Hard for me to believe, but this is my 100th blog post. That’s the equivalent of a 300 page book – with much less pain, and lots more fun interaction: thanks to all my loyal readers and responders! And hooray for the Internet for permitting easy and rich discourse about one’s ideas.
To celebrate I thought I would return to my favorite educational whipping boy* Algebra I. There are so many reasons for dumping on Algebra but a timely one (on this blog milestone) comes from the fact that one of my blog posts on math was quoted in the recently-released Publisher’s Guidelines on the Math Common Core Standards!
Before I get into my rant, let’s be very clear about a few things:
I love math. I taught math – Pre-calculus and Geometry – and I am good at it, my best grades in school. I think non-Euclidean geometry is one of the coolest ideas ever – no one should fail to learn about it in school, in my view. (The mathematician Morris Kline says that its development is one of the great ideas in modern history). I am thus not one of those Humanities types that thinks math is boring and pales in comparison with a good novel. Left to my own devices, I would rather read the Heath edition of Euclid’s Elements than any novel reviewed by Maureen Corrigan on Fresh Air.
Because of my education at St. John’s College, I have an unusually strong background in the history and philosophy of mathematics. We read Euclid, Ptolemy, Descartes, Newton, Lobachevski, Dedekind – and it was fascinating, because the meaning and purpose of the math was always in view.
So, I know, for example, one key reason why Descartes invented algebraic solutions to geometric problems and thereby gave us his coordinate system: to not only solve certain problems with conic sections more easily but to enable us to escape the arbitrary prison that you can’t have any number be greater than to the third power. Why? Because the universe only goes to 3 dimensions, so x to the 4th was nonsense in classic geometry.
I am not saying this to brag. I am saying this to make the point that I have some bona fides to justify what I am about to say:
Algebra is a dumb course.
It survives only by unthinking habit. It cannot be justified intellectually as a subject, really. It is just a set of tools, not an intellectual discipline with larger meaning and an ongoing scholarship.
Worse, it is an insidious dumb course because everyone must take it, and many people fail it – in part, because it is so dumb.
DO NOT MISUNDERSTAND ME! I said Algebra is a dumb course. I did not say the content called algebra is not worth learning. The distinction is critical.
Algebra, as we teach it, is a death march through endless disconnected technical tools and tips, out of context. It would be like signing up for carpentry and spending an entire year being taught all the tools that have ever existed in a toolbox, and being quizzed on their names – but without ever experiencing what you can craft with such tools or how to decide which tools to use when in the face of a design problem.
Algebra is thus like bad grammar teaching from yesteryear. Algebra I remains today much as it was when I took it in 1964 when I also had to slog through a year of Warriner’s Grammar, as I have found by being in many math classes in the last two years. Go over the home work, learn a new out of context tool like Systems of Inequalities, do some practice problems, quiz, repeat ad nauseum. The course has no big ideas, no direction, no purpose. And when was the last time you had to graph inequalities? (Pre-calc was worse, and I taught it: logarithms and other stuff made completely obsolete by graphing calculators.)
In fact, the course gets many things totally backwards intellectually – like graphing: why wouldn’t you graph to actually learn something that only graphing real-world messy data can reveal? Why wouldn’t you do lots of ‘best fit’ exercises with linear and non-linear data, for example, to begin to appreciate that graphing can help you find something of value? As it stands now in all algebra courses, it is the opposite: there is no reason to graph; you just have to learn set pieces with de-contextualized data already pre-made. Instead of finding cool patterns you merely learn connect-the-dots techniques for graphing.
Paul Lockhart brilliantly satirizes this routine-driven aspect of school math in A Mathematician’s Lament by imagining art taught as algebra is taught:
I was surprised to find myself in a regular school classroom— no easels, no tubes of paint. “Oh we don’t actually apply paint until high school,” I was told by the students. “In seventh grade we mostly study colors and applicators.” They showed me a worksheet. On one side were swatches of color with blank spaces next to them. They were told to write in the names. …
After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that.”…
“I see. And when do students get to paint freely, on a blank canvas?”
“You sound like one of my professors! They were always going on about expressing yourself and your feelings and things like that—really way-out-there stuff. I’ve got a degree in Painting myself, but I’ve never really worked much with blank canvasses. I just use the Paint-by-Numbers kits supplied by the school board.”
Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.
Algebra courses thus still make the same epistemological and pedagogical mistake that my grammar work once a week did 50 years ago: assuming that years of learning bits, out of performance context, is needed before you get to do the real stuff. Dan Meyer, along with others, has been debunking this idea for years on his great blog: here is an example of how problems, not tools, can drive a proper course.
We no longer make this spend-years-learning-inert-parts-out-of-context mistake in English. Even 1st graders learn to ‘write’ ideas right away. We appropriately do not now ask kids to first endlessly parse sentences, for example. So, why is algebra still doing the equivalent, with our official blessing? Learning bits of algebra out of context doesn’t make you a better mathematical thinker or problem solver – the supposed goals of math courses – any more than merely trudging through brush lessons in art makes you a better painter or going through Warriner’s, page after page, can make you, by itself, a better writer.
Again, I am not saying that the tools of algebra are useless. I am saying that all courses called ‘algebra’ are really badly designed. By divorcing content from use, by conflating exercises with problems, and by making it a tour of isolated tricks with no overall direction, we ensure that it is needlessly boring and abstract in the bad sense. Having year-long required courses called ‘algebra’ is as sterile and intellectually uninteresting as a required year-long course in C++ that never lets you actually program a computer to do anything.
To study “algebra” instead of real and interesting problems that require algebra is the nub of the issue. (Even in Greek you at least got to read great texts in the original in the end.) Read my interview with stellar mathematician (and former student!) Steve Strogatz on how unprepared HS kids are for genuine problem finding and solving in college math. And here was his NY Times column on algebra as part of his series on math.
Lets look at the matter as intellectuals. Here is a thought experiment: can you identify 4 big ideas in algebra, ideas that not only provide a powerful set of intellectual priorities for the course but that have rich connections to other fields? Doubt it. Because algebra courses, as designed, have no big ideas, as taught, just a list of topics. Look at any textbook: each chapter is just a new tool. There is no throughline to the course nor are their priority ideas that recur and go deeper, by design. In fact, no problems ever require work from many chapters simultaneously, just learning and being quizzed on each topic – a telling sign.
Here’s a simple test, if you doubt this point, to give to algebra students in order to see if they have any understanding of what they have learned:
- What can algebra do that arithmetic cannot do or does very inefficiently?
- Is the order of operations a matter of core truth or convention? How does that compare with the Associative Property? What is and isn’t arbitrary in algebra?
- Why, mathematically speaking, are imaginary numbers and the inability to divide by zero wise premises?
- What is an equation that states the proper relationship between feet and yards? (60-80% of students will wrongly write: 3F = Y, showing that they have failed to understand the difference between English and algebra)
Think the first question is esoteric? A variant of it was asked on the NY State Regents Algebra test a “few” years back:
Even the widely-praised problem-based program in math at Exeter – including by me, here – errs on the side of asking narrowly-focused questions. Most problems are pretty small bore and there is little explicit attention to overall understanding.
Pernicious gate-keeper: algebra is the Greek of the 21st century. Worse, Algebra is a nasty gate-keeper course with no justification for it playing that role, just as ancient languages once were. Did you know that to get into good colleges not long ago you had to pass a Greek exam? In 1900 to be considered ‘educated’ and to be able to go to Harvard and such, you had to know Greek and take a test in it as part of the entrance exam. (Recall, each college had its own tests: the SAT hadn’t been invented yet). But who now thinks this is a sound idea – other than as a very crude sorting device?
Ah, that’s algebra’s nasty role now.
It’s pernicious and indefensible to fail students and deny a diploma over failure to master such a sterile course. I followed with interest the massive negative response that Andrew Hacker got to his NY Times essay entitled Is Algebra Necessary? Alas, he made the mistake of not sufficiently differentiating between the value of the content and the value of the course as traditionally designed and taught. But many of his points were spot on and overlooked by critics who thought he was calling for lowered standards and anti-intellectualism – especially the importance of access and respect of diversity of career directions:
Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below “proficient,” along with 39 percent in Tennessee. Even well-endowed schools have otherwise talented students who are impeded by algebra, to say nothing of calculus and trigonometry.
California’s two university systems, for instance, consider applications only from students who have taken three years of mathematics and in that way exclude many applicants who might excel in fields like art or history. Community college students face an equally prohibitive mathematics wall. A study of two-year schools found that fewer than a quarter of their entrants passed the algebra classes they were required to take.
“There are students taking these courses three, four, five times,” says Barbara Bonham of Appalachian State University. While some ultimately pass, she adds, “many drop out.”
Another dropout statistic should cause equal chagrin. Of all who embark on higher education, only 58 percent end up with bachelor’s degrees. The main impediment to graduation: freshman math. The City University of New York, where I have taught since 1971, found that 57 percent of its students didn’t pass its mandated algebra course. The depressing conclusion of a faculty report: “failing math at all levels affects retention more than any other academic factor.” A national sample of transcripts found mathematics had twice as many F’s and D’s compared as other subjects.
Hacker and I worked (separately) on a project entitled Quantitative Literacy, an MAA intended antidote to many of these problems. The phrase “quantitative literacy” tells you much of what you need to know about a better direction for math programs.
Please do not write in saying how important algebra is for learning high-level sciences or how wonderful it was for you personally. I know it works for those who like it (me included) and we know it is needed for high-level science work. But that addresses the needs of only about 20% of the school population. Our student surveys consistently rank HS math as their least favorite course, by a wide margin – though 22% list it as their favorite. 46% don’t merely like it least: they hate it, and their words are telling: “makes me feel stupid” and “seems utterly pointless.” You can download here sample student responses as to why math is their least favorite subject here: Math HS least favorite responses.
So, the fact that some people need it and like it doesn’t justify making every American child learn it in such a dessicated and demeaning way (and fail to receive a diploma if they don’t learn it to boot). I can make a better case that learning logic and Boolean algebra is a far better pre-requisite for both professional and adult life. Even so, I would never require it of everyone. Why can’t your math requirement in high school be met by statistics and probability? Better yet, why can’t students choose math electives in high school, just as they do in college?
And please do not write in saying that this criticism of algebra will lower standards. I am happy to make the stats. course very rigorous. I am also happy to require a course in philosophy and logic as a substitute for the critical thinking that you may (falsely) claim that algebra develops. I taught philosophy at the high school level for over a decade and truly believe it deserves to be a graduation requirement more than algebra, just as it is in France and in all IB high schools today.
No, algebra hangs on like any other bad habit in schools that we then rationalize, like corporal punishment. It’s time we faced up to the fact that the course called algebra is intellectually and pedagogically bankrupt. In my next column I will propose a completely different way to embed algebraic (and geometric) tools into far better – more engaging, meaningful, and intellectual defensible – courses.
PS: Since the post, the Publisher’s Criteria for HS Math and the Common Core came out. This quote could not make the matter clearer:
“Fragmenting the Standards into individual standards, or individual bits of standards … produces a sum of parts that is decidedly less than the whole” (Appendix from the K-8 Publishers’ Criteria). Breaking down standards poses a threat to the focus and coherence of the Standards. It is sometimes helpful or necessary to isolate a part of a compound standard for instruction or assessment, but not always, and not at the expense of the Standards as a whole. A drive to break the Standards down into ‘microstandards’ risks making the checklist mentality even worse than it is today. Microstandards would also make it easier for microtasks and microlessons to drive out extended tasks and deep learning. Finally, microstandards could allow for micromanagement: Picture teachers and students being held accountable for ever more discrete performances. If it is bad today when principals force teachers to write the standard of the day on the board, think of how it would be if every single standard turns into three, six, or a dozen or more microstandards. If the Standards are like a tree, then microstandards are like twigs. You can’t build a tree out of twigs, but you can use twigs as kindling to burn down a tree.”
http://www.corestandards.org/assets/Math_Publishers_Criteria_HS_Spring%202013_FINAL.pdf
* from Wikipedia: A whipping boy was a young boy who was assigned to a young prince and was punished when the prince misbehaved or fell behind in his schooling. Whipping boys were established in the English court during the monarchies of the 15th century and 16th centuries. They were created because of the idea of the divine right of kings, which stated that kings were appointed by God, and implied that no one but the king was worthy of punishing the king’s son. Since the king was rarely around to punish his son when necessary, tutors to the young prince found it extremely difficult to enforce rules or learning.
PS: Article published after this post on just how unneeded algebra is:
http://www.theatlantic.com/business/archive/2013/04/heres-how-little-math-americans-actually-use-at-work/275260/
63 Responses
Well done. i’m wondering how you feel, then, about the Common Core Math 1-3 sequences that replace the traditional Algebra I, Geometry, and Algebra II pathway? Does the Math 1-3 pathway solve the problem you defined so well?
Not if it focuses on discrete topics and exercises as opposed to increasingly complex but always interesting problems. I’ll address this in my follow-up post. As my comment about non-Euclidean geometry suggests, I think all these patterns are far too dry and discrete intellectually for the typical student.
I was under the impression that the vast majority (almost 100%) of states using Common Core are not using the Math 1-3 as Common Core was intending when written, but rather splitting the high school math standards into their existing Algebra 1, Geometry, Algebra 2 pathway. In Oklahoma, we are aligning our High School Math courses to the PARCC Content Model Framework for the traditional A1, A2, Geometry Course Pathway.
I would love to hear from schools and teachers using Math 1-3.
What compounds this problem further is the Pre-Algebra course that is usually required before even taking Algebra 1. If Algebra 1 is a “dumb course,” “intellectually and pedagogically bankrupt,” and a “pernicious gate-keeper,” then Pre-Algebra is likely a far worse offender. In addition to the arguments you’ve made above that apply to Pre-Algebra as well, we should add the fact that it would be ridiculous if schools offered Pre-Spanish, Pre-Chemistry or Pre-Government courses. In Parkway, we at least changed the title of our Pre-Algebra course to “Mathematical Modeling: Algebra in Context.” It was a start, and helped focus conversations about the purpose of our Algebra 1 course.
Agreed! And having all 8th graders take Algebras I, a related idea, is also dumb, especially given the developmental readiness issue that such a current course runs roughshod over.
I love this piece and look forward to your next one. I have not taught Algebra I in a few years, but I have been working to make Geometry a better course for students. I have tried to incorporate historical context, the development of logic systems, and argument into the class. I spend less time with content and more time on discussion of ideas and connections (including the revolution of non-Eucliean geometries).
Cool! Do you a site or blog for sharing some of the course stuff?
No site or blog. Just reading and thinking a lot. I have recently become more internet active, following you, Grant Lichtman, Bo Adams, etc. In Geometry class I include chapters from Dunham’s Journey Through Genius. Frosh and soph get practice with close reading about math, reading proofs, learning parts of arguments. We also use the Platonic solids to explore how 2d and 3d relate and distort. This year I am adding a project where teams design and build a piece of furniture.
That’s excellent. Whenever you wish to guest blog on how it goes, let me know!
PLEASE, PLEASE, PLEASE start a blog about your geometry projects. I will follow immediately! I also teach geometry and would love to teach a blended course of geometry and woodworking. 🙂
So cool. Read my blog today and go buy yourself a copy of the Fawcett book (and the Harold Jacobs Geometry textbook)
I really enjoyed this post – thank you for stating the issues around algebra so much more eloquently and precisely than so many others who reflexively complain (including, I have to say, Andrew Hacker).
I’m a HS math teacher (and former physicist), who struggles on a yearly basis to engage students in abstract thinking through the “algebra 1” course. For what it’s worth, I think students would be much better served by a deeper investigation of geometry than of algebra; and that one could argue a deep study of probability and stats would have even more positive impact on students. Philosophy, Boolean logic, non-Euclidean geometry, and the like, would also be excellent and probably more engaging alternatives to the usual sequence.
Because of my physics background, I often wonder about learning mathematical thinking via other disciplines. I’d be curious to know your opinion of teaching math in that way – that is, using a deep study of economics, engineering, or art (for example) to not only teach students those disciplines, but also to teach them mathematical habits of mind. As a concrete example, I’ve put together a draft outline of how a course on mathematics and music would look. Do you think this cross-disciplinary approach to mathematics education has value?
I would love to see your outline. And I am totally with you in terms of the link to science. I frankly cannot understand how anyone can learn calculus well without seeing it derive from its origins in physics. I once chucked my lesson plan in English to discuss this when a bunch of kids came from calculus to my English class. “We don’t know what we’re doing or why!” one girl cried. So, I said, ok, little lesson in the history of calculus. To which a boy said – Huh? There is a history to it??? Uh, where do you think it came from, I said, God’s right hand? So, I talked about the astronomer’s need to track planetary motion that varied in time with the challenge of then calculating orbit distance distance, which led to the integral; and how seeking to know an exact planetary speed would require the derivative… wow, said the first girl. It actually makes sense. Then she said angrily: why didn’t anyone bother to tell us this??? Good question, I said. Which further endeared me to the math dept…
have you ever seen Morris Kline’s book Mathematics in Western Culture? It was actually a Sunrise Semester course on TV way back in the day but provides a great history of the math with links to the sciences – very readable for reasonably motivated HS kids.
There was a great article in the March issue of NSTA’s The Science Teacher called “The Patterns Approach” discussing an approach to physics teaching which integrates a lot of content and habits of mind from algebra in meaningful ways. Students learn and use algebraic reasoning as a tool to make sense of real data as they solve interesting engineering problems.
The article is available for free (may need to create a free account) here:
http://learningcenter.nsta.org/browse_journals.aspx?action=issue&thetype=free&id=10.2505/3/tst13_080_03
Sounds great. I had no luck downloading it, though. If you can send me a copy that would be appreciated. gwiggins @ authentic education.org
My thoughts also went to physics, but then again, I wonder if your criticisms of algebra-as-a-course wouldn’t apply to physics-as-a-course as well. The physics taught in high schools (and many “physics 101” college courses) is largely “a set of tools…” and while it’s part of “an intellectual discipline with larger meaning and an ongoing scholarship” (just as algebra is part of “math”), the most advanced topics of a typical physics course were settled physics over 100 years ago.
I don’t know what my point is. I nodded in agreement with this criticism of algebra, but I think this criticism applies to physics as well and it bothers me more there. “But you can use physics to figure stuff out!” you might say. But you can use algebra that way too. Hmm.
I am totally with you. Physics courses fall victim to the same nonsense. In fact, I just reviewed physical science textbooks at the middle schools for a curriculum project and was appalled at the vocabulary-laden dry and pointless accounts. It’s especially maddening when you think of how fascinating is the work going on as we speak to make sense of the CERN collider data to figure out the very essence of the universe. Here, too, my St. John’s education was great. We read first Galileo then Newton, then Einstein, mindful of the purpose of their work and the decidedly counter-intuitive aspect of their ideas. Big problem: physics textbooks run roughshod over the counter-inuitivity of the ideas – no wonder there are robust misconceptions that typical physics teaching doesn’t eradicate!
Years ago in PSSC physics (the 1st post-sputnik national science reform stuff) the course made a big deal out of the question: is light a wave or a particle? They understood that making things an interesting question and taking the time to explore it is a key part of learning real science. And I thought the measuring of the speed of light – and testing Michaeslon-Morley in their failed quest for relative motion in light – was a highlight for me in all my science education. The take-away? real science involves puzzles, interesting questions, and ideas to be tested. Physics as taught is like horrible baseball education: you don’t get to learn to play. You merely have to sit in the bleachers and read the box scores of the most recent past games. It’s physics as metaphysics: here’s the truth; learn it. And, no, conceptual physics didn’t solve the problem at all. It’s a lame version of textbook physics, not a fascinating exploration of important inquiries and their implications.
I will address the issue of course beginnings and the utter harm of textbooks in screwing up beginnings in my next post as I map out alternative math courses. But it all applies to science, as you note. So, for example, I would begin with Galileo and his inclined planes and pendula, and keep returning to his question: how does it move and why? and ponder his assumptions, both to see their wisdom and their counter-intuitive nature (e.g. in a vacuum things would keep continually moving) and induce Newtonian laws as much as possible by well chosen inquiries and readings.
Love the grammar analogy – maybe it is not even an analogy. It seems like many students perceive Algebra I to be much more verbally oriented than mathematically oriented. Solving problems by following steps is essentially following (or trying to follow) grammar rules. There are many, many of us that attempt to teach the usual algebra in a mathematical way with reasoning and so forth. But, despite our efforts, many students still come away viewing algebra as a strange language with a lot of grammar rules to remember. There isn’t a clear enough distinction between the language in which formal math is often expressed and mathematical reasoning.
As an example: many students continually replace x/(x + 5) with 1 + x/5. They are thinking verbally in terms of a rule for adding (or un-adding) fractions, even though they are capable of understanding the mistake if prompted. I suspect that to increase the amount of algebraic thinking (reasoning with unknowns), we probably need to spend much more time working without the symbolic language that seems to trigger this type of verbal thinking (following rules). More pictures – more diagrams – descriptive words or pictures or even “?” to represent unknowns, etc.
Learning an obscure language (variables & symbols) is not the same thing as learning algebra (reasoning with unknown values). One is basically verbal thinking and the other is mathematical thinking. It is not unusual to have a student with good mathematical reasoning, but poor ability to work with the symbolic language. Nor is it unusual to have a student with poor reasoning skills that is pretty good with the “grammar” of the symbolic language.
What is missing from your excellent article is the reason always given for why Algebra 1 is “necessary,” namely that it is the milepost on the road to Calculus. Which is also not necessary, but is very convenient for colleges, businesses and legislatures as a target to which college-bound students must be held to. By extension (according to Common Core) all high school students must be held to this standard, and so when you look at the CCS in math, going all the way back through middle school and earlier, the tools and tricks and ideas are prerequisites for what a calculus student would need to be successful. Algebra 1 is not as obvious in this regard as Precalculus or Algebra 2, which have no cohesion whatsoever, except when you look at them as the roadway to calculus. Algebra 1 is just the half-way point on that road, which is why it looks so disjointed. I love teaching geometry because it gives students a break from that road, it engages other parts of the brain and imagination than just pattern recognition and massive abstraction, namely shape and space and reasoning and drawing. There are so many directions you can take this critique–why is calculus, rather than statistics/group theory/non-Euclidean geometry/probability/topology/number theory, the goal? (hint: it solves a number of problems, like gatekeeping for college, being easily testable); why are 400 year old problems still driving a curriculum? (why study ellipses and memorize, poorly, forms and equations, if you are not an astronomer or literal rocket scientist). Why is Algebra 2 a high-school graduation requirement in my state (hint, my state is run by such businesses as Microsoft and Boeing)? This is the tyranny of calculus, which forces all of math into this exceedingly narrow channel, driving all interesting and creative aspects out of the schools in service of government and business interests. I’ll write a book on it when I’m not teaching (i.e. 20 years from now), but it is such a crucial driver of what you are critiquing. Thank you for your thoughtful arguments.
It’s missing from my piece because the value of the content was not my point, as I stated twice. Of course you need algebra for calculus (and physics and econ) but that wasn’t what I was lamenting. I was lamenting the way algebra courses are constructed and delivered so that the content actually seems pointless and disconnected from larger intellectual issues of substance.
And, yes, as I noted in one of my replies, it bugs me that decontextualized calculus is also taught the way that it is. When a struggling calculus student doesn’t get help realizing that the conceptual basis of calculus is quite concrete – wanting to compute areas under curves and finding the line tangent for a point (due to the physicist’s and astronomer’s need to compute such things) then it is no wonder that it becomes a strange ritual sorting in device to basically see who has the best math IQ. Indeed, that was also Lockhart’s point in the art analogy and at the heart of Steve Strogatz’ critique of HS math courses – no training for doing real, creative math problem finding and solving.
I’m not disputing the value of the algebraic content, pro or con. I’m contending that the reason that the content itself is disconnected is because it is not an end to itself, it is a means to get students to calculus. By contrast, take a look at Chemistry or Physics. There is no pre-chemistry course, or pre-physics, and when you’ve taken Chemistry, you can be done; there is nothing (graduation-requirement-wise or college pre-req-wise) that you need to take, so whatever you learn is sufficient. There are clearly classes that you could take afterwards, but nothing predestined. That’s just not the case with Algebra 1-Algebra 2-Precalculus. This totally distorts the topics which are covered in Algebra 1, because they are only seen as how they will be used later in Calculus. Why learn so much factoring of quadratics and special cases if not to use them in delta-epsilon proofs of derivatives? Do they hold that much significance of themselves? I don’t think you can reconstruct the algebra content so that it is connected to the issues you want (creative problem finding and solving) without having a very different course (“Problem Finding and Solving”) that couldn’t cover the vast miscellaneous materials that get shoved into Algebra 1. I’d much rather teach that course, PFS, with 3-act videos and student-defined problems. But of course, that is not testable at scale, and students wouldn’t want to take it, because they’d be behind on their ability to get into college, which is so dependent on them being ready for…Calculus.
I see. But that doesn’t explain the course as I knew it 50 years ago – and it was the same course as today’s. In my day, only a handful of kids took calculus. Indeed, like biochemistry, calc was seen as a college course that a few HS’s offered its advanced kids. Yet the algebra course then was the algebra course for now. In other words, the course has always been conceived as the parceling out of all these moves out of context, with NO intellectual focus or direction. i would almost say that conceiving of Algebra 1 as necessary for alg 2 which is necessary for… is a later development.
Perhaps a better example is Geometry. Long ago, courses taught a slightly modernized version of Euclid, with a focus on the proof and the coherence of geometry as a system. Indeed, this was the importance of geometry intellectually: it had for a thousand years served as an epistemological model of disciplinary truth (and Euclid’s Elements went somewhere BIG, recall: the final proofs related to the 5 regular solids that it was believed made up the structure of the universe). The modern ‘axiomatization’ movement (along with modern science and the discovery of non-euclidean geometries as consistent) stripped geometry of its philosophical importance.
So, in our lifetime, courses in geometry lost all the intellectual richness of a system of proofs based on givens that led to interesting places. It just became one damn topic after another, like algebra. Worse,the modern movement for axiomatic rigor introduced so many axioms that found their way into everyday HS textbooks as to destroy the idea of a logical system of insights built upon a small set of reasonable ‘givens’. Indeed, many of the ‘new’ axioms designed to fix Euclid are far less obvious than most early proofs to laypersons – not surprising given that it took experts 200 years to find them.
My view: understand that the novice needs something different than the expert. Why not have novice geometry that recaptures the interesting idea of Euclid’s project, until kids are ready to both critique Euclid and see the need for refinements of Euclid? Why not view math courses as worthy of being taught somewhat ‘developmentally’ – what used to be called the ‘recapitulation’ theory, whereby courses mimic the history of the subject in having kids learn it? (Ask kids where the axioms come from, why we have them, and what’s the difference between an axiom and a theorem and they have usually have no answer). Part of my dissertation was on this notion: even Dewey for a while supported the idea that developmentally sound courses might ‘recapitulate’ the history of the discipline as a way of making it more likely to be understood by kids since it mimicked how the subject was slowly understood by professionals over time. That idea became rigid and silly in early 20th c education but the instinct was sound: you don’t necessarily introduce novices to everything the expert knows upfront.
A long-winded bit of history to underscore a point I think we agree on: the intellectual essence and rich history has been squeezed out of these courses for many decades. If their only justification is to enable the next level courses, and calculus is the ‘top’ then they obviously fail to have any real purpose beyond isolated technique.
“Lets look at the matter as intellectuals. Here is a thought experiment: can you identify 4 big ideas in algebra, ideas that not only provide a powerful set of intellectual priorities for the course but that have rich connections to other fields?”
Yes! Big Ideas:
1. Linear change.
2. Quadratic change.
3. Exponential change.
4. The translation of words to symbols and vice versa.
“Is the order of operations a matter of core truth or convention? How does that compare with the Associative Property? What is and isn’t arbitrary in algebra?”
These are great but random questions. I wouldn’t be comfortable making an inference about a student’s understanding of algebra based on her answers to any of them.
I disagree since at least 3 of them get at basic understanding of the system.
Hmm, I’m not sure I understand. Basic understanding of what system? Algebra?
Yes. If you don’t know why you can’t divide by zero and why we wish to include imaginary numbers, for example, then you do not grasp the symmetry and reciprocal transformations that we demand to keep it consistent, plus the fact that we want to be able to solve certain problems that require inventing these ideas (e.g. square root of a negative number and the fact that dividing by zero has no unique and stable solution). If you think PEMDAS is as arbitrary as the 3 core properties, then the learner don’t likely understand that systematicity either.
I’ll grant you that your 4 big ideas reflect a good list; The last one is a strong candidate but it is not unique to algebra. Though, to be picky, I don’t think a big topic is the same as a big idea. It’s really a fact that there are different kinds of relationships between variables. If I had to come up with a big idea related to your first 3 I might say something like this: seeming correlations that appear merely contingent or coincidental often reflect underlying mathematical patterns and provide the basis of many scientific insights. I then might add something more general to yours that captures the idea essence rather than just the topic: in a vast sea of data there are a few powerful relationships between quantities that can map an infinite number of seemingly unique and finite relationships of quantities (or words to that effect).
Thanks for pressing your point – this is the kind of discussion math faculties should be having and usibng to better frame courses around priority ideas and transfer goals.
I like your big ideas although I would add: 5 Harmonic change. Now, I do not understand what you mean by answer a question but if you mean could a student construct mathematical models of physical situation utilizing these four big ideas then I must disagree with you entirely.
Your #4 is interesting as it relates to understanding the meaning of mathematical symbols. To this one I would add two more. Transformations within each representational system and knowing the symbol to referent connection. It is this later one that makes math so difficult for students to learn as it is almost completely ignored.
3F = F is helpful in numerical conversions. Can you help me understand the algebraic distinction that should be made?
3F = Y that is.
Incorrect, even after your edit. The correct answer (which is counter-intuitive) is 3Y = F. It’s actually a beautiful little insight into both the power of math and the counter-intuitive nature of much of modern math.
Let me see if this what you mean. To convert 10 yards into feet,I would multiply by 1, expressed as 3 feet/(1 yard),, namely the dimensionally correct expression
10 yards * 3 feet/(1 yards) = 30 feet
Is this the sense in which you mean
Y 3 = F ?
Yes, but I think it is clearer to say it this way: There are 3 feet in 1 yard. So to convert yards into feet you have to multiply the number of yards by 3 to get the correct number of feet. Thus Y X 3 = F. So, multiply 10 X 3 to get the right number of feet. If you think in “english” you would write 3F = Y, but that is incorrect for making the conversions. You do not multiply the number of feet by 3 to get the correct number of yards (which is what the English sentence ‘says’) but rather the opposite – multiply the number of yards by 3 to get the right number of feet. Similarly, F/3 = Y.
I think I know why I initially got it wrong: the English is indeed confusing, and there is some fictitious support by the Math: if 3Y=F, then, for positive reals F,Y, F>Y but we know that Y>(1) F. 3F=Y does indeed support that inequality, but that leads to a catastrophe. Also, there is an ambiguity, because the English is true when not stated as converstion: it is true to say that one yard is three feet, and in most numerical cases, the word “is” implies equality: x is 5 x=5.
What is an equation that states the proper relationship between feet and yards?
Grant I must disagree with you on this one. 3F=Y and 3Y = F are both true statements. The difference is that in the first case F has a referent of feet and Y the referent of a yard. In the second case F has a referent in the number of feet and Y the number of yards. A possibly better statement of the problem might be: “What is an equation that states the relationship between the number of feet compared to the number of yards where each refers to the same distance.” However, with the problem as stated I would tend to think that 3F = Y is a technically better answer. Here again is the issue of the symbol/referent connection. It is important to note also that 3 symbolizes the conversion factor yd/ft and is technically different from the counting numbers as recognized by Euclid thousands of years ago. Euclid called them ratio numbers.
I agree with you. I’ve been thinking about this issue since I wrote this post: http://davidwees.com/content/fundamental-flaw-math-education and I have been trying to make changes in my part of the world, but with little success.
Unfortunately, whether it is correct or not, people see the list of skills that is required by the standardized test (or their state curriculum) and pretty much start at the top of the list and work their way down. I actually had a teacher tell me once that she includes worksheets to make sure she “covers the required prescribed learning outcomes.” No amount of discussion from me is going to convince her that the math isn’t in the worksheets.
In order to implement this style of curriculum we need to change mindsets. I’d like educators to look at the big picture ideas in algebra (here’s my list for what it’s worth: http://davidwees.com/content/powerful-ideas-math) and for institutions to rewrite their curriculum so becomes more difficult to see the list of expected skills and teach toward those. I’d also like to be able to point at concrete examples of what this looks like in practice. As you have pointed out, Dan Meyer’s work is valuable here, but we could just as easily see 50 of his excellent 101 questions turned into the most boring set of 50 lessons ever, simply by having a teacher painfully explain why the problems are interesting and precisely how he would solve them.
Your comment about worksheets and Myers is, alas, a sad truth: no work is impervious to being bastardized and rendered dumb. I have seen it with my own in UbD: thoughtless filling in of boxes with no understanding of what the template is asking you to do in terms of intellectual goals; an utter failure to understand what an essential question is and what its role is in learning.
This is a fundamental weakness of school as an organization. Think about it: perfectly pleasant people can THINK that their job is to cover bits and pieces of things superficially and inertly and believe, at the end of the day, that this IS their job. As I have often argued, therefore, this is an utter failure of supervision and evaluation, the fallout of just hiring people and leaving them alone, left to their own devices. Medicine by contrast vigorously and unapologetically monitors young doctors for years, to the point where no major decision about treatment can be done by a resident; the sign-off comes from the senior doctor. And there are clear incentives for doctors to know and follow best practice.
At the very least, we can do a better job by writing clear job descriptions and course and program mission statements, and monitoring plans and instructions more vigorously in terms of such goals and principles. It’s ultimately the supervisor’s failure, not the teacher’s – unless there has been a deliberate resistance to doing what the job requires and supervision says must be done.
PS: Just read the post you mentioned – love the shift in the sources of the math! Nice resources, too. A quibble: in the first slide, the graphic of typical courses, you say all the things looked at are “problems” – but that’s part of the, well, problem: they are mostly not problems but exercises (see my earlier posts on this distinction). Agreed, though, that if math curricula (and texts, especially) had to be based on problems that were fresh and germane then that would help immensely.
Yeah, that is true. It would be better if they were actually problems as opposed to exercises. I see the distinction.
I’m an audodidact learning mathematics. Many of the available online resources focus on techniques and drilling. I don’t mind the drilling, as I use it to test my comprehension and memorisation, but I’m interested in establishing deeper concepts, and more fundamental connections in my understanding of mathematics.
You speak of problem-driven exploration of mathematics. This is largely how I regained interest in mathematics after high school. Specifically, I got interested in the mathematics of dice in roleplaying games, and what skill buying strategy would be most effective for a character. This lead to a very interesting exploration of combinatorics and probability.
I understand that your focus is on educating educators, but do you have any advice for me on finding more of the real and engaging algebra problems that you refer to?
Yes, go to the Phillips Exeter Academy website, math department and download their problems. If you click on my prior article on Harvard and Exeter the link is there, too.
Here are some other possible resources:
http://threeacts.mrmeyer.com
http://101qs.com
http://projecteuler.net
http://maththinking.org
http://davidwees.com/category/topic/realmath
More resources here:
http://davidwees.com/content/resources-mathematics-enrichment
Thank you kindly.
Thank you so much for sharing your wonderful ideas here! I taught high school math for 6 years in an English program in Bankok, Thaialand and now teach at a Catholic school in the US. I’ve honestly fallen in love with the Exeter curriculum to put it bluntly. I get kind of depressed when I compare the drudgery we’re forced to impose on our students with how things could be.
Sometimes I feel as if math reformers frankly expect too much out of math teachers. It would be great if our schools were filled with Dan Meyer types endlessly researching,
developing and planning fascinating lessons to engage our students, but let’s face it: it is the rare teacher indeed that has the time and energy to make this a reality.
Math teachers need some crutches! Our current popular textbooks, along with their endless supplements of worksheet handouts and premade class materials, make it INCREDIBLY easy for a teacher to follow the traditional route of shoving the required tools one-by-one into their resisting young minds.
A similar, multi-authored, supplement-rich textbook and accompanying materials needs to be developed to assist teachers, who aren’t and are never going to be superheroes, accomplish this goal. Could Exeter be talked into offering their question sets as a base for such a project? As a further aid in convincing administrators to accept, it could be shown how this curriculum actually does, as a whole, meet core standards…just not in discrete units one after another as in a traditional curriculum.
I can dream, can’t I? Thanks again!
This absolutely hits some much needed discussion about learning algebra. However, I can’t imagine having this conversation with the math teachers in my school. In the past I’ve brought up things like focusing in the slope of a graph being a ratio between two measurements and using units in calculations and equations and I’ve met quite a bit of resistance. There is a definite bit of snobbery (at least in my school) that math is too pure to soil itself with real world application. I can see that point of view from a fundamental standpoint in a university, but when teaching teenagers who have no initial interest, I find it to be very counterproductive. I’m seriously contemplating printing out this article and anonymously hanging it up in the teachers lounge just to watch the fireworks.
You are so right: I have seen this snobbery on display many times, especially in “good” schools. That snobbery also makes it very difficult for other teachers and admins (who lack the math background) to go toe to toe with them on the error of their ways. The argument here is basic: HS teachers in an introductory course have an obligation to make a required course interesting and accessible. That does not mean dumbing it down. It means, though, that they can’t design and teach for the top 20% which is what this course now does.
The solution is to require the math department to develop a valid and vetted mission statement, and to oversee that courses reflect the Mission. In my experience this is the best way to make teachers own the problem without feeling put upon. That was precisely how Exeter developed its problem-based program: it found a fundamental disconnect between their goal and methods.
Hi Grant, Thanks for a wonderful post as always. I am a MS/HS math teacher, and have spent most of my life surrounded by mathematicians (my father was a math professor). I completely understand and support your premise that students need to make connections through broader and more open-ended questions. This is what mathematicians do! Students often look upon mathematics as something that has always been… a set of facts that were uncovered and now have to be learned. This is obviously not the case. Mathematicians continue pursuing new and exciting questions to this day!
Your post (and the release of your book on EQ’s) was timed well for us. My math department is undergoing a revising of our curriculum. Several years ago, we started following the work of the Assessment Training Institute and Rick Stiggins. We created student-friendly learning targets, and adjusted our assessment methods to match. However, as I read your book, I realize that we have certainly been focused too much on content mastery, and not enough on inspiring inquiry! I am excited to look at things from this fresh perspective. We are a K-12 school (my focus in 6-12), so we have a real chance to reframe our curriculum in terms of overarching essential questions.
Do you have any advice for us on this process? Where would you start if you were designing a mathematics program/curriculum from scratch?
Great question! I would start with the Mission statement of the math program. (You might want to look at Exeter’s). Then, with a Mission statement in hand, consider the basic backward design question: what would count as evidence of ‘mission accomplished’ (STAGE 2)? What approaches to teaching and learning are required by the Mission (STAGE 3). Here, too, the Exeter approach is worth looking at: their discussion of the problem-based pedagogy that now supports their mission is exemplary.And, of course, you’ll want to download their problem sets.
A parallel place to start is with student, alumni, and professor surveys, something that I have addressed numerous times in this blog and in my writing.
Good luck! Keep me abreast of it.
Hi Grant. Thanks very much. I took a look at Exeter’s program. It looks wonderful!
My colleagues and I met this past Wednesday to start the curriculum review process. We focused our day on developing a set of overarching essential questions to guide our program development. We were lucky enough to work at Loyola Marymount in the math department, and I was able to have some math professors give us some insights into what they believed were some of the most important mathematical inquiries for grades 6-12.
Here are the questions that we came up with. Some of these we took directly from your book.
– How can we use mathematics to describe relationships?
– How can use mathematics to measure, model, and calculate change?
– What rationale could give for beliving things to be true?
– When are numbers useful, and what are the limitations of using numbers?
– What arguments or styles of reasoning characterize mathematics, that are not concerned with number?
– How can use what know to help us learn what we don’t understand?
– To what extent can we accurately predict the future?
– What do effective problem solvers do, and what do they do when they get stuck?
This is still a partial list. There some important inquiries and ideas that we have not addressed (e.g. spatial/geometric understandings, math modeling).
What do you think of these questions so far?
Excellent list! I think each one has a place in guiding learning. And that’s the challenge, of course: to have teachers keep returning to these questions as a way of generalizing from and applying insights gained from considering these questions as they work. I also strongly recommend that everyone on the committee but a copy of Polya’s How to Solve It and that you have a little seminar/book club/plc around it. I’d be curious to know what inquiries the profs thought important to undertake, too. Keep us informed!
Reblogged this on that MADDENing teacher and commented:
The way we teach Algebra is stupid!
I would like to make an observation. We “demand” that all teenagers graduate from high school. This is the same fallacy as “all children are above average”. We readily accept that in sports, writing, arts, music there is a wide range of talents, but somehow we have not addressed the fact that in the abstract sciences there is an equally large range of abilities.
I studied math in high school in Hungary, many years ago. I was very good at it and spent much of my time trying to solve problems. No, I did not come up with a new proof of the prime number theorem, like Paul Erdos. We cannot base education on people who are one in a million.
Please, please try to teach kids things that are important and interesting.
Geometry was the only math course I thoroughly enjoyed when I took it in the late ’50s. When our kids were taking algebra I knew I couldn’t help them with their homework because I’d never really understood the subject myself. Their mother, a sixth-grade English teacher helped them with algebra. When the first of our three took geometry I was ready to take the baton. But their “geometry” bore little resemblance to the “plane geometry” I had loved. There were no theorems to solve, that I could find. The language was foreign to me. It seemed as if geometry had been merged with something else — trigonometry, maybe. I looked up terms and used the index, but found no way out of the morass. I’d finally failed the only math course I ever truly enjoyed.
Thanks so much for this. I wish our three could have seen it when they were still in school.
There’s a book to be written here. What once was taught with an intellectual point of view has been ruined by the same thing that happened in algebra – geometry became a directionless mass of dozens upon dozens of axioms and no beautiful direction (such as Euclid ending up with the 5 regular solids, the universe, as it were). I for the life of me don’t understand why people keep buying books that have nothing but hundreds of out of context mini-lessons and pointless assessments. Modern math courses are like really bad foreign language courses from 60 years ago where you learned pointless bits that never added up to any basic ability to speak or use the language.
With relatively little change, you could lob the same arguments against high school history courses taught with no overarching direction. This looks like an argument against an algebra requirement largely because you love algebra the subject, and seeing its dessicated form is worse for you than seeing a dessicated history course is.
Indeed, I have made the same arguments about high school history, often. Not only here but in Understanding by Design. Algebra is the worst offender, though, because the course has no larger purpose or prioritization. It’s just a run through isolated ‘moves’.
When I was in 7th grade I was very disappointed because I wasn’t going to be in Algebra in 8th grade and consequently wouldn’t make it to Calculus in my senior year of high school. My brother did and I always wanted to emulate him so I came up with a plan. I would take Algebra in summer school and be on track for Calculus my senior year. I really had no idea what Algebra was or why I would want to take it. I just knew that it meant that I was smart if I took it, so I wanted to.
Looking back I still couldn’t really tell you what Algebra is about. If pressed I would say that it is about ‘problem solving’ and finding the unknown number. After some thought I would alter that list to 1) a systematic approach to finding unknown numbers 2) an introduction to graphing 3) learning to move from specific ideas to a generalization 4)… I really can’t come up with a fourth.
The way that I learned Algebra certainly wasn’t the best. I was covering a week’s worth of material in one day. It really left me lacking in skills that I needed later. I did finally make it to Calculus in high school, which was extremely watered down to the point of absurdity. Then I went on to take Calculus my freshman year of college and failed miserably. I went to my teacher and was told that “You know the Calculus better than most of the people in the class, but your Algebra skills are so bad that you can’t finish solving the problem.” If I had been taught in a meaningful way that took into account why we need to have these skills I would have been much better prepared.
I teach 6th grade at an elementary school. One of the comments made by educators is that the students are not developmentally ready for some of the algebraic skills that are taught at that level. My thought is that the students struggle because the skills are taught in discrete bits without reference to the larger picture and without any purpose or application. Do you think the argument about developmental readiness changes if the focus of study is turned to conceptual understanding rather than the acquisition of skills?
I think there is no doubt that the way algebra is taught is needlessly abstract, making it highly unlikely that middle schoolers will grasp the point of algebra, its value, and its meaning. I always ask algebra teachers this question: do your students know what algebra does that arithmetic can’t do? Do your students know what analyses algebra enables that can’t be done with basic arithmetic? If the answer is no then it is very unlikely that the mindless plug and chug will last or permit understanding, hence transfer.
Bravo Grant!
Without Algebra you cannot do basic programming.
Without Algebra you cannot do basic physics problems.
Without Algebra you cannot do basic finance and accounting problems.
Modern mathematics has two pillars: Algebra and Geometry. These modes of thinking have to be developed from an early age in order to fully rip the benefits. The benefits mentioned above.
“But that addresses the needs of only about 20% of the school population. ”
Without Algebra you cannot do basic programming: In c++ and python a variable will make no sense.
Without Algebra you cannot do basic physics problems: no comment.
Without Algebra you cannot do basic finance and accounting problems.
These professions represent far more than 20%.
Modern mathematics has two pillars: Algebra and Geometry. These modes of thinking have to be developed from an early age in order to fully rip the benefits.
I was one of those students who struggled all the way to calculus II to finally realize that I just didn’t understand Mathematics. I got lost in the parts but could not see the whole. It is like you are just chasing around these parts which have no real meaning to you.
I am just sharing what I have come to believe is very important in the pursuit of Mathematics. I absolutely agree with the ideas expressed here that students just don’t understand what they are doing, except manipulating numbers according to certain rules which has no real meaning to them.
I was able to make A’s & B’s but as Paul Lockhart stated in his essay, like so many students, I simply just learned how to follow directions, rather than to think. It seems like you never have time to think. It is like being on a merry go round of which you just can’t get off till you finally decide to just get off and call it quits. I jut realized I was done. I wasn’t going any farther. I was hopelessly lost because I didn’t know how to apply those ideas.
But it bothered me that I just didn’t know what I was doing, so I went on this long journey, off and on, to try to figure it out over the years; even though I pursued a non-mathematical career. I finally started to realize that there were many important concepts that I just didn’t understand. If I were to pick out the most important concept to fully understand, I would say it is the concept of a “ratio” and how it relates to “proportions”. It was an idea that just pulled it all together for me when I finally grasped it’s importance. It is such a simple idea and I just didn’t understand how I could not have understood it.
Because without understanding and comprehending the concept of a ratio fully, you just will not get anywhere in mathematics. You could randomly ask people what “pi” is and they won’t know because they just don’t understand this idea of relative lengths. How the diameter of a circle relates to the circumference?
So once you understand ratios, you can grasp the ideas of trigonometry by realizing that most of trigonometry is comparing the relative lengths of a right triangle. You can now solve those Chemistry problems where you need to convert inches to millimeters. Or even to correctly bake more than one loaf of bread or even one half of a loaf.
However, the odd thing is that once I realized the importance of this concept, you find that it is not really that easy to get someone to understand it but it is important for a student to grasp this idea fully as they are going into Algebra because they will most certainly get lost. A lot of exercises should be devoted to making sure they fully grasp this idea of a ratio, so that it actually has meaning to them and not just some abstract rule. They should be comfortable with it.
There just has to be better ways to teach mathematics because it really isn’t that difficult in a basic sense, even though it does get very complex. But students should be able to understand those simple ideas to solve for real problems without being Mathematicians.
Congratulations. Finally, a mathematics academic that tells it like it is. A few decades too late for folks like me, but you nail it nonetheless. I haven’t posted anywhere else on this subject because there seem to be an over-abundance of math snobs out there that are too stupid to understand those that don’t get it. I can see the progressions in my head, and empirically tell you by experience what physical structures work and what won’t. Arithmetic went wonderfully, but algebra is a brick wall to me.
If math was somehow purely image-based, I’d be the next Einstein (hah!) but I have never ever been able to translate anything into (or out of) the mess – sorry, indecipherable language – that is algebra. Anything algebraic in physics, chemistry, geometry, and God forbid, calculus was a never-ending mess of circular mazes that I never made it through. I still don’t know how I passed my basic qualifications, but the last year of high school and my first (and only) year of college was a math nightmare.
All I can say is to lose the weirdness in algebra teaching. Lose the disconnect with reality. Get real-world problems like ship propellers, hopping frogs, or bridges into early algebra teaching, not the abstract disjointed concepts that pass for an algebra education at the present time. I mean REAL propellers, frogs and bridges; show a real problem with hands-on solutions. I wish I could make that clearer, but I’ve never understood algebra…….
Invaluable blog post ! I learned a lot from the details ! Does anyone know if my assistant would be able to obtain a sample a form version to work with ?