MathNotations

Much has been happening in the world of mathematics and mathematics education. I'm only scratching the surface here.

• The passing of Professor Mandelbrot -- There is no question that this man has left an eternal "singularity" in the profession. Who among us has not been mesmerized by the computer images generated by one of his creations. He dared to think different and was not always recognized or lauded for his uncanny knack of seeing patterns no one else could. When asked to look back on his career, Dr. Mandelbrot compared his own trajectory to the rough outlines of clouds and coastlines that drew him into the study of fractals in the 1950s.

“If you take the beginning and the end, I have had a conventional career,” he said, referring to his prestigious appointments in Paris and at Yale. “But it was not a straight line between the beginning and the end. It was a very crooked line.”

[埋込みオブジェクト:http://www.youtube.com/v/aIj30SOoIDM?fs=1&hl=en_US]

• I read so many tweets/posts on Twitter, Facebook and in blogs discussing whether or not to continue teaching particular traditional topics in an algebra, geometry or precalculus class. Curriculum supervisors, math department chairs/supervisors and therefore teachers should now be guided by two key documents, particularly for Algebra 2 content:
• The Common Core State Standards Initiative for Mathematics
• ADP/Achieve Algebra II Test Overview
• You may also want to download the document from Glencoe which does a nice comparison of required content to the their text.

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

"You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

Posted by Dave Marain at 8:41 AM 1 comments

Labels: algebra 2 standards, mandelbrot, update

The deadline for registering for the First MathNotations Math Contest on Tue Feb 3rd is drawing to a close but you still have an opportunity to register! Look here for details and email me if you want a team of your students to participate!


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

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

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....

Posted by Dave Marain at 7:44 PM 5 comments

Labels: achieve, algebra 2, algebra 2 standards, american diploma project, standardized assessment

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