Building Representations of Children's Meanings

Maher, C. (1990). Building representations of children's meanings. Journal for research in mathematics education, 6, 79-90.

Summary:

This chapter walks you through a lesson in a fifth grade math classroom where the teacher is including problem solving group work on a regular basis in her classroom. Many of the tasks she gives her students encourage them to make models of their solutions.

Much of the chapter focuses on two students discussing possible solutions to the following problem: "At Pizza Hut each large pizza is cut into 12 slices. Mrs. Elson ordered two large pizzas. Seven students from Mrs. Elson's class are to eat one piece from each of the pizzas. What fraction of the two pizzas was eaten?" The two boys in this scenario are at odds because one chooses to solve the problem using models and one uses paper and pencil computations. They can not seem to understand each other's interpretations of the problem. To further complicate the situation, the teacher misinterprets one of the boy, Brian's thinking on the problem even though he was correct. In the teacher's mind, one pizza was "the whole". In the Brian's mind, both pizzas formed "the whole" together. When Brian answered that the class would eat 14/24 of the pizza, the teacher viewed that as him incorrectly adding the denominators instead of viewing the problem a different way. The teacher led the boys through a discussion to get them to come around to her way of thinking instead of attempting to understand where they were coming from. The boys eventually changed their answer, but were still unclear as to why their original answer was "wrong".

A year later, the boys were given the same problem. Both of them remarked that they were familiar with the problem, but didn't remember how they solved it. When Brian solved the problem this time, he still got the answer 14/24 indicating that the teacher's reasoning had not stuck with him. This is because her answer had no meaning to him and it did not make sense. The chapter asserts that, "a more promising approach is for the teacher to try to understand what the students were doing and why, and then to provide them with an opportunity to see their own faulty reasoning."

A 

Quotes:
"...one of the tasks that a teacher faces is to construct in her or his own mind a mental representation that matches the student's mental representation." -page 82

"A teacher's failure to recognize the way a student is thinking about a problem can at the very least end up wasting time in mutual misunderstanding." -page 90

Su

White Privilege: Unpacking the Invisible Knapsack

McIntosh, P.  (1990)  White privilege:  Unpacking the invisible knapsack.  Independent School, 49(2), 31-35.

Summary:

This article takes a perspective on racial disadvantage as a "white privilege".  It discusses how people in the majority group are willing to accept that certain races, genders, sexual orientations, etc, are socially disadvantaged in a way.  These same individuals are hesitant though to except the reflective statement, that majority groups receive unearned privileges due to their race, gender, etc.  In the article, McIntosh lists several “privileges “ that she has received by merely being white. Some jump out as something other than “privileges”, and more as standards that we should be striving towards as a society. An example is, “I can go shopping alone most of the time, pretty well assured that I will not be followed or harassed.“ Through white-privilege we have acquired these standards more easily and do not need to give anything up to spread the wealth. McIntosh describes these as “positive advantages”, but then there are “negative advantages” that instill a structure of power and is holding down a disadvantaged group.  She also differentiates between "earned strength" and "unearned power".  Unearned power through privilege can look like strength when it is in fact a form of oppression.  The ultimate message from this piece is that not all types of oppression are clearly visible.  In order to make true change in this oppression, we must accept these unseen factors and tackle them head on.

Quotes:

(1) “I have often noticed men's unwillingness to grant that they are overprivileged, even though they may grant that women are disadvantaged. They may say they will work to improve women's status, in the society, the university, or the curriculum, but they can't or won't support the idea of lessening men's. Denials that amount to taboos surround the subject of advantages that men gain from women's disadvantages. These denials protect male privilege from being fully acknowledged, lessened, or ended.” (McIntosh, 1990)

(2) “One factor seems clear about all of the interlocking oppressions. They take both active forms, which we can see, and embedded forms, which as a member of the dominant group one is taught not to see. In my class and place, I did not see myself as a racist because I was taught to recognize racism only in individual acts of meanness by members of my group, never in invisible systems conferring unsought racial dominance on my group from birth.” (McIntosh, 1990)

Personalized Inquiry: Help your students classify, generate, and answer questions based on their own interests or common materials

Simpson, P. (2010). Personalized Inquiry: Help your students classify, generate, and answer questions based on their own interests or common materials. Science and Children, 48(4), 36-40.


Summary/Analysis

This article on personalized inquiry provides a model for supporting students in the development of researchable questions based on their own interests.  It begins by asking students to create sketches of something familiar, like an ant.  Then by asking students to carefully observe real ants they discover the many features they did not notice/include in the sketches.  This leads to the establishment of criteria for careful observation.  After sharing and critiquing observations, students are asked to generate lists of questions about ants (~5 questions each).  Students share out their questions and create a class list of diverse questions.  

The questions can then be categorized by the class based on the type question and how it could be answered (experiment, literature-based research, observation, question not answered in science).  This process supports students in understanding the many ways that questions can be answered and the many strategies used to gather data.  Next, students create researchable questions for experiment.  

In the final activity, students work through a series of stations to support them in fine-tuning their ability to pose questions for research.  Each station contains a familiar item (packing peanuts, paper plate, gobstopper, etc) and a list of materials available in the classroom that can be used for experimentation (water, salt, light, ruler, scale, etc).  Students spend 20-30 min at each station generating lists of questions for research.

This article seems to be a practical way to use inquiry in the classroom and could be customized to any subject.  It is similar to the idea of critical exploration popularized by Eleanor Duckworth.  It’s also a student-centered way to develop the ideals of scientific thinking and experimental design.  It could be a great way to start the year and create an awareness of the possibilities that exist for scientific research.  The skills could later be applied to larger research projects.    

Relevant Quotes/Concepts

“Familiarity with the topic is key”

“It seems to work equally well with students of all abilities as long as the students are initially familiar with the object or phenomenon they are working with.”

This is similar to work Eleanor Duckworth did with graduate students.  She asked them to observe common phenomena such as the moon.  Through the observations, over time, new insights develop that deepen the understanding of the phenomenon. 

Introductory Paper on Critical Explorations in Teaching Art, Science, and Teacher Education

Cavicchi, E., Chiu, S., & McDonnell, F. (2009). Introductory Paper on Critical Explorations in Teaching Art, Science, and Teacher Education. New Educator, 5(3), 189-204.

Summary/Analysis:

This paper serves as an introduction to a series of three more papers that examine critical exploration in three separate classrooms (art, science and teacher education).  The art teacher explored Chinese brush painting with middle school girls, the undergraduate science class investigated mirrors and the teacher education students explored seeds, pendulums and the moon.  It is focused mainly on the theory behind critical exploration, the dynamics of a classroom where critical exploration is happening and the potential for student and teacher growth that can result.    

Critical exploration is a classroom practice introduced by Eleanor Duckworth based largely on research of Jean Piaget and Barbel Inhelder.  The work of psychologists Piaget and Barbel indicates that children construct their knowledge by relating actions to outcomes associated with the actions.  Through exploration/experimentation a child may reach a “stage” where, temporarily, the outcomes associated with their actions cohere with their expectations.  Then, however, unexpected outcomes may destabilize this coherence.  Piaget called this destabilization “disequilibrium.”  He saw this as a time of rich growth where children try new ways of acting or thinking and try to synthesize them with previous actions.  Critical exploration is the idea of using this process actively and consciously in the classroom.

Putting this idea to work in the classroom can be disconcerting to the teacher as he/she must
“...[break] with the role of providing answers to students or telling them what to do.”  Critical exploration is the idea of raising questions, unpacking ideas, tolerating spontaneity, wondering and discussing.  These types of activities often “[compound] the risk that teacher and students experience” in the classroom.  However, they can also lead to the type of “disequilibrium” that is the harbinger of meaningful growth.  

The paper claims that, in order for critical exploration to be effective in the classroom there must be an triangular relationship between learner, subject matter and teacher.  This idea of a triangular relationship was first expressed by David Hawkins.  For the learning to be most effective the relationship between the three should be equal and none shall be dominant.  For example, if the teacher takes a lead role, the contributions from the student and from the subject matter may suffer.  If the subject matter and teacher dominate (as in a lecture for instance) the student’s role is diminished.  

The subject matter itself is very important in this process.  The subject matter must have many facets and “problems veining through it” in order to provide room for critical exploration.  Duckworth refers to this as the “complexity” of the subject matter.  Students must be able to explore the subject matter endlessly to give rise to “new ways to act, observe and reflect...”  It must have dimension.  I imagine this as one of the more difficult jobs of the teacher.  Selecting the subject matter with the appropriate level of “complexity” is crucial in critical exploration.             

Relevant Quotes/Concepts:

“Classrooms have potential to be places for taking the risk of discovering how much is unknown within what we thought we knew.”  

“Duckworth associates the ‘essence’ of teaching with providing opportunities by which students may ‘have wonderful ideas’ that are expressions of their learning.”

“...learners’ confusion is often a prelude to their keener involvement and the making of connections or analyses that deal with significant puzzles in the subject and the learners’ grasp of it.”

“It appears ‘slow’ to teach by involving students in explorations by which they may, or may not, generate understandings that are consistent with those already formulated.  Such objections to exploratory learning privilege a certain type of efficiency and uniformity in information transfer while ignoring the processes by which anyone’s learning becomes sufficiently deep as to be usable in new, evolving situations.”

“That multiplicity of paths, which a complex subject matter can sustain but a simplified one cannot, is a means at the teacher’s disposal for accommodating diverse learners in noticing for themselves its multiple possibilities.”

Groupthink: The brainstorming myth

Lehrer, J. (2012, January 30). Groupthink: The brainstorming myth. The New Yorker. 22-27.

The ideas presented in Jonah Lehrer's article were very interesting, especially for someone interested in collaboration and creativity.  The focus of the article is looking at different factors in group dynamics that lead to creativity and innovation. First off, Lehrer attempts to prove why brainstorming, as we know it, does not lead to the most creative results possible.  Lehrer's definition of brainstorming is the rapid listing of ideas without criticism or negative feedback.  He points to a 2003 study by Charlan Nemeth, a professor of psychology at the University of California at Berkeley, to show that brainstorming with criticism actually gives better creative results!  Groups that debated ideas generated nearly 20% more ideas than the group that brainstormed without giving criticism.
***I thought this was an interesting finding because I have always followed the "no criticism" brainstorm method. I wonder about the term "creative" here though.  I agree that the team that debated had more ideas but how did he judge that they were more creative?  That seems awfully subjective to me.  On the other hand, I can see how debate could allow for more ideas.  If I put out an idea and then someone challenged me on it, I may adjust one part of the idea to fix the problem, thus creating a second idea.
Overall I am a bit skeptical so far.  It would be interesting to try this both ways sometime and see what happens. 

Next, Lehrer suggests an ideal group formation to achieve the most creative output.   Brian Uzzi, a sociologist at Northwestern, studied Broadway musicals as examples of group creativity.  In trying to find the ideal composition of a team; he discovered that the relationship among the collaborators was a reliable predictor of the commercial success of a musical.  If the collaborators knew each other too well and had worked together a lot, or, if the collaborators didn’t know each other at all, then the likelihood of success was low.  If the level of relationships was at its “bliss point” or had a mix of relationships; “These teams had some old friends, but they also had newbies,” then the musical was three times more likely as the others to be a commercial success.

***Again, super interesting but this study also left me wondering about the author's measure of creativity.  Does a commercially successful Broadway musical equal creativity?  I do appreciate the insight however on what makes a successful group.  It makes complete sense that a "mix of old friends and newbies" would make the best group composition.  I will definitely keep this in mind as I focus on collaboration in my classroom.

Finally, Lehrer describes physical spaces where creativity has been proven to thrive.  The first example of this is the Pixar headquarters.  Steve Jobs designed this space around a central atrium and moved essential rooms such as the cafeteria, gift shop, meeting rooms, and bathrooms to one central spot so staff from different departments would have to run into each other throughout the day.  Jobs felt that when people who don't normally work together meet up in these "chance encounters," more creativity and innovation takes place.

Another incubator for creativity was Building 20 on the M.I.T. campus.  This building was quick to plan and built out of the cheapest of materials.  The building was originally planned to be a short term fix to give the Radiation Laboratory more space but was never taken down and over the years housed many professional, scientific and intellectual groups such as biologists, psychologists, computer scientists, and linguists.  The mixing of these seemingly unrelated groups through the random numbering of floors and rooms and the ability to reconstruct the building to create a more collaborative space, caused the inhabitants to be more innovative.  Some of the outcomes were Chomsky's studies of linguistics, the Bose speakers, advances in high-speed photography, and the first video game.

***This section made me think of the High Tech High buildings and how they push teachers to collaborate across disciplines. I love how teachers share offices and sometimes share spaces with non teaching HTH staff.  I bet we could come up with our own stories about creative innovations that came from a chance meeting in a hallway or between schools in the village.  This also applies to students.  How can we continue to innovate around physical space our buildings and classrooms even more collaborative  for students.  Also, how can we cross pollinate students from different grades or different schools to facilitate more creative outputs.  

Quotes:

"The most creative spaces are those which hurl us together. It is the human friction that makes the sparks" (27).

"The best Broadway teams, by far, were those with a mix of relationships...These teams had some old friends, but they also had newbies.  This mixture meant that the artists could interact efficiently-they had a familiar structure to fall back on-but they also managed to incorporate some new ideas.  They were comfortable with each other, but they weren't too comfortable" (25).

"The lesson of Building 20 is that when the composition of the group is right-enough people with different perspectives running into one another in unpredictable ways-the group dynamic will take care of itself" (27).

Berger, R. (2003). An ethic of excellence. Portsmouth, NH: Heinemann.

pgs 1-63 (Intro-ch. 2)

Berger's book is of special interest to me because he is interested in the relationship between critique and school culture. He often uses stories, quite successfully, to illustrate his points.  In the introduction to his book, Berger describes the importance of young people creating 'beautiful work:' "they need to be proud of their work and their work needs to be worthy of pride" (2). This is not about false praise. It's about young people doing quality work that matters beyond the realm of the classroom. Berger believes that creating a culture where it is cool to care about school and do good work is essential. A culture is built from the kids, the teaching, the curriculum, and the school conditions all put together. He sees quick fixes to the "crisis in education" as dangerous and deceptive, much like quick weight loss plans. The focus on testing and tracking, he believes, is off-based. Rather, how do we bring the best out in kids? For Berger, it is about creating an ethic of excellence.

I was struck by Berger's precision in keeping, documenting, and caring for models of student work. In the many examples he gave, if it weren't for the models of carefully crafted student work, he could not have convinced his audience of the talents of the young people he worked with. Except it was not about talent, it was about hard work, revision, and peer critique. It was amazing hearing about the audience's disbelief that the models were made by such young people with such care. I was also struck by the significance of the work the young people in his class were doing. For example, when testing sites for radon, he reported: "Sometimes I need to be a tyrant for accuracy and quality in my classroom. Not this time. The students were scared to death. Scared that any possible error in their math would jeopardize the safety of a real family in town...The students checked their math, their spelling, their language and their reasoning twenty times before they rested easy...this was not an exercise. It was real. It was work that mattered to the world. Anything short of excellence would be intolerable" (17). This really drove home the importance of doing work that matters, outside of the classroom and school even, for me. I was also engaged by the fact that the students did not come from privilege, nor were they gifted. The quality of their work came out of the revision process and the expectation they were held to.

As a teacher leader, I was invested in Berger's process in helping a school in dismal conditions with a resistant principle build on what they were already successful at. Berger talks about having the teachers tell their stories-not just the 'war stories' (love this term) but also what's going well, what victories they have had, what excites them. Time and again, across racial and socio economic lines, the biggest obstacle schools must overcome is a culture where it is not cool (in fact social suicide) to care about school. Fitting in is the number one things kids are concerned with. So what if peer pressure could be positive instead of negative? I wonder how you can get kids to care about quality, particularly quality in other students' work, without it being about the grade.

The second piece of creating school culture is to get the community on board. Berger states, "When I've visited effective schools I've been struck with the realization that though the settings and resources are often widely different, every effective school I've seen has a strong sense of community" (41). When Berger talks about building strong school communities, he is insistent that smaller, rather than bigger is better. Smaller schools catch kids who might otherwise have fallen through the cracks and hold both teachers and students accountable. While it might make sense politically and economically to combine schools and create bigger ones, it doesn't make sense educationally. Sense of community only happens in close-knit groups where people know each other. This made me think of Chapin, my school in Princeton of 300 PreK-8, where everyone knows everyone else and there is a sense that the school is like a family. "The loss of community brought on by the "bigger is better" mentality is evolving as a painful chapter for American education (44).
Berger also writes about the aesthetics of schools. When schools are run down, dirty, trash filled places, it teaches children that they are not valued. "When kids walk into run-down, ugly buildings constructed as cheaply as possible and often falling apart, what message do these children get? We don't care about you. We don't value you...Are parents and children less important than business clients?" (46). I was really struck by the honesty of this statement. It's striking to me where we are and are not willing to put effort and money in this country. However, I love that Berger admits that a clean, well-kept building cannot guarantee anything. But it does help build an ethic of care. Berger expresses that what a community values will ultimately affect what school children value.
I am touched by the depth, quality, and care Berger puts into his teaching, and not the least surprised that this is the ethic he passes on to his students.

Other interesting Quotes:
"When considering how to improve education for children, people tend to focus on what's being delivered to students, and how to refine the package. I think it's more useful to consider schooling not as a delivery system but as an experience. What does a student go through in the course of a day? How does a student behave in this school in order to fit in? Where do students feel safe? What are the opportunities for students to contribute, create, and to be recognized for his or her talents or efforts? What motivates a student to care? This exercise is particularly powerful if the focus is on a marginal student, one whose race, background, or academic of physical needs label him or her different, therefore out of the mainstream" (44).

"...positive peer pressure was often the primary reason my classroom was a safe, supportive environment for student learning. Peer pressure wasn't something to be afraid of, to be avoided, but rather to be cultivated in a positive direction" (36).

"If peer culture ridicules academic effort and achievement-- it isn't cool to raise your handc in class, to do homework, to care openly about school-- this is a powerful force" (34).

"I have a particular interest in understanding what it takes to fit in, socially and academically, in different school cultures. I find that students from Kindergarten through high school, seem interested in discussing this question" (35).

Comments:
I have always realized that the school culture was important in creating quality work, and I have begun to suspect that there is a very distinct relationship between the school's culture and the quality of critique. I am so excited to read the rest of this book. I am very interested in how the culture of the 7th grade, or the culture of the drama room might be shifted to create an ethic of excellence.

Other Resources:
Steve Seidel
A Culture of Quality, Annenberg Foundation publication, 1996
Expeditionary Learning Outward Bound school network
Scott Hartl, Kathy Greeley, Steven Levy

The Believing Game and How to Make Conflicting Opinions More Fruitfu

Elbow, P. (2006). The believing game and how to make conflicting opinions more fruitful.   In C. Weber (Ed.), Nurturing the Peacemakers in Our Students: A Guide to Teaching Peace, Empathy and Understanding (pp. 13-25). Heinnman.

This article researches the idea behind teaching controversial and highly debatable topics in classrooms.  Elbow’s major point is the  process of thinking behind the “critical thinking” .  He addresses in 2 different ways.  First he describes the four ways in which people argue:

“This explains a lot about how most people deal with differences of opinion:
• Some people love to argue and disagree, and they do it for fun in a friendly way. They enjoy the disagreement and the give-and-take and they let criticisms and even attacks roll right off their backs. It’s good intellectual sport for them.
• Some people look like they enjoy the sport of argument. They stay friendly and rational---they’re “cool”---because they’ve been trained well. “Don’t let your feelings cloud your thinking.” But inside they feel hurt when others attack ideas they care about. They hunker down into their ideas behind hidden walls.
• Some people actually get mad, raise their voices, dig in, stop listening, and even call each other names. Perhaps they realize that language and logic have no power to make their listeners change their minds--- so they give in to shouting or anger.
• And some people---seeing that nothing can be proven with words--- just give up on argument. They retreat. “Let’s just not argue. You see it
your way, I’ll see it my way. That’s the end of it. There’s no use talking.” They sidestep arguments and take a relativist position: any opinion is as good as any other opinion. (It’s worth pondering why so many students fall into this attitude.)" (Elbow,13)

Elbow continues on to write about the major differences between what he describes as two ways of thinking about other's opinions. He titles them: The  Doubting Game and The Believing Game. The Doubting Game is the way most people have been taught to think critically.  This is mostly analytical; following reason and logic to any idea that is presented to a person. When one hears or reads something, it has been widely taught to find as much of the facts on that information and then decipher one’s own opinion and thoughts based on evidence.  His argument is that there is one major flaw with The Doubting Game:

“This blind spot in the doubting game shows up frequently in classrooms and other meetings. When smart people are trained only in critical thinking, they get better and better at doubting and criticizing other people’s ideas. They use this skill particularly well when they feel a threat to their own ideas or their unexamined assumptions. Yet they feel justified in fending-off what they disagree with because they feel that this doubting activity is “critical thinking.” They take refuge in the feeling that they would be “unintellectual” if they said to an opponent what in fact they ought to say: “Wow, your idea sounds really wrong to me. It must be alien to how I think. Let me try to enter into it and see if there’s something important that I’m missing. Let me see if I can get a better perspective on my own thinking.” In short, if we want to be good at finding flaws in our own thinking (a goal that doubters constantly trumpet), we need the believing game." (Elbow, 19)

Believing is the ability to takes someone else’s views and try to believe them.  That is, giving every idea value, yet proceed with caution. He explains that this can have more value because "we can use the tool of believing to scrutinize not for flaws but to find hidden virtues in ideas that are unfashionable or repellent. Often we cannot see what’s good in someone else’s idea (or in our own!)" (Elbow, 19)

He also explains the relationship between the two is synergenic.  That is to say, you can’t have one without the other.

He writes:  “In short, we must indeed continue to resist the pull to believe what's easy to believe. But believing what’s easy to believe is far different from using the disciplined effort to believe as an intellectual methodological tool in order to find hidden strengths in ideas that people want to ignore”.  (Elbow, 20)

The last section of this article explains how to effectively engage students in The Believing.  He lists 8 very useful and practical ways for teachers to use in their classrooms when students are having a philosophical/controversial debate.  They are:

1. Active Listening

2. 3 minute/five minute rule

3. Testimony session

4. Using specific language

5. Silence

6. Private writing

           7.

Using Physical voice

8. Non adversarial argument

I found this article very enlightening as well as progressive. Though it mostly focuses on the philosophical core of conflict within classrooms, It can definitely help in fostering a safe classroom culture to include rich and fulfilling conversations among students.

Teaching Problems and the Problems of Teaching

Magdalene Lampert, (2001). Teaching Problems and the Problems of Teaching. 1st ed. United States: Yale University .

“In this fascinating book, an experienced classroom teacher takes us into her fifth-grade math class through the course of a year and shows how classroom dynamics—the complex relationship of teacher, student, and content—are critical in improving student performance. Magdalene Lampert offers an original model of teaching practice that casts new light on the ways teachers can successfully deal with teaching problems.”

Thoughts:  Need to purchase this book soon, recommended and referenced by Carmel Schettino on her blog and website.

Update 4/6/2013:  Bought this book, arrived yesterday, will read and post detailed annotation

When Is PBL More Effective? A Meta-Synthesis of Meta-Analyses Comparing PBL to Conventional Classrooms

Strobel, Johannes, and Angela van Barneveld. 2009. “When Is PBL More Effective? A Meta-Synthesis of Meta-Analyses Comparing PBL to Conventional Classrooms.” The Interdisciplinary Journal of Problem-Based Learning 3 (Spring): 44–58.

Abstract

Problem-based learning (PBL) has been utilized for over 40 years in a variety of different disciplines. Although extensively researched, there is heated debate about the effectiveness of PBL. Several meta-analyses were conducted that provided a synthesis of the effects of PBL in comparison to traditional forms of instruction. This study used a qualitative meta-synthesis approach to compare and contrast the assumptions and findings of the meta-analytical research on the effectiveness of PBL. Findings indicated that PBL was superior when it comes to long-term retention, skill development and satisfaction of students and teachers, while traditional approaches were more effective for short-term retention as measured by standardized board exams. Implications are discussed.

Learning Mathematics via a Problem-Centered Approach: A Two-Year Study

Ridlon, Candace L, 1999.  Learning Mathematics via a Problem-Centered Approach: A Two-Year Study. Mathematical Thinking and Learning. 11 (1), pp.188-225

Eric Abstract

For nine-week periods during two consecutive years, sixth-grade students at the same school were taught identical mathematics content using two different instructional approaches. Year 1 involved low achievers, whereas Year 2 was mixed ability students. The experimental treatment was a problem-centered approach (PCL) where potentially meaningful tasks were posed to the class and solved in collaborative groups. The groups presented and defended their solution strategies to their peers. A statistically similar control group learned via the traditional explain-practice (E-P) approach. The teacher demonstrated procedures and students individually practiced what had been illustrated. Regardless of perceived ability level, the PCL approach appeared to significantly enhance achievement and improve attitude towards mathematics. Low achievers seemed to gain the most, narrowing the gap between them and their mixed ability peers. PCL appeared useful with any curriculum if the teacher understood and properly implemented the components of the approach.

Thoughts

Have not read this article yet, haven't found a free copy online, will need to check UCSD.  Looks to be one of the few recent controlled studies of PBL in a middel school environment.