty of time to complete testing without rushing. Ask the Testing Center staff for assistance if needed.


 * Visit our online Assessment Testing Workshop for help and test preparation advice. Also, please visit the Student Learning Center to access their extensive Compass Placement Test Preparation materials. If you would like to attend a Test Taking Workshop please visit College Placement Test Workshop for times and locations.


 * You can also download or look at these free study guides to see the kinds of questions you can expect and to help you get ready for the tests:
 * Mathematics
 * Mathematics: Basic [pdf] (14 pages; 643KB)
 * Mathematics: Advanced [pdf] (17 pages; 663KB)
 * Mathematics: Advanced [pdf] (17 pages; 663KB)

In reading these math materials from Portland Community College, I was disappointed to see that they merely consisted of sample tests and answers. Students need and deserve more help than this. Some indication of what is needed is provided in discussions about formative assessment. Quoting from this website:


 * Noting that in a widely cited review, the term formative assessment "does not have a tightly defined and widely accepted meaning," Black and William operate an umbrella definition of "all those activities undertaken by teachers, and/or by students, which provide information to be used as feedback to modify the teaching and learning activities in which they are engaged." Along similar lines, Cowie and Bell define formative assessment as "the process used by teachers and students to recognise and respond to student learning in order to enhance that learning, during the learning". Nicol and Macfarlane-Dick, who emphasise the role students can play in producing formative assessments state that "formative assessment aids learning by generating feedback information that is of benefit to students and to teachers. Feedback on performance, in class or on assignments, enables students to restructure their understanding/skills and build more powerful ideas and capabilities.

In summary, I believe that a College Math Placement Test should be a vehicle for math course placement, for student self-assessment, for assessment of the "Math qualities" of students entering the institution, and as an aid to these students to improve their math education. It should serve precollege students and their teachers, as well as students entering a particular college.

The next part of this document contains a sampling of College Math Placement Tests.

College Math Placement Tests
A College Math Placement Test is designed to determine how well prepared a student is for possible enrollment in and successful completion of particular college math courses. Tests must address the full range of students, from those beginning a community college program to those entering a prestigious research university.

It is important to distinguish between: (A) the year-after-year progress that a student makes by taking a sequence of secondary school math courses, and (B) the possible time spent in a short-term concentrated effort of review and practice in getting ready for a College Math Placement Test.

In (A), the unifying and long term goal is on "deep" learning that grows year after year through taking courses and using the math knowledge that is gained. This type of math knowledge and skills will serve the student for a lifetime. Descriptors of the desired learning outcomes include: being able to comfortably use math in all disciplines and areas a student studies; long-term retention; learning with understanding; math maturity; learning how to learn math; "good" math habits of mind; and learning to think in and to communicate in the language of math.

A college or university expects that a student will do a limited amount of practice and review before taking a Math Placement Test. This should be a "refresh your mind, shake out the cobwebs" level of review. This is quite different from taking a test-prep course and doing extensive studying designed to help improve one's score on a specific test.

The difference between preparing for a College Math Placement Test and studying to get a high score on an ACT or SAT test is subtle. The ACT or SAT test helps to determine what college or university will admit a student. The College Math Placement Test is taken after admission and helps to determine what college math course a student begins with.

Achieving an extra high math placement score may well lead to a student taking a college math course that will be "over his/her head." For most students, this a poor way to start a college math program of study.

Of course, another poor way to start one's college math studies is to take remedial courses that do not carry college credit toward a degree and/or that are far too easy to present any intellectual challenge or increase one's level of knowledge and skills. So, some preparation is highly desirable. This reminds me of the "Three Bears" story in which one bowl of porridge is too hot, one is too cold, while the third is just right. A good College Math Placement Test system will help a student to begin in a course that is "just right."

In a particular college or university, math placement might be based only on a placement test. But it might also be based on a combination of any of the following: high school records; discussions with an advisor; the strengths, weaknesses, and interests of the student; performance on the math components of widely used college entrance exams such as the ACT or SAT; and College Math Placement Test scores.

Accurate and appropriate math course placement is not an easy process. Secondary schools vary considerably in the math courses they offer and the standards they set. The Common Core State Standards are leading to substantial changes in the PreK-12 math curriculum. So, higher education institutions are faced by students who vary considerably, both in the extent of the math coursework they have taken and in the nature or rigor of this coursework. A college or university's Math Course Placement Test process might well take into consideration the secondary school math programs in its major feeder schools.

Who Creates College Math Placement Tests?
Some colleges and universities create and administer their own placement tests, others make use of tests that are commercially prepared and administered, and some do a combination of both. An example is provided in information about the University of Oregon quoted from http://testing.uoregon.edu/PlacementTesting/MathPlacement/GeneralInformation/tabid/94/Default.aspx:


 * New students receive an initial math placement recommendation based on their SAT/ACT Math score.


 * SAT/ACT Math Score Placement Recommendations:


 * * SAT-M 460 or below/ACT-Math 20 or below = MATH 070/Elementary Algebra


 * * SAT-M 470 – 540/ACT-Math 21-24 = MATH 095/Intermediate Algebra


 * * SAT-M 550 or higher/ACT-Math 25 or higher = MATH 105/106/107 University Math I, II, III or MATH 111/College Algebra, MATH 243/Introduction to Methods of Probability and Statistics


 * Students who are satisfied with their placement based on SAT/ACT-Math scores are not required to take a math placement test. If you feel that your SAT/ACT-Math score underestimates your math abilities there is one of 2 versions of the Math Placement Test that you can take to demonstrate readiness for a higher level class. Math placement testing will be available during IntroDUCKtion, Week of Welcome, and throughout the year by appointment Mon. - Fri., 9am - 5pm, at the Testing Center. Calculators are permitted for the Math Placement Test.


 * Which Math Placement Test should you take?


 * Standard Math Placement Test. Click here for a Sample Test. Click here for answers to the Sample Test.


 * You take the Standard Math Placement Test if: 1) your SAT-Math score is 540 or below (ACT-Math 24 or below) or, 2) you do not have SAT or ACT scores. The Standard Math Placement Test is designed to allow students to place anywhere within the continuum of introductory math classes (100-200 level). Students whose SAT/ACT-Math score has placed them into the developmental classes (MATH 070 or MATH 095) can demonstrate readiness for introductory university level math classes by taking and receiving a qualifying score on the Standard Math Placement Test.


 * Advanced Math Placement Test. Click here for a Sample Test. Click here for answers to the Sample Test.


 * The Advanced Math Placement Test is designed specifically for the student who has an SAT-Math score of 550 or higher (ACT-Math score of 25 or higher) but who believes he/she is prepared to begin taking math at a level higher than MATH 105 or MATH 111. Typically such a student has taken Trigonometry, Elementary Functions, and/or Calculus at the high school level.

Accuplacer (College Board) Pre-algebra Sample Math Questions</Center>
Accuplacer Sample Questions for Students contains sample tests on Sentence Skills, Reading Comprehension, Writing, Arithmetic, Elementary Algebra, College-level Math, ESL Reading, ESL Sentence Meaning, ESL and Language Use. Answer keys are given at the end of the 25-page PDF document.

Quoting from the Arithmetic section of the Accuplacer materials:


 * This test measures your ability to perform basic arithmetic operations and to solve problems that involve fundamental arithmetic concepts. There are 18 questions on the Arithmetic tests, divided into three types.


 * • Operations with whole numbers and fractions.
 * • Operations with decimals and percents.
 * • Applications and problem solving.


 * For each of the questions below, choose the best answer from the four choices given. You may use the paper you received as scratch paper.


 * Sample question #1.
 * 1. 2.75 + .003 + .158 =
 * A. 4.36
 * B. 2.911
 * C. 0.436
 * D. 2.938

Discussion. This is a "pure" decimal numbers addition problem. Note that the directions for the test did not specify that calculators were allowed. This problem is easily solved by use of a calculator.


 * Sample question #6.
 * Which of the following is closest to 27.8 × 9.6?
 * A. 280
 * B. 300
 * C. 2,800
 * D. 3,000

Discussion. This problem can be solved by doing the decimal numbers multiplication. However, it can also easily be solved by merely estimating the product of those two numbers. An estimated product is certainly less than 28 x 10, so the answer is certainly A.


 * Sample question #15.
 * If 3/2 ÷ 1/4 = n, then n is between
 * A. 1 and 3
 * B. 3 and 5
 * C. 5 and 7
 * D. 7 and 9

Discussion. This problem involves dividing one fraction by another fraction. It turns out that many students have a great deal of difficulty in learning to do arithmetic (in particular, division) with fractions. The problem is easily solved by "common sense." The division problem is asking how many (1/4)s are in 3/2. A person with good number sense translates this into finding how many (1/4)s are in 1 and how many (1/4)s are in 1/2. Thus, the whole problem is easily and quickly done mentally. A "rote memory" student remembers that division of fractions involves "invert and multipy." The issue is what to invert and what to multiply.


 * Sample question #17.
 * A box in a college bookstore contains books, and each book in the box is a history book, an English book, or a science book. If one-third of these books are history books and one-sixth are English books, what fraction of the books are science books?
 * A. 1/3
 * B. 1/2
 * C. 2/3
 * D. 3/4

Discussion. Part of the challenge of this problem is dealing with the vocabulary. For a student who does not get bogged down in the verbiage, the problem is simply to add 1/3 and 1/6, and subtract the result from 1. This is easily done mentally by a student who has good number sense.


 * Sample question #18.
 * The measures of two angles of a triangle are 35° and 45°. What is the measure of the third angle of the triangle?
 * A. 95°
 * B. 100°
 * C. 105°
 * D. 110°

Discussion. To solve this problem, a student needs to know that the sum of the angles in a triangle is 180 degrees. With that knowledge in mind, the arithmetic is simple enough

The sample test questions illustrate the breadth of arithmetic topics that can be covered in a short, machine-scoreable test. The term "applications and problem solving" means that the statement of the problem uses a number of "natural language" words. A student has to extract from the "word problem" an arithmetic computation to be performed, and then select the one correct answer from the four that are available.

In this type of testing, a student may make an error in deciding on the computation to be performed and/or in doing the calculation. If the result is not one of the four available answers, the student can reason that the error is in one or both of the sub-tasks that s/he performed.

For most students, it takes years of instruction and practice to gain speed and accuracy in carrying out "fast, accurate, by hand, rote" arithmetic.

A calculator can do all of the calculations, but it introduces a different type of difficulty. It takes quite a bit of instruction and practice to effectively deal with the "data" entry, notation, "memory," and decimal output format of a simple calculator. If you don't believe this statement, find yourself a simple 6-function decimal notation calculator that has the three memory keys MR, M-, and M+. You may find that you don't know how to use the three memory keys. Pose some simple fraction calculation problems such as 1/2 + 1/3 - 1/4 where the list of possible answers are all given in fraction notation. The fraction 1/3 has a "repeating" decimal expression. When you make use of the memory keys to add these three fractions the result is the decimal.4166666 with repeating 6's. The fraction answer 5/12.

This Accuplacer Arithmetic Test provides answers at the end of the document. In its "by hand" format, there is no easy way for a student to get the type of feedback needed to understand his or her weaknesses and possible lack of understanding. This type of test is poor in measuring student math maturity. In addition, the answer key provides no information on what constitutes an acceptable level of performance from the point of view of college placement in a remedial arithmetic class.

ACT Compass Pre-algebra Sample Math Questions</Center>
Quoting from the ACT site:


 * Welcome to the ACT Compass® Sample Mathematics Test!


 * You are about to look at some sample test questions as you prepare to take the actual ACT Compass test. The examples in this booklet are similar to the kinds of test questions you are likely to see when you take the actual ACT Compass test. Since this is a practice exercise, you will answer just a few questions and you won’t receive a real test score. The answer key follows the sample questions.


 * Once you are ready to take the actual ACT Compass test, you need to know that the test is computer-delivered and untimed—that is, you may work at your own pace. After you complete the test, you can get a score report to help you make good choices when you register for college classes. [Bold added for emphasis.]
 * The ACT Compass Mathematics Tests are organized around five principal content domains: numerical skills/pre-algebra, algebra, college algebra, geometry, and trigonometry. To ensure variety in the content and complexity of items within each domain, ACT Compass includes mathematics items of three general levels of cognitive complexity: basic skills, application, and analysis. A basic skills item can be solved by performing a sequence of basic operations. An application item involves applying sequences of basic operations to novel settings or in complex ways. An analysis item requires students to demonstrate a conceptual understanding of the principles and relationships relevant to particular mathematical operations. Items in each of the content domains sample extensively from these three cognitive levels.
 * The ACT Compass Mathematics Tests are organized around five principal content domains: numerical skills/pre-algebra, algebra, college algebra, geometry, and trigonometry. To ensure variety in the content and complexity of items within each domain, ACT Compass includes mathematics items of three general levels of cognitive complexity: basic skills, application, and analysis. A basic skills item can be solved by performing a sequence of basic operations. An application item involves applying sequences of basic operations to novel settings or in complex ways. An analysis item requires students to demonstrate a conceptual understanding of the principles and relationships relevant to particular mathematical operations. Items in each of the content domains sample extensively from these three cognitive levels.


 * Students are permitted to use calculators on all current versions of ACT Compass Mathematics Tests. Calculators must, however, meet ACT specifications, which are the same for ACT Compass and the ACT® college readiness assessment. These specifications are are updated periodically and can be found at http://actstudent.org/faq/calculator.html.

The calculator site referenced just above indicates:


 * You may use a calculator on the ACT Mathematics Test but not on any of the other tests in the ACT. You are not required to use a calculator. All problems on the Mathematics Test can be solved without a calculator. [Bold added for emphasis.]

This design rather thwarts quite a bit of the purpose of integrating calculators into the math curriculum. The ideas of computational thinking and "calculator thinking" are learned through being in a learning environment that routinely makes effective use of computers and calculators.

Here are a few questions from the Numerical Skills/Pre-algebra test.


 * Sample question #1.
 * 54 – 6 ÷ 2 + 6 = ?
 * A. 6
 * B. 24
 * C. 27
 * D. 30
 * E. 57

Discussion. This is a "straight" arithmetic calculation problem mainly designed to test whether a student knows the proper order of operations. My personal opinion is that anyone who writes an arithmetic expression that is ambiguous unless a student has memorized an order of precedence rule should be ashamed of him/herself.
 * Sample question #2.
 * The lowest temperature on a winter morning was –8°F. Later that same day the temperature reached a high of 24°F. By how many degrees Fahrenheit did the temperature increase?
 * A. 3°
 * B. 8°
 * C. 16°
 * D. 24°
 * E. 32°

Discussion. For a student with "number sense," the problem is one of seeing how many degrees does the temperature have to go up to get from -8 degrees to 0 degrees, and from 0 degrees to 24 degrees. This is easily solved mentally. The challenge is not the arithmetic—the challenge is what arithmetic calculations to perform.


 * Sample question #6.
 * Four students about to purchase concert tickets for $18.50 for each ticket discover that they may purchase a block of 5 tickets for $80.00. How much would each of the 4 save if they can get a fifth person to join them and the 5 people equally divide the price of the 5-ticket block?
 * A. $ 1.50
 * B. $ 2.50
 * C. $ 3.13
 * D. $10.00
 * E. $12.50

Discussion. The challenge is figuring out how to solve the problem. The information is buried in two long sentences that may prove to be a reading challenge to some students. And, of course, the general idea of paying (the huge amount of) $18.50 for a ticket or getting a reduced price if a group buys a block of tickets may be quite unfamiliar to some students.


 * Sample question #10.
 * This year, 75% of the graduating class of Harriet Tubman High School had taken at least 8 math courses. Of the remaining class members, 60% had taken 6 or 7 math courses. What percent of the graduating class had taken fewer than 6 math courses?
 * A. 0%
 * B. 10%
 * C. 15%
 * D. 30%
 * E. 45%

Discussion. The arithmetic in this problem is not overly challenging. The issue is what arithmetic calculations to preform. By now you should see the pattern. The placement test is targeting a student's ability to "understand" the math communication and accurately extract a "pure" math calculation problem from it. The emphasis is on the "figuring it out" process rather than on the calculations.


 * Sample question #14.
 * A total of 50 juniors and seniors were given a mathematics test. The 35 juniors attained an average score of 80 while the 15 seniors attained an average of 70. What was the average score for all 50 students who took the test?
 * A. 73
 * B. 75
 * C. 76
 * D. 77
 * E. 78

Discussion. The word "average" in math has the three meanings of mean, median, and mode. Thus, the question is poorly stated. The person who posed the question assumes average means "mean." (Isn't the English language wonderful!)

In my opinion, the ACT sample test is more challenging than the Accuplacer sample test. In both tests, look carefully at the "word" problems. It is relatively easy to teach computational arithmetic, and it is easy to generate arithmetic test questions. It is a greater challenge to generate "good" word problems, and students (on average) find word problems difficult. A student can be quite skilled in the speed and accuracy of rote arithmetic processes, and still be baffled by word problems.

Good quality institutions of higher education want their students to achieve content understanding and problem-solving skills in whatever disciplines they choose to study. Math is a challenging discipline because it is difficult to achieve a high level of understanding and good skills in solving novel, challenging problems. From a student's point of view, word problems are such novel, challenging problems. Students who have not made substantial progress in solving word problems may find that non-remedial precollege math coursework is over their heads. It is not that they have an inherent, inborn lack of ability to gain a high level of math maturity. Rather, their informal and formal math education and/or their own dedication to the task are inadequate.

Think about how well average adults would do on the two sample tests discussed above. Research suggests that the average adult in the U.S. functions at about the 8th or 9th grade level in math. I suspect that this is an overly optimistic estimation.

Click here for five additional ACT Math practice tests. These sample tests each contain 12 questions. An actual ACT math test is 60 questions in length, with a time limit of 60 minutes. So, in some sense, these five sample tests altogether are equivalent to a full ACT test. The tests contain a wide range of questions from Pre-Algebra up through Algebra 1, Geometry, Algebra 2, and some probability and trigonometry.

SAT Pre-algebra and Higher Level Sample Math Questions</Center>
Click here for 12 SAT Practice Problem Solving Tests. Quoting from the document:


 * Practice your math problem solving skills with our tests. Use a calculator only where necessary. You shouldn't need more than three lines of working for any problem. Redraw geometry figures to include the information in the question.


 * Each test has ten questions and should take 12 minutes.


 * Initially don't worry too much about the time until you have a feel for the type of questions. But, by the time you have done two or three tests you should start getting tough about the time you take.


 * If you don't get a problem right on either our tests or a real SAT test, first try to solve the problem yourself. If you still can't get it right, ask a friend or a teacher. If your fundamentals are weak and you need extra help check out our recommended resources.

Here are the first three questions from the first of the 12 practice tests:


 * Sample question #1.
 * Of the following, which is greater than ½ ?
 * A. 2/5
 * B. 4/7
 * C. 4/9
 * D. 5/11
 * E. 6/13

Discussion. A student who understands fractions knows that if the numerator is more than half of the denominator, the fraction is greater than 1/2. A different way of saying this is that, if twice the numerator is greater than the denominator, the fraction is greater than 1/2. That provides a simple and quick way to solve the problem when the reference number is 1/2.

However, suppose that a student faces a somewhat similar problem, but the goal is to determine which of a set of fractions is greater that 3/16? Hmm. Not so easy, right?

Alternatively, one can consider a more general method for solving such problems. How does one decide whether one number is larger than another? Well, a memorized rule might be to subtract the second from the first. If the result is positive, the first is larger. If the result is zero, the two numbers are the same. If the result is negative, the first number is smaller than the second.

In the original problem, this requires calculating 1/2 - 2/5, etc. These are "challenging" calculations that involve finding a common denominator, representing each fraction in terms of the common denominator, subtracting, and examining the result.

An alternative is to determine whether the inequalities such as the following are correct: 1/2 - 2/5 > 0? If a student has learned to work with inequalities, this type of problem is simpler (but somewhat similar to) working with common denominators.

My suspicion is that the people making up this test question assumed that it would not be solved by either of the two "brute force" methods described immediately above. Rather, this question helps to separate students who had developed a good mental model or intuitive feeling for fractions on a number line from those who memorized and used a general purpose algorithm.


 * Sample question #2.
 * If an object travels at five feet per second, how many feet does it travel in one hour?
 * A. 30
 * B. 300
 * C. 720
 * D. 1800
 * E. 18000

Discussion. This problem requires a student to figure out that there are 3,600 seconds in an hour. The student also has to know that distance = rate X time. So, this can be thought of as a physics problem.


 * Sample question #3.
 * What is the average (arithmetic mean) of all the multiples of ten from 10 to 190 inclusive?
 * A. 90
 * B. 95
 * C. 100
 * D. 105
 * E. 110

Discussion. The third question also allows brute force versus using deeper thinking and understanding. A brute force approach uses the definition of mean and no deep thinking as a student adds up the 19 numbers and divides by 19. This becomes a test of speed and accuracy in addition and division.

Students of math history will recognize the following story about the mathematician Carl Friedrich Gauss. Quoting from the website:


 * There’s a popular story that Gauss, mathematician extraordinaire, had a lazy teacher. The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to 100.


 * Gauss approached with his answer: 5,050. So soon? The teacher suspected a cheat, but no. Manual addition was for suckers, and Gauss found a formula to sidestep the problem.…

Sample question #3 can quickly be solved mentally by those who have discovered or been taught that the average of the numbers in such an evenly spaced sequence of increasing numbers is merely the sum of the first number and the last number divided by 2. In the Gauss example, the average of the 100 numbers is 50.5, so the sum is a hundred times 50.5.

Free Sample SAT Tests
Lewis, Darcy (9/12/2014.) How the New SAT Is Trying to Redefine College Readiness. U.S. News and World Report. Retrieved 9/15/2014 from http://www.usnews.com/education/best-colleges/articles/2014/09/12/how-the-new-sat-is-trying-to-redefine-college-readiness. Quoting from the article:


 * If you are checking out the Best Colleges rankings as a junior or senior, you’re preparing to sit – or have already sat – for the 2400-point SAT, complete with its fancy vocabulary words and mandatory essay. But members of the class of 2017 will begin prepping next year for a completely overhauled test.


 * Last March, College Board President David Coleman announced major revisions to the fall 2015 PSAT and the 2016 SAT, saying the SAT had "become disconnected from the work of high schools."


 * The changes, which include going back to the 1600-point composite score based on 800-point math and "evidence-based reading and writing" sections, and making the essay optional, are intended to better reflect the material students should be learning in high school and improve the SAT’s reliability as an indicator of how prepared applicants are to tackle college work.

Click here for free sample SAT tests and materials.


 * Official SAT Questions
 * Free sample SAT questions
 * Free full SAT practice test
 * Official SAT Online Course
 * Official SAT Study Guide
 * Real SAT Subject Tests Questions
 * Free sample SAT Subject Test questions
 * Official SAT Subject Tests Study Guides

College Algebra-level Placement Tests
In a college or university that sets high standards for its math courses, College Algebra is roughly equivalent to a rigorous secondary school Algebra 2 course. Thus, a student entering college who has done well in a rigorous secondary school Algebra 2 course should "place out of" College Algebra. Unfortunately, many secondary school Algebra 2 and prior math courses are not what mathematicians would call "rigorous."

University of Texas at Arlington</Center>
Click here for the 9-question College Algebra Placement Sample Test from the University of Texas at Arlington.

Here are two questions from that sample test:


 * Sample question #2.
 * A manufacturing company processes raw ore. The number of tons of refined material the company can produce during t days using Process A is A (t ) = t ^2 + 2t and using Process B is B (t ) =  10t . The company has only 7 days to process ore and must choose 1 of the processes. What is the maximum output of refined material, in tons, for this time period?


 * A. 8
 * B. 10
 * C. 51
 * D. 63
 * E. 70

Discussion. This question contains words and formulas that obfuscate the actual problem to be solved. The math question is: Which is larger, A(7) or B(7)? For me, this type of question helps to give math education word problems a bad name. I cannot imagine that a company that processes raw ore would have to decide between two manufacturing processes whose output is represented by the two formulas. In no real-world sense is this a real-world problem.

As in many word problems, the words merely constitute a barrier to be overcome and have nothing to do with the math problem to be solved. So, a student may miss this problem because of an inability to "figure out" what math problem is to be solved, and/or from an inability to correctly do the math.

Just out of curiosity, I did a Google search on the quoted phrase, "A manufacturing company processes raw ore." I got 761 hits. Evidently this is a frequently used sample test question.


 * Sample question #8.
 * The imaginary number i is defined such that i^2 = − 1. What does i + i^2 + i^3 +…+ i^23 equal?
 * A. i
 * B. -i
 * C. -1
 * D. 0
 * E. 1

''Discussion. ''I consider this to be another poor problem. The (imaginary) number i is quite important in math as well as in engineering, physics, and so on. But this question defines i and can be solved with no understanding whatsoever of its use/value in pure and applied math.

From the definition, we can see the the sequence of numbers to be added consists of: i, -1, -i, 1, i, -1, -i, 1, and so on.

The sequence of the first four numbers repeats itself over and over. The sum of the four numbers is 0. This easily leads us to the answer to the problem, that it is the sum of the first three of the four numbers, which is -1. So the problem is solved by doing quite simple arithmetic that involves multiplying by i and doing addition, and also noting the pattern in order to decrease the amount of arithmetic that needs to be done. My complaint is not that a student faces the challenge (or its equivalent) of the solution procedure I have provided, but that the problem requires no insight into the math and the importance of the imaginary i.

Barton College</Center>
Here is some information about the Barton College Math Placement Test. Quoting from the website:


 * The Mathematics Placement Test (MPT) is administered to all incoming freshman and transfer students who do not bring an advanced placement or college-level transfer credit for a mathematics course to Barton College. The test covers topics from Pre-algebra, Algebra I, Algebra II, and Pre-Calculus. The MPT consists of 40 questions, and students will have 60 minutes to complete the test.


 * The first 20 questions cover material in a standard Algebra I course. The next ten questions cover material in a standard Algebra II course, and the final ten questions cover material from pre-calculus.

The practice test contains 80 questions, which is twice the number on the actual test. They are all 4-part multiple-choice questions.

The exam site includes information about the scores needed on various parts of the test to qualify for placement in various math courses. I think this type of "disclosure" is quite useful to students; unfortunately, it does not seem to be common practice.

University of Washington
Click here to learn about the University of Washington Practice Math Tests. They make use of the Academic Placement Test Program. A 55-question lower-level and a 45-question higher-level sample test are available. Quoting from the website:


 * The Academic Placement Testing Program (APTP) is a cooperative program of Washington State public colleges and universities. Faculty from participating institutions have created the Mathematics Placement Test (MPT) to help students, with the assistance of their academic advisers, select first-year mathematics courses for which they are best prepared. The program is managed by the Office of Educational Assessment on behalf of participating institutions.

Final Remarks
Both teachers and students can browse the websites of their local and regional college math departments. They can build a collection of sample College Math Placement Tests and other local and nationally-used math tests. They can also build a collection of good free tutorials that are available on the Web.

Keep in mind that the goal is to develop skills in self-assessment and in making use of the results of self-assessment.

As you discover good sites, please send their links to moursund@uoregon.edu. They will be added to this website.

Additional Resources
Albrecht, R., & Moursund, D. (2011-2013). Math word problems divorced from reality. IAE-pedia. Retrieved 6/6/2914 from http://iae-pedia.org/Math_word_problems_divorced_from_reality.

California Institute of Technology (2014). Computer Science 1 Placement Exam. Retrieved 4/7/2024 from http://courses.cms.caltech.edu/cs1/placement/placement-exam-cs1.html. This reference was included even though it is not a math placement test. Cal Tech's Computer Science Placement Exam consists of two questions that a students can work on over a period of time. Quoting from the document:


 * We shouldn't have to say this, but we will: All of your code must be written by you and you alone. You are not allowed to get any kind of help from anyone else while writing the placement exam. You can consult on-line or printed documentation for the language(s) you use, but you are not allowed to ask questions about the problems either in-person or online before submitting your placement exam. If we find out that you have violated this rule, you will fail the placement exam.

''
 * Quoting from the Cal Tech Daily Caller http://dailycaller.com/2013/02/20/so-you-want-to-go-to-a-college-that-doesnt-offer-advanced-placement-credit/caltech-facebook/:


 * The legendary California Institute of Technology offers a few hundred hardcore and overwhelmingly male nerds the opportunity to tackle a generally crippling curriculum. The mandatory core courses include a load of math and the hard sciences (as well as humanities and physical education requirements). Caltech does not grant credit for AP, IB, or similar tests. However, every student takes a math and physics placement exam before enrolling. If you do well enough on that test, you can get automatic credit for some lower-level Caltech courses. [Bold added for emphasis.]


 * Click here to read more about Cal Tech.

Mathematics exams with solutions (n.d.) Retrieved 6/7/2014 from http://examswithsolutions.com/Subjects/math_exams.html. Provides links to: 14 university Calculus Diagnostic and Placement Exams with Solutions; 10 university Precalculus Exams with Solutions; 4 sites containing AP Calculus Exams with Solutions; 20 university Calculus 1 Exams with Solutions; 23 university Calculus 2 Exams with Solutions; 15 Calculus 3 Exams with Solutions; and additional higher level exams. Also provides access to a variety of lower-level college math course tests.

Moursund, D. (12/27/2010). A serious problem situation with math word problems. IAE Blog. Retrieved 6/6//2014 from http://i-a-e.org/iae-blog/entry/a-serious-problem-situation-with-math-word-problems.html.

Pellegrino, J.W.; Wilson, M.R.; Koenig, J.A.; & Beatty, A.S. (eds.) (2013). Developing assessments for the next generation science standards. Washington, D.C.: National Academies. Free PDF downloaded from http://www.nap.edu/download.php?record_id=18409.

Author or Authors
This IAE-pedia page was created by David Moursund.