AMATYC Presentation: Missing Link — Replace Intermediate Algebra

Here are the files and documents for the November 14 presentation called “The Missing Link: Algebraic Literacy to Replace Intermediate Algebra”.

main presentation: The Missing Link presentation AMATYC2014

References and Curricular Model: References_NewLifeSession_AMATYC2014

Algebraic Literacy Goals and Outcomes: Algebraic Literacy Goals and Outcomes Oct2013 cross referenced 2 by 2

Sample Lesson on rates of change: Algebraic Literacy Sample Lesson Rate of Change Exponential 2 by 2

Here is that sample lesson as full-page: Algebraic Literacy Sample Lesson Rate of Change Exponential

Extra handout: MLCS Goals and Outcomes Oct2013 cross referenced 2 by 2

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A Natural Approach to Negative Exponents

For today’s class in our beginning algebra course, we took a different approach to negative exponents.  The decision to do something different is partially rooted in my conviction that most of our textbooks are wrong about what negative exponents mean.

To set the stage, the first thing we did was a little activity on basic properties of exponents.   The activity is based on this document Class 22 Group Activity Exponents

This activity uses the type of approach many of us use for a more active learning classroom.  I suspect mine is not as polished as many; several students found the ‘long way’ a bit confusing.  As usual, I did not present any of the ideas before students got the activity and worked in their small groups.

One of the problems on this activity Example for negative exponents Nov2014is the problem shown here.  In the ‘long way’ method, students easily wrote out the factors and found the answer.  Quite a few of them used the subtraction method to create a negative exponent.  In a natural way, we noticed that m^-4 is the same as having m^4 in the denominator.


Negative exponents indicate division!

We did not create negative exponents in order to write reciprocals.  We started using negative exponents in order to report that we divided by some factors.  I find it troubling that we have focused on a secondary use for the notation, when the primary use makes more sense to students.

If you want to see what is so important about this, give a problem like this to your students.

Negative Exponent Divide not Reciprocal example Nov2014

Direction: Write without negative exponents.


Almost every student focusing on the reciprocal meaning will invert the fraction — making the 4 a multiplier instead of the divisor it really is.  Most students focusing on the division meaning will see that the m cubed needs to be in the denominator.

In part of this activity, students also dealt with a zero power.  In doing the long way (write it all out), quite a few students wrote that variable in their work; it made sense, though, to omit that factor because it said “zero factors” … and then we can talk about what value that ‘zero factor’ has in a product (one).

As we shift towards more work with exponential functions, it becomes critical that students understand the meaning of all kinds of powers.  A core understanding of negative exponents is part of this; fractional exponents are important too (though we tend not to cover these in either our Math Lit course or beginning algebra).

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Intermediate Algebra … the Bridge to Nowhere

Yes, I am using an emotional label being used about developmental education … yes, I am saying intermediate algebra might be such a thing.  A bit of a cheap trick, but I hope that you will continue reading anyway!

The content of our intermediate algebra courses is usually based on topics that were once covered in a second year high school algebra course.  That course, in turn, was created by companies and teams of authors (often a combination of university mathematics educators and high school math teachers).  I have not seen any documents relating to how the companies and authors determined the content; I suspect that much was based on a view “well, this topic would be good for them”.

All of this work occurred long before a general emphasis was placed on understanding, application, and cognitive science.  Procedural accuracy is the hallmark of our intermediate algebra courses — even more so than the high school algebra II course; it’s like we copied the content but limited our work to the lowest levels of learning.

We actually have some helpful stuff in there, if students can remember it later when (and if) they take more advanced courses (whether a pre-calculus/analysis course or in calculus).  The better students may do this; most do not, because the material is not usually taught in a way to create long term use.

So, here is an initial list of reasons why intermediate algebra is the biggest ‘bridge to nowhere':

  • content created over 50 years ago outside of our curricular process
  • textbooks focus on procedural accuracy
  • learning heavily weighted towards lowest levels of learning

Students who pass an intermediate algebra course meet the prerequisite for some college math courses; however, the intermediate algebra course did not prepare students for that course.  Nor does the intermediate algebra course contribute to mathematical understanding, nor to positive attitudes about mathematics.

Fortunately, we have a model for replacing intermediate algebra — the Algebraic Literacy course from the New Life model.  The outcomes for this course were extracted from what students need in subsequent courses, and these outcomes include both procedural and understanding emphases.   In addition, the Algebraic Literacy course includes the use of mathematics to understand the quantitative components of issues in the world — such as the spread of infectious disease.

The Dana Center work on a Stem Path is also involved in creating a replacement for intermediate algebra.  Those teams are approaching the problem from a similar viewpoint, so I expect their results to be compatible with the New Life Algebraic Literacy course even if their content has some significant differences.

To learn more about the Algebraic Literacy course, I encourage you to come to my session next week at the AMATYC Conference (Nashville); this session is at 8am on Friday (November 14).  [I am also doing a general session on the New Life model that Saturday (November 15) at 2:15pm; this session will include basic information about Algebraic Literacy.]

If you are not able to be at the AMATYC conference, take a look at the Instant Presentations page on this blog .  After the conference, I will be posted the materials from the session on that page.

Of course, if you have any questions about the Algebraic Literacy course, just contact me!

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Even Our Puzzles Are Outdated … Mathematics for 2025 (and today)

Earlier this month, the Conference Board of Mathematical Sciences (CBMS) held a forum on mathematics in the first two years; many of the presentations are available on the web site (

As part of one of the first plenary sessions, Eric Friedlander commented …  Students in the Biological Sciences now outnumber those in the Physical Sciences in the standard calculus 1 course.  (David Bressoud shared some specific data on those enrollment patterns.)

Historically, the developmental mathematics curriculum was all about getting students ready for pre-calculus.  Our “applications” tended to be puzzles created with physical sciences in mind — bridges, satellites, pendulums, and the like.   Few problems in our developmental courses draw the attention of those in biologically-oriented fields (including nursing).

We could include:

  • Surge functions to model drug levels
  • Functions to estimate the proportion of a population needed to be immunized to prevent epidemics (P_sub_c = 1 – R_sub_0)
  • Models for spread of cancer … and for treatments
  • Pollution prediction (simplified for closed systems)

This list is a ‘bad list’ because there is no common property (except being related to biology) … and because I do not know enough to provide a better list.  Take a look at books in applied calculus for the biological sciences; you will see applications that are perhaps better than those above.

There is a trend in the new models for developmental mathematics (AMATYC New Life, Dana Center New Mathways, and Carnegie Foundation Pathways) to include a balance of applications — including more from biology.  We need to bring in more of these applications throughout our curriculum (from the first developmental course up to calculus).

Most of us realize that the ‘applications’ in our courses and textbooks are puzzles created by somebody who knew the answer; generally, these problems do not represent the use of mathematics to solve problems and answer questions in the world around us.  Sometimes, we are not able to provide enough non-mathematical information to provide representative problems … in those cases, some reduction to the ‘puzzle state’ is acceptable.

Our puzzles should represent the diversity in the uses of mathematics, with a significant portion of applications being realistic in nature.

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