Learning theories such as behaviourism, Piagetian theories and cognitive psychology, have been dominant influences in education this century. This article discusses and supports the recent claim that Constructivism is an alternative paradigm, that has rich and significant consequences for mathematics education. In the United States there is a growing body of published research that claims to demonstrate the distinct nature of the implications of this view. There are, however, many critics who maintain that this is not the case, and that the research is within the current paradigm of cognitive psychology. The nature and tone of the dispute certainly at times appears to describe a paradigm shift in the Kuhnian model. In an attempt to analyse the meaning of Constructivism as a learning theory, and its implications for mathematics education, the use of the term by the intuitionist philosophers of mathematics is compared and contrasted. In particular, it is proposed that Constructivism in learning theory does not bring with it the same ontological commitment as the Intuitionists' use of the term, and that it is in fact a relativist thesis. Some of the potential consequences for the teaching of mathematics of a relativist view of mathematical knowledge are discussed here.
What is true is that the Common Core Standards were hi-jacked by constructivists from the start. While the content of the CCSS matches traditional mathematics fairly closely, the emphasis from the beginning on the Standards of Mathematical Practices portion of the standards. The practices are nothing more than a continuation of the 1989 NCTM practices and principles which have been disastrous for math education in the United States.
Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching
Read the Study: http://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_1
The irony is that the CCSS have been pushed as standards that will unite the nation in math education. However, the emphasis on the nebulous and non-measurable standards of mathematical practice ensure that every locality will do their own thing in their own way. The bottom line is that the CCSS will be a total bust and will not accomplish any of the claimed goals. They will not lead to higher achievement but instead will continue the same nonsense that has lead to the present chaos.
Additionally, the CCSS are built on the notion that all students will become foundationally sufficient and therefore will succeed at the higher levels. This is just as ridiculous as “No Child Left Behind” claiming that 100% of students will reach proficiency. The truth is, huge percentages of students fall further behind grade level every year and by the time they reach high school they are nearly DOA. As long as we have a system that requires all students to become proficient in Algebra and Geometry but passes students on via social promotion regardless of their achievement we will never a system that works.
Mathematics Education Constructivism in the Classroom
The Forum's Internet Mathematics Library provides a page of links to sites on Constructivism. A few selections are offered below, and we also offer a page of selected Internet sites with a constructivist orientation or that offer readings on constructivism.
What is Constructivism?
"Students need to construct their own understanding of each mathematical concept, so that the primary role of teaching is not to lecture, explain, or otherwise attempt to 'transfer' mathematical knowledge, but to create situations for students that will foster their making the necessary mental constructions. A critical aspect of the approach is a decomposition of each mathematical concept into developmental steps following a Piagetian theory of knowledge based on observation of, and interviews with, students as they attempt to learn a concept."- Calculus, Concepts, Computers, and Cooperative Learning (C4L)
It's not surprising that constructivism has a strong voice in the current dialogue on math education. Many are concerned about the success - or lack of success - of math education. Constructivism cuts a nice path between the main ideas that have influenced how math has been taught: the concept of math as facts to be transmitted to the student, and the view that some people have it and some people don't, where the educator's task is to figure out how "smart" students are and choose the right tasks for them to perform. Questions remain, however, about whether these offer rich information for developing different ways of teaching. And what's to be done for the students who aren't succeeding? In contrast, constructivism focuses our attention on how people learn. It suggests that math knowledge results from people forming models in response to the questions and challenges that come from actively engaging math problems and environments - not from simply taking in information, nor as merely the blossoming of an innate gift. The challenge in teaching is to create experiences that engage the student and support his or her own explanation, evaluation, communication, and application of the mathematical models needed to make sense of these experiences. Given this view, there are many approaches to improving teaching: look for different ways to engage individual students, develop rich environments for exploration, prepare coherent problem sets and challenges that focus the model building effort, elicit and communicate student perceptions and interpretations, and so on. We'd like to explore here the theory and applications of constructivism in math education. We invite you to submit your favorite readings, projects, and classroom materials that either point out the pitfalls or demonstrate the opportunities of this theoretical framework.
Constructivism in Math Education
Applications and Misapplications of Cognitive Psychology to Math Education - Anderson, Reder, Simon Constructivism and Education: A Shopper's Guide - Moses A. Boudourides Constructivism - Classroom Compass Constructivism in Science and Mathematics Education - Michael R. Matthews Constructivism: Vygotsky and the Internet - Mathematics Education at The Math Forum Constructivist Mathematics and Unicorns - Lee V. Stiff Constructivist Views of Learning in Science and Mathematics (ERIC Digest) - Drew K. Ishii Essays on Constructivism and Education - Tom O'Haver; Maryland Collaborative for Teacher Preparation Kinds of Constructivism: An Annotated Bibliography - Annie & John Selden; The Math Forum Mathwire.com - Terry Kawas http://mathforum.org/mathed/constructivism.html
Constructivism v. Instruction’
Math Teacher Stephanie Sawyer was quoted on Diane Ravitch’s website saying the following about Common Core:
“…They pay lip service to actually practicing standard algorithms.
Seriously, students don’t have to be fluent in addition and subtraction with the standard algorithms until 4th grade?
I teach high school math. I took a break to work in the private sector from 2002 to 2009. Since my return, I have been stunned by my students’ lack of basic skills. How can I teach algebra 2 students about rational expressions when they can’t even deal with fractions with numbers?
Please don’t tell me this is a result of the rote learning that goes on in grade- and middle-school math classes, because I’m pretty sure that’s not what is happening at all. If that were true, I would have a room full of students who could divide fractions. But for some reason, most of them can’t, and don’t even know where to start.
I find it fascinating that students who have been looking at fractions from 3rd grade through 8th grade still can’t actually do anything with them. Yet I can ask adults over 35 how to add fractions and most can tell me. And do it. And I’m fairly certain they get the concept. There is something to be said for “traditional” methods and curriculum when looked at from this perspective.
Grade schools have been using Everyday Math and other incarnations for a good 5 to 10 years now, even more in some parts of the country. These are kids who have been taught the concept way before the algorithm, which is basically what the Common Core seems to promote. I have a 4th grade son who attends a school using Everyday Math. Luckily, he’s sharp enough to overcome the deficits inherent in the program. When asked to convert 568 inches to feet, he told me he needed to divide by 12, since he had to split the 568 into groups of 12. Yippee. He gets the concept. So I said to him, well, do it already! He explained that he couldn’t, since he only knew up to 12 times 12. But he did, after 7 agonizing minutes of developing his own iterated-subtraction-while-tallying system, tell me that 568 inches was 47 feet, 4 inches. Well, he got it right. But to be honest, I was mad; he could’ve done in a minute what ended up taking 7. And he already got the concept, since he knew he had to divide; he just needed to know how to actually do it. From my reading of the common core, that’s a great story. I can’t say I feel the same.
If Everyday Math and similar programs are what is in store for implementing the common core standards for math, then I think we will continue to see an increase in remedial math instruction in high schools and colleges. Or at least an increase in the clientele of the private tutoring centers, which do teach basic math skills.”
http://whatiscommoncore.wordpress.com/tag/constructivism-v-instruction/
Let me start by saying that I share most of Jay Greene’s reservations about the Common Core State Standards. Over the past couple of years, I have had the opportunity to discuss these concerns with many Common Core supporters. Although I typically disagree with their conclusions or their logic, I believe Common Core supporters are for the most part sincere in their belief that these standards are rigorous and will improve outcomes for students. However, I find claims that the Common Core State Standards will not influence instructional practices downright disingenuous and obviously false.
In a recent Twitter exchange, the Missouri Department of Elementary and Secondary Education informed me that the CCSS don’t “tell teachers how to teach.” This is a phrase that has been echoing across the country as the Common Core has come under attack from the left and the right.
The fact is that curriculum standards don’t tell teachers how to teach in the same way that a high jump bar doesn’t tell a jumper how to jump. You could theoretically jump over a high jump bar in whatever way you would like; but because of how the jump is structured there is a clear advantage to doing the old Fosbury Flop.
It is clear from documents on the Common Core website and from the discourse throughout the country that these new standards encourage constructivist teaching practices. Take for example these two quotes from a Key Points in Common Core Math document.
- The standards stress not only procedural skill but also conceptual understanding, to make sure students are learning and absorbing the critical information they need to succeed at higher levels ‐ rather than the current practices by which many students learn enough to get by on the next test, but forget it shortly thereafter, only to review again the following year.
- Having built a strong foundation K‐5, students can do hands on learning in geometry, algebra and probability and statistics. Students who have completed 7th grade and mastered the content and skills through the 7th grade will be well‐ prepared for algebra in grade 8.
I have written extensively about what constructivist teaching looked like in my child’s classroom, where students were supposed to discover how to solve math problems rather than learn to use standard algorithms. My kid’s school is not the exception, it seems to be the rule. Across the country schools are beginning to understand that the Common Core standards will require a more constructivist based form of instruction.
In California, teachers will be “encouraging critical thinking over memorization, focusing on collaboration and integrating technological advances in the classroom.” We are told that teachers “are attending workshops and training sessions to rethink the way they relay information to students.”
A Virginia newspaper reports, “Discovery, guided math, problem-based learning, project-based learning – call it what you like, it’s here.”
Even in Massachusetts, a state that had arguably better standards than the Common Core, teachers are moving more towards constructivist teaching practices. In the Wrentham School District, “the first and second grade math programs are already implementing new methods for teaching basic math skills that are designed to create deeper understanding of math among the students.” One teacher commented, “Our job as teachers is to guide through questioning.” If that doesn’t sound constructivist, then I don’t know what does.
I am aware that some non-constructivist based curricula, like Saxon Math, are aligning to the Common Core. They are doing so because they have to or they will be at a competitive disadvantage in the marketplace. It remains to be seen if these more traditional models will resist constructivist influences. Much of that depends on how the Common Core assessments are structured.
I am also aware that the standards do not dictate which pedagogical approach a teacher must take. Although, to me it feels a bit like when my mom used to say, “You can do what you want.” Which never really meant that I could do what I wanted.
The bottom line is that the Common Core State Standards are built on constructivist principles and are being implemented, by and large, by constructivist means. If supporters like constructivism, which I suspect most do, then they should just come out and say so. That is not such a difficult position to defend. But don’t attempt to tell me these standards won’t tell teachers how to teach.
No comments:
Post a Comment