Starting Algebra too Early?

June 13, 2012

There’s been a push for students to take algebra earlier and earlier, yet there are some serious pedagogic arguments that early algebra might not be a great idea for many, if not most, students. A fascinating paper by Clotfelter et al., (2012) (pdf) showed pretty clearly that for a large number of students, taking algebra earlier actually resulted in worse performance in not just algebra, but the follow-up classes as well (geometry and pre-calculus for example), compared to students who waited to take the subject. Indeed the Charlotte-Mecklenburg School District (the district studied in the article) actually reversed their policy of having students take algebra in 8th grade.

Students affected by the acceleration initiative scored significantly lower on end-of-course tests in Algebra I, and were either no more likely or significantly less likely to pass standard follow-up courses, Geometry and Algebra II

— Clotfelter et al., (2012): The Aftermath of Accelerating Algebra: Evidence from a District Policy Initiative (pdf) via NY Fed.

The argument for early algebra comes from the correlation between early algebra and better performance on standardized tests, and more advanced math classes in high school. But the authors here indicate that forcing students to take algebra early does not result in the same outcomes.

The argument against early algebra is based on the research that shows formal thinking develops during adolescence, and the belief that to do well in algebra requires the abstract thinking skills that are seated in the maturing prefrontal cortex. Until students are ready for the abstract thinking required (which happens at different times for each student), they will struggle with algebra.

Algebra provides an essential foundation for further mathematics, which is why it is my strong preference that students progress by demonstrating mastery of the topics at their own pace rather than struggling through the class.

Citing this post: Urbano, L., 2012. Starting Algebra too Early?, Retrieved January 20th, 2017, from Montessori Muddle: .
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Building a Metaphor (Actually a Grill)

April 27, 2012

The grill entering the final stages of construction by Ryan V. and Robert M.. Photograph by Autumn F.

It took us a little more than half a day to build a grill. It’s a simple thing of cinder blocks and sand, located near the soccer field so it’ll be convenient for bbq’s next year.

It took the highschoolers all morning to dig an outline for the base of the grill and lay in the foundation, despite it being a small, three-quarters of a rectangle shape, and only ten centimeters (4 inches) deep at maximum. The local clay is extremely dense and hard.

The foundations took the longest time to build.

But the foundations were firm, secure, and level.

When the base was done, two middle-schoolers — ably documented by a peer photographer — finished all the visible parts of the structure in just half an hour.

The next day, after I’d given them a presentation on cognitive development during the teenage years that I realized how nice a metaphor the grill construction was for the training of the brain during adolescence. The extensive pruning and myelination that typifies adolescence establish neural pathways are the foundation for future mental growth.

Good, strong, level foundations are the basis for a rich and fulfilling life.

Good foundations require some effort, but they're worth it.

Citing this post: Urbano, L., 2012. Building a Metaphor (Actually a Grill), Retrieved January 20th, 2017, from Montessori Muddle: .
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Control your Destiny: How the Adolescent Brain Works

April 25, 2012

During your adolescence, which lasts from your early teens into your 20’s, the brain changes rapidly, you develop new abilities and capacities, and the habits of mind and skills you develop will last long into adulthood.

Abilities: The last part of the brain to develop is the Frontal Lobe. It’s responsible for reasoning and judgement — aka Executive Function. So, it’s somewhat understandable that teens often have poor impulse control — their Frontal Lobe (the prefrontal cortex in particular) is still developing.

The parts of the adolescent brain.

However that’s not an excuse. It is essential for adolescents to be held to account, because it’s only by practicing responsibility that they get to learn how to use their Executive thinking skills.

Because that’s how we learn — by practicing.

When we’re learning something new, brain cells, called neurons, reach out and connect to form networks. As we practice and focus on specific things — certain patterns of movement or certain ways of thought — some of the unused connections get pruned away, while others become stronger. The axons that connect the most-used pathways get coated in myelin, which acts as an insulator to make sure signals can pass quickly and efficiently.

Neurons in the brain transmit information to each other along long axons and across the synaptic gap.

By reorganizing the connections between brain cells, the brain learns and becomes better at what you’re practicing. Thus we gradually transition from novices to experts.

However, there is a cost.

Making strong pathways makes for quicker thinking about the things we’ve practiced, but makes our brains somewhat less flexible at learning new things. We develop habits of mind that stay with us for a long time.

Some of those habits we might not actually want to keep; and there’s also the possibility that we might not develop some habits of mind that we really would like to have.

The development of the frontal lobe during adolescence opens a window of opportunity for learning good judgement/executive function, but it does not mean we actually will learn it. We need to actually practice it.

So, if you would like to know yourself, want to be able to control yourself, and, especially, want to shape the future person you will become, then you’re going to have to figure out: which habits of mind you want to be practicing and which ones you don’t.

Citing this post: Urbano, L., 2012. Control your Destiny: How the Adolescent Brain Works, Retrieved January 20th, 2017, from Montessori Muddle: .
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Inside the Teenage Brain

April 24, 2012

PBS’s Inside the Teenage Brain is now online.

Citing this post: Urbano, L., 2012. Inside the Teenage Brain, Retrieved January 20th, 2017, from Montessori Muddle: .
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Creativity, Depression and Anger

September 11, 2011

[A]nger … triggers a less systematic and structured approach to the creativity task, and leads to initially higher levels of creativity. … [However] creative performance should decline over time more for angry than for sad people.

— Bass et al (2011): Creative production by angry people peaks early on, decreases over time, and is relatively unstructured

Here are a couple of studies on the interaction between negative emotions and creativity whose implications require some very careful consideration. We want to encourage creativity, but how and at what cost to the student?

Social rejection was associated with greater artistic creativity

— Akinola and Mendes (2011): The Dark Side of Creativity: Biological Vulnerability and Negative Emotions Lead to Greater Artistic Creativity


Anger, it appears, leads to more unstructured thinking, thinking that is more flexible and able to make new connections among different categories of information. However, anger’s creativity boost does not last that long – strong emotions take a toll – and people soon revert back to a more normal baseline.

These results come from an initial study, and there are a lot of unanswered questions. In particular, I wonder just how much anger is useful for this beneficial outcome. I find it hard to believe that too much anger is terrible useful. And, I’m also curious about the negative consequences in terms of group interactions. Brett Ford points out that some studies have found that anger is useful in negotiation, but only when that negotiation is confrontational. Another study found that angry leaders were better at motivating groups of less agreeable people. Conversely, more agreeable people responded better to less angry leaders.

In a scenario study, participants with lower levels of agreeableness responded more favorably to an angry leader, whereas participants with higher levels of agreeableness responded more favorably to a neutral leader.

— Kleef et al. (2010): On Angry Leaders and Agreeable Followers
How Leaders’ Emotions and Followers’ Personalities Shape Motivation and Team Performance

It seems that the ability to project anger may be a useful skill to have in one’s toolbox, given the variety of people we will have to deal with in life.

Depression and Creativity

Modupe Akinola and Wendy Berry Mendes point out that highly creative people tend to introversion, emotional sensitivity and, at the extreme, depression and other mood disorders. Unfortunately:

[M]ood disorders are 8 to 10 times more prevalent in writers and artists than in the general population (Jamison, 1993).

— Akinola and Mendes (2011): The Dark Side of Creativity

On top of the general mood, strong, more transient, activating moods, like anger and happiness, also affect a person’s ability to be creative. Both positive and negative activating moods (the hedonic tone) enhance creativity, but in different ways:

  • negative activating moods, like anger and fear, increase perseverance;
  • positive activating moods, like happiness and elatedness, increase mental flexibility.

Curiously enough, although creativity is associated with a baseline of sadness and depression, these two are not among the activating moods that can spur the creativity of the moment.

A Matter of Control

The implications of these studies are complex. I certainly need to think about them a lot more, but it would seem reasonable, or perhaps responsible, to encourage students to carefully monitor their moods and to help them better understand themselves and their behavior. Ultimately, it is probably better if we are able to control how we use our emotions, rather than the other way around.

The pre-frontal lobe, which is responsible for formal thinking, is the part of the brain that can put the brakes on impulsive emotional behavior. It can also, to a degree, modulate how emotions are expressed. As adolescents’ pre-frontal cortex develop, they should be better able to control and use their emotions to their benefit. But to do so, they need to be aware of their emotions and the power of their emotions, which would suggest training in emotional awareness and control.

I’m not aware of any programs or curricula that delve all the way into how to use your emotions proactively, but I’d like to see something that particularly discusses how to use the different activating moods.

Citing this post: Urbano, L., 2011. Creativity, Depression and Anger, Retrieved January 20th, 2017, from Montessori Muddle: .
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Searching for Answers: Earlier Puberty Over the Last 200 Years

August 21, 2011

Puberty starts somewhere in the age range of 8 to 13 years for girls and 9 to 14 years for boys. However, in Norway, in 1850, girls hit puberty at around seventeen. Over the next 100 years that age decreased to thirteen and a half, where it has stabilized, but a similar trend has been seen in pretty much all the industrialized countries, including the U.S.. No one knows quite why, but there are a number of theories, including:

  • Better nutrition,
  • increased stress, and
  • artificial chemicals in the environment or in the diet.

The trend in the timing of puberty in girls (menarche) for four western countries, from 1850 to 1950. Figure via NIH via INSERM.

It seems clear that this trend has something to do with improving living conditions. Rapidly developing countries like China are experiencing the same trend in earlier puberty right now. Wealthier areas in developing countries have girls starting puberty at the same age as girls in “privileged” countries, while their compatriots in the poorer areas do not. Also, Overweight kids tend to start puberty earlier.

However, explaining the earlier puberty is difficult because no one knows for sure what exactly triggers puberty to begin with. The genetic switch that tells the hypothalamus to start the process probably involves multiple genes that are affected in complicated ways by how and where a person grows up (when the environment affects how genes are expressed it’s called epigenetics; NOVA has a nice little program that explains how epigenetics results in differences in identical twins.).

Increased stress might be another explanation. Girls in Bosnia and Croatia started having puberty later and later during the war in the 1990’s. However, it appears that other types of stress, such as from insecure relationships with parents and adoption, can do the opposite and trigger even earlier puberty (note: really early puberty in kids as young as 9 is called precocious puberty and is a growing problem).

Certain artificial chemicals that disrupt the endocrine system, which is responsible for hormone production, have also come under suspicion, but their effects have been hard to prove.

Whatever the reason, the earlier onset of puberty has lead to an increase in the length of adolescence (which tends to start with puberty and ends somewhere in the mid-twenties). It’s hard to say though, if all the extra time is beneficial, since it does give the developing brain extra time to adjust to a more complex society, or if it just makes for a longer period of trying times.

Citing this post: Urbano, L., 2011. Searching for Answers: Earlier Puberty Over the Last 200 Years, Retrieved January 20th, 2017, from Montessori Muddle: .
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The Spirit of the Law

July 23, 2011

A You are the Ref strip by Paul Trevillion.

Every week, artist Paul Trevillon poses, in text and cartoon form, some truly idiosyncratic situations that might come up in a soccer match in his You are the Ref strip on the Guardian website. Readers get a week to propose their solutions and then referee Keith Hackett give his official answers.

It’s a fascinating series, the subtext of which is that, while there is a lot of minutiae to remember – the actual diameter of a soccer ball is important for one question – the game official is really out there to enforce the spirit of the laws, enabling fair and fluid play to the best of their ability. This is a useful lesson for adolescents who tend toward being extremely literal, and have to work on their abstract thinking skills, especially when they relate to questions of justice. For this reason, I find that when refereeing their games it’s useful to take the time during the game, and afterward in our post-match discussions, to talk about the more controversial calls.

Citing this post: Urbano, L., 2011. The Spirit of the Law, Retrieved January 20th, 2017, from Montessori Muddle: .
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The Myth of Adolescent Angst

February 24, 2011

Fortunately, we also know from extensive research both in the U.S. and elsewhere that when we treat teens like adults, they almost immediately rise to the challenge.
— Epstein (2007): The Myth of the Teen Brain in Scientific American Mind.

Is the angst and turmoil we usually associate with adolescence just a result of the way human brains develop, or is it something learned, and depends on the society that shapes our kids? Robert Epstein argues (Epstein, 2007) it’s the latter not the former, and, despite a lot of other research to the contrary, he may have a point. He believes the main problem is that western teens are treated more as children than young-adults, and they spend most of their time socializing with other teens and not with adults.

Cerebral Lobes

Cerebral lobes (image via Wikimeida Commons).

Alex Chediak posts a good overview of the work.

We’ve seen that one of the major problems with most psychological studies is that they only focus on WEIRD people, typically represented by college students in the Western world, who are the easiest people for university researchers to study. Using any such subset must, necessarily, be unrepresentative of the full range of human behavior. Furthermore, since society influences brain development, even studies that focus less on behavior and more on neurological imaging are likely to be affected by the some bias.

A similar argument can be made for studies of adolescence since most studies of adolescence focus on western teens. As a result, separating behavior learned via social interaction, from the regularly progression of genetically programmed brain development is going to be difficult.

Much of Epstein’s argument is based on the book Adolescence: An Anthropological Inquiry (Schlegel and Barry, 1991), which compared teens in almost 200 pre-industrial societies. Epstein summarizes this and other work to indicate that in pre-industrial cultures:

  • about 60 percent had no word for “adolescence,”
  • teens spent almost all their time with adults,
  • teens showed almost no signs of psychopathology
  • antisocial behavior in young males was completely absent in more than half these cultures and extremely mild in cultures in which it did occur.
  • teen trouble begins to appear in other cultures soon after the intro- duction of certain Western influences, especially Western-style schooling, television programs and movies.

— Epstein (2007) (my bulleting): The Myth of the Teen Brain in Scientific American Mind.

As a result, teens:

learn virtually everything they know from one another rather than from the people they are about to become. Isolated from adults and wrongly treated like children, it is no wonder that some teens behave, by adult standards, recklessly or irresponsibly.

Apprenticeship (image by Emile Adan via Wikimedia Commons).

Epstein’s antidote is to treat teens like adults. I agree. However, it’s essential to keep in mind what type of adults we want them to be: responsible and logical, while retaining the creativity we usually associate with childhood. This is something that typifies the ideal of Montessori education, all the way from early-childhood up.

Citing this post: Urbano, L., 2011. The Myth of Adolescent Angst, Retrieved January 20th, 2017, from Montessori Muddle: .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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