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Session Title

Thoughts On Rationalizing Algebra In Ways That Serve Kids, Not Universities


Steven Leinwand, Principal Research Analyst, American Institutes for Research


The day before CMC-North I was trading notes with our lead counselor, just swapping stories about kids, when she mentioned a student who was at the end of her turn at the local community college. She'd be transferring to a state college to complete a liberal arts degree if it weren't for a failing grade in Algebra II. Because she can't yet perform long division on polynomials, she'll have to postpone her degree in (just guessing here) linguistics a full year.

Leinwand opened his talk: "The great divider of our time is the Algebra II final exam. Algebra II squeezes off options for so many kids. Algebra II is anathema to all but the top 20% of the population. My premise: as currently implemented, high school algebra I and II are not working and not meeting either societal or student needs."

He described the courses as "focused on increasingly obsolete and useless symbol manipulation at the expense of functions, models, applications, big ideas and statistics."

He works with schools across North America and when he's trying to get a feel for the tenor and rigor of their math programs, he asks for:

  • the courses they teach,
  • their course descriptions,
  • the books they use,
  • the balance of course enrollment,
  • last year's final exams for every class.

He said they give him unrestricted access to the first four but balk at the fifth. He said, "if you want to engage people in discussion, go and get those finals."

Leinwand asked, why are most Algebra II final exams balanced towards the verbs:

  • Simplify,
  • Solve,
  • Factor,
  • Graph.

… when math is ever so much more about being able to:

  • Find,
  • Display,
  • Represent,
  • Predict,
  • Express,
  • Model,
  • Solve,
  • Demonstrate.

Lynn Steen: As mathematics colonizes diverse fields, it develops dialects that diverge from the “King’s English” of functions, equations, definitions and theorems. These newly important dialects employ the language of search strategies, data structures, confidence intervals and decision trees.

Leinwand: "No one is saying throw out the old dialect, but what about the new dialect."

This all came across depressingly but he ended on a hopeful note, citing several promising projects. Among them, The Opportunity Equation, which aims to:

… explore the feasibility of offering a mathematics pathway to college for secondary students that is equally rigorous to the calculus pathway and that features deeper study of statistics, data analysis, and related discrete mathematics applications, beginning with a redesigned Algebra II course.

He called the forthcoming Common Core math standards "the last, best hope" for meaningful math reform. He ended with a proposal for Algebra I and Algebra II curricula, paced at one chapter per month.

Algebra I

  1. Patterns.
  2. Equations.
  3. Linear Functional Situations.
  4. Representing Functional Situations.
  5. Direct and Indirect Variation.
  6. Data.
  7. Systems of Equations.
  8. Exponential Functions.
  9. Linear Programming.

Algebra II

  1. Review and Reinforce Big Ideas and Key Skills of Algebra I.
  2. Quadratic Functions.
  3. Polynomials and Polynomial Functions.
  4. Patterns, Series, and Recursion.
  5. Exponential and Logarithmic Functions.
  6. Rational and Radical Functions.
  7. Probability and Statistics.
  8. Optimization, Graph Theory, and Topics in Discrete Mathematics.


PowerPoint. Black text on a white field. He introduced his slides with this, "These are terrible slides coming up. You want to read PowerPoint slides that break every rule of PowerPoint these are them."

I felt sick. Leinwand had attended my PowerPoint: Do No Harm talk last year and I could only hope he hadn't added that disclaimer on my account. He was wrong anyway. He used his slides as conversation pieces. Doesn't matter to me that they were monochrome.




  • There is a gentleman at the table across from me murmuring and nodding agreement at Leinwand's every line. It would not be inappropriate to describe the atmosphere in this session as something like religious conversion.
  • New rule: "Legislators can't require a test that they themselves don't take and publish the results of on their websites."
  • If you're looking for an example from Leinwand of the "old dialect," here's one: rationalizing roots in the denominator of fractions. Here's another: the conjugate in the same context. Can anyone make a case for that?
  • One of "the most honest and important documents in our business in the last five years": the $3.1 billion budget State Superintendent Jack O'Connell submitted in response to Governor Schwarzenegger's pressure to make Algebra I an eighth-grade standard.

BTW: Fantastic follow-up from Josh G.

All of this just highlights the real problem: universities and colleges want a gatekeeper. They want that extra way to filter admissions, because they have to do it somehow. Worse, they don’t want to be seen as the “easy” school to get into, because this lowers their respectability. (This also drives me crazy.) So they demand gatekeepers, whether or not those gateways are actually a more useful math education for their students.

BTW: I have attached Leinwand's slidedeck here.

22 Responses to “Asilomar #1: What Do We Do With Algebra II”

  1. on 10 Dec 2009 at 7:58 amSue VanHattum

    In college math, Algebra I becomes Beginning Algebra, and Algebra II becomes Intermediate Algebra. The state of California requires that anything which counts as a college level math course have Intermediate Algebra as a prerequisite.

    I agree with you and Leinwand – the course as it is now conceived is terrible. It acts as a filter when we claim that’s not our purpose.

  2. on 10 Dec 2009 at 8:10 amtoss255

    Are there any examples of what a person who couldn’t pass Algebra II is capable of doing in the new paradigm? We’ve already seen tests dumbed down, but statistics is no joke either. I have seen people with advanced degrees make unfounded claims because they didn’t realize one can’t always fit a straight line to a set of points. Just handing someone a graphing and statistics package doesn’t make them a qualified analyst.

  3. on 10 Dec 2009 at 9:17 amElissa

    Let’s just all get together and make our own curriculum. All your blog lurkers and commenters. We throw every state’s standards in there. Take this guys premise of a chapter per month. Split up into groups. Start building on what we already have and creating what we don’t. Then we teach it. We instantly have tons of resources to go to for help, ideas, extensions, feedback, and critique. Let’s do it.

  4. on 10 Dec 2009 at 10:13 amSteven Peters

    I love the part about Legislators not being able to require a test that they can’t take and pass. Is that a good metric for testing in general? Should we be able to validate our testing by giving it to “professionals” and seeing how they score? For example, giving an algebra II final exam to mathematician, physicist, chemist, biologist, psychologist, and sociologist (drawn from the strip at ). Each of those fields uses different types of math more frequently than others. If the professional sociologist or psychologist can’t pass an algebra II exam, is that a suitable requirement for their field? (No offense intended to those professions, it’s just my understanding that they use statistics much more frequently than polynomial long division).

    That would be a good experiment. Of course the professionals might not want to take the test and be embarrassed by a low score, but it could perhaps be anonymized. What do you think?

  5. [...] 10, 2009 From Dan Meyer’s report on the “Thoughts On Rationalizing Algebra In Ways That Serve Kids, Not Universities” session from a recent math conference: The day before CMC-North I was trading notes with our lead [...]

  6. on 10 Dec 2009 at 11:17 amGlenn

    Can we get a copy of those slides? It would be interesting to compare my bad slides with those bad slides and see where I am at in comparison.

    I am realizing I am stuck in the “paradigm” of Algebra II, and I must find a way out!

  7. on 10 Dec 2009 at 4:19 pmKate Nowak

    I agree! Consider me convinced. Thanks for disseminating his very sensible plan to a wide audience. Algebra 1 and 2 have been in sore need of a sanity check, and better alignment, less overlap for – I don’t even know. Decades, I would bet.

    I just don’t know who actually needs to be convinced in order to make it happen! In NY in 2005, it was supposedly these people. But from what I hear, the input of the classroom teachers involved was more or less ignored. In NY we can’t just design and teach whatever courses we want, because in Alg 1, Alg2, and Geometry there are statewide final assessments, and a kid’s score determines the fanciness of the diploma she is awarded. I don’t have a list or anything, but I imagine many other states have moved to such a model since NCLB.

    So who really makes those decisions? Does anyone know? I would like to know. I imagine a shadowy network of bureaucrats and test publishing executives.

  8. on 10 Dec 2009 at 5:39 pmClint H

    I’ve been out of the California teaching game since 2000, but what has happened to the Integrated Mathematics revolution? Have all/most districts gone back to a traditional model of Algebra 1 – Geometry – Algebra 2 – PreCalc/Trig – Calc?

    I, too, agree with the idea of requiring legislators or other decisions makers (especially college admissions folk) to take the tests that they require. If they don’t need it in their position, then why do students need it to in to their college or university, particularly when that course has no bearing on their proposed plan of study?

  9. on 11 Dec 2009 at 7:11 amMr. H

    Leinwand asked, why are most Algebra II final exams balanced towards the verbs:

    * Simplify,
    * Solve,
    * Factor,
    * Graph.

    … when math is ever so much more about being able to:

    * Find,
    * Display,
    * Represent,
    * Predict,
    * Express,
    * Model,
    * Solve,
    * Demonstrate.

    Is solve supposed to be on both lists?

    My guess is there’s a verb-focus on finals because it’s easier to grade the first 4. Even if you don’t go for multiple choice, it’s still easy to give partial credit for work shown. The 2 lists above is more about convergent and divergent thinking/production. Students should be developing both types of skills in the classroom. Convergent production limits the outputs (easy to grade). Divergent production can produce many outputs (in too many formats to easily grade reliably).

    If the question is why algebra 2 classes have a verb focus. My guess would be that algebra are the foundational skills for the more interesting application questions in calculus.

    I think your example gives a good reason for why algebra 2 shouldn’t be a requirement for attending a state college (for some majors). The solution is to change the admission requirements not dropping long division from algebra 2.

    I do agree with finding alternative rigorous mathematics pathways to college.

    New rule: “Legislators can’t require a test that they themselves don’t take and publish the results of on their websites.”

    It’s easy to beat up the legislators, but that is not a good policy. I did smile when I first read it.

    If you’re looking for an example from Leinwand of the “old dialect,” here’s one: rationalizing roots in the denominator of fractions. Here’s another: the conjugate in the same context. Can anyone make a case for that?

    Easy for the teacher to grade :D

    Gotta run to work. Might have a post down the line.

  10. on 11 Dec 2009 at 7:15 amMr. H

    I didn’t mean to imply that you’re suggesting dropping long division from algebra 2.

  11. on 11 Dec 2009 at 8:57 amGlenn Kenyon

    Great Post! I couldn’t go to Asilomar this year, but you really brought it home to me. I particularly liked the Algebra 1 and II syllabi as they reflect a lot of what I have been thinking about for these past years as a teacher of 8th grade algebra.

    I hate teaching rational expressions, and yet, if I don’t, my students aren’t placed in the next class after Algebra (Geometry or Math 2) in their freshman year. They then feel like they weren’t taught correctly.

    The vicious circle of demanding more and more out of developmentally in sync kids needs to be challenged.


  12. on 11 Dec 2009 at 10:11 amRiggins

    Harvard recently revamped their required math course for non-math and non-science majors. Instead of a theory of numbers type of course, their curriculum review suggested they implement a statistics/probability curriculum instead. The suggestion was made based on input from what was needed in non-mathematics based careers.

    I wonder how often high school curriculum review is done at the state level. And the question is by whom? Seasoned math educators or industry leaders in conjunction with human resources and hiring personnel?

    I do agree that there has been curriculum modifications in Algebra 2. Here in TX, the first semester of Algebra 2 is a repeat of major Algebra 1 concepts. And the 2nd semester doesn’t handle Algebra 2 concepts until midway starting with Rational & Functions in March.

  13. on 11 Dec 2009 at 7:50 pmClint H

    If the question is why algebra 2 classes have a verb focus. My guess would be that algebra are the foundational skills for the more interesting application questions in calculus.

    For a large number of students, Algebra 2 is the end of their mathematical careers. Shouldn’t it be looked at as more than just foundational? And can’t the course be interesting enough on its own, rather than just be a prerequisite for calculus?

    @Riggins Have you seen Arthur Benjamin’s recent TED talk about changing math education?

  14. on 12 Dec 2009 at 10:25 ammonika hardy

    great post.

    i like elissa’s idea. i’m there.
    at least as long as we’re still making a standard curriculum. (i’m bent on kids making their own. i know – i know – just a bent i have.)

    kate is smart – we need to figure that out.
    the power we now have access to via networking – can’t that make a big enough impression – to enough of those people we are going to need to impress?

  15. on 12 Dec 2009 at 2:55 pmNewteach

    I’m with Mr. H:
    I think your example gives a good reason for why algebra 2 shouldn’t be a requirement for attending a state college (for some majors). The solution is to change the admission requirements not dropping long division from algebra 2.

    We’re still trying to make everybody good at everything. That wasn’t the model for most of history. We accepted that some people like and do better at some things than others. We encouraged some sort of foundational skills in all areas and then…let them advance and excel further in the things that interested them, whether it was math or science (providing the motivation for math too) or literature or basket-weaving or inventing or…

    Then again, perhaps this is 3.5 months of middle school math in a high poverty, inner city school speaking. Right now, I’m most interested in developing *thinking* skills. And by that, I’m talking about basics of thinking: taking a minute to picture the problem, or really understand what the question is asking, etc. Harder, of course, when many kids are reading below grade level by several grades and their ability to read aloud is not an indication of their comprehension, which is usually lower. Of course, I’m just as concerned with their scores on the state test, by which we’ll all be judged. The thinking skills certainly can’t hurt them on that, but there are those who would prefer not muddying the water with too much thought.


  16. on 13 Dec 2009 at 4:57 pmDan Meyer

    If anyone is interested, I have attached Leinwand’s slidedeck.

  17. on 13 Dec 2009 at 5:48 pmA. Mercer

    Math tests for legislators is so funny! I would be surprised if 25% could pass the state Algebra I exam, let alone II.

    My husband went to Boalt Hall. Law school is the top pick for those on the policy and politics track, which was why he went there. He’s also a math-phobe claiming an inability to do math once he got to fractions. The only math required for law school admissions at that time (LSAT) was a section of logic problems similar to those you find in Dell Puzzle magazines. It was all logic, no numbers or symbols.

    One lovely spring day in Berkeley the students were “off-task” in some truly boring first year class (Civil Procedure, Tax Law, or Contracts). The teacher asked what they were all doing there at one of the top ranked law schools in the country if they weren’t interested in law? One student raised his hand to share this nugget, ‘I didn’t have to do math.”

    That will be a super majority of elected officials. If they went to school beyond their Bachelors (likely in PoliSci, with just a glossing of statistics), they went to law school, and they went there to become a politician, and because they didn’t have to do math.

  18. on 14 Dec 2009 at 11:38 amjosh g.

    Hey, thanks for the highlight!

    I hope people are at least mostly tongue-in-cheek about testing legislators … I mean, if we want to just say “stop mandating standardized tests”, I’m all for it, but let’s not doom our students to receiving only enough math knowledge to be a *politician*.


  19. on 15 Dec 2009 at 8:41 pmAndrew

    Leinwand’s slides talk about the Common Core State Standards Initiative.

    Dan, was your impression that he liked those or was he using them as an example of the old school.

    Was there a set of standards he suggested using as a model?

  20. on 16 Dec 2009 at 4:40 pmDan Meyer

    From the post:

    He called the forthcoming Common Core math standards “the last, best hope” for meaningful math reform.

  21. on 04 Jan 2010 at 2:07 pmBurt

    Bad Algebra curriculum and bad college requirements are on one end of the issue. There is another pole. I am reading an amazing book,Mindsets by Carol Dweck. She quotes Benjamin Bloom, “After forty years of intensive research on school learning in the United States as well as abroad, my major conclusion is: What any person in the world can learn, almost all persons can learn, if provided with the appropriate prior and current conditions of learning.”

    I did volunteer tutoring at a community college a few years ago–non-credit math classes for those who didn’t pass the placement exam for Math 100. It sure seemed like a waste of time for those students to have to take the Algebra classes “learning” things that they won’t ever think about again. I decided I would have a greater impact tutoring elementary school students instead. This is the other pole.

    I don’t think we are addressing the real cause of why so many students are failing math placement exams at the college level; in my state it was 84% in 2007! If we had math entrance exams for middle school (not something I am endorsing), we would see the same problem; in my state, 63% of fourth graders and 75% of eighth graders were below grade level on the NAEP Math test. The majority of elementary school teachers are poor at math. We need to push for math specialists in elementary schools and open up math to a far more students.

  22. [...] yeah, I can't freaking believe they counter-programmed me against Steve Leinwand, who has never disappointed whenever he's turned up on my conference schedule. If you're flipping a coin between [...]