As a frequent blogger about education, I got to thinking about blogging for education. A blog is an ideal way for students to converse, share ideas, and provide each other with feedback. A class can share a blog, writing posts back and forth, or students can each be assigned to write their own throughout the year and to comment on each others’. Often, teachers tend to think of blogging as a medium for English or humanities classes to practice writing skills and critical thinking. But maybe students can benefit from math-focused blogs as well.
An article from the August 2010 issue of THE Journal recounts the story of a math teacher who suggested that one struggling student try asking for help from her peers through the school’s learning management system. Online, the girl, who had felt too embarrassed to participate in class, found it much easier to communicate and ask for clarification on algebra concepts. She even managed to build up her confidence in the classroom.
Why shouldn’t students be able to consult each other from home to share strategies and ideas for problem solving? In the humanities, students are regularly asked to critique each others’ work because collaboration is constructive. Since technology is such an engaging tool, students would probably relish the opportunity to consult with classmates online about challenging math concepts and problems. How have you integrated technology and web-based discussion into your math classroom?
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I was a reserved but enthusiastic nine-year-old when my parents decided to send me to a private all-girls school rather than back to the local public school. We had strong public schools in our town, and I had always liked the elementary school I attended. I suspect their primary motivation was to increase my self-confidence and outspokenness in the classroom in the absence of boys.
There is ongoing debate about the merits and drawbacks of single-gender education, especially as it enters the realm of public schools. The Pittsburgh Post-Gazette reports that the Pittsburgh Public Schools plan to open two single-gender high school programs this coming year, but the initiative faces strong opposition. Single-gender education can offer a more focused classroom environment free of the social pressures and distractions provided by the opposite sex. On the other hand, a single-sex classroom is not a real-world environment, and students may miss out on learning some important social skills.
In an educational system in which students are so often categorized—by RTI tier, income level, race, English proficiency, and now gender—differentiating instruction and closing the achievement gap clearly take precedence. But is classifying students really that simple? Perhaps each student’s needs are more complex and personal than these categories reveal. While single-gender education certainly has its benefits (and I, for one, loved my experience), it offers only one remedy for our struggling students. How can we provide individualized attention and support necessary in order to help every student succeed?
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As a Massachusetts resident, I have been watching the heated debate over the adoption of the Common Core State Standards in a state that prides itself on its consistently strong performance on national assessments. On Wednesday, Massachusetts education officials voted 9-0 to adopt the new national standards (see Wednesday’s Boston Globe article). With so many states suddenly sharing a core curriculum, a common assessment seems bound to follow.
It looks as though the United States is heading toward a much more unified education system. On the forefront of this movement is MetaMetrics with their Lexile and Quantile Frameworks for reading comprehension and mathematics, respectively. These measurement frameworks use a developmental scale, revealing both what students know and how their achievement compares to their peers’. They are ideal for use on a broad scale because schools, districts, and states can easily compare their results.
Already several states have begun using Quantiles to score their state assessments, and our newest program, Scholastic Math Inventory, uses the Quantile Framework to report student scores. The program, released today, is a fast and accurate computer adaptive math assessment system that tests student achievement by adjusting to student performance. It provides immediate, actionable reports to help inform instruction, so teachers can spend more time making sure their students are mastering the concepts and skills included in the Common Core Standards (or state standards).
What are your thoughts on the Common Core? For an interesting debate, check out http://www.nytimes.com/roomfordebate/2010/7/21/who-will-benefit-from-national-education-standards?ref=education.
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A recent Newsweek report addresses the apparent decrease in creativity among American students over the past twenty years. The growth of standardized testing as a measure of achievement and the implementation of core curriculums have left teachers with little flexibility or time to incorporate creative thinking into their classrooms. However, studies have shown that the correlation between lifetime creative accomplishment and creativity is three times stronger than its correlation with IQ.
The Torrance Test (much like an IQ test for creativity), developed by Professor E. Paul Torrance in 1958, measures a person’s “creativity quotient,” and those who perform well on the test have proven to be both successful and accomplished in later life. Although U.S. scores on the Torrance test were increasing through about 1990, they have decreased steadily since then.
There is no doubt that students must still master the skills and facts that our current curriculum emphasizes, but perhaps the manner of teaching the material could be more creatively focused. Fifth graders at a public school in Akron, Ohio were challenged to find a way to reduce noise in their library and to present proposals outlining their strategies. Students performed research, investigating methods for masking disruptive noise as well as the best materials for blocking sound. They also researched the challenges and costs associated with these ideas. Teachers helped guide them in the process, providing instruction and feedback along the way. Students became totally engaged in the project because they were in charge of their own learning. And furthermore, the school performed very well on the state standardized assessment, scoring among the top three schools in Akron.
Creativity requires both divergent and convergent thinking as students generate a variety of ideas and then combine them to find the optimal solution. However, skill-based learning is not necessarily excluded from creativity-focused models like the Akron project. Students are required to test and develop their ideas, and in doing so, master topics required by state and national curricula.
So how can we foster creativity while still meeting standards that focus on the mastery of many concepts and skills? Is there an appropriate balance that can renew America’s creativity quotient?
Photo Credit: http://www.coe.uga.edu/torrance/
Here are a couple of books I’ve found interesting that you might want to put on your list of reads for the remainder of the summer. Daniel Pink’s Drive: The Surprising Truth about What Motivates Us makes the case that it’s time to shift to what he calls Motivation 3.0. The carrot and stick approach to motivation may work well for repetitive tasks like those on an assembly line, but for 21st century tasks that involve real problem solving reward and punishment can get in the way. Pink points to research that support the importance of three critical elements of motivation: autonomy, purpose, and mastery. You can find a fun summary of his main points on YouTube (http://www.youtube.com/watch?v=u6XAPnuFjJc). The style of presentation is worth a look on its own. Pink makes some suggestions in his book about the application of his ideas to education, but it’s pretty cursory. It’s up to educators to take his compelling ideas and expression of the research and figure out what it can mean for driving students forward in their math learning and elsewhere.
And Alex Bellos’ Here’s Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math is a fabulous reminder about how incredibly cool math is. Bellos’ recounting of the origins of counting and arithmetic, along with the continued push by a few stalwarts (like the The Dozenal Society of America) for a switch to a duodecimal system, illuminates how our number system works. From that base (pardon the pun) Bellos travels the world to bring us engaging stories that unravel the working of mathematics. For a glimpse into the content, check out http://books.simonandschuster.com/Here%27s-Looking-at-Euclid/Alex-Bellos/9781416588252.
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Jon Star, educational psychologist and professor at the Harvard Graduate School of Education, stopped by our office to talk about math teaching practices and how kids learn.
1. Which should come first--“conceptual understanding” or “procedural fluency”?
It’s a good question and people have very strong opinions about this issue. Some would say that the whole math wars that we were in—some would say we might still be in—were based around this issue. Based on the evidence that we know, neither the only concepts first or the only procedures first is the way to go. We know that procedural knowledge and conceptual knowledge develop iteratively, that they sort of feed into one another. So it might be that for certain topics we might say, ‘well, we should start with the procedure then go to the concept then back to the procedure and back to the concept,’ and for other topics, it’s the other way around. It doesn’t seem to matter so much where you start, as long as we’re going back and forth and recognizing they develop iteratively, and they inform each other…so when they learn about a concept and that informs by subsequent learning about the procedure which informs by subsequent learning another concept, and…I think that’s the message we want to say. It’s not so much that if you only do procedures then the concepts sort of develop by themselves or if you only do concepts for a long time then when you get around to the procedures, they’re super easy. We haven’t found that to be the case.
There are in other countries certain cultural norms that come into play here; it’s unclear how applicable they are to the U.S. context. In some cultures there is this sense that you practice something a thousand times and then you’ll understand it. It’s sort of a Confucian saying some people say. There might be something there. It might be built into certain cultures that that’s the way learning occurs and that there’s a certain amount of fluxion when you’re practicing something a thousand times that really helps, but the evidence to date that we’ve done in psychological research suggests that it doesn’t impact development.
2. How can we measure conceptual understanding?
Good question, so in the past, say, fifty, a hundred years ago, there’s a sense that we weren’t really asking this question at all; we were just having students demonstrate procedures and that was the outcome we were interested in. They could do the procedure, and we sort of assumed that they knew what they were doing.
In relatively recent times there was a push against that when people said, ‘well, I don’t think that tells us the whole story, we don’t really know what they know just by what they do, we need to ask them questions,’ and so we’re kind of in this phase now where many people feel that the way to access student understanding is by getting students to talk and what they say. So you have this link between what students say and what they know. The verbalization of knowledge has become prioritized. We only know if they really understand if they can explain it. There is some merit in that. Certainly, this maxim that you really don’t know it until you teach someone else, kind of idea—there’s some truth to that. If I can explain to you really what I was thinking about and what I was going through, then it does reflect a level of understanding, but I guess I would caution from going too far in that direction.
People who are at the extreme of that view might argue that no written test that doesn’t have a verbalization component can ever truly tap understanding. For example, some people might say there’s no way a multiple choice test could ever tap what understanding is, that understanding is really only assessed by asking students questions and probing. And I would disagree with that. It’s challenging to design prompts that assess things that we think are understanding, written prompts, like multiple choice tests, or free response items. I’m not saying that’s easy—that’s challenging—but it’s doable. That’s what we need to consider. It’s really not feasible, if we’re interested in understanding, to be able to interview everyone all the time to assess understanding, even in my class of thirty or in my school or in the state. That’s just not feasible. So maybe we can just be more creative about items that assess understanding. In the research literature I think there’s a little more movement on that. When you look at a standardized test, even though they might have sections that are labeled conceptual knowledge, some people would question whether they really are assessing conceptual knowledge or whether they’re just sort of a different kind of performance.
One last thing I’ll say about this is that from a philosophical point of view, I’m not comfortable separating the knowing from the doing, which is sort of the direction that this sometimes goes. People say, ‘well, how do I know that you really know it?’ It’s because you can talk about it, but in real life we look at people’s performances, what they do, and we can make a judgment about what they know just from what they do. So in the philosophical literature there’s different examples of this. I can look at musicians. So you have a brilliant musician, and they’re playing a piece. Do you only judge their intelligence or the intelligence of the performance if they can verbalize what they did and break it apart verbally? Or is there something in the performance itself, in the act of doing, that you can say, that was an intelligent performance. It represents that they understand, just from what they did. Or similarly for a chef, if you have a chef, and the chef is doing something brilliant that indicates real understanding of the way flavors and ingredients fit together and build something, then do you depend on their ability to verbalize that and to essentially explain to you what they did to judge their performance as being really superlative or intelligent? Or is it in the act of doing that you really see intelligent performance?
I feel like I’m hesitant to move in the pure verbalization realm because I feel like it separates knowing and doing, and I feel like intelligent performance, intelligent action should be something we are looking for, and we have to design careful prompts and look for it, but that’s what we should be looking out for.
3. 2/3 of students are below proficient in math. How much of that is because math is inherently hard vs. the way we instruct students?
The cop-out answer is that both are important, and that’s probably true. We’ve certainly learned a lot in the past twenty years about the mathematical capacities of very, very young children and we know that kids can do a lot more at a very young age than we thought they could do. So we do seem to have this sort of built-in ability to do certain kinds of math, which is very powerful. What perhaps is missing is our ability to connect school experiences with that intuitive foundation. Students come in knowing a lot. Can we somehow use what they know and build on it so they can experience greater success when they’re learning school math? Is there a way to kind of leverage intuitive math toward the learning of school math? And I’m not sure that we’ve completely figured that out yet, although I think that in the elementary school curricula, the newer curriculum that have come out in the past twenty years are really trying hard to do that, to look at students’ strengths and the way that we understand people learn and to connect school math with that. Whether or not that helps us understand, say, algebra is a different question. I’m not sure that we have intuitive knowledge, or sort of pre-school knowledge, about algebra or algebraic concepts. People might disagree about that, but I’m not so sure. Algebra is abstract and maybe it takes a while before we’re comfortable with abstraction. How do we leverage certain intuitive arithmetic, concrete knowledge about mathematics as we move into more abstract problems like algebra and stuff?
When weighing the benefits of conceptual understanding versus procedural fluency, it is interesting to consider what procedural fluency means today. The authors of Children’s Understanding of Mathematics: 11-16, which describes the results of a study of British secondary school students, think that paper-and-pencil calculations are becoming superfluous.
In a world in which pocket calculators are readily available there is likely to be a shift in emphasis in the mathematics curriculum away from laborious pencil and paper methods of computation and towards the selection of the correct buttons to press in a given problem. (p. 23)
The book was originally published in 1981, almost thirty years ago, but the algorithms used for basic math computations still receive a lot of classroom time. Is it worth giving these methods so much emphasis when students almost always have a calculator readily available (on computers, phones, etc)? The study found that students who demonstrated significant conceptual understanding in interviews did not necessarily have a firm grasp of the algorithms. And some students who were able to perform rote calculations did not demonstrate an understanding of why they were using that operation.
Determining what should be plugged into the calculator for a given problem requires conceptual understanding. Can this skill also be considered procedural fluency? In a society that is increasingly reliant on technology, knowing how to calculate by hand still has some value, but perhaps the definition of procedural fluency is changing.
Photo Credit: http://www.amazon.co.uk/Childrens-Understanding-Mathematics-11-16-Kuchermann/dp/1905200021
What if school no longer meant sitting in grade-level classrooms, but instead learning with peers of a comparable ability level (regardless of age)? A recent article from the Associated Press reports that several elementary schools in Kansas City will be adopting this type of system, and schools in Denver and Maine have already implemented such a program. Not all students of the same age are able to master concepts at the same time, so eliminating grades allows students to move through the curriculum at their own pace. Teachers can check in and decide what individual students are ready to learn and then place them into small groups by ability level.
This flexible program ensures that students are constantly encountering material that is manageable for them. Even within a classroom, small groups of students can be working on similar topics with activities tailored to the appropriate ability level. High-achieving, motivated students will be able to complete college coursework if they finish the high school material early. Struggling students, on the other hand, can spend an extra year in high school if necessary.
The idea of teaching based on ability is reminiscent of many of the new differentiated instruction and Response to Intervention programs being used in math classrooms. Our Scholastic Math Inventory program, for example, is a fast and accurate computer adaptive assessment based on the Quantile Framework that tests students’ knowledge by adjusting to student performance. Students can then be taught at the level dictated by their math achievement.
With the growing use of online tutors and personalized practice that cater to each child’s needs, students are saved the frustration of falling behind because they lack the essential math skills necessary to move forward. Instead, they learn concepts they are ready to tackle, thereby increasing engagement and motivation. Could grades be a convention of the past and could ability-based instruction be the new frontier of education?
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The Atlantic reported that this September, New York City public schools plan to launch the largest prototype yet of the innovative School of One program for 1200 6th and 7th graders at three schools. The program envisions classrooms where students alternately learn in a variety of contexts: traditional teacher-led instruction, small group collaborative activities, online software-based instruction, online live instruction, independent learning, and one-on-one tutoring. Students will have the opportunity to learn at their own pace and in the manner that best caters to their learning style.
In launching School of One, New York has not eliminated its grade level standards or the sequence of content. However, it is allowing students to grasp content in their own way, thus hopefully reaching even the most reluctant and struggling students. Every student has different needs, and for some, the traditional classroom approach will never be effective.
Technology facilitates the simultaneous nature of the different learning modalities. At the end of each day, students complete short progress assessments that the program uses to generate tentative individualized lesson plans. Since some students can work with online tutors, instructional software, or in small group settings while others participate in teacher-led instruction, educators have more flexibility to tailor instruction to different students as needed.
The author of The Atlantic article, Ta-Nehisi Coates, speculates that having access to a more fun and personalized learning environment would have made school an exciting rather than a dreaded experience for him and could have deterred him from ultimately dropping out of college. Can technology be the key to sparking both the interest and understanding of underachieving students?
Photo Credit: http://schools.nyc.gov/community/innovation/SchoolofOne/default.htm