Wednesday, April 29, 2009

NY Times Article on AP Courses Raises Critical Issues

The article published 4-29-09 is entitled, Many Teachers in Advanced Placement Voice Concern at Its Rapid Growth. Look here for the complete text. With the AP Tests only a few days away this is certainly timely and will surely generate many reader comments on the NYT website. I'm sure it will also appear in Education Week and many other print and online media. The article refers to a study done by the prestigious Thomas B. Fordham Institute and the opening paragraph really says it all:

A survey of more than 1,000 teachers of Advanced Placement courses in American high schools has found that more than half are concerned that the program’s effectiveness is being threatened as districts loosen restrictions on who can take such rigorous courses and as students flock to them to polish their résumés.
Having taught AP classes for a few decades I have strong feelings about the equity vs. excellence issues surrounding these classes but I would rather raise a series of questions and ask my readers to provide some data as well as voice their opinions.

Trevor Packer, the Vice President of the College Board, was very pleased by the "questions the report asks" and welcomes the dialog. I'm not so sure, however, that he is pleased by the majority sentiment of AP teachers that the quality of these classes has been compromised by encouraging more and more students to take classes for which they may not be prepared.

My sense is that like most complex issues we need to first determine who is doing what out there. I believe that there is wide disparity among districts regarding entrance into these classes and whether AP tests are required or not. The College Board has attempted to address some of these issues with their new AP Audit process but I wonder how effective that really is.

These questions are being posed to AP Calculus teachers or those familiar with how their district handles these decisions.

1) What stated prerequisites (e.g., in a Program of Studies), if any, are there in your district for entrance into an AP class?
For example:
"Students must earn a grade of B or better in Precalculus and/or be recommended by their teacher."

2) Is the test mandatory for all students in the class? If a student does not take the test do they still receive AP credit for the class? (This is more relevant of course, transcript-wise, for Juniors, Sophs and Freshmen).

3) Who pays the $86 for each AP test?
Note: I'm guessing that parents in most cases pick up the tab but, if a hardship can be documented, the district would pay. I've also read that in some states the cost is covered by the state or district.

Click Read more... for questions regarding your opinions.

Some more questions/thoughts...

4) Do you believe that strict entrance requirements are needed to insure the quality of the course or that enrollment will ultimately be affected by the reputation (quality, expectations and grading standards) of the teacher?

5) Your other thoughts...

...Read more

Tuesday, April 28, 2009

Strong Opinions from Curmudgeon Re Math Education...


An innocent recent post, Two SAT-Type Percent Problems Appropriate for Middle School as well..., generated some intense and passionate opinions about the current state of US education, math in particular, from one of our readers, affectionately known as Curmudgeon (I'll abbreviate his name as CM).
Unless you subscribe to my Comments feed (see link in sidebar), you might not have followed this dialog so I've decided to post a link to these comments and to copy some of the more provocative ones in this post.

Based on his comments I find CM to be highly informed, articulate and not intimidated by anyone. He eloquently challenged my difficulty rating system for the SAT-type problems I have posted and he made some excellent points. Re curriculum/instruction, his perspective is somewhat similar to mine although I lean a bit more toward a balanced view. A direct quote describing his blog summarizes his beliefs very well.

"I believe that mathematics should be taught, not collaboratively explored; algebra and geometry are better than a vague course of Integrated Math; spiraling doesn't work nearly as well as learning it properly the first time; "I don't DO math" should be an incentive rather than an excuse. "I don't DO English" should be treated the same way."

I encourage you to visit his blog, Curmudgeon.

Below is CM's long response to my remarks about the teaching of the part:part construct which, IMO, receives far less emphasis in classrooms than part:whole approaches.

Rarely taught? Probably, for three reasons. The first is that teachers are under pressure to "finish" things, to check off the standard and move on, to "get through" the material. As soon as the kids "know" the material, it's time to move on. There is rarely time for the extensive exploration that I seem to recall from my own days. Standards-based educational theory says that you need to pick and choose your topics until you get each kid to the understanding point but says nothing about total mastery. "Drill and kill" is an epithet. "Drill and Practice" is unknown. Many people have also fallen for the "spiraling" fad and never quite complete a thought before they're on to the next one. "We'll spiral around to this again in that module in next year's course" is probably the dumbest thing ever to come out of edumacation colleges. I can't tell you how many times (because I've lost count) I and the other teachers have been told that we needed to get our "bubble" kids over the line into the passing zone. "To hell with knowing math, just know enough for the test" seems to be the rallying cry. The upshot is that you can mention these things to the better students who will get it easily and gain an even better understanding. The weaker students just trundle along. The second reason is that weaker students are resistant to trying a second or third idea. They want to understand THIS one. A second approach is confusing. It takes a while before they get comfortable with multiple approaches and some never get much beyond "I'm not a good math student. I'll learn this but only to a point." It takes a determined teacher to ease them into this, but she can't have anyone breathing down her neck to do test-prep. The worst reason is that a fairly large percentage of our teachers don't really understand math to the level required. They've either bought into failed and worthless education theory or they simply are stupid. I've related the story of the fourth grade teacher in the in-service this year ... Me: "You say you want to be a guide on the side not a sage on the stage, yet you're teaching 4th graders. You still need to teach them things. They have to memorize 4*3 for example." Her: "No. We should be teaching them how to look that up on Google. Did you know that Google will give you the answer to that? It's true." She leaned back, satisfied that she had put one over on me. Is it any wonder that her kids arrive with no understanding of fractions, decimals, percents, operations? "Where's the fraction key?" "Sorry, that calculator doesn't have one, the problem you're doing doesn't need one and you wasted more time than if you just looked at it and solved it." And some elementary and middle-school teachers "just don't DO math, tee-hee-hee." They SEEM to do math - they have lots of test-prep bubbling exercises from the publisher of the math book, but they fundamentally don't understand the nuances of the material. "Let's see what the calculator says." In a different in-service this year, the presenter was showing how pre-schoolers and elementary kids learn math. She had lots of visuals and was "teaching" the teachers as if they were students. The aides were getting questions wrong and the elementary teachers weren't doing too much better. This explains a lot.

To read my response (aka, "rant'), click on
To read all the comments to this post and/or post a comment, look here.

May I call you C.M.??
I'm not going to say something inane like "I feel your pain." Most of the angst you expressed is a reflection of what I was feeling when I retired. You're describing a system that is in dire need of systemic change, not tweaking.

I will not apologize however for my advocacy of national standards in math. It is unconscionable that students across the country are not learning the same content - concepts, skills, procedures, terms, definitions,... This is truly inequitable.

However, I have also been preaching "LESS IS MORE" for the same period of time. Finally, NCTM has taken this position with their Curriculum Focal Points document. William Schmidt (of TIMSS renown) stated this obvious fact 15 years ago when he described our math curriculum as "one inch deep, one mile wide."

We cannot expect students to learn math well if we fill our textbooks each year with every topic under the sun. If, for example, our teachers could focus on the essential ideas of ratios and ALL students were required to solve a range of problems from the basic to the more challenging, then most students would eventually learn ratios and be able to handle fractions. BUT facility with ratios and fractions requires facility with division which requires mastery of multiplication, etc.

"Jack, you will learn the times tables. The facts you got wrong in class today, you will have to write five times each for homework tonight. Tomorrow, I will ask you to answer just those." I know there are some teachers out there who are doing this. Is everyone?

While base ten blocks, unifix cubes and learning software have their place, we all know that nothing replaces repetition. Some students need far less than others but all students need some.

Yes, C.M., there are serious teacher preparation issues out there. Read my comments at the bottom about Finland. Yes, C.M., I share your feelings about spiraling, although there are aspects of spiraling which make sense.

I am also concerned as you are that standards-based learning has devolved into learning only for the state assessment. While we are moving toward national standards, we have to rethink how we will evaluate student learning and the bottom line is: "WHAT IS THE REAL PURPOSE OF TESTING?" What you and I and millions of other teachers see is that testing has little to do with helping children improve. It has everything to do with POLITICS:


Remember that quote:
"In other countries, education is seen as an investment; in the US, it is seen as an expense."

Perhaps this administration will "see" it differently. I truly hope so. In Finland, teachers, have to take additional years of training beyond college before they officially are certified. This additional 2-3 years culminates in a year of working in a laboratory school with real students. A true internship. And, by the way, THE GOVERNMENT PAYS EVERY PENNY FOR ALL THIS ADDITIONAL TRAINING. This is how Finland has turned around its system in the past 20 years.

INVEST IN EDUCATION, INVEST IN OUR CHILDREN, INVEST IN OUR FUTURE. Don't look for short-cuts, folks. There are none. Expedient solutions lead to students who only care about the bottom line, the grade, not about learning. This "get results without really earning it" mentality is pervasive in our society. In the worst case, this mentality produces the AIGS, the Enrons and the Madoffs of the world.

Ok, now you got me to rant too. I guess I needed that catharsis. otherwise it sounds like I'm just pontificating about challenging our best and brightest with all these problems I'm writing. But there's so much more to it, C.M...

P.S. I have a feeling that your comments and mine are not being read by most of my readers. I'm thinking I should copy them into a separate post and really incite a riot! I think I will do that unless you state an objection! THANK YOU!

...Read more

Monday, April 20, 2009

ODDS and EVENS Week of 4-20-09

  • Don't miss the latest Math Teachers at Play #5. Denise somehow is pulling this together and producing a delectable and high-quality carnival of K-12 Math every two weeks and this one is no exception. I particularly enjoyed reading her intriguing quotes. One of my favorites came from Paul Halmos who quoted a cynic on student learning. Here's an excerpt...
    "...since, he said, students on the average remember only about 40% of what you tell them, the thing to do is to cram into each course 250% of what you hope will stick"
  • I've already received several requests from teachers for the registration form for MathNotation's 2nd FREE Online Contest. One request has made me re-think my rule about one team per school. So here is the latest revision:
    Schools may field up to TWO teams.

    A couple of other points about the contest...
    (1) Even if you're not sure you will be able to field a team, don't hesitate to email me (dmarain at gmail dot com) to request a registration form - no obligation here!
    (2) I want to reiterate that this contest is designed for the student who has completed Algebra 2 and knows some trigonometry.
    (3) Please include the phrase 2nd MathNotations Contest in the subject line of your email to me. Just copy and paste that to make it easier for me to organize all of my emails and not miss any.
  • Walter Isaacson, former managing editor of TIME magazine has written a definitive piece on the National Standards movement and the need for improved quality of standards in this week's edition of TIME. It's entitled How to Raise The Standard In America's Schools and I strongly urge all my readers to read the online version at

Click Read more for further comments on the TIME article...

This one article justifies purchasing the print version of the magazine in my opinion. Mr. Isaacson documents the history of the Standards movement but more importantly makes two compelling points which my readers will surely recognize from the dozens of essays I've written on the subject:
(1) The existing standards are not working (most are of poor quality and there is considerable inconsistency of expectations from state to state).
(2) There is, not surprisingly, a significant gap between the proficiency levels being reported by many states (based on their own assessments) and the NAEP results which set a higher and more uniform standard.

How do I feel about my passionate beliefs and advocacy for uniform standards beginning to come to fruition two decades after I ranted on every forum for these changes? Relief, not self-satisfaction. It's long overdue and we better move quickly now. Right, Mr. Duncan?
...Read more

Friday, April 17, 2009

Classic Exponent Challenge for SATs, Algebra 2, Math Contests...

Don't forget to register for the upcoming 2nd MathNotations Free Online Contest for secondary students. Click here for more info.

The first 4 terms of a sequence are 2, 6, 18, and 54.

Each term after that is three times the preceding term.
If the sum of the 49th, 50th and 51st terms of this sequence is expressed as k⋅349, then k = ?

Click Read more to see the answer, solution, discussion...

Answer: 26/3

Suggested Solution
The first three terms can be written as
2(30), 2(31) and 2(32). (***)
In general, the nth term is 2(3n-1).
The sum of the 3 desired terms would then be 2(348) + 2(349) +2(350). Factoring out 349, we obtain 349(2/3 + 2 + 6) = (26/3)(349), so k = 26/3.

(1) Too hard for SATs? Similar (but slightly easier) problems have appeared on the test.
(2) Could students use the "Make it simpler strategy" here to reduce the problem to the sum of just the first three terms? But this is the essence of geometric sequences (or exponential functions):
From (***) above, this sum would be
2(30) + 2(31) + 2(32) = 2 + 6 + 18 = 3(2/3 + 2 +6) = 3(26/3). The coefficient 26/3 would be the same for any three consecutive terms! Is this concept/technique worth developing?

...Read more

Wednesday, April 15, 2009

MathNotations Second Math Contest (FREE) Announced!

Silly Joke For Today
Have you heard the very sad "fractioned" fairy tale?
Six out of seven dwarfs are not 'happy.'

Announcing MathNotation's Second Free "Online" Team Math Contest!

If interested in participating, please send an email as soon as possible to "dmarain at gmail dot com." I will then email a registration form. Your initial email expresses only your interest. You are under no obligation.

There will be several changes from the first contest.

1) Team advisers may administer the contest at any time during the week of May 18th-22nd.
2) Advisers must email the registration form no later than Fri 5-15-09.
3) The contest is designed for secondary students who have completed Algebra 2. Some trigonometry may be necessary. I would not recommend middle schoolers take the contest unless they have completed or are completing Algebra 2.
4) Questions include topics from geometry, algebra, trigonometry, discrete math, etc. No calculus...
5) Teams must consist of from two to six members. Homeschooling and international teams are welcome!

For more details, click Read more.

Questions types include
(a) Short constructed response (students enter only numerical answers)
(b) Open-ended requiring detailed work and explanations
(c) Multi-part questions


1) Team members must complete the contest within 45 minutes on the same day.
2) Advisers must email the official answer form the same day the contest is administered. Scanned solutions will be accepted.
3) Any scientific or graphing calculator is allowed.

...Read more

Sunday, April 12, 2009

Number Theory, Logic, Proofs and Patterns for Middle School and Beyond...


The following is a series of apparently straightforward arithmetic problems for middle schoolers. However, the objective is to have students justify their reasoning beyond "guess and test" methods. Proving there is only one solution or none requires more careful logic using algebra as needed. Students will need some basic algebra for the "proofs." For the younger student, modify these questions to have them find the squares in questions 1,2,3 and 5. Take this as far as you wish...

In the following, square refers to the square of an integer. Justify your reasoning or prove each of the following.

(1) There is only one square which is 1 more than a prime.

(2) There is only one square which is 4 more than a prime.

(3) There is only one square which is 9 more than a prime.

(4) There is no square which is 16 more than a prime.

(5) There is only one square which is 25 more than a prime.

(6) Can one generalize this or not??

Click Read More for selected answers, solutions...

Selected Answers, Solutions

(2) If n2 is 4 more than some prime, p, then we can write
p = n2 - 4 = (n-2)(n+2). Since p is prime, the smaller factor must be 1, so
n-2 = 1 or n = 3. Thus, there is only one square, 9, which is 4 more than the prime, 5.

(4) p = n2 - 16 = (n-4)(n+4). There n would have to equal 5, n2 would equal 25 but 25 - 16 = 9 is not prime.

(6) If there were a general rule would that mean we'd have a formula for primes?

Your thoughts about these questions...

...Read more

Thursday, April 9, 2009

A Unique Math Blog and Website

I'm sure many of my readers out there are aware of Tanya Khovanova. Including her blog in my blogroll is long overdue and was a result of ignorance on my part. I've corrected that oversight and, now that I have discovered and enjoyed some of her essays, I want to share this with the rest of you. Brilliant (winning a Gold Medal in the IMO might suggest that!) and witty, her writing style is original and provocative. BTW, we share a passion for linguistic puzzles although I don't consider that unusual for those with a math bent. I strongly recommend you visit her website, Tanya Khovanova's Home Page as well as her blog. Her site contains a rich mine of creative and challenging problems. One of my favorite 'puzzles' is called Fly Droppings On Your Pizza. Click on Read more for further details...

Her most recent article, Multiple Choice Proofs, is a fascinating discussion of how to improve assessments (math contests in particular) which tend to be multiple choice here in the US and proof-oriented in her native Russia. Tanya suggests a compromise between those two extremes ("balance!") offering some alternatives one of which would involve artificial intelligence (AI) to evaluate proofs. I've always believed it might be possible to evaluate one's writing using an algorithmic approach but the obstacles certainly seem formidable at this time.

I also posted a couple of comments on her "About" page. I included a challenge mental math problem for her talented son, Sergei, to attempt. It may not appear for awhile as her comments are moderated. I may post it here as well...

...Read more

Wednesday, April 8, 2009

A Recurring Problem for SATs (Functions)

SAT "Grid-In" Type
Level of Difficulty: 5 (High)
Content: Algebra 2, precalculus

The function F satisfies the condition
F(N + 6) = F(N) + 8, for all integers N.
If F(7) = -2, what is the value of F(25)?

Click on Read more to see the answer, solution, discussion.

Answer: 22

Suggested Solution:

Replace N by 7 since F(7) is known:
Therefore, F(7 + 6) = F(7) + 8 or F(13) = -2 + 8 = 6

Next, replace N by 13 since F(13) is known:
F(19) = F(13) + 8 = 6 + 8 = 14

Finally, F(25) = F(19) + 8 = 14 + 8 = 22.

(1) Too difficult for the SATs? Not really! A similar problem recently appeared. There aren't that many "hard" questions (Level 5) on the SAT but, if a student wants to score over 700 they will need exposure to these types in practice.

(2) Consider writing some variations of these function-type problems for additional practice. At first, change the constants, then consider changing the operations (from addition to multiplication for example). One could raise the bar even higher by asking the question in reverse:
If F(25) = 22, what is the value of F(7)?

(3) There is considerable advanced theory in functional equations and recurrence relations underlying these problems. However, the student needs only to feel comfortable with the function symbolism (or should I call it "Math Notations!"). Starting by "plugging in' N = 7 seems simple in retrospect but most students are too intimidated to consider it. Even the precalculus student may be able to get started, but, without experience, they will often get lost. This is all about exposure, but isn't it always?

(4) One could rewrite this problem using sequence notation:
aN+6 = aN + 8. By expressing the problem in the context of the Nth term of a sequence, students may grasp it a bit better, but, in the end, it's all about interpreting function notation.

...Read more

Monday, April 6, 2009

Math Teachers at Play #4, Krypto, Updates (Odds and Evens),...

1) The latest "biweekly" edition of our new Carnival, Math Teachers at Play, is currently being hosted by Misty over at Homeschool Bytes. I enjoyed the "step by step" approach, progressing from primary math activities like Candy Math through middle school posts like Division of Fractions Conceptually to secondary articles like Ten 16th Century word problems. I contributed a post on Function Questions for the SATs. I'm not including any direct links to these articles. Go to Misty's site to enjoy this carnival!

2) I'm putting a lot of effort into the SAT Math Tips feature in the sidebar (readers of my feed won't get to see this of course unless they visit the site). I realize a good part of the country takes the ACT but the suggestions may be applicable to any standardized test and the math content is valid for anyone who wants to use it.

3) The Read More feature I recently instituted has some bugs as I'm sure you noticed. It doesn't work in RSS or Atom feeds of course and it sometimes doesn't work properly even on this site. I'm a coder at heart but I'm not sure it's worth all the effort. Clearly the Blogger developers are not interested in making things easy for us. You may see Read more... even if there is nothing else to see or it may not work all! Too late for me to migrate to Wordpress at this point... Please be patient with me here as I work through this.

4) The Math Problems of the Day in the sidebar are of good quality and are challenging but I'm seeing more repetition of questions. I will evaluate this and decide if I want to keep this feature.

5) No, I haven't forgotten about the next MathNotations contest I promised for April or May. Stay tuned...

6) Ever hear of Krypto or 24 (the game not the TV show!) or the more well-known "Four Fours Game"? I played many years ago and have always enjoyed the challenge of this tantalizing arithmetic game. I've used it effectively in upper elementary and middle school classrooms when I was a math staff developer to reinforce order of operations and basic fact recall (no calculator allowed!). Ok, so I'm now driving you crazy with another KenKen-like game!!
Click on Read more to learn more. (if this works!).

Here is a sample play of Krypto:

Suppose you're dealt the following five number cards:
5, 9, 4, 11, 1.
A 6th or objective card is turned up, say 2.
Using some or all of the four basic arithmetic operations and the 5 numbers exactly once, produce the objective number. This one is straightforward using only addition and subtraction:
11 + 5 - 9 - 4 - 1 = 2. Krypto! You've Won!

It is possible to be dealt an "impossible" hand for which there is no solution or there could be many solutions for the same hand! The original game did not allow the use of parentheses but you could choose any variation you wish, including reducing the number of cards to 4. An ordinary deck could be used modifying the face cards to be 11,12, etc.

If you want to play the online version from mphgames, go here for instructions and play. Your browser must be java-enabled but most are. For more background on the game and a discussion of the underlying combinatorial mathematics and a discussion of the computer program which generates it, look here.


...Read more

Friday, April 3, 2009

Two SAT-Type Percent Problems Appropriate for Middle School as well...

Version I (Level of difficulty 3 - medium)

With a special promotion, Al received a 60% discount on a new stereo system and paid $x. Sylvia bought the same system (same original price) but only received a 20% discount. In terms of x, how much (in dollars), did Sylvia pay? Assume x > 0 and disregard sales tax.

(A) 4x (B) 3x (C) 2x (D) 4x/3 (E) x/3

Version II (Level of Difficulty 4 - medium/hard)
Grid-In Type

Maury purchased a new electronic game system with a 25% off coupon. His friend bought the same system (same original price) with a 40% off coupon. If his friend paid $45 less for the system, how much did Maury pay (disregard sales tax)?

For the answers, suggested solutions, strategies and discussion, click Read more...

Level I problem
Answer: (C) 2x

Possible Solutions:
Method I ("Plug-in" SAT Strategy - Student-preferred?)
Let original price = $100.
Then Al's discount was $60, so he paid $40. Thus x = 40.
Sylvia's discount was $20, so she paid $80, which means she paid twice Al's price or 2x.

Method II (conceptual)

Al paid 40% of the original price, Sylvia paid 80%, therefore Sylvia paid twice as much as Al.

Method III (traditional - Algebraic)
Reasoning as in Method I, Al paid 40% and Sylvia paid 80% of the original price.
Let y = original price (before discounts).
Then Al paid 0.4y = x. Solving, y = 2.5x.
Sylvia paid (0.8)(2.5x) = 2x.

Level II Problem
Answer: $225
Note: There is a mental math method which will be discussed later.


  • What are some of the factors which might make the second problem more difficult?
  • Should these types of percent questions be included in prealgebra texts for middle schoolers?
  • The first question appears to be fairly straightforward. Which part of the problem would cause the most difficulty for students in your opinion?
  • Note that in the ever-popular 'plug in a number method', the ever-popular $100 was chosen for the original price. Can you think of an occasion where you might not want to substitute $100 in a discount problem?
  • Students also need to recognize that I plugged in $100 for the original price rather than replace x by $100. They often feel they must substitute a number for the variable given, which, in this case would lead to more work.
  • There are other approaches here. Please share different methods you have used!
  • How would we assess their learning of the concepts here? Include one of these on the next quiz or test? Use these as warmups occasionally? Develop a worksheet of similar problems for homework?

...Read more

Wednesday, April 1, 2009

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 = 2x 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