The Singapore Math Primary Curriculum adopts a concrete-pictorial-abstract progressive approach to help pupils tackle seemingly difficult and challenging word problems. Mathematics Teachers in Singapore usually make use of concrete objects to allow students to make sense of the comparison concept by comparing two or more quantities. Once the pupils can visualize how much one quantity is greater than or smaller than another quantity, they will then move on to put these relationships in rectangular bars as pictorial representations of the math models concerned.
We can first give the child concrete objects, like 5 pencils and 3 erasers, and let the child put the two groups of objects side-by-side to match the 2 types of items, i. Then, he will be able to see that there are 2 more pencils which cannot be matched with any erasers because he has run out of erasers to do that.
Eventually, the equation can be visualised as a comparison between the 2 quantities given in the question and the pupils can easily see that to find the difference, they just need to subtract the smaller quantity from the larger quantity.
We quite properly expect older middle and high school students, never mind college students, to do extensive self-directed and inductive work in reading, writing, problem solving, and research because they are no longer novices at core academic skills.
Indeed, here is research with college science students that counter their argument. Indeed, later in the article, the authors strike a somewhat different pose about the complete repertoire of pedagogies needed by good teachers:.
Small group work and independent problems and projects can be effective — not as vehicles for making discoveries but as a means of practicing recently learned skills. Though this properly expands the list of effective instructional moves, their framing is odd — and telling.
These approaches have different aims, understanding-related aims, that are never addressed in their paper. Indeed, this is just how conceptual and strategic thinking for transfer must be developed to achieve understanding: through carefully designed experiences that ask students to bring to bear past experience on present work, to connect their experiences into understanding. As Eva Brann famously said about the seminar at St.
This shows the wisdom of instructional techniques that begin with lots of guidance and then fade that guidance as students gain mastery. It also shows the wisdom of using minimal guidance techniques to reinforce or practice previously learned material.
Well, which is it, then? When does a gradual-release-of-responsibility kick in? Willingham in fact concludes his article by questioning the very novice-expert sequence laid out by Clark, Kirschner, and Sweller when the goal is conceptual understanding. Somewhat more controversial is the relative emphasis that should be given to these two types of knowledge, and the order in which students should learn them. Perhaps with sufficient practice and automaticity of algorithms, students will, with just a little support, gain a conceptual understanding of the procedures they have been executing.
Or perhaps with a solid conceptual under- standing, the procedures necessary to solve a problem will seem self-evident. There is some evidence to support both views. Conceptual knowledge sometimes seems to precede procedural knowledge or to influence its development. Then too, procedural knowledge can precede conceptual knowledge. A third point of view and today perhaps the most commonly accepted is that for most topics, it does not make sense to teach concepts first or to teach procedures first; both should be taught in concert.
As students incrementally gain knowledge and understanding of one, that knowledge supports comprehension of the other. Indeed, this stance seems like common sense. Sequence in learning is not at all settled, as Clark et al profess, when the aim is understanding as opposed to basic skills to be learned the first time. The key to understanding understanding: the ubiquity of persistent misunderstanding. Ultimately, a key lacuna in the everything-is-knowledge-through-direct-instruction view is its inability to adequately explain student misconceptions and transfer deficits that persist in the face of conventional direct teaching in science and mathematics.
A glaring weakness in the Clark, Kirschner, and Sweller paper is their one-sentence treatment of student misconceptions: they suggest that misconceptions are the likely result of allowing students to discover concepts and facts for themselves! This is surely a slanted view. There is a year history of research in science and math misconceptions that shows conclusively that traditional high school and college direct instruction leads unwittingly to persistent misconceptions, and that a more interactive concept-attainment approach works to overcome them.
Multiplication is not repeated addition. Presumably as a result of teachers not teaching for conceptual understanding and failing to think through the predictable misunderstandings that will inevitably arise when teaching novices the basics in simplified ways.
Teaching a concept as a fact simply does not work, as Willingham notes. The paradox of education. What these examples beautifully indicate is the paradox of teaching novices that so many knowledge-centric educators seem to overlook. Yes, we must simplify and scaffold the work for the novice and make direct instruction clear and enabling — but in so doing we invariably sow the seeds of misconceptions and inflexible knowledge if we do not also work to attain genuine understanding of what the basics do and do not mean.
In fact, a telling comment made by Barak Rosenshine , a leader in direct instruction, that DI has a more limited use than Clark et al acknowledge:. Rosenshine and Stevens concluded that across a number of studies, when effective teachers taught well-structured topics e. In fact, he notes that reading comprehension is a different kind of learning task than developing straightforward skills, and thus requires a different kind of direct instruction — instruction in cognitive strategies:.
DI is a method for learning and applying skills. Here we see the paradox, more clearly: no one can directly teach you to understand the meaning of a text any more than a concept can be taught as a fact. The teacher can only provide models, think-alouds, and scaffolding strategies that are practiced and debriefed, to help each learner make sense of text.
Otherwise we are left with the silly view that English is merely the learning of facts about each text taught by the teacher or that science labs are simply experiences designed to reinforce the lectures. Interestingly, in an interview Rosenshine seems a bit insensitive to the problem of inflexible knowledge in less able students who need to rely on initial scaffolds for a long time:. For example, in teaching writing there is a cognitive strategy called the five-paragraph essay.
The format for this essay suggests that students begin with an introductory paragraph containing a main idea supported by three points. These points are elaborated in the next three paragraphs, and then everything is summarized in the final paragraph. After describing a lesson on Macbeth in which the essay template and DI are used, Rosenshine says:.
The teacher told me he used this same approach with classes of varying abilities and had found that the students in the slower classes hung on to the five-step method and used it all the time. Students in the middle used the method some of the time and not others, while the brighter students expanded on it and went off on their own.
But in all cases, the five-step method served as a scaffold, as a temporary support while the students were developing their abilities.
Indeed, this is just the kind of scaffold for inferring a concept that lies at the heart of teaching for understanding: concept attainment and meaning-making via examples, non-examples, and guided inferences — mindful of prior learning experience and likely misunderstanding.
PS: Rosenshine offers a very different take on the issue that so motivated Clark, Kirschner, and Sweller, i. He laments our failure to pursue the pedagogical question of how novices become experts:. Rosenshine: One very promising area of teaching research has been to compare the knowledge structures of experts and novices. For example, the experts might be professors of physiology and the novices might be interns or graduate students.
Or the experts could be experienced lawyers and the novices were first-year lawyers. What the researchers consistently found was that the experts had more and better constructed knowledge structures and they had faster access to their background knowledge.
These findings occurred in diverse areas such as in chess, in cardiology, chemistry, and law. They also compared expert readers with poor readers and found that the expert readers used better strategies when they were given confusing passages to read. A lot of expert-novice research was done from the mids until about , but then it stopped.
But, unfortunately, the research was never used to develop an instructional package for training experts. It was never used to establish instructional goals for classes to teach all children to be like the experts.
A postscript to the initial critics of the post. No, I have NOT made a category mistake. Knowledge is necessary but not sufficient for understanding; understanding is not a direct function of knowledge.
Similarly, performance is more than the sum of skill; it requires judgment and strategy. All of us have experienced such contrasts. You explain them, then.
Susan said:. March 25, at pm. Our school has recently embraced the AIW framework to looking at teacher tasks, instruction and student work. Much of our interdisciplinary conversations involve talking about teaching about concepts, not topics. It appears that a subject like math can easily overlap those 2 words. March 26, at am. A concept is a model, theory, general principle — an idea, an inference that is used to explain and connect facts.
Mathematics contains many, some of which are crucial for understanding: congruence, equality, linear or non-linear relationship, function, derivative, imaginary number, etc.
A key concept in problem-solving: finding simpler equivalences. In short, many core concepts that kids often do not understand. So, the question becomes: what must be understood about these concepts for understanding to advance vs. What is often misunderstood about these concepts that impedes understanding?
You might also want to check out the article that Randy Charles wrote for the NCSM journal on big ideas in math about 5 years ago. Epigami said:. March 20, at am. Interesting article! It is indeed quite difficult to get concepts through to the younger minds, and educators need to find original ways to reach out to them.
The issue is that many academic systems around world will focus on pure theoretical exercise, which might attract some pupils, but others will be much less able to follow. Conceptual understanding has to do with intrinsic interest of the subject, not just the work put into it.
Keith said:. April 27, at pm. Again, thanks for the post. To be honest, I was one of the former before I became involved with the ASU modeling program where they gave me the FCI and exploited my own misconceptions. I have found that those teachers in the latter category that have been closed to understanding can be opened when they see the impact of strategies that elicit understanding from their students. To this end, I have also been trying to figure out how to elicit students understanding to a higher degree.
Of course, student discourse will naturally bring it out effectively. But on an independent level it was the use of your understanding rubric that really helped me clarify my expectations to students when they were writing independently.
Because so much emphasis has been placed on content acquisition students have been programed in schools to give vague, procedural descriptions based on content because that is all what knowing content requires.
Once I used an explanation rubric from the six facets of understanding that was framed in the context of a content based question, but with understanding expectations for their explanation, I was able to get the responses from students that showed understanding of concepts.
Showing these off a bit has, in turn, helped me advocate for the very point you are making in your blog to teachers in that latter group. Your policy is the best: do by modeling ironically. Atlas Educational said:. April 28, at am. Keith, like you, have a description rubric really helped me clarify how to assess understanding.
Knowledge and understanding are both necessary and with the dawn of the age of the internet, hopefully people will gain greater insight as to the differences between knowledge and understanding. Susan Clayton said:. April 26, at pm. The articles in this volume on math support your claims regarding teaching math for conceptual understanding.
Good examples for the concept of fractions and variables. Coverstone said:. Here is my best first attempt at writing a test that attempts to address the conceptual understanding of a topic. Comments are appreciated. April 27, at am. I agree that making it practical and more test-like is the next step just from having read your intro to your attempt. As I noted in other comments, my examples were meant to be suggestive of issues — why does seomtthing work?
What are common misconceptions? Max Ray maxmathforum said:. April 25, at pm. April 26, at am. April 24, at pm. Reblogged this on principalaim and commented: If you are following principalaim, I have shared some of my favorite educators, innovators, and creative thinkers. Among my favorites is Grant Wiggins, co-author of Understanding by Design. Wiggins, an authority on subjects dealing with assessment, student engagement, and the Common Core, he is constantly being sought out to answer many of the toughest question concerning the best practices in education.
Ultimately, it is critical that we understand how best to prepare our students to become mathematical thinkers, independent, and innovative creators. Kevin Hall said:. Humans have very high limits on how much information they can process simultaneously from their long-term memory, and very low limits for how much they can process from working memory. April 25, at am. But by hs every student should be able to read and write, so the cognitive load is diminished on those core skills.
The 7-item rule about memory is also being misused. We have learned to chunk those items in reading and writing after many years of doing it. By your argument, no one should be bale to understand complex text when they read independently. Again, their argument is surely a stretch. For example, the biggest example they describe in the paper is medical students being trained to make diagnoses in a lab setting with minimal guidance—a task with little reading involved.
Their argument is that the costs of inquiry instruction come from the way the student must search the scenario for the relevant information while simultaneously figuring out how to put that information together.
What information is extraneous? I find that Cuisenaire rods are an excellent tool for showing this as well. We line them up with one on top so you can see the number that is larger and then we find the rod that makes the difference. For those firsties ready for it, we bridge the ten to help by figuring out the distance from each to ten and add those two distances together.
That helps them understand what they need to look for. It is a very hard concept in first! Your email address will not be published. Submit Comment. How Many More? Comparison Subtraction. Written by Donna Boucher Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator.
Donna is also the co-author of Guided Math Workshop. Freebies Grades K-2 Operations. You might also like Amy B on November 16, at pm. LOVE this!!! Donna Boucher on November 17, at pm. Hi, Amy! My boy is home from college for a week…life is good.
Sandi on November 17, at pm. Literacy Minute Reply. Donna Boucher on November 18, at am. My pleasure, Sandi! Su Anna on November 17, at pm. A visual always helps! Donna Boucher on November 19, at am. Anonymous on January 26, at am. Thanks ahead of time Amy Terry Reply. Donna Boucher on January 30, at am. Teacher and Life Long Learner on January 29, at pm.
Oh, good!! Anonymous on September 12, at am.
0コメント