Bringing Needed Coherence and Focus to High School Mathematics
By Matt Larson, NCTM PresidentOctober 25, 2016
Today, it seems as if nearly everyone agrees that high school mathematics needs to change. For far too long high school mathematics has not worked for far too many students: too many students leave high school unprepared for college or a career, particularly a STEM career; too many students do not see how math is useful in their lives; too many students leave high school without an affinity for doing math; too many students leave high school without the quantitative skills necessary to make sound decisions in their personal life and in our society which is increasingly quantitative in nature. High school mathematics has not changed substantially in my lifetime, nor has it changed substantially for most students, teachers, schools, districts, and states. It is clearly an issue—and it is a critical issue of access, opportunity, and equity.It is with great excitement that NCTM announces it is embarking on the development of Pathways through High School Mathematics: Building Focus and Coherence (working title). This new publication will • Address the purpose of high school mathematics and include guiding principles such as access, equity, and empowerment; • Define math curricular pathways leading to college pathways and career readiness, as well as active participation in our democratic society; and • Provide narrative descriptions of course exemplars, including their big ideas, that could populate the pathways. The goal of high school mathematics education must always be to expand options for students in ways that appropriately accommodate the post-secondary goals of different students.The NCTM Board of Directors has appointed a nine-member task force representing the constituencies that make up the larger mathematics education community at both the K–12 and post-secondary levels. The task force’s charge is to develop and present these high school pathways with the same level of focus and coherence that currently exists in the NCTM Curriculum Focal Points and the K–8 Common Core State Standards. The formation of this task force occurs at an opportune time. The authors of A Common Vision for Undergraduate Mathematical Sciences Programs (NSF 2015) called on the community to “1) update curricula and 2) articulate clear pathways between curricula driven by changes at the K–12 level and the first courses students take in college.” The recently released 2015 Grade 12 NAEP results further reinforce the need for change at the high school level where scores have been flat for over a decade. When the Grade 12 NAEP results were released in April, NCTM made the point that the results “should serve as a call to the education community that we need to get serious about addressing the high school mathematics curriculum and the needs of students.” The task force will also consider criticisms of high school mathematics and its relevance, such as those found in Andrew Hacker’s The Math Myth and Mike Schmoker’s recent commentary in Education Week, “Math and K–12 Schools: Addressing the Historic Mismatch.” Criticisms that there is too much “legacy” content in the high school standards focused on symbolic manipulation and narrow preparation for calculus, while other topics from discrete mathematics, the use of technology, and the possible place of coding are ignored, will be addressed. As it produces its recommendations, the task force will carefully consider recent findings and information, such as the National Center of Education and the Economy report “What Does it Really Mean to be College and Work Ready?” Therefore, the task force’s work will not only focus on bringing coherence and focus to high school curricular pathways, but will also carefully look at the content itself. The implementation of the Every Student Succeeds Act (ESSA) also makes it necessary for the mathematics education community to address the high school curriculum now. ESSA returns much educational decision-making to individual states and increases the likelihood that standards might no longer have as much “in common” as states and even school districts have now as they attempt to “fix” the high school standards on their own out of necessity. NCTM has long argued that common high-quality expectations are an effective mechanism by which to begin to close one aspect of the opportunity gap by ensuring that every student at every grade level in the United States has the opportunity to learn mathematics at the same high level. It does not serve the goal of access and equity to have widely different high school expectations across the country. Defining high school curricular pathways is one effort to support state and local standards so they do not devolve into a hodgepodge of unequal expectations that do not serve our students or our society well. This work is likely to impact nearly every teacher, because of the connected nature of mathematics learning. There will be opportunities for every interested individual to provide input as the process unfolds, either through the review of a public draft document or through specific focus groups. The final product is scheduled for release at the NCTM Annual Meeting & Exposition in Washington in April of 2018, and it will reflect the input of the NCTM community at large. This work will not be easy and for some the idea of changing what we have always done in high school mathematics will be challenging. Truthfully, it is a critical area that we have been glossing over for too long to the detriment of too many, and the benefit of too few. NCTM looks forward to tackling this challenging and seemingly intractable issue with the input and expertise of our members and the broader community.
This is bound to be truly critical, invaluable and challenging work. For so many of our students, learning mathematics stalls out in high school, and I've long held that what holds them back more than anything is a lack of a personal connection to the content. And looking objectively at what we expect students to engage with – the levels of abstraction and formalism that are often included "for math's sake" – it is immediately evident why. For the scope of high school study in a discipline so grounded in rigor, logic and reason to be so beholden to tradition beggars belief.
The vestigial features of the dominant, traditional Calculus-preparatory pathway have intrinsic value in and of themselves, and are worthy of consideration for inclusion in the high school curriculum, but as education becomes more personalized and learning becomes more culturally responsive, a curriculum can no be a singular, one-size-fits-all highway facilitating drive-by-instruction in a few areas – Algebra and Geometry most particularly – while passing by other through diverse areas of study – Statistics, Combinatorics, etc.
I would hope that this work culminates in a roadmap outlining multiple pathways through a more diverse view of what high school mathematics could be. By offering a more broad view of what math is, more high school students may find something within math that motivates them to become invested in their own learning, resulting in more equitable access to mathematics for all students. An accompanying challenge, therefore, remains that any and all pathways maintain a consistently high level of rigor regardless of the specific areas of study.
This work will likewise have implications on the pathways into high school study and out of high school into higher education. I look forward to this ripple effect starting towards a further iteration and refinement of standards across and beyond the K–12 band.
Insight, be it into a specific problem or into a wider mathematical theory, often becomes most profound when the underlying assumptions and conditions are called into question. It is high time that the assumptions driving mathematics education are likewise questioned, so that our children born in this millenium are not held captive to the priorities of the last millenium. We need to ensure that they think less about mathematics as something that they learn (and, too often, learn poorly) and more about mathematics as something that they do (and, hopefully one day, do well).
It is great, most challanging as well as with great excitement that NCTM announces on the development of pathways through high school mathematics with the main theme of coherence and focus.As mentioned in present's message, it has been too late to high school mathematics to reconstruct it so as to be useful to high school education.What is common among our mathematics teachers and mathematics educators(in Nepal) is that students devote more time to mathematics in comparision to many other subjects. But their achievement and understanding are found to be not satisfactory as mentioned by many studies.It seems to be common problem.If we go into the different areas of high school mathematics, it seems to me that geometry is one such area which ought to be more interesting and applicable but has been more problematic.Proof geometry(Euclidean geometry) in our curriculum( in Nepal) even holds significient weghtage, but it seems that it has been less effective in fulfilling its goal..Due to the poor porformance and outdated curriculum,many theorems in geometry have been dropped from school math curriculums.What is to be noted is that the logical consequences of proofs(specially used in Euclidean proof) have been most useful for computer programs and compurer programs have been found much attractive to our school children. We know that mathematics is a tools as well as language to science.So,the students of twenty-first century should be expected to have minimun mastery over both commom language and mathematical language.The other thing to be noted is that the secondary school students should be expected to develop logical and quantitative arguments in their curricular activities and to use it for further study in colleage and to use for decision making in personal life.Such aims (which are the part of this megha-project of NCTM)) seems to be fulfilled by developing a balenced , coherenced and focussed curriculum of mathematics built on new ground based on the previous experiences, experiments and studies.
Among many problems, the implementation of the Every Student Succeeds Act (ESSA) without down-grading the intellectual integrity of the subject of mathematics, is one to be seriously considered by the task force. In making big changes, there may be possibility of more deflections due to current experiences.But history teaches us also that more precautions should be taken to reject a good one for the better one.
As mentioned in president's massages "this work will not be easy%
I completely agree that high school mathematics needs an overhaul, but truth to be told, education has fallen far behiind in presenting/showcasing mathematics with the joy, whimsy, and quirkiness that social media has been doing for almost a decade. Numberphile and ViHart have a million subscribers. Here mathematics is explored for the only thing that really matters--love for patterning and symmetry. When the natural and historical narrative of mathematics' exploration is reflected, a natural gravitational pull towards its magic and beauty occurs. Every mathematician who explored mathematics did so out of simple and boundless curiosity. Repeated failure, blind alleys, locked doors and broken paths were the rule and not the exception in this journey. So when education attempts to pasteurize the learning process and "package"/can content in ways that are often too inorganic, students begin to see mathematics as courses and sections in a curriculum. I have tutored too many students who when asked what they are doing, respond with something like " I am on 3.2", meaning they are on chapter 3 and the second section.
One of the dangers of mathematics education is to repeatedly play the "usefulness" card--first. Why does mathematics have to be a Clydesdale horse in education? What is the usefulness of probing Edward Albee's "Zoo Story" in school? What is the usefulness of showing kids the Blue Period Picasso pieces? Waht is the usefulness of listening to Coltrane in music class? These questions only sound ridiculous when presented to other subjects besides math. Math, for some reason needs justification and application. If we were being brutally honest about true usefulness, then high school math would be repalced with game theory, the math of voting, chess, GO and discussions as to why yahtzee and checkers have optimal strategies for winning and poker is almost mpossible to find.
It is important to reflect on all criticisms of math education, and referencing someone like Andrew Hacker is key to stir honest and open dialogue. But, Hacker was debated by James Tanton at the Museum of Mathematics in September. From all accounts, Tanton politely handed Hacker his hat. If we are going to look at the important reconstruction of the high school curriculum, then shouldn't we also reference people who are talking about the joy and wonder of math(Tanton) and not just those who see mathematics in a less soulful way?
If we as educators cannot impart the "painful beauty" that Paul Lockhart talked about mathematics back in 2004, then I personally see no reason to teach mathematics. To tether it to only career paths and functionality is a path that is completely counter to how it is being happily expressed in the "free range" platforms like YouTube and Twitter.
I wrote a Medium article addressing the "Usefulness" of math a few days ago.
Let's make sure that whatever will be the new high school curriclum, that we honor mathematics first and foremost--and those that love the subject:)
Let me answer as someone who has both devoted his life to the pursuit of beauty in mathematics, through research and teaching, and who is working along similar lines to this effort in my state.
In short: we must have both usefulness and beauty, and we are lucky that mathematics has an abundance of both. We must have usefulness because every state requires a fair amount of math (at least three years and sometimes four, to my knowledge), a requirement based on STEM readiness. If we want to be completely free from basic expectations around content and students' ability to apply it, we should get rid of such requirements. I personally wouldn't agree with that, and I don't see the community arguing much against such requirements, even though they do "tie our hands". And we must have beauty for reasons you allude to and I fully support.
I think we can manage both: follow up doodles of branching trees (ala Vi Hart) and counts related to them with calculations of total values of constant income deposited in an interest-bearing account (and ultimately the numeracy to evaluate adjustable-rate mortgages and store-financed purchases). I see these as two sides of the same (geometric series) coin, and disagree with the notion that utility preculdes beauty. Indeed, a lot of the best research mathematics these days is at the interface with science and computation.
Teaching to both beauty and utility is a challenge, but I think those who have devoted their life to the teaching of mathematics are up to it, especially if we work together.
I have no problem with utilty, and it needs to share the stage with the intrinsic beauty of patterning and symmetry in mathematics. But, currently, the utility is rather benign and lacking any teeth. Why aren't students exposed to game and graph theory early on? Why isn't expectation/utility fleshed out in more detail, so students can examine the duplicity of insurance/government lotteries? Mortgage rates? Now we are dealing with math equity issues. Home ownership is something has evaporated with the middle class up here in Canada. Financial literacy is crucial, but if it only tells the safe/soft story, then we are doing a disservice to our students. But, I agree with your general post:) !
And I agree with what you're saying:
- true utility invites a much wider range of topics than we have traditionally seen in grades 9-14 (I'm lumping in some grades I teach here). In Oregon we're exploring a "tight-loose" model where there are two years of content for everyone (containing, for example, linear and exponential functions, measures of spread in data etc etc) and then things would open up considerably. The dream, at least for larger schools, would be to have classes such as math-government which discusses topics like state budgeting, game theory, polling, and predicting/ interpreting electoral results (breaking down into different populations to analyze coalitions). I'm teaching a very entry-level math-chemistry class at my university (lots of linear function modeling including a "bungee activity"; spreadsheet chemical equations; etc) which could potentially be modified for use in HS. (Smaller schools could still rely on a statistics or precalculus course to serve everyone.)
-true utility would also reinforce an emphasis on equity issues. (But I'd be careful about rhetoric such as "insurance lotteries". Insurance companies are generally adding value, though to understand that is to also call into question "purchase insurance" for example.)
Thanks for sharing your blogpost. I found it very interesting.
Is there a representative from the Adult Basic Education Community on this task force?
All members of the mathematics and mathematics education communities will have an opportunity to provide input and feedback as the process unfolds and we look forward to hearing from the Adult Basic Education Community.
This is a movement that is long overdue, and I hope with the right people driving this bus of change more people will fight to climb on board.
Excellent! Having just observed some high school classes today (I'm in higher education), I feel stronger than ever that we need to keep the progressive momentum for equity and accessibility in mathematics! I look forward to reading the document in 2018!
Thank you, Matt. This is very encouraging to read.
My concern is that while we desperately need multiple pathways for high school mathematics, if intermediate and college algebra courses remain the primary entryway into post-secondary math, then the multiple pathways may be ultimately frustrated as the system “restores itself to its default setting.” (I am thinking, in particular, about the Core-Plus Mathematics Project in Michigan some 20+ years ago.) We also need “multiple math entryways” into post-secondary disciplines (STEM and non-STEM) so the high school pathways flow naturally into post-secondary math pathways.
Thanks Matt! For far too long, we have not focused heavily enough on transforming mathematics in high school in an effort to combat the flat data we have seen nationally for 30 years or so, while K-8 has shown consistent progress since the 1989 standards were released. I will be anxious to see this product!!
Thank you, Matt. The challenges of changeing are always upon us. It will be interesting to see what the document will be.
Thank you Matt. I am wondering if the proposed important work will have its foundation in the Learning Progressions for the High School CCSS?
The task force will consider a wide variety of resources and existing work.
This is great news.
You are tactful in not explicitly calling out the problems with the Common Core Standards for high school math. However, if we are to have a broad conversation on this topic throughout NCTM, we need less tact and more frank discussion. I hope, hope, hope that this new initiative means that the strengths and weaknesses of the CCSSM will be discussed in The Mathematics Teacher and at conferences. New policies can only be created in dialogue with the current paradigm. I'm happy to contribute to that conversation if asked.
My analysis of the CCSSM for high school has been well received by math teachers and math ed leaders. Please share this link with the task force:
Thanks for getting the ball rolling, and for the guiding principles you mentioned in your post. I wish the task force good luck in this challenging and important project.
Thanks Henri for sharing your resources. The task force certainly wants to hear from the community as the process moves forward. We appreciate your support and interest.
I recently attended a symposium on College Readiness in Mathematics at UW-Madison. As we engaged in conversation throughout the day with panel members and mathematics teachers from around the state, the same message emerged. I feel as if the tide is turning, yet we continue to struggle with how to prepare students for what is truly needed for career readiness (including post-college) while still preparing students for college math entrance exams. There is a disconnect!
Thank you. Long overdue.
Thanks for leading the charge on this Matt. It's unfortunately often easier to criticize someone for their efforts rather than to commend them for their initiative. There is certainly bound to be disagreements on priorities going forward, but thanks for making this a priority as with the current state of politics it seems like anyone else is better equipped to make a case for change.
Oops. Should have been "no one is better equipped"
Who's on the task force?
A link will soon be added to the message listing the members of the Task Force.