Making the mathematics education sausage
By Keith Devlin @profkeithdevlin
“People who like sausage should avoid watching it be made.” Old saying
There’s an old Irish joke that goes like this. A motorist pulls up in front of a pub, winds down the window, and addresses an old man sitting at an outside table sipping a glass of Guinness: “Excuse me, what’s the best way to get to Killarney?” The old man puts down his glass, and responds: “Killarney, you say. Well, now, if it’s Killarney you want, the best thing is, don’t start from here.”
Which nicely sums up the position US K-12 mathematics education finds itself in if it wants to adapt to meet the challenge presented by the newly released PISA 2022 Mathematics Framework that I wrote about last month.
Figure 1. From https://pisa2022-maths.oecd.org/ca/index.html#Overview
That framework — a living document designed to keep pace with (and as far as possible anticipate) changes in the global society and the way it uses and depends on mathematics — is the result of an ongoing process over many years, involving academic mathematicians, mathematics education researchers, teachers, and representatives from industry and government agencies that use mathematics. (I participated in a number of workshops the PISA team organized or attended as they developed the new framework.)
Coming as it does twenty years since the culmination of the biggest revolution in mathematical praxis since the Thirteenth Century, there is a lot in the new PISA framework that will appear strange (and perhaps incomprehensible) to anyone who is not in the business of developing or using mathematics. (If you don’t know which “revolutions” I am referring to here, check out this newspaper article from 2017 for the recent one, and this book for the one in the Thirteenth Century.)
Of particular note, given the way many countries, including the US, structure their education system, some math teachers will surely also be among those who find it strange. Though the revolution in mathematical praxis I referred to began way back in the 1960s, and was essentially completed by the end of the Millennium, changes in the education system have, for the most part, lagged far behind the changes in the field. (There are good scientific and educational reasons why change should lag, I should add; but system inertia and the inevitable opposition of parties who see changes as a threat to their livelihood or position have probably made it lag more than it should have.)
In the US, a major attempt to bring K-12 mathematics education up to date was the establishment of the Common Core State Standards for Mathematics (CCSSM) in 2010. I discussed that initiative briefly in my last column (link above), pointing out that it was in the eight top-level Standards for Mathematical Practice (CCSS.MP) where the CCSS overlaps PISA 2022.
To some extent, that’s an apples and oranges comparison. The CCSS are, as the name reflects, learning standards that the nation’s educators should aim for their students to achieve (cumulatively by graduation in the case of the MP, and year-by-year for the more detailed standards that follow the MP). In contrast, PISA 2022 is a framework for assessment. PISA is, after all, an assessment organization — as its name indicates (Program for International Student Assessment).
Acting for its parent organization, the Organisation for Economic Co-operation and Development (OECD), PISA seeks to provide national governments with up-to-date feedback on how well their education systems are meeting the needs of modern society. (OECD is an intergovernmental economic organization, headquartered in Paris, France, founded in 1961 to stimulate economic progress and world trade. It currently has 38 member countries.)
Since heads of state are usually highly sensitive to international comparisons, nations invariably look to PISA to guide the development of their national standards and their educational systems, and to measure how well they are doing compared with other countries.
What PISA gives them by way of guidance, is, at heart, a small set of broad themes to aim for. In PISA 2022, those themes are based on “mathematical literacy, relating mathematical reasoning and three processes of the problem-solving (mathematical modeling) cycle.” (See Figure 2 regarding the latter.)
The next PISA assessment will, they say, measure how well a nation’s students “use mathematics in every aspect of their personal, civic, and professional lives, as part of their constructive, engaged, and reflective 21st Century citizenship.”
If this does not sound like what you thought PISA would want to measure when it comes to mathematics, you haven’t paid attention to the degree to which the world has changed when it comes to mathematics.
What happened to all that (symbol-heavy) “math” you remember from your schooldays? You know, all those formulas, equations, and procedures.
Well, that’s all still there, and in the US you will find it in the (very fine-) detailed standards in the CCSS. But all of that detailed procedural stuff can now be done by readily available, and for the most part cost-free, computer systems (and is usually done that way, apart from in some school classrooms). As a result, it’s been significantly downgraded to the supporting cast; important, but no longer the leading role. The primary focus of mathematics education today, as determined by the mathematical requirements of today’s world, is to prepare the next generation of citizens with the mathematical thinking skills (as opposed to the procedural execution skills of former times) they will need to lead productive and rewarding lives as members of society.
That shift in emphasis makes mathematics education today very different, and a whole lot harder to provide effectively, than it was in the past. (Actually, it’s only a shift in implementation. Preparing the next generation for their adult lives was always the goal of education. What’s changed is what that entails. People used to have to be computers; today, they have computers, and what they need to know is how to use them effectively.)
Okay, let’s unpack that brief overview of that foundational theme of the PISA 2022 Mathematics Framework I quoted earlier: “mathematical literacy, relating mathematical reasoning and three processes of the problem-solving (mathematical modeling) cycle.”
Here is how PISA defines mathematical literacy:
“Mathematical literacy is an individual’s capacity to reason mathematically and to formulate, employ, and interpret mathematics to solve problems in a variety of real-world contexts. It includes concepts, procedures, facts, and tools to describe, explain, and predict phenomena. It helps individuals know the role that mathematics plays in the world and make the well-founded judgments and decisions needed by constructive, engaged and reflective 21st Century citizens.”
Notice that there is explicit mention here of all the apparatus of what most people think of as “mathematics”. But notice too that it is there as simply a set of tools to achieve a more important goal. The focus is on how people use those tools to do things.
The PISA Framework document goes on to elaborate:
“PISA 2022 aims to consider mathematics in a rapidly changing world driven by new technologies and trends in which citizens are creative and engaged, making non-routine judgments for themselves and the society in which they live. This brings into focus the ability to reason mathematically, which has always been a part of the PISA framework. This technology change is also creating the need for students to understand those computational thinking concepts that are part of mathematical literacy. Finally, the framework recognizes that improved computer-based assessment is available to most students within PISA.”
What about that other term in the foundational theme statement: Mathematical Reasoning? Here is PISA’s elaboration:
“The ability to reason logically and present arguments in honest and convincing ways is a skill that is becoming increasingly important in today’s world. Mathematics is a science about well-defined objects and notions that can be analyzed and transformed in different ways using mathematical reasoning to obtain certain and timeless conclusions.
In mathematics, students learn that, with proper reasoning and assumptions, they can arrive at results that they can fully trust to be true in a wide variety of real-life contexts. It is also important that these conclusions are impartial, without any need for validation by an external authority.”
And here we see reference to the idea that mathematics has an absolute nature, with eternal, context-free truths. There is a lot that can be said and discussed about that. And frequently is. But it’s worth noting that PISA thinks it is important to call out the value of mathematics as a tool that provides “impartial conclusions” when applied to real-life issues. A minority of students find the philosophy of mathematics interesting. (I was one such.) But it today’s world, it is crucial that all citizens appreciate and understand that mathematical results are not “opinions.”
Finally, the term “three processes of the problem-solving cycle” refers to the cyclical
FORMULATE — EMPLOY — INTERPRET & EVALUATE
reasoning process (shown in Figure 2 above). This is familiar to anyone who has learned how to use mathematics (and hence requires no elaboration here).
As I discussed in my post last month, the CCSS.MP was the Common Core writers’ way of establishing essentially the framework outlined above in terms of learning standards that make sense in the context of US K-12 mathematics education.* Consequently, the MP are in alignment with PISA 2022.
[* Actually, I don’t know that first hand, since I was not part of the CCSS development process, but even if they did not view it that way, that was the result. Of course, they could not have expressed it in exactly those terms, since PISA 2022 was still in the future. But this entire process was being carried out by various mathematics and mathematics-users communities around the world, all having connections to one another — often because the same individuals participated in two or more groups — so ideas flowed freely between them. PISA, in particular, casts its net widely.]
I should stress the point I made earlier, that PISA 2022 is a framework for assessment, whereas the CCSS are standards for learning. Both are based on why, and both stipulate what, but neither lays out instructions for how (to assess/teach, respectively).
In the US, with its regionally managed education system, the CCSS were developed for elective use nationwide (nine states currently do not follow CCSS). But the CCSS is not a curriculum. The development or adoption of curricula to guide education aligning with the standards was left entirely to the states (or entities within states) to organize.
Also left to the states was provision of guidance to districts, schools, and teachers on how to teach to achieve the learning outcomes desired. This is the key “rubber-hits-the-road” step, and, given the need for mathematics education to be aligned with mathematical praxis, it’s particularly challenging at the present time.
By and large, prior to the 1960-1990 technological revolution that outsourced procedural math to machines, the mathematical skillset essential to adult life could (at least in principle) be developed by the end of middle school, leaving high school to focus on the higher mathematics required for careers in science and engineering.
Preparing all of today’s students for their adult lives, in a world where the execution of the procedures of higher mathematics is available to all by way of a myriad of mathematical apps, is a much greater challenge. The nature and the degree of mathematics learning to live and operate in that world goes way beyond what we can achieve in middle school.
For instance, the PISA 2022 Framework highlights the following topics that are essential for life in a free society in the 21st Century: computer simulations, uncertainty and data science, the mathematics for handling global growth phenomena such as climate change and pandemics. All of these are computer-heavy topics. The PISA Framework also emphasizes that consideration be given to the different contexts in which mathematics is learned and used, including personal, occupational, societal, and scientific contexts.
Moreover, math educators have to be able to provide this more extensive “mathematical skillset everyone needs today” at the same time as they:
expose all students to sufficient higher mathematics so that those who find they enjoy it have an opportunity pursue it further, and
provide students who do decide to pursue it with the foundational education they need.
In other words, achieving the new (aimed at everyone) should not be at the expense of the old (which was always focused on a minority).
Threading that particular needle, and doing so in an American context that values individual freedoms of choice, is a considerable challenge.
Curriculum frameworks are the tools that offer guidance for implementing content standards (most frequently the CCSSM in the US). They describe the curriculum and instruction necessary to help students achieve proficiency; they specify the design of instructional materials and professional development; and they provide guidelines and selected research-based approaches for implementing instruction to ensure optimal benefits for all students.
In other words, deseigners of curriculum frameworks start with the what to teach and the why teach it (both determined by others), and focus on specifying the how to teach it. The resulting framework is intended to indicate how the educational rubber should hit the road.
Curriculum Frameworks: Where the rubber hits the road
Figure 3. The new California Mathematical Framework, currently nearing completion of a lengthy process of development and consultation that was delayed by the COVID pandemic. Note that the focus is on how to implement the California Content Standards. California is one of a majority of states that has adopted the national Common Core State Standards to determine content. From https://www.cde.ca.gov/ci/ma/cf/
My own state, California, is just nearing completion of a lengthy process to develop the new California Mathematics Framework (CMF) to provide that guidance. (It is just guidance. The CMF is purely advisory. Districts and schools can follow as much or as little as they choose. Coming from Europe, I think that’s crazy, but it is the American way.) See Figure 3.
Figure 4. The CMF’s three Drivers of Investigation (DIs) provide the why of learning mathematics: Making Sense of the World (Understand and Explain); Predicting What Could Happen (Predict); Impacting the Future (Affect). The DIs overlay and pair with four categories of Content Connections (CCs), which provide the what mathematics shown along the x-axis. The DIs work with the Standards for Mathematical Practice (y-axis) to propel the learning of the ideas and actions framed in the CCs in ways that are coherent, focused, and rigorous. From https://www.cde.ca.gov/ci/ma/cf/mathfwfaqs.asp
The last update was in 2013. Advancing from there to the new Framework has been a lengthy and tortuous process. (I was not involved in it, other than to participate in one workshop on how to handle data science in K-12 education. I also know two of the academic researchers who were involved in the process, and have interacted with a third on a few occasions.)
For a quick overview of the purpose, development, and content of the new CMF, the Mathematics Framework FAQs page on the California Department of Education website provides an excellent summary.
The first thing that struck me when I looked at the new CMF was how closely it aligns with PISA 2022. I queried one of my colleagues who was involved with the CMF, and was told there was no deliberate effort to coordinate, or to consult the publicly available drafts PISA made available. The strong similarity was a result of everyone working in the same global environment of present-day mathematical praxis, being aware of the latest research, and having the same overarching, learning-for-life goal in mind.
For instance, the CMT graphic shown in Figure 4 to illustrate the overarching educational theme of the California Framework could easily have been used in PISA 2022 (which has an analogous graphic).
Since I have first-hand knowledge and experience of the what and why of present-day mathematics education, but virtually no experience of the how that the CMF addresses, I found this empirical alignment very reassuring.
Certainly, fears raised during the development process by anxious parents, some higher education mathematics faculty, and other interested parties (in some cases fueled by special-interest groups), that the new CMF was going to “water down” math or “hold students back,” proved to be unsubstantiated. (Here is one example of such fears.)
A lot of the public anxiety was occasioned by the California Department of Education’s need (and obligation) to address both the poor mathematics achievement levels of the state’s K-12 students, and demonstrated racial inequities.
Figure 5. This statement, from the new CMF, Chapter 2, is in complete alignment with PISA 2022 regarding the need to account for the different contexts for learning and using mathematics. https://www.cde.ca.gov/ci/ma/cf/
One argument raised against steps proposed to meet these particular goals is that mathematics is context free. (There is a sense in which this is true, particularly if you think of mathematics as a body of established knowledge.) Hence, the argument goes, it should be taught in isolation. Now that does make sense for some parts of mathematics. And no one has ever suggested that should not continue. But mathematical praxis is anything but context-free, and in today’s world mathematical praxis is a dominant global activity, for both good and ill. That’s surely why PISA 2022, with its eye firmly on preparing tomorrow’s global citizens, makes a big deal about the contexts in which mathematics is taught and used. See Figure 5 for just one of many instances where the CMF authors paid attention to contextual issues.
Another issue that led to a lot of debate was the inclusion of calculus (as an elective) in the high school curriculum. PISA 2022 makes no mention of it. The CMF includes it (as an elective that schools or districts may want to include). As part of the required skillset for all future citizens, data science and statistical reasoning are, to my mind, the “advanced topics” that should be covered by all; not calculus. My experience as a university mathematics instructor over many years was that the majority of students entering with high school calculus credit(s) would have been better served making their first encounter with the subject at university. By and large, what they acquired at school was a superficial mastery of symbol-manipulation, without any real understanding.
On the other hand, it was my own encounter with calculus at age 16 (it was an obligatory subject for the optional last two years of high school in the UK back then) that led me to major in mathematics at university. It was, simply, the first mathematics I’d come across that I did not find dull, shallow routine. Yes, I had trouble at university, when my shallow knowledge of the subject proved woefully inadequate, but in my case I then had the motivation to work hard and fix it. I have also heard other mathematicians report a similar experience. There is undoubtedly value in exposing high school students who have mastered the required prerequisite skills to a “deep” (and important) topic like calculus.
I still despair of the “calculus is the pinnacle of high school math” mantra. In my own case, I was the only student in my high school that worked hard to understand it, and I was excused from the regular classes to study on my own from textbooks, with occasional oversight by my teachers. The rest of the class struggled hopelessly. But I think the CMT made the right call in including it as an elective that schools or districts can choose to offer (and should if possible). In general, the CMT stresses that students with a desire to move ahead should not be held back.
* * * * *
As always when a complex deliberative process is carried out openly, with a mechanism for input from all interested parties, experts in one domain can easily misinterpret what they see, and may fail to recognize the expertise of others. This was certainly the case for the formulation of the CMT, where experts in different domains all had access to the educational sausage while it was in its messy development stage.
In my own case, as a mathematician with extensive experience both in pure research and in a range of application areas, and in teaching mathematics to undergraduates, I could understand and evaluate how the CMF development handled the what and the why. I could also provide input about those to the process (explicitly for PISA in a number of workshops, indirectly through my writings and other professional activities for CCSSM and the new CMF). But I had to (and have to) rely on the expertise of others when it comes to the how.
Fortunately, I’ve interacted with, and worked with, both K-12 mathematics education scholars and K-12 mathematics teachers a number of times over my long career. As a result, I long ago came to the realization that knowing a lot about the what and the why of K-10 (sic) math ed is not remotely enough to understand the how.
In other words, I can provide experienced input and judgement regarding what kinds of meats and other ingredients should go into creating a good K-10 math ed sausage (the what), and I can provide experienced assessment of what the resulting product tastes like and whether it is sufficiently nutritious (the why). But I don’t understand how it is actually made. With regards to this essay, you should take that as a DISCLAIMER. :)