In the just-released Math Publisher’s Criteria document on the Common Core Standards, the authors say this about (bad) curricular decision-making:
“’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.”
Hallelulah! As readers and friends know, I have been harping on this problem for decades, and especially with regard to mathematics instruction and assessment. So, to have such a clear statement is welcome. Not that I am naïve enough, however, to think that a mere statement will alter some people’s wrong-headed thinking and habits. But this should catch some attention.
Longstanding problem of teaching bits out of context. This problem of turning everything into “microstandards” is a problem of long standing in education. One might even say it is the original sin in curriculum design. Take a complex whole, divide into the simplest and most reductionist bits, string them together and call it a curriculum. Though well-intentioned, it leads to fractured, boring, and useless learning of superficial bits.
Here is John Dewey on the problem – and the false analogy with physical taking apart that it is based on – writing over 100 years ago:
Only as we need to use just that aspect of the original situation as a tool of grasping something perplexing or obscure in another situation, do we abstract or detach the quality so that it becomes individualized…. If the element thus selected clears up what is otherwise obscure in the new experience, if it settles what is uncertain, it thereby itself gains in positiveness and definiteness of meaning. Even when it is definitely stated that intellectual and Mental physical analyses are different sorts of operations, intellectual analysis is often treated after the analogy of physical ; as if it were the breaking up of a whole into all its constituent parts in the mind instead of in space. As nobody can possibly tell what breaking a whole into its parts in the mind means, this conception leads to the further notion that logical analysis is a mere enumeration and listing of all conceivable qualities and relations.
The influence upon education of this conception has been very great. Every subject in the curriculum has passed through — or still remains in — what may be called the phase of anatomical or morphological method: the stage in which understanding the subject is thought to consist of multiplying distinctions of quality, form, relation, and so on, and attaching some name to each distinguished element. In normal growth, specific properties are emphasized and so individualized only when they serve to clear up a present difficulty. Only as they are involved in judging some specific situation is there any motive or use for analyses, i.e. for emphasis upon some element or relation as peculiarly significant. [emphasis added]- from How We Think
Dewey’s point is clear even if the writing is dense: so-called analysis of things into bits for the purpose of learning the whole has no basis in cognitive psychology or epistemology. The whole is not the sum of the easiest to describe bits. Indeed, as he says just after, it is a case of “putting the cart before the horse” when we ask kids to learn parts before seeing or working with the whole. Distinctions are usefully made when we need them in the service of understanding, in the service of working with the whole. (Think soccer: whole/part/whole, every day). Learning an endless array of distinctions and their names prior to (or, worse, in place of) encountering the whole and the interesting problems that then require analysis yields no meaning and merely verbal knowledge.
To put it graphically, this is how driver’s education would look if we followed such logic:
Mastery Learning projects made this mistake in droves in the 70s and 80s. The idea of backward design from competency was bastardized into a Learn-All-the-Bits, and we’ll call it mastery if you get over 80% on all the quizzes. The same thing is happening today in many projects, like RISC, that call themselves Competency-based. All these projects are is a march through endless micro-standards. In some projects, students cannot “advance” to the next “level” until and unless they test out on “interim assessments” of such knowledge of lots of little bits out of context. That’s not only dumb but immoral: lots of great performers might not have mastered some of the bits first. As I have long said, it is like not allowing a kid to play soccer until they have mastered 100 paper and pencil quizzes on each soccer bit. And my blog readers can easily understand my harangue against Algebra I courses as a textbook (!) example of such a mistake.
Related error: fixation on premature vocabulary technical vocabulary. A related problem, as Dewey noted in the above quote, is to assume that students need, first, to learn all sorts of technical vocabulary in learning the little bits. Here is a dreadful example from a middle school science book that we are working with as part of a curriculum-writing project for a client.
The book’s topic and title is Sound and Light. By page 10, Chapter 1, the following terms have been (needlessly) introduced to discuss waves: Transverse, mechanical, troughs, longitudinal, compressions, rarefactions. By page 12 they also add amplitude, wavelength, frequency. Middle School! The chapter ends with 3 formulas utterly out of context.
The chapter assessment? Recall the terms and plug in some data into the formulas, of course! No discussion of why we might be interested in waves; no discussion of the link to key physics questions, like the quest to understand light and sound; no discussion as to why or how one might use these distinctions or formulae to learn something interesting. Absurdly, all of this is introduced before any observations and experiments with waves. This is not only how waves are introduced to the student but how science is implicitly to be understood – as the picayune naming of bits of experience. Why would a young middle schooler become interested in science through such an introduction?
Or as Dewey famously described this mistake in Democracy and Education:
There is a strong temptation to assume that presenting subject matter in its perfected form provides a royal road to learning. What more natural than to suppose that the immature can be saved time and energy, and be protected from needless error by commencing where competent inquirers have left off? The outcome is written large in the history of education. Pupils begin their study . . . with texts in which the subject is organized into topics according to the order of the specialist. Technical concepts and their definitions are introduced at the outset. Laws are introduced at an early stage, with at best a few indications of the way in which they were arrived at. . . . The pupil learns symbols without the key to their meaning. He acquires a technical body of information without ability to trace its connections [to what] is familiar—often he acquires simply a vocabulary (p. 220).
So, please: let this be a warning to all course designers, curriculum writers, and (especially) textbook designers. The sum of the itty bitty parts is not a whole, ever. You need to understand that movement toward interest in and mastery of a complex whole requires designing backward from – and never losing sight of! – the complex whole and the interesting questions related to it. We do it right much of the time in soccer, immersion approaches to foreign language, art, and philosophy. Math, history, many science courses, and many foreign language courses get it hopelessly wrong – making the same mistake, yet again, that Dewey wrote about over 100 years ago. It’s way past time to avoid this unthinking error.
42 Responses
Reblogged this on Focus on Learning and Achievement and commented:
This is an outstanding article from Grant Wiggins about the thinking error (and terrible curriculum design) behind the common impulse to break standards apart. The sum of the parts in many cases is not greater than the whole!
I think we get way too caught up in the details when what really matters is the big picture. As learners we do not always understand everything during the beginning, middle, or end. How often do we suddenly “get an idea” in another topic or another class?
Isn’t the big idea to get students who are creative, who can make connections, and can apply what they are learning to help their lives?
Brilliant! Thank you! Reblogged this at http://notesfromnina.wordpress.com/
I agree completely with your assessment of the Common Core Standards document, abounding in microstandards as it is (including microstandards that, when you add them up, do not even give you a car – they are the random bits and pieces of various constructions, not necessarily compatible, that cannot even be re-assembled because they were never whole to begin with). So I find it hard to reconcile this entirely reasonable criticism of the Common Core Standards and all its endless microstandards with your contention in an earlier post (The Standards and creativity – compatible) that people who criticize the Common Core are simply proclaiming their own failure to be creative. Standards in general, and microstandards in particular, do indeed restrict creativity. WWJDD? What would John Dewey do? I imagine he would uphold the integrity of his teaching convictions and reject the CC Standards rather than accepting their constraints… as if we needed yet more challenges to overcome in our efforts to develop really creative learning environments that serve our students best. There are plenty of challenges that creative teachers face without having to shoulder the burden of the Common Core and its microstandards as well.
Laura, I think you totally missed the point of this article. I read it as a warning against breaking down the Common Core Standards into little bits and pieces since the CC Standards are meant to be a coherent whole. Reread just the first couple paragraphs with this in mind.
P.S. AS Bill McCallum said, read Grant Wiggins “spirited defense of the standards”
http://grantwiggins.wordpress.com/2013/05/01/the-common-core-standards-a-defense/
The problem is that they do come with the substandards, and when things are “aligned” to the Common Core, this can easily happen at the level of the substandards, resulting in an incoherent mess. As a writing teacher, I am esp. disappointed by the incoherent way that the substandards have been written for narrative writing – better to have no substandards at all than such gobbledy-gook. The standards would be far better off IMHO without the “itty-bitty” parts and I feel that the overwhelming number of itty-bitty parts is going to so badly distracting that what value there might have been in the standards will vanish. I’m working on a tutorial project right now where the funding agency wants the contents keyed to the Common Core – and by that they mean the itty-bitty substandards, e.g.
CCSS.ELA-Literacy.L.3.1b Form and use regular and irregular plural nouns.
I’m with you on the sub-standards but I can tell you, from lots of personal experience with standards projects in 3 states, that teachers demand the specificity of them; they complain that having only the highest-level standard is too vague. SIGH. It’s clear to me what the ELA Anchor Standards demand, and one presumes that a half-decent supervisor/writer of curriculum could figure out what to do in each grade level – apparently not, alas. It could be worse: many of the state sub-standards were absurd (e.g. TX, MS, GA)
We all read with different “lenses.” When I first read the progressions, I saw “proportional” and immediately thought, “Oh, here we go again with cross products.” It was a long time before I realized that there is a difference between “proportional reasoning” and “setting up and solving a proportion.” So it is with the Standards. It will be awhile before the beauty becomes apparent to the mathematics community. The important thing is that we leave our egos at the door as the discussion continues.
The math standards are definitely intended to be used as a coherent whole, and it would be a travesty (will be, because it will happen, I am sure) to break them down into a list of subtopics. It seems to me that the point of the math common core standards is made abundantly clear in those little grey boxes at the top of EVERY page. No chance is taken that they could be missed. Problem solving, modeling, reasoning etc. These are the focus, and should be the thread running through every lesson, every day, no exceptions, no matter what the topic.
Bravo! Yay!! Thank you!!
Grant, Bravo!! Rigor, relevance and relationships are the three R’s here. You can’t have relevance, let alone substantive rigor, with fragmentation. Thank you for your post!
BReaking math into tiny tidbits is one way to have students “get grades” regardless of what they’ve learned. THat’s awfully tempting 🙁
The motive cannot be discounted, but that doesn’t explain how dreadful the textbooks are.
But the textbooks are marketing to teachers, administrators, school board members, etc. who more often than not, want a seemingly “easy” step-by-step manual and assessments (i.e. broken down tidbits) to march through in order to “get grades.” And so the dreadful textbooks live on…
That’s why it is imperative that there be –
1. Textbook adoption criteria and protocols with teeth
2. Mission statements for each course
3. syllabi for each course in which it is clear what the text can and cannot contribute to the course, and that the syllabus cannot be just the book table of contents.
I couldn’t agree more with what you’re saying. I get somewhat caught in the trap of standards by using Standards Based Grading. I’m quite cognizant of reductionism by breaking everything into sub-tasks. We want to achieve that balance of having clear learning intentions and the ability to assess them, with bringing them forward via essential questions and big picture ideas.
Recently in physics I put together a unit in electrostatics using what (I think) I learned from UdD. While this helped with context and tied the sub-topics together, I’m not sure how successful I was with the big picture. The next positive step would be to approach the topic with interesting over-arching problems. I think this is the most difficult aspect but maybe also the one that offers the greatest rewards.
Another approach that may be worth trying in sciences is to learn through engineering. For example, a traditional physics course could be replaced with engineering physics, where the same learning outcomes are expected in both. However, whereas a physics course may fall prey to a march through textbook chapters, an engineering course would cover the outcomes via integrated projects. It’s like the image in your post: a student can learn about all the parts in a car from standards and textbooks; or a student can learn by rebuilding a car. If you had to get your car repaired, which student would you choose to do the work?
I think your instinct is sound – engineering problems, overarching physics problems are key. Think of the original physicists: Galileo doing ball down inclined planes and pendular; Newton looking at the movement of the planets. The point was to make sense of complex phenomena. It was all about the problem of motion (or forces generally) that needed an explanation. De-bugging a roller coaster or catapult would be the equivalent of auto mechanics… Thats the way to go, I believe.
I am a bit concerned about the “way to go” backslapping attitude related to this post. Indeed, the big picture is important, but not all kids are going to get there. Sorry, but true. So, what are the steps along the way that must be mastered? And where do I know a student falls short if I don’t understand the steps to the final goal. Granted, one doesn’t need to be able to define words to carry out tasks, but the teacher does. Often, teachers look at a standard and say, “Yep, I do that.” When I hear that kind of response in a struggling school, I ask about the standard’s particulars (i.e. “what does analyze mean?” “what makes a sonnet different from any other poem?”); far too often, the room falls silent.
Microknowledge of displinary learning does not need to be the center of instructional attention, but must be incorporated as a focus for planning for instruction. Please, correct me if I’m wrong here…hasn’t your design always asked that Stage 1 of planning idicate “Desired Results” which not only delineates what students will know but also, what students will be able to do. These outcomes may naturally occur in varied orders–I may in fact be able to do something before I know why or how it works. For instance, I may be able to bake a cake from scratch without knowing what ingredients make that cake rise. However, if the among my learning plan is an outcome for knowing the leavening powers of soda, salt, buttermilk, eggs, and/or the incorporation of air during the beating process, then by the end of the learning, am I fragmenting instruction by expecting students to explain the process? On the other hand, will I have students that will be able to produce the product but never explain how it happens? That is a rhetorical question.
I think you miss my point. These microstandards are NOT steps to the final goal; they are unmoored from a final complex performance goal, serving as arbitrary easy-to-measure “steps” that have no intellectual validity in far too many cases. We simply do not need arbitrary grade-level sub-standards in 5th grade when the anchor standards make crystal clear what the goal is in ELA, for example. Look at the differences between 6th, 7th, and 8th grade: arbitrary. I have seen this first hand: groups come up with arbitrary additions of adjectives and adverbs to distinguish grade-levels.
That’s precisely why we would never expect a single goal in STAGE 1 or a single assessment in STAGE 2 to suffice as evidence. Of course we must develop pathways toward a complex goal; of course we can expect varying levels of achievement. But that’s precisely the point: many of these microstandard schemes propose only a single pathway and a depressingly limited view as to what constitutes interim assessment.
My concern with cake baking is that too many students have no idea what ingredients to modify if they want a different outcome or understand how the outcome will differ when a few ingredients are changed.
I totally agree with your post. I have used the metaphor of teaching students about a greek vase. Would we smash the vase into tiny bits, then ask them to study each small fragment, before we finally show them the complexity and beauty of the whole? I think as educators we get scared of “throwing kids into the deep end” by giving them larger and more complex problems and ideas. We are scared because we see how incompetent students are at even simple tasks. Our hope is that enough practice of simple tasks will build to ability in a complex world. However, it really isn’t like that. Only practice with complexity can prepare for complexity. Of course we can still provide students with concepts and vocabulary when we see they need the term/formula to assist in their studies. http://cytochromec.net/blog/2009/11/in-defense-of-facts/
Rock on, Grant. Keep fighting the good fight.
Reblogged this on … Not the Principal's Office!!! and commented:
Are you still unpacking standards? Are you writing new curriculum this summer? If you are, read this.
Thanks, Grant – always enjoy reading your harangues [or many just ‘lectures’] – even though I am in Canada and am primarily and ELA teacher, I can relate to the ‘breaking into pieces’ problem. Teaching writing has faced similar ‘challenges’ – Teachers would say that students could not write stories if they could not form correct sentences and other such nonsense. Did I use the past tense? Ha-ha- some teachers are still saying that!
Marcy Emberger
Great article. Your thoughts are very much aligned with the motivation for my ‘Science Discovery’ curriculum I’ll be rolling out on Khan Academy. It’s based on ditching all technology in the classroom and going back to observations with historical materials. I posted a rough demo here:
https://www.khanacademy.org/science/projects/discoveries/batteries/v/zinc-copper-cell–reduction-oxidation
I can’t figure out why people think we need to teach ancient ideas (such as electricity & magnetism) using 19th/20th Century formulas and theory. Ohm’s law is the END of the story, not the beginning. I think science class needs to begin with basic observations, play and experiments driven by students. If you give a student V=IR with no context then you can be sure they have no practical understanding of electricity or batteries.
In my view students should have to build all their equipment (such as volt meters and ammeter using compasses and wire), no modern technology should ever be presented in a lesson until the student has independently realized why/how it works. (who says you need batteries to teach electromagnetism when you have coins?)
Even a compass…how many people can make a compass from scratch and know which tip of the needle will point north? Few I wager…
I am completely with you: learn to make your own tools, use them, make discoveries… very cool. I did it in physics with inclined planes and pendula in school.
The trick, of course, is to slowly build to the more advanced thinking in deliberate investigations and teachable moments. Ideally, the course builds in, at each stepping stone point, a new tool, be that tool mathematical or practical. So, for example, we wonder what light is, we build a diffraction grating to find out – hmmm, it looks wave-like, etc….
You may have already looked, but the original PSSC and BSSC materials post-Sputnik, influenced by Bruner and others, tried to do that. So do programs like ChemCom.
I have 15 hours of listening to CCSS authors-Coleman and more.. He/they too warn of the “flat list” and the urge to do twigs/branches/bits. My middle school is on the bits road and I cant move them. The SUP in my district said, as I told her, “I have 15 hours listening to CCSS authors, who else does?” Her reply, ” Who would want to?” Well my dear, someone who wants to understand the INTENT. The early deep discussion never happened. Superficial understanding prevails,with too little time to do it right.I remember the man at the hardware store looking at the broken, and rather expensive sander. “Looks like you have ditch diggers working for you.” Ray
Ditch diggers, indeed. With colleagues like that, who needs enemies? Sorry to hear it. There are more vigorous and enlightened people out there; I have met and worked with them. Maybe you need new colleagues?
Grant, terrific analysis.
A curriculum made from splintered standards is the very definition of “mile wide-inch deep” curriculum. This is the very problem the CCSS tried to help solve. The splintering pathology has always been the cause of the problem, now it is doing the same again.
Why is it so hard to get people to understand this? I think many people, when they are frightened by big responsibility, want to break their responsibility down into a list of little responsibilities.
Another monstrous danger I see from these splintered:
the digital tagging of apps, software, adaptive systems to the splinters of standards means that one can create an “aligned” curriculum merely by selecting anything from the ecosystem with the splinter-tag and checking that standard off; when the standards all have splinters, the curriculum is ‘aligned’.
This trail of splinters is no building of knowledge, let alone the development of expertise in thinking. It will be a mile wide and an inch deep.
Thanks for raising your voice on this.
Phil: great to hear from you! It’s been too long since those days of CAP and CLAS…
I honestly puzzle over this on a regular basis. (And I thought your vase analogy was perfect, too). Interestingly,the lament goes back to Hegel, Kant, Bacon, and Comenius, back to Plato’s Allegory of the Cave – so, clearly this is an intellectual fallacy with some staying power. And your fear about the software apps is right on: we had serious disagreements with folks promoting such software in the NYC iZone project.
I fear the math standards will not avoid this problem, as i wrote in my Ed Week Commentary last year. Some of the so-called standards are awfully discrete, and there is far too little in the document about how to weave practice and content standards together with some DO and DON’T cautions. Not sure if you saw my Algebra course blog post rant but that sums it all up for me.
None the less, we shall keep the faith and keep tackling this. Thanks for all your good work over the years.
I very much appreciate this discussion – so thank you, first of all. I’m a little confused about how the practice of posting “daily learning targets” figures in to the idea of granularity. As Phil argues in his great talk on Planning Chapters, Not Lessons (http://serpmedia.org/daro-talks/), mathematics is best understood at the grain size of a chapter – not a lesson. So when I post a daily target on the board – what I expect the student to be able to know or do by the end of the lesson – am I running the risk of splintering the chapter into tiny pieces that detract from the coherence of the chapter? How do I avoid that? And if it needs to be avoided, how do I approach this with my administrators who walk through our classrooms looking for these daily targets and expect us to be referring to them throughout the lesson?
Great question. Let me answer it with an athletic analogy. The goal is winning soccer. Today we are working on give-and-go in a 2 v 1 setup. The give-and-go is 1 move that is part of winning soccer – in part because it is a sub-set of 2 big ideas: fake the opposition and create space. At the end of the drill we’ll scrimmage and try to use more give-and-go in the game context. In short: small-grain-size ‘move’ in the service of 2 clear ideas, linked to our key performance goal; whole/part/whole. This is an analogue for working, say, on one of Polya’s questions in the service of non-routine problem solving – Do you know a related problem? And we could use that question when working with the distance formula (for those times when you can use the pythagorean theorem) in the chapter on graphing linear relationships.
As my example suggests, I think part of the issue is that most textbook ‘chapters’ have nothing to do with problem solving; they are just content bits organized under headings. So, it is wise, I think to do just what Tyler advised 60 years ago: plan by making a matrix of content and process, in this case Polya questions. (If you think about it, this is like planning as an English teacher: meshing texts with work in reading and writing).
I’ll have more to say on this when I finally honor my commitment to sketch out the ideal algebra course.
Before we judge the hearts and brains of bitmasters, we need to consider how many high school math teachers (who should be leading the way toward improvement), have over 100 students; many almost 200. We try to provide clear instruction, perform quality assessment and provide personal feedback. I liken the task to being in a third world country when a natural disaster hits. Would we run home and study best-practices in emergency response or would we do the best we could at the moment with CPR and bandages?
A large number of students does not logically lead to the outcome of itty bitty sub-standards with little validity – this simply does not follow. For one thing, I have seen this mistake in private schools with classes of 12, just as I have seen college courses of 200 that fail to make this mistake. Best practice is best practice, period: people need worthy tasks and teaching that prepares them for them – period. By your argument the orchestra conductor would give up on performances.
Do you remember a visit you made to a small group of item writers in Napa, CA a few decades ago? I do. We talked then of making performance assessment items practical, cross-curricular, measurable, and achievable by all students. I sure hope we get a little closer this time than last time…
I do! Great food, wine, and conversation.
This approach absolutely defines mathematics education almost everywhere. Break everything down into sections, teach y-intercepts today, x-intercepts tomorrow, graphs another day, rewriting equations in a different form on another different day. At the end of it all give a test with two or three questions about each subtopic and move on. It’s for this reason that Algebra, in this most common form, has little relevance to life, and the only students it serves well are the ones who are already able to think mathematically, reason and solve problems before they walk in the classroom door at the start of the year. For the students who need support and experiences which lead to developing skills for solving realistic, complex problems; students who don’t automatically tend to make connections between topics unless the course is designed to support this type of thinking, our mathematics classes fail miserably. Students seem to finish each year of math education at the same level that they began with regard to their ability to problem solve and make connections – to think mathematically. They may have learned new discrete skills, but with little increase in their understanding of how to understand the overall mathematics.
How do we change this? The Common Core Standards may have its heart in the right place, but it’s true that they offer very little to assist the well intentioned, hard working teacher who honestly has no idea how to effectively teach students to improve their problem solving skills. Many teachers strive and intend to teach students to think mathematically, but find it to be an extremely difficult task to accomplish. It’s not for lack of trying, but it is ineffective.
As a math teacher my greatest success has come in dramatically reducing the number of examples considered during each 50 minute class. A class which focuses on a single quadratic function, first asking students what observations can be made without doing any calculating or writing (y-intercept, direction the parabola will open, rough sketch based on these two facts), then moving on to finding x-intercepts (completing the square, factoring, quadratic formula, possibly more than one method can be used), locating the vertex and axis of symmetry (either via completing the square or by considering the locations of the x-intercepts), then drawing a graph.
This may seem like many concepts, and it is. This approach might be used with 3 different functions, each one used on a single class day, and the first day might not include all of the listed topics. But the point is to focus on only ONE function. As the students gain confidence, and a sense of how all the parts work together, the picture grows and becomes more complete. It begins to come into focus. The list above might be what the class looks like after 3 or 4 days of using this approach, and may also depend on past instruction on factoring. A day on the mechanics of completing the square would be critical. Each of these skills is then placed in context on the days spent looking at ONLY a single function for a whole class period.
It is also critical that scenarios be provided, at least by the 3rd or 4th day of using this “one function a day” approach, which can be used to overlay words and meaning: “The y-intercept at (0,5) represents the fact that the ball was thrown from a height of 5 feet,” or “The x-intercept at (2.3,0) represents the fact that the ball hit the ground after 2.3 seconds.” We must move beyond word problems as a description and some data with a single question and answer. Students should be examining whole scenarios, looking at all the aspects, modeling them with functions and graphs and connecting y-intercept, x-intercepts, vertex, domain and range all to the role they play in clarifying the scenario mathematically. Again, one scenario, with one function, per day, please. More than that means the students have not had the opportunity to make connections and reach a depth of understanding.
The main idea is to present ideas and skills in context, both mathematical context and also real world context, and to keep the big picture as the constant theme and focus. Less is more. Slow down, observe, think, make connections.
I agree with your comment that “many teachers strive and intend to teach students to think mathematically, but find it to be an extremely difficult task to accomplish. It’s not for lack of trying, but it is ineffective.” In my work with teachers as an instructional math coach, I have found that there are multiple reasons why the hard work of some teachers is ineffective.
One of the reasons is that teachers lack quality resources to use for instruction. Another is that even when they have quality resources, they don’t know how to use them effectively. In my experience, very few textbooks offer what is needed which is why the publisher’s criteria is necessary. The textbook that is used primarily in the classroom is a major influence in how a student is presented and studies mathematics. If we address the first issue and provide teachers with quality materials (not to be confused with quantity of materials) then more time can be used to focus on the use of the materials. I am thinking of the Mathematical Tasks Framework (Stein, Smith, Henningsen, and Silver, 2000) in which the first part is the task as presented in the curriculum materials and the second is how the task is set up by teachers. If the first part improves then it is more likely that the second part will improve.
From a parent’s perspective as to the textbook issues… I am very concerned about the way we choose textbooks and agree with this comment. For example, my son brought home his world history textbook and asked us what the Crusades were. According to the textbook they were, “A series of trade missions.” and that was all. Really? It’s mind boggling. A good teacher can expand on that. However, why should we have to depend on the teacher to correct their district and state mandated reading material? We pay a high price for these textbooks. The ought to be informative and a good resource to help the teacher- not something they are stuck with.
This, of course, a big-deal issue for those of us trying to get curriculum written around worthy outcomes instead of a march through a single textbook. I made this point Saturday at the ASCD Conference on teaching and learning: if you are teaching social studies or history it is simply negligent to use only one “official” source of history. (I showed excerpts from British and Russian accounts of the American Revolution to make the point). That’s why it is a mantra in UbD training that the text is one resource, not the curriculum.
Reblogged this on 21st Century Science and commented:
Been thinking about this as a pitfall to avoid as we work on new curriculum for the Next Generation Science Standards.