The SCWAMS Math Package is Really Just Two New Courses

The text description of my SCWAMS proposal for the next iteration of the CCSS-M does not allow a proper visualization of how the pieces fit together.

There are really just two courses: “School Math – Tools for Life and Work” and “College Math – Preparing for the Professions”. Both would be full-year courses, and may be delivered in a variety of methods – from traditional classes, to individualized, to online formats. They would not replace the traditional secondary courses. My emphasis is that the skills need to be reviewed, refreshed, integrated as a capstone, and assessed in the two years before high school graduation.

There should be no difficulty in offering School Math in both junior and senior years for those students who need more time to complete the assignments. That is why flexibility in delivery formats is so important. The essential thing is that ALL students meet the standards by graduation time.

Likewise, the College Math could be offered to juniors who have completed School Math as a pre-requisite. Again, the School Math could be delivered in the summer, on weekends, or online for those who might desire an accelerated approach. Again, though, ALL students need to show competency, so no one should be allowed to skip – the assessment, not a transcript, is the evidence of learning.

So the School Math course contains what ALL students NEED TO KNOW for life and work, but nothing more. That means NO “algebra”! There would be lots of “manipulation” of geometric formulas, though.

For the College Math course, I find that the first unit, “Mathematical Methods”, is largely missing from any current or traditional secondary math courses. This unit specifically sets out the topics of: Reasoning, Propositions, Sets, Logic, and Proofs. Mathematical thinking is thus developed as a single process, rather than spread around several courses and chapters, leaving it for the student to connect the dots (which rarely happens). This is the single most critical part of my SCWAMS suggestion, and is not clearly identified in the CCSS-M, although it is certainly an implied goal.

The other parts – Applied Math, and Workplace Math – are built on top of the two core courses, and include concepts related to other school courses, such as the “Communication, Creative, Cultural, and Social Arts (CCCSA)” (aka “Humanities”), and to occupational applications. These are not included in the “21st Century Core Standards (21CCS)”, since the standards are assessed separately. So they are not separate courses, as such, but integrated into the two core courses.

Ratner’s comments about Common Core in the WSJ

It’s all a matter of perspective. Considering that Marina Ratner’s view reflects the position of the Mathematics Department of the University of California – Berkeley, it would appear that the Common Core State Standards for Mathematics (CCSS-M) “were several years behind the old standards, especially in higher grades”, and that students “… would have little chance of being admitted to even an average college and would certainly struggle if they did get in.”

Beginning with my initial participation in the “National Science Foundation – Statewide Systemic Initiative (NSF-SSI)” in 1991, we repeatedly rejected the UC-driven standards as being too rigorous for ALL high school graduates. Those particular items were certainly appropriate for preparing SOME students for entry into university mathematics programs, thus relieving the collegiate faculty of teaching “remedial” classes. We did feel that they should be considered “goals” to strive for, but not as requirements for all graduates.

One remnant of that elitist view still remains in the CCSS-M as “Grade 8 » Expressions & Equations » Analyze and solve linear equations and pairs of simultaneous linear equations.”


Really? For EVERY American student? In eighth grade?

Our South Dakota group discussions over the years agreed that the initial 1989 NCTM standards were a little too fuzzy, and so appreciated the modifications published in 2000 as being in the “Goldilocks” range, containing the essentials ALL students would need for college and career, yet letting those who wish to pursue technical and professional careers to move on to more advanced courses as they wish. Our group approved a final 2009 proposal, which is closely matched with the CCSS-M. Our state legislature approved the standards in 2011, and they are currently being implemented.

Thus, my suggestion is for stating what ALL high school graduates NEED TO KNOW in a year-long course for the junior year in high school. This course is “School Math – Tools for Life and Work”, and would be required. The delivery and assessment methods are beyond my proposal, but should be flexible and appropriate for individual learners.

Those who show proficiency of the standards could then take an optional course using a “systems approach” (in comparison to the traditional “silo” separated topics), called “College Mathematics – Preparing for the Professions”.

By separating “Math for ALL” from “Mathematics for SOME”, my intent is to clearly identify topics according to how learners will use them after graduation. The Common Core people should be comfortable with “School Math”, while the post-secondary crowd (including Ms. Ratner and the UC faculty) could look for “College Mathematics” on their applicant’s transcripts.

What’s Next After Common Core for Mathematics?

To go beyond the Common Core State Standards for Mathematics (CCSS-M), we need to think in terms of the ways we use actually use math skills – as tools for daily life and work, as well as systems that help explain and predict how things function in the professions.

The next step in the real transformation of math education standards is to clearly identify what skills students MUST KNOW when they graduate from high school, the “21st Century Core Standards (21CCS)”.

These would be provided in a required full year course at the junior grade level for ALL high school students to complete. This would include an assessment of competency for each of the standards during the year, along with opportunities to refresh, update, and upgrade the topics and skills presented in prior grades. The instructional and testing methods should be flexible and appropriate for particular learner situations.

This required course, called “School Math – Math for Life and Work” would consist of eleven units, starting with pencil-paper computations of whole and decimal numbers, and common fractions. The assessed competencies of this course would correspond to the 21CCS items and provide a basis for what ALL students NEED TO KNOW as they graduate into life, work, college, and career.

For each of the topics presented in “School Math”, though, it would be NICE TO KNOW how those skills connect to related subjects, such as STEM (Science-Technology-Engineering-Mathematics) and CCCSA (Communication, Creative, Cultural, and Social Arts) within the school. Collaboration between math and other subject area specialists can generate activities and problem sets which develop “Higher Order Thinking/ Depth of Knowledge (HOT/DOK)” understanding. These “Applied Math – Building Basicl Skill Sets” activities are recommended, but optional, for all students, who would earn additional credit for completed assignments.

A further level – “Workplace Math – Applications from the Real World” – would connect the core math topics to occupational situations that learners might want to explore in advance of their years beyond high school. Again, additional credit would be earned for completed assignments.

Some learners, though, might wish to continue beyond “School Math” to  a full year “College Mathematics – Preparing for the Professions” course in the senior year. Using a systems approach, this course first develops “Mathematical Methods”, and then applies those principles and techniques to number systems, algebraic operations, and functions, including linear, non-linear, polynomial, exponential, and trigonometric functions. This course would be optional for completers of School Math, and its topics would not be part of the 21CCS.

Putting the five pieces of this puzzle together, then, ALL students should be assessed for what they MUST KNOW of the “21st Century Core Standards” in the junior or senior years of high school.

The standards would correspond directly with the topics contained in “School Math – Math for Life and Work”, as to what they NEED TO KNOW for college and career.

For learners anticipating a college experience, they SHOULD KNOW about “College Mathematics – Preparing for the Professions”, which provides a foundation for further post-secondary study.

In any case, once learners show competency of the designated skills, they can expand their understanding through “Applied Math – Building Personal Skill Sets”, and “Workplace Math – Applications from the Real World”. These include various activities and projects that learners would find are NICE TO KNOW, according to their individual interests, abilities, and preferences.

Combining these gives “SCWAMS – School, College, Workplace, and Applied Math Skills” as a comprehensive systems approach to the next step beyond Common Core.

This approach also looks beyond the historical sequences in which mathematics subjects were developed through the centuries, to focus on how they can provide useful functions at home and work in this 21st Century.

This SCWAMS approach goes beyond reform, into a new transformation for mathematics education in the 21st Century.

An Education “Transformation” in this Decade? Yes!

Throughout a variety of continued blog discussions, there appears to be a lot of repetitive bashing of the current “education system”, as though it were some dystopian governmental monolith, intentionally preserving its status quo through oppression of better ideas for teaching and learning. I suggest, though, that it is doing what it was designed to do, as a product of the Fordist assembly-line factory organization of the first half of the Twentieth Century. We should recognize that, as a vehicle without wings, traditional education cannot provide effective and relevant learning experiences for ALL students, as we wish. Further criticism won’t change the situation.


What is missing, though, are the understandings we have gained with a “PostModernist World View” which has evolved in latter half of the century. Von Bertalanffy, Boulding, and Beers (The “Three B’s” ?), among others, have given us “systems thinking” methods of looking a structures and relationships among  organizations to make them more efficient and cost-effective. Deming’s techniques of measuring quality can be also used to improve the rates of “success”, even when applied within that factory model.


Computer processing speed and displays now let us interact with modeling constructions so that we may visualize how natural, social, and education systems and processes work, to simulate alternatives, and to predict possible outcomes. Most recently, the use of digital media, wide-bandwidth communications, and data storage capacity have made quality content information available to the far reaches of the globe. We are also learning techniques of informatics for analysis and to make real-time recommendations about various choices students, teachers, and administrators can make, much as Amazon tracks and suggests our online shopping experiences. Also, the science of complexity has provided various “non-linear” ways of looking at learning and education as diverse, evolutionary, and emergent processes, utilizing effective strategies from gaming theory, graph theory, and risk management to improve the sustainability of our current and future societies.


So I see this decade as an exciting time in which a true transformation in education can occur, when the perspectives of “Systems, Quality, Modeling, Informatics, and Complexity (SQMIC)” are implemented. I feel these five perspectives are parts of “WKID Intelligence”, a substrate underlying the typical content areas  of STEM, the “Humanities, Arts, and Social Sciences (HASS), as well as “Health, Physical Education, and Recreation (HPER)”.  Access to a variety of interactive “apps” would provide tools to assist learning, as technology skills needed in a global economy, and would also be part of the organizational processes that guide their learning experiences.


Rather than an assembly of instructional components put in place and tested at certain times, learners could become designers of their own understandings of their world, by developing data into information, building that into the knowledge they need for entry into society, and, hopefully, gaining wisdom enough to become successful. Much like the “3-D printers” we are seeing these days, education could become an efficient, effective, and customized production and delivery system that morphs out of rigid traditional modes, and truly becomes a “Comprehensive STEM Curriculum Framework for the 21st Century”.

Flying John Boyds “OODA Loop” through STEM

John Boyd’s contribution to systems thinking was the “OODA Loop” – Observe, Orient, Decide, Act – with a feedback loop that brings the results back into a new cycle. As a combat fighter pilot instructor, he was known as “40 second Boyd” because he shot down every bogey within forty seconds of contact.


His key contribution was in the “Orient” thought process, in which the agent filters through “culture, genetics, ability to analyze and synthesize, and previous experience”, with a “faster tempo” than the adversary, who does not have enough time to “generate mental images”, with the effect of making the situation unpredictable. The most well-known strategic application of the OODA Loop, of course, was the Gulf War, in which the first strikes into Iraq disrupted Sadam’s communications networks, thus slowing his awareness of the actual situations.


A good description of the OODA Loop and its applications to military, business, and other aspects of systems thinking is provided on Wikipedia, as a first link to other reference information.


While it appears that many commentors to this thread express concerns about getting the whole “system of education” correct before we begin to take action, I prefer to utilize the OODA Loop, with incremental and iterative repetitions of trying new methods and evaluating the results, and then go around again, in developing an effective teaching/ learning system and process.


My descriptions of a “Comprehensive STEM Curriculum Framework for the 21st Century”, as described previously in this topic thread, identify every high school STEM topic with a three-dimensional coordinate of instructional modules and lessons in an “InfoSpace”, much like the Nevada skies Boyd flew in.


Those modules utilize the “Information Mapping” techniques of Robert E. Horn, in which the learning activities are developed using templates according to the type of instruction, such as: Fact, Concept, Structure, Procedure, Principle, Process, and System. Hyperlinks then connect the content modules to others, allowing alternative pathways through the InfoSpace, with waypoints identified for required graduation competencies. Each lesson, module, unit, and course has review and assessment components that give immediate feedback to the learner, and also provides documentation for the student, teacher, and administrators. This empowers learners to employ their own OODA Loops as they proceed through the highways and byways of the InfoSpace.


This overall plan, then, prepares the whole of STEM content in a way that allows flexible exploration, in groups or individually, using effective teaching/ learning practices, with immediate feedback to all stakeholders. I think that it is a “good enough” attempt to add to the discussion about transforming education. So while some may say that such a process may appear to be “designing and building the aircraft while flying it”, I don’t mind, since at least I’m flying – in the Boydian Way.

STEM as a Sandwich? Might be good!

Technology, as a whole, is an important part of the STEM curriculum in that it shares some “habits of mind”, as compared to the language, arts, and social science subject areas. While science utilizes the scientific method to understand principles of matter and energy in our world, we find that mathematics allows us to quantify those principles and make predictions by utilizing rigorous proofs, and engineering gives us a “built world” that gives us utility, survivability, and (hopefully) sustainability as a global society. Technology’s role is for the use, operation, and maintenance of that built world. Principles, proofs, and designs have no value to us unless they are put into action in our lives and communities. And we need technical people to keep that clockwork machinery running.


So what Technology brings to the STEM curriculum, I think, is that every high school graduate needs: to know how to put those “Owner’s Manuals” to use for all those gadgets they have; an awareness of the ‘”theory of operation” of how those devices work; recognition of dangerous situations and safety precautions; and some sense of a troubleshooting process when things aren’t working right. These skills and practices are exactly those critical-thinking and problem-solving techniques that are supposed to be part of an overall high school curriculum.


Yet these “modes of thought” should not get too tangled up in specific products or occupational fields. In my suggestions for a “Comprehensive STEM Curriculum Framework for the 21st Century”, those options would be provided in the third dimension of “Applied Career Preparation Pathways”, such as: Agriculture, Business, Communications, Construction, Health, Information Technology, Marketing, Manufacturing, and Transportation, among others.


So once learners develop “Core Content Competencies” of the STEM topics, to the step of a “Basic Workplace Skill Set” needed for entry into a desired occupational level, they can explore the career pathways of their choice as applications of where their “S, E, and M” knowledge is put to use. This would also be most appropriate for longer term projects, small-group interactions, and so forth. The technology aspects of the STEM curriculum, then, would be “capstones” to the essentials of the traditional courses, and infused throughout the programs, rather than distinct courses in themselves.


Likewise, the techniques of “information” would be tapped into where appropriate, including those involving Systems, Quality, Modeling, Informatics, and Complexity (SQMIC). We can integrate the traditional STEM subjects by building on a substrate of these “Big I = Information” tools, while overlaying them with options leading to applications in various career pathways. High school graduates would be much better prepared for life and work, I believe, with such an integrated “sandwich” curriculum, than the “silo” or “stovepipe” traditional structure in common use.

STEM Curriculum: Rearranging the Tableware?

The traditional STEM curriculum in American secondary schools was developed in the 1890s, with the apparent success of “scientific” methods in producing the industrialized and electrified “progress” of the times. Other than “modern” additions to account for developments in atomic and particle physics and such, the table of contents of most textbooks was largely unchanged over the last century. Even so, the latest “improvements” to course content generally involved topics that were too complicated for the typical high-schoolers’ mind, and were irrelevant to their daily lives, or for their college and career preparation.

I agree that this reductionist view has long ago run its course, and that the “whole-to-parts” viewpoint is really the new development of the late 20th Century. Since we largely knew how the basic elements of math and the sciences worked, we started looking at ways in which those components could be assembled and organized in different (sometimes impossible) ways.

We began looking at things as “systems”, which had properties that were not evident from the individual elements. We began to look at the products of those systems in terms of their utility and value through the measure of “quality”. We gave data and information meaning through techniques of “informatics”. We also began to look at ways to describe physical processes with “modeling” methods, and then to plan and design for optimization using virtual and simulation technologies. Most recently, we are looking at the impacts of non-linear “complexity” to those systems and processes.

What I suggest that needs to be changed for a “Comprehensive STEM curriculum framework for the 21st Century” is that we add an orthogonal overlay of “unifying perspectives” to the traditional “pipelines” of the STEM courses in math, physics, chemistry, biology, earth science, etc.

To the table of contents of the usual textbooks, each chapter might show why the topics relate to the overall “systems”, “quality”, “informatics”, “modeling”, and “complexity” perspectives. So while the “body” of the text information may not change a lot, specific sidebars and links can make connections to how these facets relate the facts to the students’ lives and to their preparation for life beyond school. It would be the curriculum designers role to make sure that such linkages are made in the overall “Content Information and Knowledge Space” (CIKS).

So the real way to STEM reform is not just to “rearrange the table (of contents)”, but to be sure that each place-setting of 19th Century chinaware includes the 20th Century utensils of the Systems, Quality, Informatics, Modeling, and Complexity (SQIMC) perspectives, so that we may enjoy a feast of knowledge and competency in the 21st Century.