Integrated School Integrated School Mathematics:Mathematics:

Pitfalls, Products, Pitfalls, Products, and Predictions for and Predictions for

the Peach Statethe Peach StateKeary Howard Ph.D., Keary Howard Ph.D.,

SUNY FredoniaSUNY Fredonia

keary.howard@fredonia.edu

Some 60 Minute ObjectivesSome 60 Minute Objectives• Define integrated mathematics via a

few good hooks.• Put you in a New York State of mind.• Discover the real Dana TeCroney.• Tell the fortune of Georgia school

mathematics.• Get you ready for some SEC football.

A truly integrated mathematics curriculum relies on rich A truly integrated mathematics curriculum relies on rich problems that:problems that:

1. Prompt solutions between and within fundamental 1. Prompt solutions between and within fundamental mathematical content areas.mathematical content areas.

2. Foster multiple representations.2. Foster multiple representations.

3. Make you wanna solve ‘em.3. Make you wanna solve ‘em.

Hook: How many solutions are there to the equation:

2x = x2

Some NYS School Math Fast FactsSome NYS School Math Fast Facts• NYS is home to some the largest school districts (New York City

School District), smallest school districts (Clymer Central School District), and the richest (the Hamptons).

• It’s called the Empire state for a reason. Driving time from Clymer to the Hamptons is approximately the same as it is from Clymer to Columbia, SC or from Clymer to Knoxville, TN.

• Since the 1950s, all school districts in NYS have been united by a common high school mathematics curriculum.

• Since the 1920s, common assessments (Regents Exams) have been a staple of the secondary school exam.

• An Integrated Mathematics Curriculum at the high school level was first piloted 30 years ago. Although significant revisions have been made to the curriculum on at least two occasions, Integrated Mathematics remains firmly in place in NYS.

Riddle me this…Riddle me this…• Rank order the following three

artifacts from greatest to least on their influence on the ‘taught’ curriculum.

A. State Standards DocumentB. National TextbooksC. State-Mandated Exit Exams

Integrated Mathematics Revisited Via a Integrated Mathematics Revisited Via a Clymer PirateClymer Pirate

• 9th-11th grade mathematics takes on weird and blatantly non-descriptive names such as Course I, II, and III or Sequential Math I, II, or III.

• The topics taught are expansive and often re-examined from year to year. In the 1990s typical NYS freshmen and sophomores were exposed to: logic, transformational geometry, combinatorics, as well as more traditional curricular material such as algebra and Euclidean Geometry.

• Texts needed to be created specifically for NYS students, since the curriculum was so different from those in other states. They varied from the traditional to very progressive. Each school district is free to choose its text. The decision often rests with the department chair or a committee headed by the department chair.

• While schools are free to choose texts, all schools must give the same exit exam. Course I, II, and III exams are created by teams of teachers, secured, and administered at the same date each year. The exam serves as the ‘final’ for each student and scores (both individual and district) are reported to a state clearinghouse

Some Examples of 1990s Integrated Some Examples of 1990s Integrated Math Regents Exams in NYSMath Regents Exams in NYS

• You can trace the history of NYS Integrated math through its exit exams:• http://www.nysl.nysed.gov/regentsexams.htm

Course I Sample 10 Point QuestionsCourse I Sample 10 Point Questions

Some Basic Theory on Curriculum Some Basic Theory on Curriculum ImplementationImplementation

Posner’s Five Concurrent Curricula:– 1. Official Curriculum– 2. Operational Curriculum (Taught and Tested)– 3. Hidden Curriculum– 4. Null Curriculum– 5. Extra Curriculum

Posner, G. (1992). Analyzing the Curriculum. McGraw-Hill.

Four Critical Tasks of Teachers:– 1. Content Coverage (Breadth)– 2. Content Mastery (Depth)– 3. Classroom Management– 4. Positive Affect

“Teachers will adapt or transform an unsuitable curriculum in such a way that they can make the curriculum fit the classroom realities.” (Posner, p.187)

NYS Integrated Math, NCTM Standards, and NYS Integrated Math, NCTM Standards, and NCLB = Bad DateNCLB = Bad Date

• The last decade has been one of critical and significant change in NYS school mathematics.

• Now all students would be required to pass the new state-mandated Math A exam to receive a high school diploma. These new exit exams would eliminate the high school level Integrated Mathematics Course I, II, and III offered annually in June. Now two exams (Math A and Math B) would cover the material of three.

• Three critical changes were made that affected the curriculum:– Standards were raised requiring all students to pass the Integrated

Math A exam.– The exams moved from criterion referenced to norm-referenced.– Choice was eliminated in what questions student could answer. In

addition, much more emphasis was placed on open-ended constructed response problems

Take the ‘L’ Out of Lover ‘Cause That’s What We Take the ‘L’ Out of Lover ‘Cause That’s What We Are: An Integrated Math A Exit Exam FiascoAre: An Integrated Math A Exit Exam Fiasco

• In June of 2003 a Math A Regents Exam produced the greatest uproar of any NYS exam to date.

• Two thirds of all students failed the exam, many of whom planned to graduate.

• The exam was poorly written, verbose, and some thought overly difficult.

• State ed commissioner Richard Mills immediately voided

the exam results and established a panel to review the exam and provide direction for the future.

• The Math A panel quickly produced a list of recommendations for Commissioner Mills. These included the development of a new exam, new piloting procedures, and (most importantly) the establishment of new mathematics standards with specific grade performance indicators.

Two Samples from the 35-Question, Two Samples from the 35-Question, Four-Part, 20-Page ExamFour-Part, 20-Page Exam

A ‘Honey Do’ List for Georgia’s Best A ‘Honey Do’ List for Georgia’s Best Math EducatorsMath Educators

• Do work hard to establish a set of math-specific standards and content performance indicators for each grade level.

• Do create state-wide curricular material that can be used in the creation of Georgia-specific texts.

• Do entrust your teachers with the creation of high-quality exit exams that accurately reflect your curricular content and philosophy.

• Do look to the GCTM for leadership in professional development and oversight of curricular materials, curricular implementation, and exam creation.

• I do like your chances – you’ve got great students, teachers, schools, and universities.

In the process, please don’t sacrifice your mathematical soul. In the process, please don’t sacrifice your mathematical soul. Remember, a truly integrated mathematics curriculum relies Remember, a truly integrated mathematics curriculum relies on rich problems that:on rich problems that:

1. Prompt solutions between and within fundamental 1. Prompt solutions between and within fundamental mathematical content areas.mathematical content areas.

2. Foster multiple representations.2. Foster multiple representations.

3. Make you wanna solve ‘em.3. Make you wanna solve ‘em.

I Challenged the Goliath

Following ‘Fright Night’ at SFOG and a midnight run on the Goliath, you ask yourself a simple question: Are the tallest coasters, the fastest?

Another McGyver Miracle?

McGyver claims that he can find the height of any object with only a protractor and mirror, while never leaving the ground. Describe, in theory, how this is possible. Use diagrams and your understanding of geometry, similarity, and proportion to provide a detailed explanation of the mirror mystery.

SEC Showdown: Defense Wins Championships

The legendary Vince Dooley has a question for you: How many points does it take to guarantee a win in SEC football? Suppose that winning 75% of SEC divisional games is typically the benchmark for big-time bowl eligibility.

Closure: Helmet Sticker QuizClosure: Helmet Sticker Quiz

• Q1: How many solutions are there to the equation: 2x = x2

• Q1 Answer: 3

• Q2: What curricular product is likely to have the greatest impact on the ‘taught’ curriculum?

• Q2 Answer: State-Mandated Exit Exams

• Q3: Posner argues that teachers will adapt the official curriculum to fit the needs of four critical teaching tasks which include coverage, mastery, positive affect, and what fourth critical component?

• Q3 Answer: Classroom Management

• Bonus: What’s the greatest ‘Break-Up’ Line of all time?

• Bonus Answer: Take the ‘L’ out of ‘Lover’ ‘cause that’s what we are…