Envisioning Our Future

Where are we going?  How will we know that we have ‘arrived’?

These questions deal with the identification of what is important to us, as mathematics educators in the first two years of college.  We can, of course, simply say “student success”.  That is fine; there is nothing ‘wrong’ with that phrase … as a phrase.  However, as a statement of where we want to go — ‘student success’ is totally inadequate: completing a math courses which totally fails to meet student needs can meet the ‘student success’ goal.  We need a goal which helps us bring the best mathematics to every student who comes to any of our classes.

Here is the goal statement I put forth for your consideration:

Our courses will provide modern mathematical content using effective teaching and efficient sequences when necessary to prepare students for a widest range of occupational options especially fields related to STEM.

The word ‘efficient’ means that the shortest possible sequences are employed; one course is preferred over two … and two courses are preferred over three.  This does not mean that we need to condense everything (dev math to Calc I) down to two courses, though I’ve heard people advocate that radical position.  Rather, the intent is to avoid longer sequences; a probability of somewhat weaker ‘preparation’ is preferred to probability of exponential attrition in a sequence.  Fifty students reaching ‘the end’ with moderate preparation is better than twenty students reaching the end with better preparation.

The word ‘modern’ in the goal simply means that the content reflects the overall context.  Part of this is the typical K-12 experience of the day; another part is the type of mathematics actually employed currently for the purpose.  For example, ‘modern’ means that numeric & computational methods are developed concurrently with symbolic methods.  Our curricula must change, on a regular basis, to remain modern.  As an analogy, consider the problem of “kidney stones”; fifty years ago, people with kidney stones were told to avoid eating the foods which shared ingredients with the stone (usually calcium) and the stone was taken out surgically.  The modern treatment is to look for systemic issues with the person’s health and then use the least-invasive option (usually involving a moderate level of medication).  The old treatment was not effective and was inefficient. To translate this to mathematics, we are still doing surgery when that method is not modern (nor effective).

The phrase ‘widest range’ means that our goal is best served by using non-specific courses in general.  Mathematics in the first two years should not involve specialized math courses for the vast majority of students.  In fact, part of this thinking is that we need to support the goals of general education.  “Alignment” and “general education” need to be balanced; as mathematicians, our preference should be on generalized not occupation specific.  The fluidity of modern students suggests that there is a cost to them when we make our courses specialized.

THEORY OF EVERYTHING: A presentation on our future (October 2018)
Theory of Everything presentation Oct2018
References:  References Theory of Everything Oct2018

PAST, PRESENT, AND FUTURE: https://www.youtube.com/watch?v=cRVt45N4BzU
This is a recording of an AMATYC webinar (April 3), which emphasizes some of the ideas mentioned on this page.

So, WHY do we need to change?

It’s simple … the traditional math courses in the first two years have too many ‘fails’ compared to our goal (see above for my goal statement).

Overall, our curriculum (in these basic courses) is obsolete AND ineffective … when we want modern and effective.

This does not meant that we have done a terrible job.  What we do is critically important, and we have contributed … we work hard to help our students.  However, we need to recognize that our content is in critical need of updating and that our sequences are too long.  While we are addressing this set of core issues, we can also work on updating our teaching methods.  All of this can be informed by professional resources (primarily from the MAA and AMATYC).

Of course, I could have just reported that “external forces demand that we change” (especially at the developmental level).  By itself, this is a weak reason to change a curriculum.  We need to change the curriculum because we believe that this change is necessary to meet our goals (which means it will also better serve  our students).

Does the future look like today, “only later”?

The basic answer is ‘somewhat’.  Just because a particular change is seen now … does not mean that it will be a significant part of our future.  On the other hand, the future is based on the now … meaning that what becomes the standard in the future is probably present now at a lower rate.  Our goal is to sort through all o of the ‘stuff’ happening now (2017-2018) to find the good stuff.  “Good stuff” supports our goals, reflects our professional values, and is seen as something we’d want to do regardless of outside pressures.

When we are required to act or react to outside pressures, we should always focus on where we wanted to go anyway.  We have significant power over the future, given a longer point of view (past the next couple of years).

The Good Stuff happening now

Here are some changes existing now that are in the ‘good stuff’ category (in my opinion).

1. Curricular reform at all levels
   Mathematics in the first two years is being updated to provide modern mathematics to our students.  The progress is slow, but the content of courses from arithmetic to calculus is being re-created to reflect both the K-12 conditions and the state of our profession.
2. Improving placement of students
   The traditional placement strategy sometimes amounted to “the student shall start at the lowest possible level to ensure that they can pass”.  This unrealistic view of our students is being replaced by a maximizing strategy: start at the highest possible level with a reasonable chance of success with support.  The future belongs to a placement system which takes several factors in to account to make this estimate — test scores, prior course work, prior grades, and even non-cognitive factors.
3. Supporting student learning of mathematics
   The good stuff here includes the use of diverse pedagogical methods and assessment focused on learning.  We are teaching in different ‘styles’, matched to the needs of the current material.  The use of active learning is becoming intentional, again aligned to support the material — different types of activity depending on the learning outcome.
   Assessment (often brief) is a every-day experience, with the frequency a function of the material and the students in the course.  Assessment is integrated with (and supports) learning, for both students and faculty, with course grades being a secondary purpose.
   Co-requisite learning is a ‘good thing’, but not so much for remediation.  Our goals are better served by providing extra support within the courses with a history of high failure rates (pre-calculus, calculus).

In addition, there are some trends that are neither good nor not good:  flipped classrooms, open educational resources, and others.  Those types of work are likely to be in the future we are envisioning but are not ‘necessary conditions’.

So, What does THAT Look Like?

If we apply the Good Stuff to the mathematics curricular needs in the first two years, the result might be the image below.

Currently, the range in mathematics is 9 courses (from the lowest level to calculus III): (1) arithmetic, (2) pre-algebra, (3) beginning algebra, (4) intermediate algebra, (5) college algebra, (6) pre-calculus/trig, (7) Calc I, (8) Calc II, and (9) Calc III.  A specific student program usually involves a subset of this range, perhaps branching off after step 3 or 4 to a ‘non-STEM’ course such as statistics.   The full sequence has: Four developmental courses … two basic college courses … and three calculus courses.  Much of this awful sequence is due to a fixation on ‘grade levels’ in a K-12 world — one that existed 50 years ago.

This future image reduces the range to FIVE:  two pre-college, one basic college, and two calculus courses.  Once we let go of a grade-level fixation, we can see better ways to design a curriculum.

• Students who would have placed in to arithmetic and pre-algebra, as well as the more basic half of beginning algebra … place in to Math Literacy.
  However, in this future, we no longer test adults on arithmetic skills — we worry more about reasoning with numbers & quantities along with a basic grasp of algebraic reasoning.
• Students who would place at the upper half of beginning algebra, as well as the lower half (60%) of intermediate algebra students … place in to Algebraic Reasoning (which is a course like ‘Algebraic Literacy” Algebraic Literacy (A Bridge to Somewhere))
  If students at this placement level don’t need calculus, they proceed to their Stat or QR course.
• Students who would place at the upper half of intermediate algebra, as well as the lower 60% of college algebra students … place into Modeling & Pre-calculus (which is a modern course combining the necessary symbolic work for calculus as well as numeric methods for basic modeling).
• Students who would place at the upper 40% of college algebra students as well as most students who place in to calculus I … place in to Calculus & Numeric Methods I  (which is a course combining the core symbolic work & concepts of derivatives and integration as well as numeric methods for science [using technology … MatLab or Mathematica for example)

The image incorporates co-support classes for each of the five courses, designed to allow a wider range of students to succeed in the course.  Colleges might assign the ‘weaker’ 30% of the students in each of the 5 courses to a required co-support class meeting an hour or two every week.

The system helps us move from 60% of students in remedial mathematics to about 30% … perhaps 20% into Math Literacy, and about 10% who need calculus and place at the algebraic reasoning level.  Something like 40% of students would be able to begin with the Modeling & Pre-calculus course with another 20% who can place at the Calculus & Numeric Methods I course level.

Imagine this:

• 70% of students do not require separate remedial mathematics courses
• Most of the other 30% need only one remedial math course (either Math Lit OR Algebraic Reasoning)
• So … at least 90% of students need zero or one remedial math course

All of this … while improving the content of our courses.  We meet our goals, focus on our priorities, and deliver modern mathematics courses to our students.

To learn more about where we have been … and where we might be headed, check out these slides.  This was the “Past, Present, Future” Webinar on April 3, 2018.
Past, Present, Future of Dev Math

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