Of course, the word ‘algebra’ itself has multiple meanings. In this post, I am referring to polynomial algebra along with the reasonable connections to geometry, trigonometry, and modeling at the curricular level of first year of college. The delivered curriculum in ‘algebra’ has degraded to the point that the primary student outcome is ‘survival’ that qualifies them to take another course.

This is not the same discussion as “Algebra II for All” in the K-12 world; we could debate the pros and cons of that issue, though in most ways that train has left the station. Our interest is in college mathematics in the first two years.

At the highest level, an observation is that the enrollments in STEM-enabling math courses is declining based on increased enrollments in courses aligned with programs (by which I mean statistics and quantitative reasoning [QR]). As a general education course for students in non-scientific programs I think a rigorous QR course is the best option. Such a rigorous QR course includes a significant focus on algebra and algebraic reasoning. We probably don’t reach that goal very often in QR courses. In any case, the STEM-enabling math courses are declining in enrollment.

Why? Why does our leadership consider these non-algebra options to be superior? Is it because they have conferred with us about the mathematical needs of students within the context of their programs and the issues of the 21st century? Have some of us taken on the anti-algebra mantle to the extent that we encourage excessive emphasis on statistics and QR?

Sometimes, algebra has been used as a filter to weed out students who “can’t make it”. Let’s be honest — that is not the nature of algebra, only the nature of algebra courses used to weed out students. A positive … and accurate … conception of algebra is this:

- Algebra provides a set of tools for representing scientific and technical knowledge
- Algebra provides a framework for dealing with quantitative problems which are not primarily computational exercises
- Algebra encourages precise communication

If students do not need to deal with scientific or technical knowledge, AND will not need to deal with quantitative problems, then the emphasis of QR and statistics is not inappropriate. As mathematicians, we value the precise communication aspect of algebra, and we might even make the case that this type of communication is just as foundational as the ‘regular’ communication areas (writing, speech, etc). That rationale is probably insufficient to require students to take an algebraic STEM-enabling course.

Let’s just consider the first feature of algebra — representing knowledge. Take a look at the occupations with the best employment prospects (above minimum wage), and I think you will find primarily scientific and technical fields (including health careers). Some of the very best employment prospects are in highly quantitative professions.

We don’t need all of our students to declare a STEM major (though we can always dream of what this would be like). However, I wonder if the rush to completion is putting a large portion of our students in programs for which they are either not prepared for the jobs available OR not prepared to handle the quantitative demands of those jobs. That statement might not be clear; here’s an example of the latter condition: students in an associate degree nursing program take a statistics class to meet their math requirement, but they are not prepared to deal with problems requiring algebraic representations or algebraic reasoning.

The ‘elephant’ in the room is how poorly we have been delivering algebra-based courses in college. In spite of fundamental changes in both the mathematics profession and in K-12 mathematics, we still emphasize courses which might be called “death by algebra” … which serve to weed out students rather than prepare students. How could we, in good conscience, suggest to our leadership that these algebra courses should be used instead of the QR or statistics course?

The changes in college mathematics, so far, have been at the edges — developmental mathematics reform and co-requisites (usually for QR or statistics). I believe that the external pressure will come to our algebra-based STEM-enabling courses: either we make fundamental changes to those courses OR the leadership will make curricular changes that take our courses out of the normal set of student programs. Within 10 years, we could be dealing with a situation in which the only students taking STEM-enabling math courses are those in ‘high’ STEM fields (physics, engineering, perhaps a few math majors).

What’s the future you want to see? What’s the role of STEM-enabling math courses in your vision?

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Here at Lansing CC, we are working on a different narrative, with a story of student success within a department willing to make fundamental changes because we think those changes will result in a better experience for our students — as opposed to ‘because the state (or chancellor) told us we had to’.

Here is a representation of the progress we have made:

This chart is showing the proportion of students enrolled in credit level mathematics out of the total (including developmental). Within 5 years, we have doubled the rate of students taking credit math courses.

Here is a chart of our basic curriculum:

The progress is the result of several changes and decisions:

- Eliminating pre-algebra as a course
- Replacing beginning algebra with math literacy
- Using math literacy as the prerequisite to the quantitative reasoning (QR) and statistics courses
- Removing intermediate algebra from the list of general education courses for an associate degree

The only co-requisite work involved (so far) is within developmental courses (Math Lit with Review; Fast Track Algebra).

Another piece of good news is that we have slightly more students in the initial STEM path courses (college algebra & pre-calculus) than we do in the QR and statistics courses.

Early in April, I will be delivering an AMATYC Webinar on “Dev Math: Past, Present, and Future”. In that webinar, the conclusion will be some thoughts on what a brighter future could be for us … including a specific vision for the curriculum in the first two years. I hope you will consider being a part of that webinar (tentatively scheduled for April 3, afternoon).

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However, we are still being subjected to pressures from within the institution. Our president made several remarks critical of our work at a college-wide event, partially based on not understanding what we have already done.

Last week, my college brought in an ‘expert’ who gave a presentation on “success and equity”. By ‘expert’ I mean that the qualifications were (A) employed (B) PhD in hand and (C) agreed with college leadership (the president in particular). I refer to this type of expert as a “cheerleader” — their task (based on what was presented) was to motivate us to implement a different solution, just like cheerleaders in sports try to get everybody motivated.

The question is this:

Can cheerleaders be effective change agents in academic work?

I’ve actually thought about these issues for a number of years. When I began this blog as part of the AMATYC “New Life” project, I needed to understand what forces and conditions are necessary for ‘change’ … as well as what we mean by ‘change’. I’ve been involved with a variety of ‘change’ in my life, and have learned a bit about other scientific fields; ‘change’ is studied in several — though I have focused on sociology and anthropology specifically (groups) as opposed to psychology (individual).

Change is not just a question of ‘being different from the past’. The concept of productive change is more like “progress” — change directed towards a goal in a manner such that the trajectory of the work reflects the values and goals of those doing the work. When change is accomplished without these conditions, the resulting system is often unstable, as well as requiring significant resources to push people in a direction in which they did not want to travel.

However, we can’t remain content with what we have done. Changes and progress are a reflection of the people involved, so we often see our current efforts as being more productive than they are (for our students). A group requires leadership to make the connection between where we are now and where we want to go. There is a quote by Dr. Martin Luther King relative to this (during an interview where he was asked about consensus and leadership).

So, back to ‘cheerleaders’ — can cheerleaders be effective leaders, connecting the present and the future? I think this deals with issues of perception; do we perceive cheerleaders as providing information, or do we perceive them as motivation and anecdotes? I suppose that there might be some highly skilled folks who can combine the cheerleader function with a leadership function. Certainly, the person who came to our campus did not deliver this combination; people generally left the presentation with either no internal change or a decline in their optimism. Most ‘policy influencers’ are cheerleaders — Complete College America, Jobs for the Future, foundations, etc.

Cheerleaders are not effective change agents — even if they have a PhD and a pocketful of data. We need leadership willing to work with us over an extended period of time to achieve progress … with this collaboration, we can go further than the cheerleaders can imagine.

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This doom has two primary sources, one objective and one subjective. The objective doom is our historical ties to grade levels in K-12 mathematics of a prior era; the subjective doom is the perception that we have created an exponential decay function experienced by our students which prevents them from completing their degree. These ‘dooms’ of developmental mathematics can not be removed by debate nor by data. That does not mean our work will end; we can create a model which avoids these dooms … and (more importantly) works for our students.

First, the objective doom: historical ties to grade levels in K-12 mathematics. The traditional developmental mathematics courses were created as clones of high school courses — 8th grade math copied as pre-algebra or basic math, 9th grade Algebra I copied as our ‘beginning algebra’, 11th grade Algebra II copied as our ‘intermediate algebra’ with some copying 6th-7th grade math as ‘arithmetic. All this copying was based on facts from the 1960’s: not all students completed Algebra II in high school, and we needed to get them ready for college algebra.

The result is a sequence of courses prior to college mathematics which exhibits the exponential decay function property … no matter how high the pass rate in a given course, the net result of the sequence is that only a small minority can finish. Even ignoring that, we have a curriculum which is inconsistent with current course taking patterns in high school; we are not serving current needs — the ‘need’ is an image from 50 years ago. This doom, this conflict with reality, is as obvious as it is deterministic.

Second, the subjective doom: developmental mathematics prevents students from completing their degrees. Why is this ‘subjective’ when we have seen data supporting it? Perhaps it would be more accurate to say that this is a hypotheses which has not yet been statistically supported by data. The data used to support this viewpoint uses observed correlations to support a conclusion — a large portion of non-degree-completers did not complete their developmental math courses, therefore the developmental math courses caused the non-completions.

Nobody has (yet) shown that removing the ‘dev math barrier’ results in a significant increase in degree completion. Yes, there have been reports that removing dev math results in more students completing a college math course. By itself, that is a small improvement. We don’t know if more students are completing degrees … or if we are just changing which courses students complete on their way to non-degree status.

The fact that this is an unproven hypotheses does not matter for our purposes. For our purposes, the doom is present regardless of evidence: our presidents and provosts and chancellors generally accept this conjecture as ‘the truth’. We would need years of effort to counter this subjective truth; we lack the time. Once accepted, this conjecture causes a failure in the patience circuits. We have passed the tipping point, and few of our collegiate leaders support traditional developmental mathematics.

How do we avoid the dooms?

- Replace traditional developmental math courses with new courses which do not clone K-12 mathematics
- Avoid the word “algebra” in all course titles (ALL … including college level)
- Avoid the words “developmental” and “remedial” when describing our work; perhaps use the label “pre-college mathematics”
- Allow no more than two pre-college courses at any institution
- Establish placement processes which allow at least 90% of all students to reach their college math course with one year

It is my view that traditional developmental math courses will NOT survive; within 5 to 7 years, they will be eliminated … either by our planning or by external directives. We can not escape the doom of dev math. However, we can greatly help our students by re-creating pre-college math courses which provide modern content in an efficient curriculum.

If we fail to create a new curriculum, stand-alone dev math courses will become extinct. Is that what we want? I hope not!

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The presentation slides: Forty Five Years of Dev Math in 50 minutes web

The handout: 45 years of dev math in 50 minutes AMATYC 2017 S137

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Presentation slides (all): Using Data for Improve Curriculum

The Handout (shorter): Using Data for Curriculum AMATYC 2017 S116

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I am hoping that you (if you are attending) will consider both of my sessions. Information on each is provided below.

**“Using Data to Improve a Curriculum” (S116, Friday at 2:55 pm)**

This session focuses on two key issues in our curriculum: the transition to pre-calculus, and equity. We will look at how to use data to help understand the issues and then monitor for program improvement. You will learn about methods and variables that can be used at your campus.

Preview: Using Data to Improve a College Math Curriculum and Equity S116 Preview

**“45 Years of Dev Math in 50 Minutes” (S137, Saturday at 11:55am)**

The goal will be understanding our history and the current issues sufficiently to see the path forward. Dev Math has been nudged, prodded, and attacked in the last few years, and some of us are dismayed; it may be difficult to see where we are headed. By the end of this session, I hope to show you how we can move forward … towards a goal we can be excited about. Do not come to this session to hear whining; come to hear a positive message focused on what we can (and should) do.

Preview: Forty Five Years of Dev Math in 50 minutes Preview

As many of you know, this will be my last AMATYC conference. In one way, the ’45 years’ session is my farewell … and my thanks … to AMATYC. [If you are curious, I will be teaching for another year or two.]

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One of the sessions I attended focused on lower levels of dev math — pre-algebra and beginning algebra. The presenter shared some strategies which had resulted in improved results for students; those improved results were (1) correct answers and (2) understanding. That sounded good.

However, the algebra portion was pretty bad. The context was solving simple linear equations, and the presenter showed this sequence:

- one step equations (adding/subtracting; dividing)
- two step equations (two terms on one side, one on other)
- equations with parentheses, resulting in equations already seen

All equations were designed to have integer answers; the presenter’s rationale was that students (and instructor) would know that a messy answer meant there had been a mistake. All equations were solved with one series of steps (simplify, move terms, divide) — even if there was an easier solution in a different order.

When asked about the prescriptive nature of the work, the presenter responded that students understood that it was reversing PEMDAS (which, of course, makes it even worse for me).

The BAD PART of dev math is:

- Locking down procedures to one sequence
- Building on memorized incomplete information (like PEMDAS)

As soon as students move from linear equations taught in this way to any other type (quadratic, exponential, rational) they have no way to connect prior knowledge to new situations. In other words, the student will seem to ‘not know anything’ in a subsequent class.

To the extent that this type of teaching is common practice, developmental mathematics DESERVES to be eliminated. Causing damage is worse than not having the opportunity to help students. When we offer a class on arithmetic (even pre-algebra), the course is very likely to suffer from the BAD PART; offering Math Literacy to meet the needs in ‘pre-algebra’ and ‘basic algebra’ will tend to avoid the problem — but is no guarantee.

All of us have course syllabi with learning outcomes. Those outcomes need to focus on learning that helps students, not learning that harms students. Reasoning and applying need to be emphasized, so that students seldom experience the BAD PART.

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I am thinking about how well our Math Literacy course is doing in the Math Lab format. The Math Lab format creates a learning environment by establishing assignments and a structure for students to work through those assignments. The instructor ‘stays out of the way’ as long as learning is successful. This format has been used with very traditional content, and is now being used with a modern developmental course — Math Literacy.

Although some students struggle longer, and do not initially ‘get’ new ideas, the vast majority of students in the Math Lab Math Literacy course have been successful with:

- identifying linear and exponential patterns in sequence
- using dimensional analysis for unit conversions
- identifying the type of calculation for geometry (perimeter, area, volume)
- writing expressions for verbal statements

What’s been tougher? Anything dealing with percents — applications, simple & compound interest, etc. Of course, these are weak spots for students in any math class; over the years, I have not seen anything that ‘fixes’ these in the short term; the fix involves unlearning bad or incomplete ideas, and this takes time and long-term ‘exposure’ to errors (along with support from an expert). Direct instruction or group activities have limited effectiveness against the force of pre-existing bad knowledge.

The instructional materials form the basis for the learning in this Math Lab format. If the ‘textbook’ is focused on problems to do, contexts to explore, with the expectation that the instructor will provide ‘the mathematics’, then the learner centered approach requires that we use specialized processes in the classroom. The classroom becomes the focus, and we spend resources & energy on tactical decisions such as ‘homogeneous groupings’ or ‘group responsibilities’ or ‘flipping the classroom’. The materials we use in this course are well crafted to support learning; the authors ‘expected’ the classroom to be the focus, though our Math Lab ‘classroom’ is working quite well with the materials.

What if we could offer a true “student at the center of learning” design? Seems to me that this goal would lead us to use methods like our Math Lab, where students interact with the learning materials without an instructor mediating (as much as possible). Students in our Math Literacy course have been successful in learning new mathematics with decent reasoning skills in this format. Although initially confusing to students, the classroom is lower stress than a ‘regular’ classroom; there are no artificial social processes used to ‘facilitate’ the learning. Think of it as being more like a student as an apprentice, where direct engagement with the objects of the occupation is the key for learning.

Of course, we are not normally able to offer all math courses in this format of active learning. For me, the approach is to design my ‘lecture’ classes to be more like workshops. In a 2-hour class, I might deliver 45 minutes of very focused presentations (direct instruction) distributed in a deliberate manner through the class time. The length of ‘lecturing’ is varied according to the course and somewhat according to the needs of the students in a given class.

The point of this post is …

Stay out of the way of learning.

Students can learn by interacting directly with the learning environment.

We want students who are independent, and able to learn without a special structure. Prepare your students for the real world by creating learning environments where they develop those skills while they are learning mathematics.

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In a basic way, the answer to the last question is ‘yes’, due to the fact that all of the forces shaping the future are known at the present time. We don’t know precisely which forces will have a larger influence, and that is fundamental since the forces are not operating in the same direction. Imagine yourself in an n-dimensional force field where you can see the vectors around you. Although the wind varies over time, some types of vectors dominate your environment.

These vectors around us originate from power sources. Professional standards (MAA, AMATYC, etc) send out vectors in the direction of higher levels of reasoning, modern content, more diverse content, and more sophisticated instructional methodologies. The K-12 educational system, the Common Core in particular, send out vectors in very similar directions. Policy influencers, higher education provosts and chancellors, and state legislators send out vectors representing forces in different directions from those in the prior lists.

In the short term, this latter set of forces will dominate … because some of the individuals involved have sufficient decision making power that they can impose a set of practices on portions of our work. However, these practices will not survive long term except to the extent that they support the prevailing set of forces around us. As the people in authority change faces, the practices will tend to revert … either to the pre-existing conditions (bad) or to a condition making progress in the direction of the prevailing forces.

Here is a description, a picture, of where we will be in 10 to 15 years.

- Remediation will be smaller than in the past, but still normally discrete (not combined with college courses as in co-requisite models). Arithmetic will be ‘taught’ but never as a separate course and never will be a barrier to a college education. Content will focus on the primary domains of basic mathematical reasoning — algebra, geometry, trigonometry, statistics, and modeling. No more than two remedial courses will ever be required of students, regardless of their ‘starting condition’.
- “College Algebra” will not be used as a course title. Similar courses for non-STEM majors will have titles such as “Functions and Modeling in a Modern World. The content of this course, never used as a prerequisite to standard calculus, will be from the same domains as remedial mathematics — algebra, geometry, trigonometry, statistics, and modeling.
- “Pre-calculus” courses will be replaced by a one-semester “Intro to Math Analysis” course which focuses on the primary issue for success in calculus: reasoning with flexibility supported by procedural understanding. This course will have a very strategic focus in terms of objects and skills involved, with a shorter topic list than prior courses … taught in a way which results in a true readiness for calculus.
- “Calculus” courses will be re-structured to focus on a combination of symbolic and numeric work. The first semester of the two-semester sequence will include derivatives and integration for basic forms, as well as an introduction to scientific modeling using matrices such as those encountered in the client disciplines; this eliminates the need for our client disciplines to teach basic quantitative methods, and provides modern content to serve those disciplines. The second (and final) semester calculus course focuses on multi-variable processes combined with a more complete approach to scientific modeling — appropriate for students who may eventually conduct their own research in a client discipline
- “Liberal Arts Math” and “Quantitative Reasoning” will have merged in to a new QR course at most institutions. At some institutions, these courses are replaced by the “Functions and Modeling” course (which is fundamentally a QR course). Where QR exists as a separate course, the ‘practical’ content will be de-emphasized relative to today’s courses, with an increase in symbolic mathematics. The primary distinction between QR and Functions and Modeling is that QR does not include as much trigonometry.
- “Intro Statistics” will exist with similar content to the best of today’s courses. The primary change will be a relative decrease in the number of students taking a Stat course to meet a degree requirement, as program planners realize that their mathematical needs are more diverse than statistics … and that requiring statistics should not be based on just a desire to avoid college algebra (which does not exist in this ‘now’).
- Students will become inspired to consider a major in mathematical sciences by the diverse quality content along with the effective methods used within the courses. Instead of a focus on weeding out students not ‘worthy’ of majoring in mathematics, we will focus on including all students on the mathematical road to maximize the distance covered.

I see an exiting future, once we get past the relatively short-term impacts of changes imposed from outside. In the long-term, nothing can stop us from achieving a desired goal … except for our own doubts and lack of clarity.

My hope is that you see something in this image of the future to get excited about, something that plays the role of a beautiful sunrise in the forest. If you can SEE where you want to go, you can get there … and it is a lot easier to survive temporary struggles along the way.

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