More Challenges/SAT Practice, Core Curriculum Standards, Reminders, Comments...

Additional SAT/Contest/Challenges

Challenge 1:

HOW MANY DIGITS OF 1000¹⁰⁰⁰ - 1 WILL BE EQUAL TO 9 WHEN THIS EXPRESSION IS EXPANDED?

Challenge 2:

HOW MANY 5-DIGIT POSITIVE INTEGERS HAVE A SUM OF DIGITS EQUAL TO 43?

Challenge 3:

Jorge can run a 6-minute mile while Alex can run a 5-minute mile. If they start at the same time, how much less distance, in miles, will Jorge run in 10 minutes?

(Yes, you can respond with answers and solutions to these in the comments!)
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Tired of hearing about THIRD MATHNOTATIONS FREE ONLINE MATH CONTEST!? IF I RECEIVE 10 MORE REGISTRATIONS, I MAY JUST STOP!
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The Common Core State Standards Initiative
First look here for a quick overview and here for an index to the latest draft of the standards. Of course, this blog only discusses the mathematics part of the document.

Overview

The Common Core State Standards Initiative is a joint effort by the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) in partnership with Achieve, ACT and the College Board. Governors and state commissioners of education from across the country committed to joining a state-led process to develop a common core of state standards in English-language arts and mathematics for grades K-12.

These standards will be research and evidence-based, internationally benchmarked, aligned with college and work expectations and include rigorous content and skills. The NGA Center and CCSSO are coordinating the process to develop these standards and have created an expert validation committee to provide an independent review of the common core state standards, as well as the grade-by-grade standards.

HIGHLIGHTS

• Core Concepts and Core Skills
• 11 Core Standards including the new "Mathematical Practice":
  
• Mathematical Practice
• Number
• Quantity
• Expressions
• Equations
• Functions
• Modeling
• Shape
• Coordinates
• Probability
• Statistics

• Each standard is broken into Core Concepts and Skills, provides research-based evidence and many illustrative examples to clarify the language
  
• Alignment of these standards to those of 5 representative states: California, Florida, Georgia, Massachusetts and Minnesota
• Standards reduce the number of Core Concepts and Skills in accordance with many recommendations to pare down the number of required topics to allow for greater depth

Example of a Standard (Standard 5)

Equations | see evidence
An equation is a statement that two expressions are equal. Solutions to an equation are the values of the variables in it that make it true. If the equation is true for all values of the variables, then we call it an identity; identities are often discovered by manipulating one expression into another.

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs, which can be graphed in the plane. Equations can be combined into systems to be solved simultaneously.

An equation can be solved by successively transforming it into one or more simpler equations. The process is governed by deductions based on the properties of equality. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, stimulating the formation of expanded number systems (integers, rational numbers, real numbers and complex numbers).

A formula is a type of equation. The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b₁ + b₂)/2) h, can be solved for h using the same deductive process.

Inequalities can be solved in much the same way as equations. Many, but not all, of the properties of equality extend to the solution of inequalities.

Connections to Functions, Coordinates, and Modeling. Equations in two variables may define functions. Asking when two functions have the same value leads to an equation; graphing the two functions allows for the approximate solution of the equation. Equations of lines involve coordinates, and converting verbal descriptions to equations is an essential skill in modeling.

Core Concepts
Students understand that:

1. An equation is a statement that two expressions are equal.
   see examples

2. The solutions of an equation are the values of the variables that make the resulting numerical statement true.
   see examples

3. The steps in solving an equation are guided by understanding and justified by logical reasoning.
   see examples

4. Equations not solvable in one number system may have solutions in a larger number system.
   see examples

Core Skills
Students can and do:

1. Understand a problem and formulate an equation to solve it.
   see examples

2. Solve equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality.
   see examples

3. Rearrange formulas to isolate a quantity of interest.
   see examples

4. Solve systems of equations.
   see examples

5. Solve linear inequalities in one variable and graph the solution set on a number line.
   see examples

6. Graph the solution set of a linear inequality in two variables on the coordinate plane.
   see examples

FUNDAMENTAL ASSUMPTIONS AND CONSIDERATIONS

Very Important!
(Click on image to see a clearer view)




























INITIAL MATHNOTATIONS REACTIONS

1. Exceptionally clear and definitive document
2. Influenced by NCTM (Curriculum Focal Points), Achieve, College Board, ACT
3. Illustrative examples are of high quality
4. Will serve as a basis for states' revisions of current standards hopefully creating more consistency than currently exists
5. Leaving curriculum to local districts and states was a politically necessary decision, however, in my opinion, developing a reasonably consistent curriculum by grade level and/or course across districts and states from these standards may prove to be difficult and may again lead to considerable disparity. Hopefully, this will be self-correcting when standardized assessments are created as is currently being done with the End of Course Tests from Achieve
   

ADP/Achieve Algebra I Practice Test Now Online! Links, Discussion...

This is not an April Fool's joke! As anticipated and mentioned previously on this blog, a complete practice test is now available for you to download in pdf format from the Achieve web site. Click on the bottom link on the right sidebar.

Student expectations are shown above (you may need to click the image to see a larger version).



From the site:

The ADP Algebra I End-of-Course Exam consists of algebraic topics which will be taken by students across all participating states. These topics are typically taught in an Algebra I course, and fall into four strands: 1) Operations on Numbers and Expressions 2) Linear Relationships 3) Non-linear Relationships and 4) Data, Statistics and Probability.



Composition of Test
47 operational items

• 40 multiple-choice (1 pt ea)

• 5 short answer ( 2 pts ea)

• 2 extended response (4 pts ea)

I'm not permitted to reproduce any of the questions here but I will enumerate some of the topics tested:
Identifying linear functions (y = 2x vs. y = 2^x vs. y = 2x²)
Choosing effective data displays
Absolute value graphs
Rationalizing denominators in radical expressions (square roots only)
Exponential functions (simple)
Graphical interpretation of systems of equations

For further discussion, click Read more below.


and some more...
Solve quadratic equation by factoring
Graph of linear inequality
Probability of independent events
Determine vertex of graph of a quadratic function
Interpretation of slope from problem situation (rate of change)
Solving linear equations
Identifying irrational numbers (square root form)
Multiplication of radical expressions
Solving absolute value equations

Download the entire document to see examples of the multiple-choice, short answer and extended response type questions. This should prove very useful for teachers and students as they prepare for the first administration of this test. From the partial listing of topics above you can see that this is a test reflecting an ambitious first year algebra curriculum which will surely raise the bar for those states and districts participating. I would expect most students will struggle the first time around with the content, format and a level of difficulty they may not have yet experienced. Over time and with experience students should perform at higher and higher levels. This is a learning experience for all of us and it should be viewed as an opportunity to be part of an exciting change in American mathematics education...

Additional Comments

• Even if your district is not participating, this test is an excellent way to measure your Algebra curriculum against a highly regarded world-class standard. I strongly encourage you to include these sample questions in warmups, for classroom discussion or review for other assessments.
  
• As nontraditional or difficult as some of the earlier problems may appear to be, read through the entire sample test. You will definitely see many traditional algebra exercises with which your students should feel comfortable.
• Problem 31 is an interesting application of the Pythagorean Theorem and quadratic equations. Some students have a knack for guessing 'special' right triangles for these however. After discussing the algebra, remind them to consider multiples of 3-4-5. Since the legs differ by 4, it's worth trying sides of 3x4 and 4x4, then checking if the hypotenuse works. It does!
• The test reflects an excellent blend of tradfitional and reform. The extended response and short answer questions should have a definite impact on instruction and assessment in your classes whether or not your district is on board with this test. Ultimately, I predict EVERY state will be on board!
  
  Your comments...
  

...Read more

Updates Squared...

Update: My RSS feed seems to have been repaired. I want to thank Denise for contacting me with helpful info...

• I think there may be a problem with my RSS Feed. Please email me (dmarain at geeemail dot com) or post a comment here to let me know if you're still getting my feed or if there has been any interruption. Also indicate which feed reader/aggregator you are using (Live Bookmarks, Google reader, Bloglines, etc.). Since Google purchased Feedburner I've noted some quirky behaviors of late...

• I decided to place an SAT Math Tip of the Day in the sidebar. If any of your students (or students who happen to visit) find it helpful, let me know. I'm also working on an SAT Problem of the Day (different from the College Board), but that will require some work to make the math symbols appear correctly.

• Blogger, unlike Wordpress, doesn't make it easy to have expandable posts (aka, Read More...). There are some solutions out there from techies so I hope to have this feature up soon. This would be particularly helpful for challenge problems where I can "hide" the solution until you click on something. This is done very well in the Math Problems of the Day in the sidebar but the solution is not easily transferable to the layout in Blogger.

• Have you looked at the Math Problem of the Day in the sidebar recently? Some excellent algebra and geometry problems which are quite challenging. They're along the lines of math contest problems, some close to AIME questions or even higher. Some require careful derivation or justification and the solutions are done well. I have noticed a few repetitions in the problems but, overall, the quality is good. I'll have more to say about the source of these later on...
  
  
• Achieve/American Diploma Project/Algebra I
  As soon as released items/practice test for Algebra I is posted, I will let my readers know. My understanding is that this should be coming very soon, considering that the test will be operational in May! Anyone in your state participating? I would strongly recommend looking into this for your district as the benefits will be great IMO:
  
  • Helps to standardize your Algebra curriculum and insure consistency of content and instruction; helpful for programmatic review
    
  • Aligned with most state standards and may well serve as a basis for national algebra standards
    
  • May become a mandated test in the future in some states
    
  • Individual report forms can identify students with potential weaknesses who may require additional remediation or can even be used for placement purposes
  • My understanding is that levels of proficiency will be set this summer and that reporting will be more timely the following year
  • Excellent practice for students in taking standardized tests
  • Early exposure to this type of testing usually results in better performance later on
    
  • Contains both calculator and non-calculator sections consistent with current recommendations for assessment
  • Contains objective, short-answer and open-ended questions, again consistent with most recommendations for assessments
    

Odds and Evens 2-6-09: Contest Updates, ADP Algebra EOC Tests, "Numb and Numberer", a MathNotations Podcast?

MathNotation's First Math Contest
The Feb 3rd contest is 'virtually' over! We had a dozen schools participate including high schools, middle schools, a HomeSchooling team, a math fraternity team and a team from the UK! Our British team has had to delay the contest until they dig out of the deep snow that's been blanketing and paralyzing the UK for several days. I sure hope I am not in some way responsible for that!

The contest has been well-received although much of it proved too advanced for our middle schoolers. I will post results and selected problems in a few days after I sort it all out. I am planning on one more free contest in the spring. This one will probably be restricted to secondary teams. If interested, let me know early on. I am also planning 4 contests for the 2009-10 school year. Details about dates, registration and fee will be forthcoming.

"Is trillion the new billion?"
What are the odds that a CNN feature story appeared a few days after the Super Bowl post I published - Was the First Super Bowl More or Less Than a BILLION Seconds Ago?

Heres' an excerpt from this wonderful piece on CNN.com:

"To put a trillion dollars in context, if you spend a million dollars every day since Jesus was born, you still wouldn't have spent a trillion," McConnell said.

CNN checked McConnell's numbers with noted Temple University math professor and author John Allen Paulos.

"A million dollars a day for 2,000 years is only three-quarters of a trillion dollars. It's a big number no matter how you slice it," Paulos said.

Here's another way to look at it.

"A million seconds is about 11½ days. A billion seconds is about 32 years, and a trillion seconds is 32,000 years," Paulos said. "People tend to lump them together, perhaps because they rhyme, but if you think of it in terms of a jail sentence, do you want to go to jail for 11½ days or 32 years or maybe 32,000 years? So, they're vastly different, and people generally don't really have a real visceral grasp of the differences among them."

Everyone is tossing around the words million, billion and trillion. With the national debt now topping $10 trillion, following a $700 billion bank rescue and proposed $800 billion-plus stimulus package, have we become numb to the numbers?

Prof. Paulos is making an excellent case here for our middle schoolers developing better sense about large numbers. This is another view of the Number Sense standards which have been promoted for over two decades. Ah, the Power of Ten! Gee, that would be a cool name for a game show...


Update on ADP Algebra I, II EOC Tests
Do you frequently check the Achieve website to get the latest info on these tests, revised benchmarks and latest developments. For a current overview on the Algebra I test, look here.
I don't yet see released items or a set of sample questions but there is some very helpful information there, in particular the Algebra I Expected Knowledge document you can download. This can serve as the basis for curriculum development and provides sorely needed consistency. It will help to address questions like: Should we still teach those mixture problems? How much emphasis should we place on factoring quadratic trinomials?



MathNotations Podcast
Ok, what's this nonsense about podcasting? Another venture that I will try and not follow through on? Perhaps, but I'm never afraid to challenge bold frontiers that people my age are not supposed to consider! I would like to offer MathNotations visitors another way to connect with issues in math education. Once I get this set up, I'll let you know. There are a few technical obstacles I need to overcome but I'm getting close.

A sample of what I'm considering -- A 10-15 minute 'broadcast' you can subscribe to and download to your iPod or computer.

Here's the first topic I'm preparing:
Using current technologies to provide supports for our students outside the classroom.

What does that mean? Well, we all know that many students, who actually want to complete their math homework, become frustrated and give up when they can't get started or hit a wall. OR it's the night before a quiz, test, midterm, final, SAT and there are a couple of problems they can't do. To whom do they turn? Parents? Friends?
What are the Web 2.0 options?
Email, chats, forums, free help sites, mathcasts, videos on YouTube? Parents and students want to know what is available and I'd like to open up a dialog here and make some recommendations of those avenues that are cost-effective (aka 'free'!) and are effective for learners (i.e., high quality). Not every parent can afford high-cost tutoring, so where to turn? I can post recommended links but a podcast allows more informal discussion. I may also be able to have an audio dialog with some of the more knowledgeable people in these areas.

Our next Mathanagram? Prof. Rademacher to Infinity and Beyond?
Well maybe....
It's getting harder to keep this going so, for now, enjoy his picture!
Of course, you could visit MathNEXUS, and the Math Person of the Week challenge in the sidebar. Prof. Johnson, of Western Washington University, who maintains this exceptional site, provides new weekly goodies for all math educators. A truly great resource...

ADP Algebra 2 EOC Exam Released Items Update Oct 2008

[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...

Achieve/ADP Algebra 2 End of Course Exam Report/Findings and MathNotations Commentary - Part I

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 Advanced Algebra 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...

Discussion of Algebra 2 End of Course Exam from ADP/Achieve/Pearson - Continued...

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...

Released Items for Upcoming End-of-Course Exam for Algebra 2- Achieve/ADP

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.

Algebra 2 End of Course Exam - Latest Info

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

Nine States to Administer Standardized Algebra 2 Exam

NJ Commissioner of Education, Lucille E. Davy, announced on 4-10-07, that New Jersey is one of nine states that will administer a common exam for Algebra 2 students in May 2008. A link to the actual Algebra standards that will be used is given near the end of this post.

Here is an excerpt of her statement:

New Jersey Joins Nine-State Partnership to Administer New Algebra II Exam

New Jersey has agreed to join a partnership of nine states in the American Diploma Project Secondary Math Partnership to administer a common exam with common standards for Algebra II students beginning in May 2008. The eight states joining New Jersey in the partnership are Arkansas, Indiana, Kentucky, Maryland, Massachusetts, Ohio, Pennsylvania and Rhode Island. The project is an initiative of the ADP Network, a group of 29 states that educate more than 60 percent of all American public school students.

“This new exam will help to ensure that our children are learning the math skills that are becoming more and more essential in an increasingly competitive job and secondary education marketplace,” said Commissioner of Education Lucille E. Davy. “Our work in the American Diploma Project (ADP) Secondary Math Partnership complements our efforts in New Jersey to re-design our schools to meet the challenges of preparing our young people for the demands of the 21st century.”

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Those of you who have been reading my posts for the past four months know that I am elated at this news. I had mentioned the possibilities of this happening a couple of months ago and now it's gaining momentum. Common exam, common standards, common curriculum,...

Of course, we will need to see how this plays out. When it becomes mandatory, will all NJ high schoolers be required to pass this in order to get credit for the course? Does this mean that all NJ students will eventually be expected to take Algebra 2 in high school and does this imply that those students taking more basic level mathematics will have to reach much higher than now? Will it follow a Regents model as in NY? If a student does not pass the exam, what appears on their transcript? NYS educators can probably provide insight into how this is handled in their state. From my understanding of the American Diploma Project, developed by Achieve.org, ALL youngsters need to and will be strongly urged to take Algebra 2. For those intending to continue their education, this is not an option.

I do believe this is a big step in the right direction, however, Commissioner Davy must surely recognize that there is a demographic of youngsters who, at this time, do not yet have the skills to tackle this course. Perhaps she and the others in the consortium believe that this will force the math curricula in Grades K-8 to be significantly upgraded and compel high schools to begin to phase out low-level math courses for most students. Not that there's anything wrong with that but we're certainly not there yet!

For those interested in the Algebra benchmarks (standards) developed by Achieve.org, read this. I looked it over and I like its structure, clarity, examples and content. I need to consider more carefully whether it encompasses all of the important topics in Algebra 2, but, on first glance, it looks good. Their goal and mine is to make our children more competitive and to upgrade the quality of education for ALL of our students, particularly underrepresented groups in our society. After looking these over, let me know if you see any omissions in these benchmarks.

Update on my view of the benchmarks: After reading both the geometry and algebra 'standards' and some of the sample postsecondary problems more carefully, I detect a more traditional flavor with some newer content sprinkled throughout. Those who remember learning from or teaching from the Houghton Mifflin Dolciani series for Algebra 1/2 (Structure and Method if I recall correctly) may feel nostalgic. This is more of an instinct than a careful analysis, so feel free to correct me....