A number of math teachers have complained either directly or indirectly recently that I have offered criticisms but no solutions to the problem of poorly-designed math courses, especially at the secondary level. For example:
“In my opinion the writer suggests that textbooks are merely a collection of topics with examples of exercises under each and that teachers merely race through a textbook to get to the end. In a sense I agree with this but my problem/concern is that he offers no alternative/answer to what we should be doing instead…. It seems there are so many people out there saying that this is not what we should be teaching our students and that us Math teachers are in fact wasting students time with our outdated teaching methods. My question is then what should we be teaching them? What am I missing? He offers no answer to that question.
“What I find lacking in your rants are specifics. What types of “broad questions” would you suggest to stimulate the interest of hormone driven 14-year old boys? or incredibly self-conscious 14-year old girls many of whom lack basic computational skills, the ability to read critically or who are afraid to take chances or to explore into areas in which they are not familiar, yet who are required to sit in my Algebra I class?
Leaving aside the fact that I have indeed offered numerous resources under the Understanding by Design name (including many math units such as this one: Algebra Unit – before and after) over many years, let me offer a short list of print and web sources for problems, assessments, and pedagogical advice on teaching mathematics more meaningfully that every secondary math teacher ought to have in their library (or at least know about). Math teachers and supervisors: please post others and I’ll add them to this list. I am also building a Storify page of tweets that propose such sites.
WEBSITES
The first go-to resource is Dan Meyer’s video, blog and his open-source collection of problems. The second go-to site is the archive of Car Talk Puzzlers. Here’s my favorite, and make sure to read all the follow-up posts and listen to the next week’s radio show). Here’s my next favorite, and the great kids from E Tipp MS had fun with it.
Another helpful source is from the United Kingdom (and has served as a partner to Common Core developers) – the Shell Center.
I contributed to a big volume for MAA on Quantitative Literacy (my article begins on p. 121) and you can find many examples not only in my article but in those of others in the volume.
NCTM publishes resources under the Illuminations banner. Here are lessons in algebra.
Mathalicious has some great resources for real-world lessons. So does PBS. Buck Institute and Edutopia have long been known for their materials on problem-based learning.
BOOKS
To build courses around worthy performance tasks, the series entitled Balanced Assessment, edited by Judah Schwartz, is excellent. (You can find free resources from it here.) Good tasks, good rubrics, and samples of student work. There are books for middle school, high school, and advanced high school.
The 20 year-old book from NCTM entitled Teaching and Assessing Problem Solving is probably the best of it s kind, a great mix of theory and practice, filled with helpful examples. A newer NCTM book, Teaching Mathematics Through Problem Solving, is equally helpful.
An edited volume entitled Real-World Problems for Secondary School Mathematics Students has lots of great examples from different countries.
One of the better textbooks in math is by Harold Jacobs called Geometry: Seeing, Doing, Understanding.
Beyond Formulas in Mathematics and Teaching is a bit text heavy but provides a solid perspective on such an aim. For a more general text on the meaning of mathematics, highly readable and usable with HS students, nothing beats Morris Kline’s old book Mathematics in Western Culture.
And as I have noted numerous times in this blog, arguably the best course ever designed, from the 1930’s, was Harold Fawcett’s course later written up as an NCTM Yearbook, and republished 20 years ago. And surely the most seminal and vital book in a math teacher’s library is Polya’s classic How To Solve It. Here is a great old video of Polya at work. Stick with it: there is a dramatic conclusion to the inquiry.
A blunt postscript: All of these resources are not new. I find it a bit depressing that so many math teachers such as the ones I quoted above are seemingly unaware of the materials that are available to ensure better engagement and outcomes in mathematics. BTW, it is ONLY math teachers who routinely make complaints in high numbers (such as the ones up above) that they lack resources to develop better courses, instruction, and assessment. At a certain point, I simply must say: isn’t it your professional obligation to know about these resources rather than vent at me for not providing more resources?
Please also note the comment posted by Rose in which she identifies many great resources (as well as provides important stories related to my main point).
35 Responses
Many great resources have emerged. Here is a link to some: http://coreessentials.wordpress.com/2013/05/22/problem-based-learning-and-the-common-core/
A couple places of note for secondary math are the Mathematics Vision Project (http://www.mathematicsvisionproject.org) and the work of Howard County Public Schools (https://secondarymathcommoncore.wikispaces.hcpss.org). There are many more great things happening too!
Thanks for the blogpost,
@dgburris
Thank YOU!
Ouch! Your backhanded “help” for teachers (math in particular) who are struggling to find ways to improve in this very difficult climate for teaching is baffling. Is it not unbelievable to you that there may be some who do not have the same background in education as you?
I also must point out that this gets to some of the weaknesses in our programs to educate future teachers. I know (since I have an ed degree) that you do get the foundations for education in the masters degree coursework- at least in a good one. But, many dismiss these as unhelpful to “real world teaching”. It’s not. It’s a foundation and should give the teacher the ability to research themselves how to improve their educational skills- themselves! But, when we focus on rerouting the education of teachers – via TFA programs or “real world experienced” people becoming teachers in half the time, this is what you get.
This type of learning starts early. Teaching our kids in school how to think critically, handle learning and being responsible learners, how to access resources, and how to choose resources are skills that must be more highly valued.
In the meantime, we should do all we can to make sure we arm teachers with the resources and opportunities to create this kind of climate in their classrooms so the vicious cycle is not repeated. Backhanded help is kind of cruel.
I go out of my way to help people who seek assistance. I think if you ask the folks I have worked with over the years they would say so. (This blog is an example: I get no money from it, and I reply to almost every post). I love helping those who want to improve their craft. But that’s not what those commenters were doing. They were lashing out at ME for not providing THEM with resources on how to do their job better. That’s just passive-aggressive – and unprofessional. A 1-hour Google search under problem solving math HS would have turned up most of my links. It’s unthinkable that a doctor would do something like that – ask me to read the journals for them. We can only improve our sorry math situation if HS math teachers take more responsibility than they currently do for widespread student disengagement and lack of in-depth understanding – especially in such a gateway course.
Ok, fair enough. I think that it would be helpful to be more direct with your message then rather than slapping hands and then tossing them the info anyway. Plus, those who did not complain to you could benefit from this article. It’s just a tad irritable in delivery. But, I hear you.
I am not certain the majority of classroom teachers see it as their responsibility to keep up w/their discipline.
I think that math teachers get the brunt of the criticism of teaching in secondary schools, because it is fashionable in the US (and, I’ve heard in the UK and Australia) to hate math. Even very good math teachers struggle to keep the attention of kids who have been told that no one needs math.
It is also harder in math classes to bring in “fun” stuff in a relevant way (no trebuchets and projectiles like in physics, no “bangs and stinks” like chemistry, no “current events” like social sciences, no reading stuff originally written for entertainment like English, …).
So I think that the math teachers get a bit defensive about the unrelenting barrage of criticism.
I agree with you, though, that it is unprofessional of them to criticize *you*, as you have been helpful in providing resources.
I don’t buy your causality. Secondary math teachers, in general, have simply stiffed the feedback that has come from surveys, interviews, conversations, student feedback and test results for years. I have many times had to make this case in the face of great resistance on the part of HS math teachers to examine their practices – in fact, they are always the toughest group in every workshop of a HS faculty. More telling, I think, is the reason in our student surveys that kids hate math: they are made to feel stupid. In other subjects that are most disliked, the complaint is that they aren’t interested in the work or in English the books. Even in a show of hands in a faculty workshop: which of you disliked math a lot and felt it was pointless? on average about half the hands go up. And that’s educators! In over 15 years of asking people to identify the best designed courses they were ever in i can recall only a handful of math examples, and I have heard answers from thousands of educators, all over the world.
What will it take before HS people stop resisting the tide of evidence and get on with revamping their courses to make them more interesting and effective for more than just the already able and interested 15% of students?
For me the problem is not lack of awareness of these resources. On the contrary, I am overwhelmed by the sheer number of them. The problem is that these resources are scattered and without focus or coherence. A course as you have defined it is EXTREMELY hard to put together so that it has quality and depth. I began my (still young) teaching career frustrated with the mainstream textbook options so I thought naively that I would research and find the best resources and create exciting, deep courses from scratch. I gradually realized that such an undertaking would require several years of work (at least) in which I would have to focus only on that task.
In my view, full-time classroom teachers should not have to collate resources the way you describe. That is the job of textbook writers. Admittedly the mainstream options are deficient, but that should translate into a demand for better textbooks, not telling teachers that they should be curating thousands of resources and streamlining them into a dynamite, understanding-driven course from scratch.
I disagree. Every teacher in every other subject supplements textbooks – indeed, you must, as I argued. Why isn’t this a departmental obligation in your school? Why isn’t this the supervisor’s job, too? I just can’t understand why math teachers should get a pass on this. I never just used a textbook; I always supplemented what I did with helpful and interesting resources.
The fact that teachers have to supplement textbooks suggests that those textbooks aren’t very good, not that there is some perpetual moral obligation to supplement, regardless of textbook quality. How long did it take you to write Understanding by Design and its accompanying resources? Were you teaching full time as you wrote it? To reiterate, it takes a staggering amount of time to conceive of and flesh out a course, as you have described it. Trying to arrange concepts in a logical sequence of instruction, identify big ideas and recurring themes, researching helpful analogies and visual organizers, writing or adapting problems so that they progress from easier to more complex, ensuring that they do not presuppose more background knowledge than is reasonable to expect at a certain level, multiple versions of quizzes, tests and projects, where and how technology can enhance instruction…
I would add that I don’t mean this to apply just to math teachers. This is not about giving anyone a free pass. This is about what is the most productive division of labor among the different workers in the education field. Classroom teachers should be able to focus mainly on getting to know their students and give them personalized feedback and coaching, as well as thinking deeply about the content and how to deliver it (or provoke its construction through student inquiry, if you prefer), rather than curating thousands of resources and creating materials from scratch.
I am certainly sympathetic to the overload issue and the proper division of labor issue. I am merely saying that in the absence of good materials, you supplement. What I find odd is that math seems to be one of the few areas – Chemistry is another – where people define the course as a book. I think if you poll your colleagues in English, History, and Art that they do just as I suggest you guys need to do as a department – if, in fact, the textbooks are inadequate to meet your goals. For that matter, how carefully has your department reviewed and adopted texts? And why not do what numerous people are now doing – including in math – which is writing their own e-books? (If memory serves, a district in so. CT, maybe Westport, is doing it, as is a group of districts in Utah or Wyoming.)
My other thought on this is that making such materials makes you a better teacher, as I have seen in working with teachers on units and courses as a group. I am not saying we should do this alone or reinvent the wheel. I am saying the obvious: 2 heads are better than one, resources DO exist, and I think it is always the teacher’s job to think through alignment of course goals, resources, and pedagogy.
Mr. Wiggins. Thanks for those links. Funny, the link you provided for the old algebra/new algebra after Ubd,( the discussion on the presentation of why we use the order of operations as a convention) was similar to one I dreamed up on my own my 2nd year of teaching. I think if you actually went into the classrooms you so loudly denigrate you might find teachers doing exactly what you’re asking them to do. You’re right, you’ve listed nothing I haven’t found on my own with a Google search – but those don’t seem like deep, probing, engaging tasks to me – they’re more like word problems. Admittedly, they’re good word problems, but still, in the eyes of my students, just more word problems. My students would rather look at their cell phones to be sure they’re popular, and honestly, they could give a rip about the order of operations, whether or not they talk about it. But thanks for trying. Maybe you should spend some time in the classroom yourself for an entire year or two. The kids I teach today aren’t the same as the ones I taught 15 years ago, and I bet they aren’t the ones you taught either. They’re coddled, protected and afraid to take risks or responsibility because of it. I suggest you stop harping on math teachers and get back out there and do it yourself for a different perspective.
I’m happy to supply more performance-based units and will do so. I am in classrooms all year including in Trenton, NYC, and Toledo, so there is no need to give me snark. I have guest taught in those places, too. And worked with teachers to build the kinds of units we are describing. I watched a way cool problem-based unit in Camden County technical School with as tough a student crowd as I have seen – kids all engaged and doing interesting stuff.
So venting at me just doesn’t work. This is a problem for you and your math colleagues to solve, as a group. Period. I’ll do my part by adding some more units to the resources; the rest is up to you.
Singapore math is very good for elementary grades. The Exeter problem set for Algebra I is good. James Tanton has a really outstanding unit on quadratics. Released items for AP Calculus and AP Statistics can sometimes be modified to make them appropriate for an Algebra II or even Algebra I class.
The majority of our textbooks are lousy – traditional & reform. Maybe we are not meeting our professional obligations in allowing that to happen. There is a market for those lousy books. Somebody wants them. If enough of us demanded something better, it would be written, and we would get it.
Of course it is our professional obligation to supplement or refine lessons from the textbooks. It is our responsibility to find and create materials. But, you can easily end up with an incoherent Frankenstein curriculum of mismatched pieces doing this. Asking each teacher to heavily, heavily supplement is a grossly inefficient and ineffective response to lousy books.
My department was in the position of selecting a textbook for geometry last school year and we agonized over the decision. In the end, we realized that there just was no such thing as the “perfect” book we hoped was out there and that we would have to supplement/ design our course around the shortcomings of any text we selected. My favorite text was the Harold Jacobs book you mentioned – in part because it offers a bit of a narrative, in part because the problems that reference “real” things do so in a meaningful and interesting way (not artificial pseudo-math!). Even so, the task of designing a course is huge – the textbook is not my course.
Aside from resources mentioned in the original post and previous comments, here are some others I have found useful:
Drexel’s mathforum is a really good resource, and I also found the PD course I took through the mathforum to be very helpful. There are also a number of good blogs written by math teachers. Going to Dan Meyer’s blog (mentioned in the original post) and visiting those he has linked to is a good place to start. I am really excited about using Andrew Stadel’s estimation180.com this coming year (and adding my own estimation challenges for my students) as well as the problems at http://fivetriangles.blogspot.com (not sure who curates these, but they are great!). I will probably also utilize this as a resource: https://docs.google.com/document/d/19R0BNVRIL2tTE586L4EQJkkO7o1P2BrP-a7YNuV1neo/edit
Some other print resources that I’ve used/have had an impact on my teaching are:
Writing to Learn Mathematics (Joan Countryman)
Thinking Mathematically (Mason, Burton, and Stacey)
Using Formative Assessment to Differentiate Mathematics Instruction (Leslie Laud)
The Open-Ended Approach: A New Proposal for Teaching Mathematics (NCTM publication ed. Becker & Shimada)
Math for Smarty Pants and The I Hate Mathematics! Book (Marilyn Burns)
A Visual Approach to Functions (Frances Van Dyke)
Constructive Assessment in Mathematics: Practical Steps for Classroom Teachers (David Clarke)
Perhaps most importantly I began my teaching career working alongside a woman who had taught for 30+ years and encouraged me to become a member of professional organizations, to read, to try new things, attend conferences, and even to present at a conference in my 2nd year of teaching. Here was a veteran (and beloved!) teacher who worked to constantly improve her courses 30 years in – it was striking. Connecting with colleagues (many outside my own school building, and many teaching in other subject areas) has had an incredible impact on my teaching and I would encourage math teachers to find communities of colleagues who are also working to improve their practice. This kind of professional development may not “count” towards maintaining certification, but it is absolutely invaluable.
Rose, thank you so much for taking the time to share these wonderful resources and your commentary on them. This is an inspiration! By the way, I asked Patrick Higgins about who was behind 5triangles and he said that he did not know – and was a bit concerned by their anonymity, thinking perhaps that they were using materials without permission or something else to continually hide the fact as to who they are.
It strikes me as a sad state of affairs when some teachers (and luckily they seem fewer and farther between than they once were) are unwilling to be students of the what and perhaps more importantly, the how of teaching. For instance, balking at the idea of needing to thoughtfully prepare for instruction seems antithetical to the definition of educator. Isn’t an educator charged with organizing material and choreographing the learning opportunities with the goal of moving the students from “the dark” to “the light” of a particular discipline? Aren’t we supposed to be experts in our discipline? If so, they why does the idea of researching not only the what of what we teach but the how seem so threatening?
Really folks. Asking textbook editors and authors to do that work for us is like pining for the olden days of recorded or scripted curriculum where the teacher was just the one who pushed play on the tape recorder.
Show students that we are co-learners. Take risks. Try to do something that engages them. Put them first. Dispense with the expectation that some publisher somewhere has distilled all of the important information and written in the margins of our over-sized teacher’s editions.
And before I get flayed for shooting from the sidelines, yes, I’m an administrator but I’m also a classroom teacher as are all of the administrators in my school. Each year I’m lucky enough to teach one to three high school math and/or science classes in addition to supervising teaching and working to support curriculum development in my school. If the admin types in your schools don’t teach, get them teaching. You’ll soon have all the support you need.
So I say, anyone working on behalf of collecting and collating resources for classroom teachers is doing good work but that this doesn’t supplant the teacher’s awesome responsibility for deciding which of those to use and when. The teacher must realize that he or she is the most important person in any student’s opportunity for success and exercise this responsibility liberally.
Stacey,
It is a sad state of affairs when educators, especially experienced ones, leap to judgment without trying to charitably discern what is actually being said. This is not about abdication of responsibility, it’s about what the best use of each education worker’s time is. I have plenty of content expertise (although I’m sure you’re aware that experts in a discipline actually often suffer from a blind spot when it comes to instructing novices). I won a recognition of excellence for outstanding performance on the Praxis content exam in mathematics. During observations I was consistently cited for ‘expertly crafted materials’. In my first year of teaching I was put in charge of the vertical articulation committee in the mathematics department.
I could without a doubt sit down and write a textbook, or course materials packet if you will, that is light years ahead of mainstream options. BUT it would take me years of patient work focusing full attention on that. As a full-time teacher, is that really what I should be focusing on?
Even with an excellent textbook, there are still plenty of decisions about choosing and adapting that legitimately fall within the classroom teacher’s domain of responsibility. This comes about as the teacher gets to know the students well. But where is the time for that if teachers, especially beginner teachers, are spending hours each night poring through pages and pages of hyperlinks like the ones that are filling the comment box, trying to figure out if and how different problems and activities fit together?
Grant Wiggins himself has pointed to an example of a truly excellent textbook, Jacobs’ Geometry, and a teacher would not go wrong designing a course around it. I myself am using his Elementary Algebra, which is also excellent, in my 8th grade Algebra I class. In the introduction to the teacher guide to the latter book he said that one of the reasons he wrote his textbooks in the first place was that figuring out how to make all lessons fun and engaging and make sure there was a logical progression of objectives took a lot of time and energy and he wanted his fellow teachers to have a resource that would spare them a lot of that effort, although he does also caution against following the plans too rigidly. But they can and should serve as a template which a teacher can modify or expand on rather than creating full courses out of Internet raw materials.
My only reply here is that you beg a key question: what, exactly, is the ‘job’ of being a full-time teacher? To what extent does that job include or not include being a good designer of courses, units, lessons? You make the mistake here of assuming, without making a case for it, that the best use of your time is mostly ‘teaching’. But I think a strong case can be made that gains come from a robust and engaging curriculum, not just the ‘teaching’. After all, a good teacher with a lousy curriculum can’t cause much. I am not saying my job definition is right and yours is wrong. I just don’t think you should assume that, as a teacher, you can say that limited time designing is optimal. My work for years has shown the opposite: the more teachers use non-contact time more effectively, especially as teams, the better the results when with kids. And that finding is supported by Marzano and Hattie (and at the college level, by Ken Bain).
Even PBS has some great math resources. http://www.pbs.org/teachers/mathline/concepts/designandmath/activity1.shtm is a great activity where students measure various bike styles to form conclusions. We should be doing more of this type of thing in math.
This activity opens the door to other questions: Are cars designed this way? Motorcycles? Planes? Computers/tablets? My students have loved this activity when we hit angles.
The material is out there, we just need to supplement it and tailor it towards our students more and more.
Hi Grant,
Thanks for taking the time to share terrific resources and to provoke contentious discourse. Great learning opportunities on both counts. We’ve done a great deal of work at the Georgia Dept. of Ed, working with teacher teams to develop K-HS math units, and grade level and course overviews, found here: https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx, https://www.georgiastandards.org/Common-Core/Pages/Math-6-8.aspx, https://www.georgiastandards.org/Common-Core/Pages/Math-9-12.aspx.
We provide accompanying professional learning opportunities, here: https://www.georgiastandards.org/Common-Core/Pages/Math-PL-Sessions.aspx . We also facilitate collaboration and share further resources on our 3 mathematics wikis, linked here: https://www.georgiastandards.org/Common-Core/Pages/Math.aspx
All are free teacher-developed and supported resources, which we happily share with anyone interested in availing themselves of the materials. While we do not provide any of this with the intent it serve as a teacher’s sole resource, the materials go a long way towards supporting mathematics learning in a problem-based context. We work diligently to ensure we are doing enough to support Georgia teachers (facing criticism that we have not)…while each day folks from other states and countries ask if they may use our resources, which we happily share.
Hope someone finds this useful! That’s our intent.
Best,
Turtle
Wonderful! Thanks for sharing. I have used some of these in my own workshops as good examples available for free.
Agree with Turtle Toms–Such a great resource available to us all by the teachers of Georgia. Thank you Georgia. I have learned a great deal from your work and look forward to my Georgia moments in my Boston classroom this year.
Reblogged this on Curiouser and Curiouser.
Mr. Wiggins, I wholeheartedly agree. I would adore to teach a geometry course that breaks the mold. I am tired of being trapped by textbooks and forced to assign problems that students know *how* to do but not what they doing *means*. I am, however, a purist at heart. I enjoy math-y type problems and stress the puzzle-like quality of them. I do some real world applications, but they are, to quote Dan Meyer, “pseudo contextual” at best. I feel most geometry textbooks are a severe bastardisation of what geometry could really be.
I recently rewrote our district’s geometry curriculum to link topics and create mathematical flow while adhering to the challenging, yet vague common core standards. I have been told to rewrite it to match the “flow” of the textbook and that my curriculum “sucks”. I have fought this battle for weeks, and have no ammo left in the chamber.
My comment is this- maybe, perhaps, it’s not the teachers, but the willful ignorance of administration to allow is the freedom and creativity that requires straying out of the bounds of the traditional text.
For sure this work is made far more difficult by dunderhead admins. To assume that the curriculum is the text is bankrupt. Nor am I seeking only real-world problems. As I have often said, some of the best problems are about pattern finding, e.g. what other figures drawn on the legs of a right triangle can make the pythagorean theorem hold true? (e.g. not just draw squares on the legs, but other figures, such as pentagons or hexagons.) See my earlier blog entry on the great kids and teacher from E Tipp middle school who tackled this and shared their work. And send me your original idea!
I will definitely reread that post about E Tipp. Change is hard, and I would love to send you my ideas for some constructive criticism. I have taught geometry traditionally for almost ten years and I was quite excited to try focusing on the big ideas in the standards and address those. Please let me know what information you would like; big ideas, skills, standards, etc. I am so passionate about math education and it will be an honor to have you look over my work and offer your advice.
E-mail me whatever you wish to get my feedback on. And, if you haven’t already done so, search for my blog entries on Harold Fawcett’s famous book The Concept of Proof from the 1930s, arguably the best geometry course ever.
I actually tried to buy Fawcett’s NCTM yearbook after reading your posts! It’s a little out of my price range right now, but I always keep my eyes open for a used copy. Geometry for Enjoyment and Challenge is out of print, and a little dry, but its closer to a proof course than more recent textbooks.
Send me an e-mail: I have a pdf of the original. A bit hard to read, but legal and free.
I sent you the first unit with standards, complaints, reasoning, and skills. To call it “original” is a huge overstatement- it’s nontraditional at best. (I hope I sent it to the correct email address.) Even though my district is unwilling to use this particular curriculum, I am going to continue to work on it as a tool to learn how to write effective curriculum. Thank you for your time and patience. Please let me know if you would like to see more.
Will do, once I return from Europe. (What email address did you send it to? Just reply with the first 4 letters, so as not to expose it to spammers).
I sent it to gran* at ubde******* and got a bounce back email. My email is glynna at yahoo.***- and its always had spam. I would really be interested in that book, plus I just got the revised curriculum, so I’ll send you that with the one I wrote as pdfs.
I also have to comment: I am so impressed with how accessible you are- most educational gurus are a face on the back of a book, but your responses to the posts on your blog (as well as other blogs) are evidence of your concern and determination to change the face of curriculum. I thank you for your time in advance, because as a hs math teacher, I know how precious every second of time is.
I don’t think the address you sent it to works, but now I have yours, so we’re good to go. I return from Europe tomorrow and will send it out soon after, ok? And thanks for the kind words. i thrive on dialogue, so it’s really a labor of love not a burden in a busy world.