by Grant Wiggins, Ph.D, Authentic Education
In my 100th blog post I complained about the course called ‘algebra’. Some commenters misunderstood the complaint. Though I said a few times in the article that my critique was not about the content called algebra but the aimless march through stuff that makes up almost every algebra course in existence, some thought I was bashing the value of the content. Not so. Another commenter said: you might have ranted, then, about many history courses! Indeed I could have–and have done so multiple times in my career.
The issue, then, is not ‘algebra’ or ‘history’ but what we mean by ‘course of study’. I am claiming that to be a valid course, there has to be more than just a list of valued stuff that we cover–even if that list seems valuable to me, the teacher. Rather, a course must seem coherent and meaningful from the learner’s perspective. There must be a narrative, if you will; there must be a throughline; there must be engaging and stimulating inquiries and performances that provide direction, priorities, and incentives.
Notice that I haven’t merely defined a course. What I have just done is identify some criteria by which any so-called ‘course’ can be designed and critiqued. And such criteria are vital: I know from 35 years at this work that very few teachers ever self-assess a course as a course against explicit criteria–with unfortunate consequences. They may tweak lessons and even units but they rarely dispassionately critique the design of the entire course against criteria such as mine; or receive feedback against criteria about their design.
Textbooks Are Tools, Not Courses or Content Areas
Next time I will say a bit more about my criteria, but we can’t ignore the other lurking issue in this discussion: ‘coverage’, i.e. teachers marching through the pages in a textbook. I wish to claim that defining a course as a tour through the textbook, page by page, is simply not a course by any valid set of criteria. A textbook is merely a collection of topics, with exercises and text under each topic.
The textbook does not know your personal or school priorities; the textbook does not know your students; the textbook doesn’t identify any priorities or through lines that unite all the chapters, etc. So, a march through a book is a non-design. It would be like learning English through a page by page tour of the dictionary and grammar book; it would be like learning history by reading through the encyclopedia page by page.
It doesn’t matter how good the textbook is. My critique is not a critique of textbooks. (I have worked on over a half-dozen for Pearson, to infuse UbD). My critique is the use of books. A text–be it an algebra textbook or Catcher in the Rye–is a resource in support of clear and learning-focused goals. Goals cannot be supplied by a text, they are supplied by purposeful teachers.
7 Prompts That Every Teacher Of A Well-Designed Course Should Be Able To Answer
Here are some simple prompts that a teacher who has really thought through the course as a course should be able to answer:
- By the end of the year students should be able to…. and grasp that…
- The course builds toward…
- The recurring big ideas about which we will go into depth are…
- The following chapters and sequence support my goal of…
- Given my long-term priority goals, the assessments need to determine if students can…
- Given my goals, the following activities need to build insight and incentive…
- If I have been successful, students will be able to transfer their learning to… and avoid such common misconceptions and habits as…
So, even before spelling out the meaning of and rationale for my course criteria, you should be able to realize from these prompts why almost all algebra courses are complete failures as courses–i.e. purposefully designed learning in support of clear intellectual goals. No, almost every algebra course (and, yes, history and science course) is a mere march through a textbook, page by page. Rarely do explicit overarching goals and priorities inform the sequence, the activities, assessments, and choice of topics.
Most importantly, the assessments almost never require students to synthesize learning across many chapters and transfer their understandings and skills to priority performance tasks. And it is therefore no accident that students uniformly find high school math to be their most boring and difficult course, as our student surveys show.
In my follow-up post I will say more about the criteria mentioned above as well as walk the talk: I’ll share some design work that my colleagues and I have done over a 10 year period to build better high school math courses. Here’s a hint: if we want understanding instead of mere dutiful learning, we must begin in a very different place than almost every math course I have ever seen.
We have to begin with giving the learners intellectual reasons and incentives for taking such a course. And, thus, we have to justify both the content and the overall direction of the course.
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 theUnderstanding 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.
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, as I blogged here.
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.
To build courses around worthy performance tasks, the series entitled Balanced Assessment, edited by Judah Schwartz, is excellent. (You can find free resources from ithere.) 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 MathematicsStudents 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 bookMathematics 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 these complaints such as the ones up above in high numbers 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?