MathNotations

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As I've reported for some time (look here), here in NJ the Commissioner of Education has been promoting higher standards and more ambitious graduation requirements, choosing Algebra 2 as the cornerstone. I've had misgivings about this as a requirement for all students for several reasons although I'm a strong supporter of the American Diploma Project's Algebra 2 benchmarks and the End of Course Test for all students who choose to take the course because of their educational goals.

A recent article (1-27-09) on the website pressofAtlanticCity.com gives an excellent account of the debate raging over this topic at the State Legislative level. I will reprint a good portion of the article and then reprint the comment I posted on the site. I strongly encourage my readers to read the entire article and all of the comments posted thus far. It is a microcosm of much of the current debate in math education. Several of the commenters provided a commonsense view of these issues and gave me food for thought.

Education Commissioner Lucille Davy and a panel of education and business professionals appeared Monday before the Assembly Edu-cation Committee to discuss the Department of Education's High School Reform project.

The requirement that all students take algebra II has been controversial, and on Monday it dominated a discussion that attempted to identify just what students need to know to succeed and compete in the 21st century.

Davy insisted the algebra II requirement would not be so rigorous that it would lead to high rates of failure or students dropping out.

She said it would be a continuation of algebra I,butschools could offer more rigorous honors courses to those who would need them.

But Rutgers math professor Joseph Rosenstein, of the New Jersey Math and Science Coalition, wondered if the proposed courses might then get so watered down they would no longer really be algebra II.

"Most of our students don't need algebra II," said Rosenstein, who supports requiring more practical applied math courses.

Rosenstein said if courses were tailored just to meet state requirements, students who should take a true algebra II course might not get the higher level of work they need.

The algebra II issue has also frustrated vocational high school officials, who worry that too many requirements will make it impossible for students to complete programs in high school.

"These are students who benefit from applied learning," said Thomas Bistocchi, superintendent of the Union County Vocational School, adding that their goal is to have students graduate as industry-credentialed professionals."We just want students who want to become plumbers have the time to do it," he said.

Davy said there will be flexibility in how the coursework is offered, so that it could be integrated into vocational coursework, but opponents wonder if that could be done and still teach what would be tested.

Stan Karp, of the Education Law Center, said reform is needed, but the state needs better education, not just more requirements. He said teachers and students will need better preparation to meet the new requirements.

"Less than half of the high schools now require those courses," he said. "What is it going to take to get there?"

Asked about the cost of reforms, Davy said the state already spends the most of any state and should not need more money, just a better reallocation of existing funds.

Business representatives said they just need students who can perform modern jobs.

Dennis Bone, president of Verizon, said students need the foundation of skills to be able to adapt to new and changing technology.

"We are being revolutionized by technology," he said. "Billboards now are electronic, run by someone sitting at a computer, not climbing a ladder."

"So what does algebra II have to do with that?" Education Committee Chairman Joseph Cryan, D-Union, asked.

Dana Egresky, of the New Jersey Chamber of Commerce, said that if taking algebra II can help a carpenter solve more problems on the job, then that is the carpenter who would get the job.

Assemblyman Joseph Malone, R-Ocean, Monmouth, Burlington, suggested asking professionals ranging from carpenters to doctors how they actually use algebra skills.

"We need to do a better job at finding out what people actually need to know, not what we think they should know," he said.

Here was my comment:

I thoroughly agree with Prof. Rosenstein that not all students will need the skills/concepts of a more advanced algebra class. While I admire Commissioner Davy's desire to significantly raise the bar for NJ students there are some underlying issues that must be addressed first. How many of you believe that the majority of NJ students have demonstrated proficiency in the foundational arithmetic and prealgebra skills needed to be successful in a legitimate Algebra 1 course, never mind Algebra 2? As a retired math supervisor, believe me, that question was rhetorical!
However, we must clearly distinguish between the issue of a graduation requirement for all and the need for consistent, clearly stated and rigorous standards for a 2nd year Algebra course. Despite opinions to the contrary, I believe the latter is necessary for most college-intending students. The American Diploma Project (NJ is a member of this consortium) has developed precisely those kinds of world-class standards and the result is the new End of Course Test in Algebra 2. This test, which many NJ students have already taken, requires a deeper conceptual understanding of topics such as mathematical modeling which separates the Algebra 2 of the 21st century from the Algebra 2 course many of us remember. And, yes, there are still some mechanical skills which students need to master away from the calculator!
I strongly advocate that NJ adopt these higher standards for those students who will go on to take more advanced math courses. Clearly, it isn't for everyone and therefore we should reexamine it as a grad requirement for all.
Dave Marain

I felt it was important to make a clear distinction between Algebra 2 as a high school graduation requirement and the need for a high-quality curriculum which should be uniform for all students who need to take the course. Many commenter ranted about the evils of testing, the "who really needs algebra anyway" argument, allowing politicians to make educational decisions they know little about (imagine acknowledging that it should be left to math education professionals!) , the skills needed for the 21st century, etc. Fascinating stuff...

Is this same discussion happening in your district or state? Your thoughts are important to me. Do you take strong exception to my comments? Do you agree with the NJ Commissioner of Education or has she gone too far? What do you say to the many adults who argue that, in their occupation, they haven't ever used any of the 'stuff 'they learned in Algebra 2?

Posted by Dave Marain at 7:49 AM 9 comments

Labels: algebra 2 end of course test, algebra 2 standards, issues in math education

[PLS NOTE: Several edits have been made. This version should be accurate!]

Two new documents are now available on Achieve's Algebra 2 Test Overview web site:
Released Items Oct. 2008
Released Items Oct. 2008 Commentaries

I strongly encourage our readers to download these pdf documents. The 2nd document is particularly useful since it contains excellent discussions of each question, scoring rubrics, sample student solutions and detailed explanations and alternate methods.

There will be endless arguments about over-testing, open and fair testing, who's making the profit from these tests, quality and authenticity of assessments, the politics of how results will be used (programmatic vs. comparing teachers, schools, states). In the end, we need to get past the rhetoric. For me the bottom line is that these released questions are high quality and require youngsters to demonstrate both mechanical skill and conceptual understanding. Further, they include several open-ended (extended response each counting 4 pts) and short constructed response items (each counting 2 pts) that give students the opportunity to display what they know, not how skillful they are at eliminating answer choices.

Released items often contain more difficult questions. I found many of these questions required some sophisticated thinking and analysis.

I'm not permitted to reproduce any of the items however I will attempt to categorize each problem and enumerate topics (you may disagree with some of these classifications so please read the documents).

Non-Calculator Section

• Imaginary solutions of quadratic equation

• Graph of inverse of a linear function

• Meaning of rational exponents

• Solving absolute value equation which includes a linear expression outside the absolute values (leading to an extraneous solution)

• Relationship between a polynomial function and its graph

• (Short answer) Solving a quadratic equation resulting from a Pythagorean application

• (Short answer) Constructing the graph of a simple rational function, e.g., f(x) = k/x²

• Recognizing an exponential function from its characteristics (domain, range, intercepts, asymptotes, etc.)

• Determine the slope of a linear-type function involving absolute values

• Graph of a system of linear inequalities

• Associating a function involving the greatest integer function with a problem situation

• Roots of a quadratic equation with a negative discriminant

• Matching the graph of a quadratic function with characteristics involving its coefficients

Calculator Active Section

• (Short answer) Determining an expression for the volume of a cube whose original dimension is increased by a variable amount (also, expand the expression).

• Determining the zeros of an exponential function

• Solving for a variable in a literal equation involving a radical

• Solving an applied problem (physics-type) involving a given quadratic function

• Domain of a composite function

• Simplify a 'complex' fraction (Mechanical skill)

• Analyzing functions of the form cx^d (including end behavior)

• (Short answer) Construct a piecewise function to model a given problem situation

• (Extended Response) Applied problem involving interpretation of a given quadratic model

• Simplifying rational expression (Mechanical skill)

• Recognizing graph of a linear programming application (simple)

• Matching a given exponential function (involving a parameter) to a function table (conceptual)

• Application of concept that the product of a complex number and its conjugate is real

• Exponential growth application

• Analyzing the effect on the zeros of a quadratic subjected to different transformations (conceptual)

• (Extended response) Applied problem involving percent increase and an exponential model

After reading the above and scrutinizing the actual items, I hope you will agree with me that this kind of assessment will have a positive impact on curriculum, instruction and assessment. For those teaching the course, I hope you will find this useful. Let me know...

Link

Posted by Dave Marain at 10:14 AM 0 comments

Labels: achieve, algebra 2 end of course test, american diploma project, curriculum

Addendum: This commentary will shortly be followed by Part II which will focus on some of the following issues:
(a) Why does Achieve stress that the content is AdvancedAlgebra when it appears to be primarily standard Algebra II.
(b) Significant discrepancy in student performance between multiple choice vs. student-constructed and open-ended questions; implications for other standardized tests (do students do better or worse on student-constructed questions on SATs?)
(c) Do the results on this test suggest that Algebra 1 should have been the first such "standardized" test? In other words is the real issue here weaknesses in Algebra 1 background?

Note: Any facts or figures cited below come from the recently released report from Achieve. You will find a link to the full report below. For further background on the exam and links to released questions, link to my post from April 15, 2008.

If your school district participated this past May or June in the first administration of the Algebra 2 End of Course Exam developed by Pearson for the American Diploma Project you already know the results have been published. Nearly 90,000 students from 12 of the 14 states in the ADP partnership participated.

This post will provide an overview of the full report and some commentary. For general information regarding the exam, look here. Click on the next to last link in the right sidebar - it will take you to a new page which provides an overview of the Annual Report for this exam. The first link will give you the full pdf report. If you're familiar with the Exam, go directly to this new page. Also, for an excellent overview and objective commentary, the Achieve group obtained permission to link to the article in a recent Education Week (3rd link down on the report page). You must adhere to the restrictions about reproduction of this article but it's well worth reading.

OVERVIEW/BACKGROUND
When the Calculus Reform group wanted to impact curriculum and instruction in high school (and undergraduate) calculus, how did they do it? They knew the key was to change the AP Calculus Exam: the format, the content, the emphasis (less mechanics, more conceptual, more data-based/modeling open-ended questions, more use of graphing calculator technology).

If NCTM's reforms have not fully been felt K-12 (particularly 7-12), perhaps it's because there is no standardized assessment out there that truly reflects these reforms. It's true that some standardized tests now reflect more problem-solving, data analysis and conceptual understanding, but there's no single powerful test for grades 6-7-8 that will drive change in the classroom. Each individual state has its own independently developed and scored assessment for each grade level now, but the content, difficulty and quality of these tests vary widely. This is why I felt the benefits from the Achieve program far outweighed the potential risks.

Predictably, each time there is a significant change in the AP Exams or the SATs, scores initially drop. This is to be expected and desirable since the appropriate response to this is to understand what needs to be changed in content and instruction. All of the reports and recommendations from the most esteemed mathematics groups/panels have had little effect compared to the more immediate results that follow a drop in scores on some standardized test.

I read the report thoroughly. Passing scores or cutoffs were not determined at this point. Average raw scores and percents were reported for each grade level. It is very hard to draw informed conclusions without an analysis of the questions themselves since the level of difficulty, content and format of these questions are critical factors in performance. Further, scores on a first administration of any standardized test are expected to be lower.

I have not received permission from Achieve to reproduce excerpts so I will summarize major findings.

First I will provide some additional background on the format of the exam itself that you will need to make sense of the results below:

Three types of questions: Multiple-Choice, Short Answer and Extended Response.
A total of 76 raw score points, broken down as follows:

Multiple Choice: 46 questions - 1 pt. ea.
Short Answer: 7 questions - 2 pts. ea.
Extended Response: 4 questions - 4 pts. ea.

Further, the questions are broken into 3 cognitive levels with the majority of questions at Level 2 which "requires students to make some decisions as to how to approaqch the problem or activity."

There was a calculator and a non-calculator part.

For more info regarding the actual topics tested, refer to my link in the first paragraph of this post.

PARTIAL RESULTS OF SPRING 2008 ADMINISTRATION
Based on a max of 76 raw score points, the average number of points scored ranged from a high of 39 points (about 50%) for 8th graders to a low of 16 points (about 20%) for 12th graders with a fairly steady decline from 8th through 12th.

MathNotation Commentary: The decrease from 8th to 12th is easy to explain as the more capable students take the course earlier in accelerated classes. The 8th grade population was of course a very small sample but you get the idea. More significant is the average 24% correct for grade 11, the most common grade for students to take this course (in fact, the number of juniors nearly equaled all of the other grades combined). I'm not surprised by this low percentage for several reasons:
(a) First administration of the test
(b) We already knew there was an issue here or there would have been no impetus for developing uniform standards and a standardized assessment. Are these results so dramatically different from the TIMSS findings? I don't think so. However, there is no cause for alarm. The appropriate response is to provide the data to the states and local districts so that deficiencies can be addressed. I'm not at all concerned about the "Now they'll start teaching to the test" critiques. Those arguments were leveled at AP teachers as well. However, good assessments drive change in content and instruction. Excellent tests can enhance learning -- that's all I ever care about. If this Algebra 2 exam leads to more consistency and higher quality of curriculum and instruction, then everyone should be elated. Unfortunately, each side in the Math Wars will spin the results to make a case for their position. Similarly, Achieve, individual states (governors, state ed departments) will put their spin on it as well. It's up to the reader to become as highly informed as possible to draw her/his own conclusions. Overall, I'm not surprised by the initial outcome.

To be continued...

Posted by Dave Marain at 8:24 AM 1 comments

Labels: achieve, algebra 2 end of course test, american diploma project

Unfortunately, I did not receive permission from Pearson Educational Measurement to reproduce any of the released items from the Achieve website. I will respect their wishes. However, they understand that I plan to discuss some of the items indirectly without specifics. For this to make sense to my readers, you will need to download the pdf document as suggested in an earlier post and have that in front of you as I refer to individual items. Here is the link to the Achieve website that contains the released items (in the sidebar).

For this post, however, I plan to discuss the implications of this exam and related issues.

I would like to invite comments about the issues raised by a common exam that will be administered to students in 14 states (up from the original 9). I consider this to be a highly significant development in the movement toward more standardization for all students. Up to this point, the only similar kind of interstate standardized test covering Algebra 2 topics has been the SAT Subject Test - Math I. Several states now give their own end of course exams in Algebra I and Algebra 2, but the Algebra 2/ADP exam from Pearson is impacting on students from many states and this is just the beginning of this trend. For example, a similar exam for Algebra I is already under development (if not already completed) and, here in New Jersey, it will become operational shortly.

Furthermore, textbook publishers such as Pearson have already begun to publish texts (e.g., Pearson, Algebra I, 2007) that correlate with the American Diploma Project's Algebra standards (download this). Imagine that! Instead of inserts in the text that correlate to fifty different sets of state standards, we are now going to see some consistency. Glory Hallelujah!

Here are just a few of the issues that each state will have to confront as these exams proliferate:
1) How will a student's grade be determined in the course? Will the exam be worth a percentage of the final grade? What if the results do not come back in a timely fashion?
2) If a student falls below the minimum level of proficiency on the exam (great euphemism/edu-jargon for 'failing'!), how will this be recorded on their transcript?
3) Will the exam be required for graduation just as current graduation tests do in some states?
4) Can a student re-take the exam if they don't make it the first time?
5) Should the Algebra I exam have been developed and implemented before this exam?
6) Will different groups and consortia now compete to develop curricula and assessments independently as a result of the recommendations from Achieve/ADP, the National Math Panel and NCTM's Curriculum Focal Points? More splintering?
7) Will there be more than one administration of these exams each year? For the Algebra 2 Exam, the answer is found on the Achieve website:

The exam will be administered at the end of fall—December and January and at the end of spring—May and June beginning with the 2008-2009 school year.

Ironically, while this exam improves consistency of curriculum, there might not be as much consistency about how these issues will be addressed. Of course, one can hope...

Are these exams intended to impact on instructional methods, emphases, strategies, techniques just as currently occurs on the AP Calculus Exam? If you read through the pdf document and examine the Released Items document (p.4), the answer is clear for the Algebra 2 Exam (the following is an image which will appear blurry);













Your thoughts...

Posted by Dave Marain at 7:07 AM 4 comments

Labels: achieve, algebra 2 end of course test, american diploma project, pearson, standardized assessment

With the first operational test running from May 1-June 13, 2008, I'm sure that the participating districts are being given regular updates and materials to help their students prepare for the exam. Just in case you haven't checked the exam website recently (the released items appeared in March I believe) or you are curious about seeing some sample items, just click here.

You will need to click on Released Items in the right sidebar and a pdf document will be downloaded to your desktop. This is a well-designed 45-page document providing a wealth of useful information for teachers, administrators and students. It is far more than a collection of sample exercises. In particular, page 45 provides an actual breakdown of the exam in matrix form, showing how many of each of the 3 types of questions (Multiple-Choice, Short Answer and Extended Response) there are for each module tested. A total of 76 raw score points are possible broken down as follows:

Multiple Choice: 46 questions - 1 pt. ea.
Short Answer: 7 questions - 2 pts. ea.
Extended Response: 4 questions - 4 pts. ea.

Further, the questions are broken into 3 cognitive levels with the majority of questions at Level 2 which "requires students to make some decisions as to how to approaqch the problem or activity."

Here's a quick overview of the content covered in both the core exam and the optional modules:

The Algebra II end-of-course exam will consist of a common core, which will be taken by students across all participating states. This core will cover a range of algebraic topics that are typically taught in an Algebra II course, and fall into five content standards: 1) Operations and Expressions 2) Equations and Inequalities 3) Polynomial and Rational Functions 4) Exponential Functions and 5) Function Operations and Inverses.

In addition, seven optional modules will be available to states to enrich the core with content that is important to colleges and employers alike. These include: 1) Data and Statistics 2) Probability 3) Trigonometric Functions 4) Logarithmic Functions 5) Matrices 6) Conic Sections 7) Sequences and Series.

Initial Reaction from MathNotations

Since I've always been a firm believer that required exams have a major impact on what is covered and how a course will be taught (e.g., the AP Calculus Exam), the released items in this document will be scrutinized by instructors, supervisors, etc. From my experience with many other standardized tests (e.g., state tests), released questions tend to reflect the more challenging aspects of the test.

That being said, my immediate feeling was that the questions reflected considerable traditional content.

However, the impact of reform was felt strongly in the extensive discussion following each item. In addition to traditional approaches, solutions were provided that demonstrated the use of multiple representations, solutions by graphs and tables (using a graphing calculator). The discussion following each item is the most important part of this document, IMO.

NOTE: In its stated calculator policy, Achieve recommends the use of a graphing calculator. They were discreet in not requiring it, however, as that would get into equity issues. Advanced QWERTY-type calculators (such as the TI-89) are not permitted.

The released questions included many that I would rate of average to above-average difficulty.
They also include some challenging items along the lines of Math I from the SAT Subject Tests. I've contacted Pearson, asking for permission to reprint some of these items for discussion purposes. I'm awaiting their response...

I urge my readers to download the document and share their reactions. Again, specific items should not be stated verbatim in your comments as I do not yet have permission for this.

Posted by Dave Marain at 2:54 PM 4 comments

Labels: achieve, algebra 2 end of course test, pearson, released items

Math teachers, math supervisors/curriculum leaders/specialists as well as students and parents should be interested in reading the latest information regarding this assessment that can be downloaded from the achieve.org website. School leaders should have already received this during the summer when it became available. The links are given below. Several months ago I posted information on this historic consortium that was formed via the American Diploma Project and Achieve. Nine states signed up to participate in this project, including my home state of NJ. Several districts in my state signed up to be part of the pilot project and students in these districts will be taking the assessment in the spring of 2008. There will also be a Field Test this October. As anticipated, a detailed enumeration of the core content for this assessment is now available and can serve as the basis of a syllabus for Algebra 2. From a cursory reading of the core content and the test specifications, I am impressed by the thoroughness of the development team. Moreover, I am struck by the balance struck between a traditional and reformed view of math curriculum and instruction. For example, the test will be in 2 parts, one allowing a graphing calculator and one without. Further, the core content repeatedly refers to student proficiency with multiple representations of functions - graphs, tables, verbal and symbolic expressions - what I've been calling the Rule of Four. Function modeling and problem-solving are considered central themes in the curriculum and the assessment will reflect this.

The core exam will have 60 questions, 50 of which are one-point multiple choice questions, 7 short answer (2 pts. each) and 3 free-response questions (4 pts. each). The non-multiple choice questions therefore account for about one-third of the point value of the test, a significant portion. In addition, schools, districts or states can select optional modules which will expand the assessment. These include Data and Statistics, Probability, Logarithmic functions, Trigonometric Functions, Matrices, Conic Sections and Sequences & Series. Since most algebra 2 curricula include several of these additional topics, I would expect some states to eventually select some of these.

There is no doubt in my mind that this is a positive step for raising the level of mathematics curriculum and instruction in these states and, hopefully, across the country if other states sign on (which I'm sure they will). The test will be developed by Pearson. I will be very interested in reading sample assessment questions when they become available. They will have a definite impact on change in curriculum and instruction. Teachers in courses leading up to Algebra 2 as well as the courses depending on Algebra 2 skills and concepts should also be vitally interested in this project. Algebra 2 is viewed as the gateway course to higher mathematics. The purposes of this project and the assessment are clearly stated in the document you can download (see link below).

Important Links:
Algebra 2 Exam Content and Overview from Achieve
Note: Click on the links under Related Content on the right side. The second link will download the complete pdf document enumerating the standards and detailed test specs. Save this document and share it with your colleagues at upcoming department and school meetings!

ADP Algebra 2 Benchmarks (with examples)

Achieve and the American Diploma Project Home Page

Posted by Dave Marain at 6:21 AM 0 comments

Labels: achieve, algebra 2 end of course test, algebra 2 standards, american diploma project

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