Conversation II: Herb & Jack on the Why — Practicality of Theory

In response to a post about STEM students and the traditional developmental mathematics curriculum, Herb Gross began with this quote (from a prior talk he gave):

The music is not in the guitar.

I think Herb is saying that mathematics is not in the visible tools used, whether these tools are procedures written down or technology used to answer questions.  This is a great point, and it suggests that we question any suggestion that we limit the content of mathematics courses to just those things seen as ‘practical’.  Seeing mathematics as being bound by the practical (for STEM or non-STEM) is a self-defeating behavior; a health profession is based on continuing growth, and growth depends upon research both applied and theoretical (the two work together in surprising ways).  Our students are future policy makers — do we want them to only value mathematics that is practical NOW?  (Think University of Wisconsin budget cuts.)

The music, and the mathematics, is based on connections among concepts.  This speaks to the growth of mathematical reasoning and critical thinking.  Herb adds this comment:

So I am not overly impressed with the pass rate improving as much as I am in seeing what the effect is further down the road.  In fact one of the reasons I don’t like non-algebra/calculus based courses is that even the students who are most successful in these courses tend to know how to crunch numbers into the calculator but have little feel as to what to do when the distribution is anything other than normal.

I think Herb is speaking to a basic goal of education — the improvements retained over a longer period of time, meaning improved capabilities.  The comment Herb makes is important, and I think it applies to most algebra based courses; I also wonder about calculus based courses.  Look at this re-phrasing of a critical part of Herb’s comment:

Students tend to know how to manipulate symbols or numbers often with the use of tools but have little understanding as to what to do with mathematical concepts applied to a new situation. (JR)

Creating scalable change within an individual involves some of the same work as creating scalable change in a profession.  A more complete view of learning is required, with less focus on ‘passing’; passing is a great thing, but it can not be the core measure of our success.  We seek to create mathematical abilities, including the willingness to apply existing knowledge to new situations where this knowledge is not sufficient.

Students in STEM programs need a broad foundation in mathematics, combining procedural and conceptual fluency.  To some of us, we follow that statement with “Non-STEM students to not”; this is where we can make large mistakes.  The mathematical needs of citizens and the mathematical needs of our partner disciplines are not different in a basic way — they need procedural and conceptual fluency as well.  The difference, overall, is a matter of degree and extent.  STEM students need MORE, not so much ‘different’.

Our work in the AMATYC New Life project supports this single-source approach to mathematics — the Mathematical Literacy course serves the needs of all students.  The initial uses of the course have often been for non-STEM students; however, the outcomes of the course were designed back from the needs of all students.  I agree with the design of the New Mathways Project (Dana Center), which has a similar course serving all students.

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STEM Students and Developmental Mathematics

Pathways … Mathways … creating alternatives for non-STEM students.

The changes in pre-college mathematics are significant, and I am incredibly pleased with the work of my colleagues in dozens of institutions across the United States.  The modified curriculum for these students has dramatically increased the proportion who achieve their goals, with a significant increase in the number passing their required college-level math course (statistics or quantitative reasoning).  These gains have been achieved by putting thousands of students into a different path, wherein they avoid beginning and intermediate algebra.

We need to get over a myth about other students — the students who need at least a college algebra course, often because they are pursuing a STEM or STEM related field.  [STEM refers to Science, Technology, Engineering and Mathematics.]

Myth: STEM students are well served by the traditional developmental mathematics curriculum.

Alternate hypotheses: STEM students are ill-served by the traditional developmental mathematics curriculum.

As you may know, I have been in this work for over 40 years.  Much has changed.  However, developmental mathematics is currently bound by these constraints (which has been true for over 40 years):

Constraint 1: The pedagogy and content is limited by the preponderance of adjunct faculty assigned to developmental math courses.

Constraint 2: The content is based on textbooks reflecting a set of topics which were copied from a typical high school curriculum of 1965.

These constraints interact within our curriculum, including at my own college.  The rigor of a developmental algebra course is most often established by the complexity of the procedure students would use to solve problems; these ‘problems’ are copies or slight variations of exercises seen in the homework.  These exercises, in turn focus on the achievement of a correct answer to a well-defined problem either stated symbolically or in the disguise of a verbal puzzle (where such puzzles lack both value in the real world and value in their structure).  We have a sense of pride if OUR algebra course includes conic sections or inverse functions, based on knowing that these topics await students in their college algebra course.

Some people might wonder if I think the presence of adjunct faculty in a classroom results in lower quality; definitely not — some of my adjunct colleagues are better instructors than I am.  The constraint is based on the fact that these courses need to be ‘teachable’ by the pool of adjuncts available; the issues deal with the expectations that are reasonable for a group, rather than individuals.  Full-time faculty may, in some cases, face similar limitations in the knowledge and skills they bring to a developmental algebra course; the difference is that full-time faculty have greatly enhanced access to professional development and networking.

In terms of data, the pathways work is fueled by the low pass rates in traditional courses (50 to 55%) compared to the typical 65% to 70% seen in the reform models (Mathematical Literacy, Fundamentals of Mathematical Reasoning, Quantway I).  By saying that STEM students are well-served by traditional developmental mathematics:

We are apparently comfortable with 25% (or less) of students completing two semesters of developmental algebra.

The improved outcomes for the reform models is likely due to the fact that all 3 address both constraints — professional development for faculty AND improved content.  By saying that STEM students are well-served by traditional developmental mathematics:

We are apparently comfortable with STEM students having to survive lower quality pedagogy and outdated content.

I see other issues, as well — such as the relative lack of technology in developmental algebra courses as a basic part of the content; calculators are banned … we avoid numerical methods … and remain out-of-touch with the world around us.

Again, I say that STEM students are ill-served by the traditional developmental mathematics course.  The content is inappropriate, pedagogy is not supportive, and little inspiration is ever seen for why a student would persevere in their STEM field.  STEM students need a reformed curriculum just as much as non-STEM students; the needs of society would suggest, in fact, that STEM students have a greater need for a reformed curriculum.

Take a look at the reform curriculum; it’s actually not that complicated.  Instead of beginning algebra, use the SAME reform course as non-STEM (Mathematical Literacy, Foundations of Mathematical Reasoning, or Quantway I).  Then … replace your intermediate algebra course with a reform course.  In the New Life work, that reform course is called Algebraic Reasoning; you can see some information at https://www.devmathrevival.net/?page_id=1807 , or head over to the wiki http://dm-live.wikispaces.com/

The New Mathways project is starting their work on STEM path — take a look at this post https://www.devmathrevival.net/?p=1935

I hope to do a presentation on the Algebraic Literacy course at this fall’s AMATYC conference … as a ‘bridge to somewhere’!  I believe that the Dana Center will also be there.  I encourage you to learn more about the reform curriculum for STEM students.  The work is important, students need it, and we will find it very rewarding.

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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 https://www.devmathrevival.net/?page_id=116 .  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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Basic Math or Pre Algebra or Nothing

What do students need before a ‘beginning algebra’ course?  Several of us (math faculty at my college) are working on this problem, with a goal of helping more students make a good transition to algebra while being aware of other expectations or demands.

My college does not have a basic math class, having eliminated that quite a few years ago.  There is still a prerequisite for the pre-algebra course (a placement test) though the cutoff is not very high, which means that one of the issues is students with extensive gaps in numeracy.  Our pre-algebra course has these components:

• variables and expressions used from the first chapter
• signed numbers start next
• solving first degree equations (some with simplifying first)
• fractions
• geometry (formulas primarily)
• units and conversions (the only math course doing this, for most students)
• percents and applications (tends to be uncomplicated)

One of the issues I see us dealing with is our own views on “what students should know”.  In our course, we designate the first part ‘calculator free’ because students “should know” their basic facts about numbers; the remainder of the course allows a calculator.  We also expect students to use arithmetic procedures for fractions, though we do not check to see if they understand ‘why’.  We cover classic percent problems, because students “should know” these.

So, what essentials are needed to help students succeed in basic algebra?  In some ways, the answer has been “do some basic algebra”; the last course revision integrated algebra throughout.  We’ve looked at the data for the progression, and it is my opinion that the alumna of the newer course have similar struggles in basic algebra compared to the older course (with less algebra).  One observation is that the students struggle with the expressions and first degree equations that they ‘had’ in the pre-algebra course, whether the algebra was integrated or covered separately.

Here is the basic need I would identify for success in basic algebra:

Students need a core of understanding about numbers and properties, and need a sound beginning on procedural flexibility.

The traditional percent material focuses on correct answers, often using memorized procedures.  I would shift to questions about equivalence and multiple solution methods … because these are core issues in algebra.  My class work and assessments would focus on creating as well as identifying alternate correct methods.  The traditional geometry work in this course also tends to have a focus on correct answers (though we do not memorize formulas).  I would instead deal with how parts of shapes relate to the whole, and concepts of perimeter/area/volume; the same focus on multiple solutions would be appropriate.

The numerical demands of a basic algebra course are quite limited; we are not going to solve a lifetime of numeracy problems in 15 weeks of a basic math course.  A pre-algebra course gains little by making the attempt.   A reasonable goal is to develop a significant set of understandings about numbers and objects, along with the flexibility that this understanding supports.  Deliberate design, sophisticated pedagogy, and faculty expertise are required for this … just as is the case for most math courses that we should place in front of our students.

One of my colleagues used to say:

The student’s fragile understanding of mathematics begins in the pre-algebra course.

We need to shift our focus.  Without understanding, any math course becomes just a barrier to student success.  Without understanding, math is that subject that everybody says they are bad at.  With a focus on understanding, we offer an honest math course that can provide real benefits for students. With a focus on understanding, we demonstrate our commitment and respect towards all students … starting from the first day of our first math course.

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