Tag: Pedagogy

How to Think Like a Mathematician

The epistemological approach to education suggests that the best way to learn a subject is to learn how to think like the experts in the field: how to think like a scientist; how to think like an historian; how to think like an engineer; etc.

How to think like a mathematician is Kevin Houston‘s attempt to explain how one mathematician at least approaches problems. To whet your appetite, he has a free pamphlet, 10 Ways To Think Like a Mathematician, which starts off with:

  1. Question everything, and
  2. Write in sentences

Logic is, apparently, quite important.

If you want to understand mathematics and to think clearly, then the discipline of writing in sentences forces you to think very carefully about your arguments.

— Kevin Houston: 10 Ways To Think Like a Mathematician

It’s an interesting introduction to how mathematicians see the world, and its a useful reminder that many of the ways of thought that apply to any field can be useful in other places, or even in life in general.

National Academy’s Books are Now Free

How People Learn from the National Academies Press.

The National Academy Press has just made all of its publications free for downloading (as pdf’s). Of particular interest is the Education section which includes titles such as:

There are 20 books in the Education section, and while many of the books in the other sections are quite technical, there are some gems among the 4,000 available titles.

Block Schedules

Jenny Anderson has an interesting article on a New York school that changed from the typical 45 minute class periods to longer 130 minute “blocks” (thanks to Kara D. for pointing this one out). The whole idea of set class periods is one I’m having to get used to again as I move out of my one-teacher, one-classroom middle school at Lamplighter where time management is a lot more fluid.

Block time allows for in-depth student presentations.

Apart from Spanish class (30 minute class period), everything was pretty flexible at Lamplighter. In theory, I would have a short, spark-the-imagination type lesson at the beginning of each week for math, language, and social/physical science. Then, for the rest of the week, students would use the two hour morning and afternoon blocks to do whatever individual or group work they needed to get done.

In theory at least.

In practice, my lessons would tend to last a lot longer; I like to promote discussion, and once you get them going, adolescents often find it difficult to stop talking. So, almost inevitably, a quick review of the novel chapter they read last night would devolve into a discussion of something like Mutually Assured Destruction and continue onto the potential for accelerated evolution due to nuclear fallout, and then to the parts of atoms and why some elements are radioactive and some are not.

I also found that I would have to assign specific blocks that prioritized group work. It’s hard, apparently, to arrange everyone’s schedules to do group work, even when you’re with the same group of people, in the same classroom, all day at school. Part of this though is that, in giving students so much control of their time, students find their own rhythms to the days, with, say, math in the mornings and language after noon, that may not match up well with each other. So I’d find myself saying, when we restarted after lunch, “Remember your group projects are due tomorrow, so you might want to get on that.” And, typically, they did, getting the group work done before going back to their individual work.

The key difference with a set time period for each class, is the tendency for the teacher to feel that they have to stay on subject for the entire period. There are, of course, different topics and subjects that need the full period or block, and I certainly favor having a longer time period to work with, but there are times when you might feel the urge to artificially stretch the work just to fill in the time. This problem was mentioned in the article:

Another complaint: boring 45-minute classes became boring two-hour classes. Robert Ronan, a senior, said, “There are some classes that lend themselves more easily to 2-hour-and-15-minute classes and teachers that can do that, but I sort of feel like a lot of the classes are the same, just stretched.”

–Anderson, 2011: At Elite School, Longer Classes to Go Deeper

It seems to me, that if you don’t want to lecture or have a discussion for entire two hours, which could get boring if not done really well, and you want individual or group work, which some students will complete faster than others, you are going to be faced with students who have time on their hands. You’d prefer that they spent that time productively, and definitely don’t want them distracting other students, so there need to be clear expectations about what they should be doing in these interstitial moments.

Sometimes you just need a nap to rest and recuperate.

I’d lean toward making sure they know how much time they have before you need their attention again (or they have to leave) and then giving them the choice of what to do: they could start on homework for this or even another class; they could take a quiet break to relax and recuperate (journaling might be a good idea); or they could do what you’re probably doing, and go around helping their peers with the work at hand.

Within the same block of time:

Within the same block of time, peer-teaching (math in this case) occurs at one end of the classroom.

The group splits and reforms around individual work, but with company close by. These students are working on the same subject, and will occasionally ask each other small questions about the work, but are working individually.
At the other end of the classroom a pair of students work on their small group project (science).
And another student works on something completely different (his research project) in a different, quieter space.

Building the Machine: The Role of the Teacher in a Montessori Middle School

With students working on different things at the same time, sometimes collaborating, sometimes working individually, a fluidly function Montessori classroom is somewhat akin to a complex but well-oiled machine: there are lots of individually moving parts that sometimes interact and sometimes not, in an ever-changing configuration. As a result, the job of the middle school teacher is less to convey information than it is to develop a successful classroom culture and ensure its efficient working.

Building the machine starts with the teacher as a role model. The teacher is a role model at all levels, but in middle school this takes on a slightly different color. After all, your adolescents are furiously figuring out how to be adults, so they’re taking a lot of clues for their behavior from the adults in their presence. The key things they’re looking for are, in Montessori’s (1948) words, “a sense of justice and a sense of personal dignity.” The trick is that underneath all the cynicism, they’re all idealists.

Justice is a particularly important and delicate concept because students want justice badly, but they tend to see it as distributive justice, where everyone is equal and get equal rewards and punishments. Unfortunately, this view tends to lead to an over-expansive expectation of rights and often to a sense of entitlement: the belief that if that person is getting something, I should get the same thing too. What’s too often missing, is the recognition that beyond the basic human rights, rights and privileges have to be earned.

This is something I find that I have to explain again and again for everyone to internalize what it means. It does not help that adolescents’ frontal cortex, the part of the brain responsible for critical thinking and impulse control, are not yet fully developed. What makes things even more interesting is the fact that girls tend to cognitively mature a lot faster than boys.

In addition, my own philosophy is that there are two key things I want to impart to my students: a love of learning and the willingness to try new things. This is somewhat contrary to where students are going developmentally. Adolescents tend to chase certainty as they change physically and mentally, all the while trying to establish their personal identities and place in the world; their focus tends to narrow toward what they’re good at and where their interests lie; there is “an unexpected decrease in intellectual capacity” (Montessori, 1948).

To encourage independence and creativity, and to build the sense of personal dignity through accomplishment, I sometimes break the pattern of the Montessori three-part Lesson (introduction, practice, application), and throw them assignments that they should have most of the background tools and knowledge to deal with, but have never encountered in this particular way. The Student Run Business is great for this, as unexpected problems are always cropping up, and, in case of emergency, it’s easy to create extra problems and challenges if you need to. When our bread-baking ovens started acting up, the oven calibration provided a great opportunity. It needed to be done, and students could figure it out on their own. Mostly. Eventually.

Developing a good classroom culture is probably the hardest challenge for those new to Montessori, especially with early adolescents who tend to have their own ideas and know everything already. However, with a few carefully designed lessons and exercises, the machine takes care of most of the teaching and learning of math and science and social studies and whatever else the curriculum requires students learn, peer-teaching and collaborative learning are all part of the classroom culture. The best part though, is that, once well established, the teacher ends up with a lot less work to do, and with a culture that propagates itself from year to year in our multi-aged classrooms.

So while we want to create a well organized, fluidly functioning classroom, it’s sometimes useful to introduce a little extra friction to keep things interesting. Of course, most often you don’t have to do this yourself. A lot of friction will come from the students themselves, and then the trick is anticipating it, allowing students the chance to deal with it, and then finally using it as a lesson so students learn from their experience if it gets beyond them. All the while, you must recognize that your every word and action is being carefully scrutinized with an eye for justice, even though you and they may not have the same definition for the word.

References

Grazzini, C., 1996. The Four Planes of Development, The NAMTA Journal, 21 (2), 208-241.

Montessori, M., 2005. From Childhood to Adolescence, The Montessori Educational Research Center, Trans. New York: Schocken. (Originally published in 1948).

Finnish Schools and Montessori Education

The BBC has a fascinating article on the Finnish educational system; specifically, why it consistently ranks among the best in the world despite the lack of standardized testing. A couple things stand out to me as a Montessori educator.

The first is the use of peer-teaching. There’s a broad mix of abilities in each class, and more talented students in a particular subject area help teach the ones having more difficulty. It’s something I’ve found to be powerful tool. The advanced students improve their own learning by having to teach — it’s axiomatic that you never learn anything really well until you have to teach it to someone else. The struggling students benefit, in turn, from the opportunity to get explanations from peers using a much more familiar figurative language than a teacher, which can make a great difference. I give what I think are great math lessons and individual instruction, but when students have trouble they go first to one of their peers who has a reputation for excelling at math. In addition to the aforementioned advantages, this also frees me up to work on other things.

A second thing that stands out from the BBC article is how the immense flexibility the teachers have in designing their teaching around the basic curriculum coincides with a very progressive curriculum. This seems an intimate consequence of the lack of assessment tests; teachers don’t have to focus on teaching to the test and don’t face the same moral dilemmas. Also, this allows teachers to apply their individual strengths much more in the classroom, making them more interested and excited about what they’re teaching.

E.D. Kain has an excellent post on the video The Finland Phenomenon that deals with the issue specifically. It’s full of frustration at the false choices offered by the test-driven U.S. system.

(links via The Dish)

The Moral Dilemmas of High-Stakes Tests

Just in time for the standardized testing season, Gillum and Bello have a damning article on irregularities in the testing at some Washington D.C. schools. NPR has a good summary of the situation and the investigation.

Sadly, with the fates of their schools and their jobs depending on the outcome, the faculty and staff administering these tests to their own students face an unfortunate conflict of interests and are placed in a serious moral hazzard. It’s also not hard to imagine the potential for ramped-up pressure on the students.

Standardized tests can play an important role in maintaining quality in the vast network of schools that make up the US’s educational system. They also help maintain consistency, of which a certain amount is probably good, but can be awfully restrictive. But the most unfortunate aspect about the way they’re actually used, is that they create intense pressure on students and faculty that is deleterious to student performance on the tests themselves, and severely restricts the way students think about what it means to learn.

Social Loafing: Getting Groups to Work Well Together

PsyBlog has an excellent summary of the research on social loafing, the phenomena where people working in a group work less compared to when they work alone. Because we do so much group work, this is sometimes an issue.

The first research on social loafing came from Max Ringelmann way back in 1913 (Ringelmann, 1913). He had people pulling on a rope, and compared the maximum they could have pulled, based on individual test, to how much each person actually pulled. The results were, kind of, sad; with eight people, each one only pulled half as much as their maximum potential strength. A graph of Ringelmann’s data is shown below. If everyone pulled at their maximum the line would have stayed horizontal at 1.

The relative loafing of people working in a group. As the group gets larger, the amount of work per person decreases from its maximum of 1. Data from Ringelmann (1913)

The PsyBlog article points out three reasons why people tend to loaf in groups:

  • We expect others to loaf so we do it, too.
  • We feel more anonymous the larger the group, so we feel less need to put in the effort.
  • We often don’t have a clear idea about how much we need to contribute, so we don’t put in as much as we could.

This can be summed up in Latane’s Social Theory:

If a person is the target of social forces, increasing the number of other persons diminishes the relative social pressure on each person.

— Latane et al., 1979: Many hands make light the work: The causes and consequences of social loafing in the Journal of Personality and Social Psycology. Quote via Keith Rolag’s Website.

How do we deal with this

The key is making sure students are motivated to do the work. We want self-motivated students, but creating the right environment, especially by training students in how to work in a group will help.

  • Make sure students realize the importance of their work; this makes them more motivated.
  • Build group cohesion; team members contribute more if they value the group they’re in.
  • Make sure the group clearly and fairly divides the work. Let everyone be part of the decision making process so students have choices in what to do will help them be more invested in their part of the work.
  • Make sure each group member feels accountable for their share of the work.

A Brief Excursion into Mathematics

Ringelmann’s data falls on a remarkably straight line, so I used Excel to plot a trendline. As my algebra students know, you only need two points to write the equation of a line, however, Excel uses linear regression to get the best-fit line through all the data. Not all the data points will be on the line (sometimes none of them will be on the line) but the sum of the distance from each point to the line is minimized.

Curiously, since the data is pretty close to a straight line, you can extend the line to the x-axis to find out how many people it would take for no-one to be exerting any force at all. Students should be able to determine the equation of the line on their own, but you can get Excel to give you the equation of the trendline. From the plot we see:

y = -0.0732 x + 1.0707

At the x-axis, y = 0, so;

0 = -0.0732 x + 1.0707

solving for x we first subtract the constant, 1.0707 from both sides to get:

0 – 1.0707 = -0.0732 x + 1.0707 – 1.0707

giving:

-1.0707 = -0.0732 x

then divide by -0.0732 to isolate x:

! \frac{-1.0707}{-0.0732} = \frac{-0.0732 x}{-0.0732}

which yields:

x = 14.63

This means that with 15 people, no-one will be pulling on the rope. In fact, according to this equation, they’ll actually start pushing on the rope.

It’s an amazing result, but if you can find flaws with my argument or math, please let me know.

A Diversity of Education

Pluralism … allows individual schools, educators, and providers to excel at something, rather than asking every school to excel at everything.

–Hess (2010): Doing the Same Thing Over and Over in AEI Outlook Series.

Frederic Hess’ new book advocates a diversity in educational formats. Steven Teles has a detailed review.

The Same Thing Over and Over: How School Reformers Get Stuck in Yesterday's Ideas by Frederic Hess.

Hess shares the same basic premise of most progressive, constructivist, educational approaches like Montessori’s, that students learn differently so they need different educational approaches. However, he takes this need for diverse educational environments further with the recognition that teachers are different so they will have their own educational philosophies and methods that work best for them, and that parents are different, with very different expectations about what education should be and what it should accomplish.

… the basic components of schooling—parents, children, school leaders, and teachers—are irreducibly diverse. Parents have different ideas about what a “well-educated” child is, and children differ quite significantly in temperament, aptitude, habits, and interests. School leaders vary as to how they think schools should be run, while teachers have different skill levels, enthusiasm for different tasks, and ideas about what children should learn and know.

… Educators will always be less effective if they are made to teach in a way that they believe is wrongheaded or that they haven’t bought into. Students will have difficulty learning if they are forced to work at a pace that is too fast or too slow, or if they are taught in a manner that doesn’t match their individual learning styles. Parents can be disengaged or hostile if the pedagogy, discipline, or school culture differ fundamentally from what they think is right for their child. And schools as a whole will be incoherent and disorganized if they cannot count on some baseline of agreement as to what—and who—the school is for.

— Teles (2011): One Size Doesn’t Fit All in Washington Monthly.

Although Hess works for the conservative American Enterprise Institute his own thought on education are far from traditional:

[T]here is value in nurturing diverse intellectual traditions, models of thought, bodies of knowledge, and modes of learning. It is prudent to embrace a system of schooling that nurtures a diverse set of skills, knowledge, and habits of mind. This allows us to foster intellectual diversity that enriches civil society and … [i]t allows individual schools, educators, and providers to excel at something, rather than asking every school to excel at everything.

–Hess (2010): Doing the Same Thing Over and Over in AEI Outlook Series.

Furthermore, Hess argues, the world has changed since the inception of universal education, but the educational system has not adapted to the changing needs and technology. He points out new innovations allowed by technology, like the School of One program in New York.

All of this is hard to argue with. It’s almost the standard constructivist critique of the current educational system, although constructivists tend to focus on how we’ve not applied all the stuff we’ve learned about pedagogy since the 19th century (Lillard, 2005 lays out this argument eloquently in Montessori: The Science Behind the Genius).

Hopefully, this book broadens and advances the arguments for reforming the educational system. It is a progressive view from a conservative organization. Yet it still begs the question of how do we get there from here, while dealing the serious concerns that greater diversity may well lead to some failures as well as successes. Ultimately, we end up with the same fairly intractable problem. However, how do you measure success where there is such a diversity of expectations for education?