In previous posts, I have made a brief case for why new standards in high school math are needed. I also proposed some guiding principles to outline a vision for what these new standards might look like. Below, I try to respond to some anticipated questions and concerns:
- We don’t need another round of training, curriculum development, textbook realignment, and massive shifts from the top down.
- What’s described here isn’t really standards.
- The problems cited with the current standards are solved when teachers use Power (or Essential) Standards.
- Don’t the Standards for Mathematical Practice (SMP) already address most of the goals in the guiding principles?
- These new standards seem like they really minimize the importance of content.
- With standards written like this, how will teachers know what to teach?
- Will this mean math teachers will not teach skills?
- How will the new standards achieve depth if it is not used as a guiding principle?
- Yeah…I still don’t get eQPTC.
- These guiding principles fly in the face a national standards movement that sought uniformity in what children were learning across districts and even states.
We don’t need another round of training, curriculum development, textbook realignment, and massive shifts from the top down.
Agreed! The new standards are intended to be inclusive. They could allow for teachers to continue using current texts and curricula. The new standards proposed here are intended to be written so that they cannot be wielded with political power to impose massive and immediate changes to curriculum and instruction. These proposed standards cast a very different vision for high school mathematics, but they do so in a way that does not exclude more traditional approaches. Any new standards written along these guiding principles should include a statement saying as much.
The guiding principles described here are intended to create standards that can only inspire, not require, change from the top-down. They are intended to open possibility from the bottom up and mitigate the ways standards are used to coerce and systematize the transmission model of education.
What’s described here isn’t really standards.
The first sign that the work of developing new standards has been done correctly is that it will be immediately met with a challenge: those are not even standards. To which there is only one appropriate response: that is exactly the point. New standards for high school mathematics should conceive of standards as something fundamentally different from what they currently are.
The current standards were developed in, and continue to contribute to, an education culture that has been steeped in the transmission model. A ‘good’ standard explicitly states the knowledge transmitted to the student; the knowledge is then be reproduced by the student; and finally, a determination is made about how well the reproduction aligns with the original standards.
If we want to move beyond the transmission model, as often professed by movements for deeper learning, personalized learning, constructivism, inquiry, project-based learning, etc., we need standards that are not bound to the transmission model. As such, these guiding principles seek to provide a framework that supports more learner-centered, critical, and humanizing approaches to education.
Perhaps, put another way, the current standards are written as the perfect partner for standardized testing. Thus, our view of what standards are has been molded by standardized testing, even if unconsciously. It is incongruent to decry teaching to the test while simultaneously embracing teaching to standards designed for testing. Standards would look very different if they were not designed to empower testing but were instead designed to facilitate rich, multi-faceted learning experiences.
Ultimately this is a question of values. Those who value a transmission model of education and who view learning as the mere accumulation of facts will certainly struggle to accept the vision laid out here for new high school mathematics standards. Many educators, however, know that educative experiences are much more than this. Standards hold a great deal of weight in our system. They can be written to anchor us to the transmission model or to point us toward deeper, more meaningful learning experiences.
The problems cited with the current standards are solved when teachers use Power (or Essential) Standards.
No. First, the most practical concern: is a single Power Standard selected from the current standards worthy of deep exploration? That may be debatable and certainly depends on the standard selected and the way it is implemented. But I would argue that the bent of the current standards does not favor a deep mathematics. Consider answering the first question by asking a series of other questions about any of the current content standards:
- Does it provoke tension and ambiguity that could resolve in unexpected ways (or even remain unresolved) OR does it point more toward pre-determined knowledge and procedures?
- Is it giving students an idea to wrestle with or merely something to do?
- Does it frame mathematics as something that is constantly developing OR as something that is already developed?
- Does it present mathematical knowledge in historical, critical context OR does it present isolated knowledge?
- Does it immediately suggest multiple expressions, interpretations, or enactments of a mathematics OR does clearly indicate a single, traditional mathematical process?
- Is an meaningful concept that could engage anyone with any level of mathematical knowledge OR is it merely a scaffolding fact that requires prior technical knowledge and skill and whose inclusion is only necessary as prerequisite for acquiring additional technical knowledge and skill?
It is hard to argue that many of the current standards would answer anything other than the latter. Thus, even when selected as Power standards they are not inherently ripe for creating mathematical experiences that are fundamentally different from what we have always had.
Additionally, the selection of Power standards is not always the purview of the teachers. Standards are political documents also us by administrators, bureaucrats, instructional coaches, etc.… Many times, educators in these roles value the standards that are most easily replicable and assessable for data. Thus, systemic pressure can tend towards Power standards that are the least accommodating to rich, divergent mathematical experiences.
Most importantly, the Power standards practice creates a momentum toward the transmission model of education and a rote practice of mathematics. A large list of knowledge is consulted. Educators select which items they deem most ‘important.’ Those items are then broken down into more granular learning targets. The process creates a funnel of knowledge – the direction of knowledge flowing down and narrowing as it reaches the student.
Power standards do not resolve the major problems I find with the current standards. Power standards are a byproduct of those problems that in turn reinforce them. The current standards are not written to directly support a model of teaching and learning that begins with the student and opens out.
Don’t the Standards for Mathematical Practice (SMP) already address most of the goals in the guiding principles?
Indeed, the SMP are probably the closest model for what I am proposing in the Guiding Principles for New High School Mathematics Standards. If learning experiences were developed almost entirely around the SMP, we would certainly see some very different high school mathematics classrooms. While there may be places that prioritize the SMP over content standards, the SMP cannot compensate for the failings of the current standards.
First, using the SMP as the central focus of planning is not widespread — SMP is most often supplemental or secondary to content. More often educators begin with a content standard and then tag it with an SMP, as opposed to beginning with an SMP and asking how to further that practice. This is not the fault of educators, but the fault the current standards. The current standards attempt to denote the importance of the SMP, but that is overshadowed by the sheer number of content standards. Standards reveal their values in what they give attention to.
Additionally, standards are political documents also us by administrators, bureaucrats, instructional coaches, etc.… Many times, educators in these roles value standards that are easily replicable and assessable for data. Thus, teachers often face pressure to create an easily measurable assessment of A.3.T4 instead of having deep conversations about sense-making or reasoning.
If the SMP were the totality of the current standards, many of the issues above would be resolved. Teachers would be able to create rich learning experiences and still explore important mathematical content. They may even be more likely and certainly more politically able to focus planning on the bigger, more enduring capacities and questions within the discipline of mathematics.
Yet, while the SMP resemble elements of the proposed new standards, they alone do not provide a full or accurate vision for the new standards. The SMP do not encourage a mathematics that is historically aware and critical. They do not make space for the human and mathematical experiences of joy, wonder, play…. They are also still rather narrowly focused – they do not open a learner up to big questions, ideas, or tensions. There are ways in which they provide a biased and even false view of the practice of mathematics, and they do not give learners space to shape that practice. These guiding principles intend to set out much bolder and broader vision than the current SMP affords.
These new standards seem like they really minimize the importance of content.
Quiet the contrary. The eQPTC defined in the guiding principles cannot be explored by students without mathematical content. eQPTC necessitates and motivates the need for mathematics content, and in fact will probably do so more effectively than the mere mandating of content. Moreover, the guiding principles also state that suggested concepts be included in the standards document. That content, however, should not be mandated at the standard level.
The mathematics that matters most in the world is constantly changing and is likely a matter of subjective opinion anyway. As a political document, standards cannot keep up with or anticipate what mathematics will matter nor can they identify the mathematics that matters in every local context. When standards attempt to prescribe high school mathematics content, they risk becoming outdated quickly and irrelevant immediately. Worse, they bind educators to hyper-specific mathematical ideas and procedures and inhibit mathematical classrooms that seek more creativity and relevance for both teachers and students. These are the classrooms we need to incubate the mathematics education of the future.
On some level, this also recognizes a trust in educators as professionals, as people who can recognize the vying contextual interests they and their students face and then make responsible decisions about what and how to teach. New standards can help redefine our notion of achievement in mathematics. They do so by being rooted in a value of difference rather than an ethic of sameness. Educators and students can decide locally how to navigate systems that are designed otherwise, but the pursuit of sameness need not be perpetuated further with the political power of standards documents.
Finally, the current standards have offered a trade-off: meticulously prescribed content that fails to draw bigger, engaging picture of the discipline of mathematics. The new standards seek to remedy this, and not by reversing the trade-off. But, by leading with big ideas that situate, illuminate, and motivate mathematics content. Moreover, if the new standards can develop larger mathematical capacities and dispositions, students will be far more likely to be successful when they encounter new content and also more likely to become generators of new content themselves.
With standards written like this, how will teachers know what to teach?
The broad nature of the new standards also makes them inherently inclusive. In short, the new standards would allow teachers to teach the content they are currently teaching using their current curriculum and resources. The new standards may pose questions that might challenge traditional teachers to think differently, but the new standards should be written so that they cannot be used to force change.
Many teachers, however, see more value in prioritizing places other than technical documents when deciding what to teach. More and more teachers are adopting a learner-centered model that begins with the student’s gifts and interests, or are building mathematics out of contexts immediately relevant in their community, or are answering the calls to re-humanizing mathematics. Such efforts are hampered by current standards that can easily wield power to invoke a strict and narrow practice of mathematics. These classes need an autonomy often not afforded by the current standards.
Will this mean math teachers will not teach skills?
Absolutely Not. These guiding principles should not be interpreted as saying technical procedures and skills should be removed from the high school mathematics classroom. Rather, it is identifying the right place to articulate which skills are to be learned. Standards documents should not be so granular if they desire to move education toward to deeper, humanizing approaches.
This is large shift from a systemic perspective. Current standards are predicated on conception of growth in mathematics as progression through more and further stops along specific mathematical trajectories. In this perspective, it makes a great deal of sense to prescribe skills so that as is commonly said, “each builds on the next.” Thus, there is framework of dependency.
Yet, this framework is not reflective of actual learning. Nearly every math teacher can attest to examples of students working through complex skill (like solving a system of equations) with prerequisite skills that are lacking. Sometimes the acquisition of a new skill even adversely affects an old one, like the student who learns to solve proportions and then applies the same methods to multiplying fractions. Even in a highly skill-based approach, learning is not linear.
The new standards conceive of growth in mathematics in an entirely different way: through a deepening cycle of engagement with big questions and ambiguities. That process still requires the development of skill, but not in a linear or prescribed fashion. The new standards accept that the goal of mathematics education is not to define a fool-proof path for ensuring that all students can divide polynomials, but to open a student’s mind to explore many paths or even create their own. We have too long privileged specific expressions and progressions of mathematic thinking, for example triangle congruence proofs or advanced algebraic manipulation. Rich skill can be developed without constraining ourselves to a very narrow set and path of skills.
Moreover, the inertia of our system is toward the traditional. So, any enumeration of skills and procedures within in a standards document activates a collective confirmation bias toward the traditional, no matter how other aspects of the standards may attempt to mitigate this. For those worried about the presence of skill in the mathematics classroom our traditional inertia should be comforting. Regardless of the standards, our current culture and resources will still reserve a place for skill and procedure in mathematics classrooms.
Nevertheless, and importantly, the new standards do not call for the abandoning of skill, merely the distinction that skills and procedures need not be enumerated in high-level standards documents. The questions posed by the new standards intend to motivate the need for skill and hope to do so in a way that makes more conceptual and pedagogical sense.
How will the new standards achieve depth if it is not used as a guiding principle?
Depth of understanding is perhaps one of the more universally agreed upon goals of educators. But two things must be considered when determining the role of standards in achieving that goal. First, to ask what is likely to happen when ‘depth’ is used a guiding principle for writing standards. And second, to acknowledge how depth is actually pursued.
We need look no further than our current standards in high school mathematics to see what happens when ‘depth’ is articulated as a goal for writing standards. ‘Depth’ as an explicit end encourages the enumeration of detailed and explicit pathways for achieving ‘depth’. The momentum of enumeration then overtakes any previously defined goal of having fewer things to learn. ‘Depth’ as a goal in writing standards becomes self-defeating.
The current standards ultimately develop ‘depth’ as a trek though increasing technicality, for example from solving linear equations, to quadratic equations, to systems of equations, to systems of linear and quadratic equations, etc… OR proofs about lines to proofs about triangles to proofs about parallelograms. Structurally this creates a mathematics that is more about how far down the path a student can go than how deep and rich their understand is. Thus, we ae more likely to produce students who can write an accurate two-column proof about parallelograms but have no real understanding abstraction, deduction, or argument. In the current standards depth as become too conflated with distance.
The current standards have so thoroughly enumerated the ‘what’ that they not only implicitly define depth but create political structure for enforcing a ‘depth as coverage’ mentality. They create a system that is more likely to ask teachers, “how can we get students to pass an assessment about proving theorems about parallelograms?” than to ask them “how can get student to demonstrate their actual capacity for abstraction, deduction, and argument?”
Identifying ‘depth’ as a guiding principle to developing new standards is more likely to produce standards that actively work against the goal of producing depth in mathematics classrooms. Depth occurs within the student and is developed at the intersection of their personal and mathematical experiences and their interactions with teacher, curriculum, and other students. Depth cannot be manufactured at the standard level. Standards can only point toward depth, but they cannot prescribe it. The new standards seek the make the space that is needed for students to purse genuine depth by drastically decreasing their quantity and by framing them around big and important Questions, Pursuits, Tensions, and Capacities.
Yeah…I still don’t get eQPTC.
This will undoubtedly be the biggest challenge and the bulk of the work in writing new standards. It is, perhaps, the big discussion that needs to happen. I avoid giving examples of potential eQPTC at this moment to avoid biasing any new and creative thoughts and to avoid any bias associated with an already known quantity. I intentionally use the perhaps confusing acronym eQPTC to further emphasize the distinction I am trying to make between new standards and the current ones. They should be vastly different. Moreover, the eQPTC inherently should be co-created in the process of writing new standards.
These guiding principles fly in the face a national standards movement that sought uniformity in what children were learning across districts and even states.
True. It is a tempting and easily marketable sentiment to say that a 10th grader should be able to move from Kansas to California and be learning the same things in math class. It resonates with the commonly held production-line view of education and our tendency to oversimplify learning. Yet, I think after interrogation many would find the goal of such conformity unrealistic and undesirable.
First, re-consider the 10th grader. Does that 10th grader have the same level of understanding as all the other 10th graders in their immediate vicinity? In other words, even within a single classroom, are all students ready to wrestle with the same level of content? Are some able to devour the mathematics at a blistering rate while others will need more time to process? The model that presumes learning occurs in a neatly prescribed and linear order is tempting because it is so convenient. In March we do x so that in April we can do y. It does not really matter if humans we are doing x and y to move, swap out, etc… In addition to not reflecting the complex nature of learning, systems (and standards) built on this assumption de-humanize the children in our schools.
Overly prescribed standards do not benefit the child who moves from place to place, they make it more difficult to meet every child where they are. They ask educators to first consider what arbitrary facts/skills have been assigned to the 10th grade instead of first considering the needs and gifts of an individual child and asking what they can do next.
Moreover, children have different goals, aspirations, and dispositions. Overly prescribed standards assume all children have the same desired future. Certainly, there is a level of mathematical literacy all students need to fully contribute an engage in the world. Students need the power to understand, interrogate, and criticize the way mathematics is used in the world. The current standards are not written to develop this kind of rich, empowered mathematical citizenship. Rather, they more develop students to serve a strict and narrow gatekeeping version of mathematics.
It is correct that these guiding principles do not share the goal of uniformity of experience, because they recognize that uniformity can only ever be achieved on the surface (and at a costly price). Underneath is always a set diverse and wonderful humans each teeming with unique brilliance. These guiding principles do not seek to standards that drive that diversity into conformity, but instead they seek to elevate the full and unique potential of each child.
