Category: Algebraic Literacy course

Texts for “After Math Literacy”

A recent post here did a brief review of the 3 available Math Literacy textbooks (http://www.devmathrevival.net/?p=2804).

That post quickly led to the natural question:  What about books for courses AFTER math literacy?  (Thanks, Eric!)

Ideally, we would have 3 books for “Algebraic Literacy” … a course designed to replace intermediate algebra.  However, much of the Math Literacy work is still stuck in a pathways approach, where Math Literacy is only used for “non-STEM” students.  I don’t think this pathways emphasis can survive that long (see http://www.devmathrevival.net/?p=2779) In this period, however, most uses of math literacy courses is as an alternative to beginning algebra for those who “don’t need algebra” (as if that was possible or desirable).

To review, here is the New Life vision of basic mathematics courses at colleges & universities:

 

 

 

 

 

 

 

 

 

 

 

 

 

So far, the reform work in our college curriculum has been limited, with the most systemic work being done at the Math Literacy level.  Many people are holding off on Algebraic Literacy until there is a textbook, and publishers are interested in creating those texts.  We need to achieve a higher level of interest before those “AL” books will be developed and published.  Authors want to write them, publishers are willing to support them … IF the market interest is there.  Lesson:

Always tell publishers that you want to see textbooks for Algebraic Literacy, and that Algebraic Literacy is not an intermediate algebra with a new ‘cover’.

There are colleges who are implementing Math Literacy for all students, replacing beginning algebra in their curriculum (mine, for example).  Most often, this means the use of a typical intermediate algebra book for the course following math literacy … a bit like getting to use an iPhone 7 one semester and then being handed a rotary phone the next semester.  If only there were better options!!  [Some folks use the “Math in Action” materials, which are not Algebraic Literacy at all … they just provide great context and applications.]

Both publishers with good Math Literacy texts (McGraw Hill, Pearson) have considered algebraic literacy books; they may even have them ‘under contract’ (I would not necessarily know about that).  Keep telling them that you really want an algebraic literacy book, and they will develop them.

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Why We Will Stop Doing Pathways in Mathematics

Currently, and for the past few years, “pathways” has been a big thing in community college mathematics education.  For students not needing calculus or similar courses, alternate paths have been established — with a focus on courses such as Statway™, Mathematical Literacy, and Foundations of Mathematical Reasoning.  The fact that all three of those courses are very similar in content is not an accident, and the fact that the three organizations involved collaborated is a key reason for their success.

The reasoning behind the creation of pathways is essentially “give them what they need, not what they don’t need”.  Students with a pre-calculus target are still placed into the old-fashioned developmental math courses, and students with other targets are placed into a ‘pathway’.  All students are generally required to meet some arithmetic criteria before starting at the Math Literacy level or beginning algebra.

My own work has certainly played a role in this creation of pathways.  However, that was not the intent of the efforts beginning this work.  Neither do pathways have a good prognosis for long-term survival.

Let’s go through some of the reasons why “pathways” are not a long-term strategy.

Reason 1: Pathways are a dis-service to “STEM” (calculus-bound) students!
The original design of the major pathways courses (Quantway™, Math Literacy and Foundations of Mathematical Reasoning) was based on identifying what all students needed in college-level mathematics — statistics, quantitative reasoning, AND pre-calculus.  These outcomes were then categorized in two clusters … those needed by ALL students became the core of the Math Literacy course, and those primarily needed by pre-calculus students became the core of Algebraic Literacy.  [Algebraic Literacy also includes some outcomes needed for technical programs.]

In effect, “pathways” is preventing STEM (calculus-bound) students from getting the learning they need for success.  We have accumulated data showing the the traditional developmental algebra courses do not add significant vale for these students when they take pre-calculus.  In addition, we also know that the traditional courses were not designed for this purpose — they were designed to replicate the 9th to 11th grade content of a 1970’s high school.

Pathways create a better experience for non-STEM students, at the price of harming (relatively) those bound for pre-calculus.

Reason 2: Curricular complexity costs too much
One of the extreme cases I have seen is a college with SIX different courses at the Math Literacy level.  Clearly, half of these are quite specialized for students in particular occupational programs.  However, half were general in nature — a Math Literacy course, and two basic algebra courses.

Curricular complexity raises the cost of support functions at an institution, advising in particular.  Few colleges can support this extra work in the long-term, even when the initial launch of those efforts is strongly supported by the then-current administration & governing board.  As time goes on, the focus on advising slips … mistakes are made … and a later administration will question why things are so complicated.

This curricular complexity also raises costs within the mathematics department.  More courses at the same level means more difficult scheduling, less predictable enrollments in each course, and a host of faculty coordination issues.  Unless an institution has excess resources not needed for other situations, the mathematics department will realize in a few years that they can not support the complex curriculum.

Reason 3: Pathways allow the continuation of arithmetic courses at colleges
The presence of arithmetic courses at a college involves several problems and costs; the fact that our profession has not accepted these are overwhelming rationales for discontinuing arithmetic courses is a failure with moral and economic dimensions.

First of all, these extra courses at the developmental level are primarily taken by students of poverty and minorities.  This is the moral dimension for us:  these are the students coming to college to get out of poverty, who are then required to take one or more courses prior to the course that is a prerequisite to their required course.  No possible benefit from learning arithmetic can justify this process; in fact, there is no evidence of any significant benefit for taking such arithmetic courses in college.

Secondly, arithmetic courses in a college create costs for the mathematics department. We often have a fairly discreet set of faculty (heavily adjunct), and these faculty are seldom qualified to teach a college mathematics course.  In many colleges, the arithmetic courses are administered in a separate department.  As faculty, we should want to design a curriculum that does not depend on a course at the arithmetic level.

Thirdly, the presence of arithmetic courses at a college will tend to perpetuate the outdated focus on procedures and answers.  This conflicts with the design of Math Literacy, and impedes development of basic reasoning needed even in a traditional basic algebra course.

Reason 4: External Forces Will Continue to Push Us To Change
So far, the evaluation of ‘pathways’ has focused exclusively on the impact for students taking Math Literacy (or companion course) as preparation for statistics or quantitative reasoning courses — specifically, students who enroll in stat or QR after passing Math Literacy.

Curricular complexity means that there will be a less successful experience for students needing pre-calculus … by definition, because those students need two courses (beginning algebra, intermediate algebra) compared to the one & done of Math Lit.  There are also operational causes for other ‘bad’ data to show up — students taking Math Literacy instead of the course they were supposed to take, for example.

In addition, we can predict that these change agents will critique our developmental math courses compared to modern standards (whether Common Core, or NCTM standards).  We are not ready for this critique, and have no response for the results that are bound to come from such a critique — that developmental mathematics operates as if the year is still 1975, ignorant of the fundamental changes in our students’ experiences in K-12 mathematics.

 

In a way, I am reminded of something I learned at a conference session on graph theory and traffic design.  Our intuition might say that it is better to have more options in street designs, where there are several north-south options and several east-west options.  The traffic design results were the opposite … that the best throughput for a traffic system is the fewest possible streets.

A pathways curricular design presumes the presence of at least two courses at the same level in a sequence.  This design is not particularly stable, as a system.  In the long term, I think the system will collapse down to one of the options.

We need to be prepared for the demise of pathways so that we can maintain the improvements from those efforts.  The danger is in assuming that both Math Literacy AND the old courses will ‘always’ be there.  Within a few years, one of them will be gone.  Which type of course do YOU want to survive?

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Intermediate Algebra … the Barrier Preventing Progress

The traditional math curriculum in colleges is significantly resistant to change and progress; I talked about some of the reasons for this condition in a recent post about the Common Core & the Common Vision related to the future of college mathematics (see http://www.devmathrevival.net/?m=201703  )  We carry some historical baggage which creates additional forces resisting efforts to make progress in the curriculum at the college level.

Our “Intermediate Algebra” course occupies a position of power.  First … it has long served as the only accepted demarcation between “college level” and courses which are not.  AMATYC recently approved a position statement to help clarify this demarcation (see http://www.amatyc.org/?page=PositionInterAlg )   Second … it has been used as the prerequisite to both college algebra and pre-calculus, which contradicts the origin of intermediate algebra as a copy of HS algebra II (which was never designed for this prerequisite role).

I’ve written previously about the need for Intermediate Algebra to be intentionally removed from the college curriculum; see http://www.devmathrevival.net/?p=2347

Intermediate Algebra must die … now!

Recently, we’ve had some email discussion in my state about the credential requirements for faculty … especially those teaching “intermediate algebra”.  Although we all want to provide students with quality faculty for every math course, we don’t agree on what this means.  Like most accrediting bodies, ours makes a distinction between developmental courses and general education courses; developmental courses require that faculty have a degree at least one level above what they teach … while general education courses require that faculty have 18 graduate credits in the field they are teaching.

Because of that credentialing difference, faculty teaching college mathematics courses tend to be functionally separate from those teaching developmental math courses (unless at a small institution).  A consequence of this faculty split is that the interface zone (intermediate algebra to college algebra in particular) is difficult to change in basic ways.  Faculty with a STEM focus are more concerned with their ‘upper level’ courses (calculus, linear algebra, etc), while those with a developmental focus are often more concerned with the beginning algebra level.

Intermediate algebra, just by its presence in our curriculum, is a barrier to making progress in modernizing our work.  If we were to remove Intermediate Algebra as a course, both levels of mathematics faculty would (by necessity) work together to create a more reasonable replacement.  If Intermediate Algebra had never existed, do you think we would create that same course now?  Obviously, no … we would do something much more reasonable.

Intermediate Algebra must die … now!

Efforts to ‘improve’ intermediate algebra typically involve micro-adjustments (different mix of skills).  Changes of this type have been tried over the past 30 years (or more) with almost no impact on any problem or outcome.  Our problems have become severe enough that no set of micro-changes will create a solution … we need macro-changes.

We need to remove the barrier — get rid of your intermediate algebra course (and mine!).  Replace it with a modern course like Algebraic Literacy (http://www.devmathrevival.net/?page_id=2312) if that makes sense to you.  Or, create a different solution for the problems.  Of course, part of the solution is to keep some of the students out of any course at the intermediate algebra level — developmental but preparing for college algebra.  Intermediate algebra is certainly not needed as preparation for statistics or quantitative reasoning at the college level.

Some of us are having a strong response to this proposal (of removing the intermediate algebra barrier).  If you live in a state that has a policy of ‘intermediate algebra for general education in college’, or your institution has such a policy, you are experiencing another reason why intermediate algebra is a barrier that must be removed.  Intermediate algebra is a copy (sometimes quite weak) of an old high school mathematics course in an era when the overwhelming majority of our students have experienced more advanced mathematics in their high school.  This was true before ‘the Common Core’, and is becoming more true as time goes on.

Intermediate Algebra must die … now!

We can create viable solutions, with modern courses about current mathematical needs, if we are just willing to toss this one course from our curriculum.  Intermediate Algebra must die, and die soon.  It is a barrier to progress that we … and our students … need urgently.  Don’t wait for a replacement to be ‘ready’ — the solution will be ready when we are committed to make a change.

Which of these is your choice?

  • Eliminate intermediate algebra at your institution effective Fall 2018
  • Eliminate intermediate algebra at your institution effective Fall 2019
  • Eliminate intermediate algebra at your institution effective Fall 2020
  • Ignore the intermediate algebra problem, and hope it goes away by itself.

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The Student Quandary About Functions

For students heading in a “STEM-ward” direction, understanding functions will become critical.  Unfortunately, a combination of a prior procedural emphasis and some innate cognitive challenges tends to result in a condition where students lack some basic understandings.

For example, in my intermediate algebra class, we provide problems such as:

For f(x) shown in the graph below, (A) find the value of f(0), (B) find the value of f(1), and
(C) find x so that f(x)=0.

 

 

 

 

 

 

 

Since there is no equation stating how to calculate function values, students need to use the information in the graph.  The vast majority of students make 2 novice errors:

  • Error of x-y equivalence:  providing the same answer for (A) and (C)
  • Error of symmetry: Since the answer for (A) is x=1, stating the answer for (C) as x=1

To improve this understanding, I use the longest (time measured) group activity in the course.  This is definitely a situation where “Telling” does not correct the errors [I’ve tried that 🙁  ], and the small group process helps dismantle some of the errors.  Clearly, the correct understanding for reading function graphs is critical for success in pre-calculus and eventually in calculus.

Another function concept we dealt with this week is ‘domain’.  Now, once students have found a domain, there is a tendency for some students to think they should find the domain of any and all functions, regardless of the directions for the situation.  This “inertia error” (what was started … continues) is not a long-term problem.  Here is a typical problem for the long-term problem:

Find the domain for the function graphed below:

 

 

 

 

 

 

In this particular class, I provide a fair amount of scaffolding … in a small group project, we explored the behavior of rational functions (without using that label) including what the “undefined” x-value means on the graph.  We don’t use the word asymptote; rather, we talk about the fact that some x-value results in division by zero, and the graph of the function can not show any ‘point’ for such inputs.  This leads to the graphing of the function, including the behavior around the ‘gap’.

Students struggle quite a bit with this type of problem.  Sometimes, they continue the ‘function values from graph’ thinking, and latch on to x=0 or y=0 to make some statement about a ‘domain’.  Many students will correctly identify the x-values for the gaps (yay) but make illogical statements about the domain.  The typical student error is:

  • (-infinity, -2) ∪ (-2, infinity)  … or even just one interval (-2, infinity)

This type of error usually follows from a process-focus, detached from the underlying meaning.  I am trying to get them to see:

  • gap on graph equates to excluded values in the domain

The process focus looks at the first part of  this.  Like the function value errors, the effective treatment of this problem requires time and individual conversations.

This type of function work is not typical for an intermediate algebra course.  However, it would be typical for an algebraic literacy course.  As we transition from traditional content to modern content in our courses, I am expecting that our intermediate algebra courses will fade away … to be replaced by variations of the algebraic literacy course.

 

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