Learning Algebra, In Steps

Today's study breaks down the different skills needed by a student to be able to complete an algebra problem. It turns out that there are four separate types of knowledge needed, and a person must master the easiest before progressing to the next, etc. So, without further ado, here is:

"A Developmental Model for Algebra Symbolization: The Results of a Difficulty Factors Assessment", by Neil T. Heffernan and Kenneth R. Koedinger. Published in Proceedings of the Twentieth Annual Conference of the Cognitive Science Society, 1998, edited by Morton Ann Gernsbacher and Sharon J. Derry, pp 484-489.

The Experiment
During the first month of school, 74 urban ninth graders enrolled in a regular-level algebra class were given a test made by the experimenters that included word problems of different levels of complexity.

The first type asked for each student to make a single-operation formula out of numbers. Example: "Sue made 72 dollars by washing cars to buy holiday presents. She decided to spend 32 dollars on a present for her mom and then use the remainder to buy presents for her sisters. How much can she spend on her sisters?" Answer: 72-32.

The second type asked for each student to make a single-operation formula out of a combination of numbers and variables. Example: "Sue made 72 dollars by washing cars to buy holiday presents. She decided to spend 'm' dollars on a present for her mom and then use the remainder to buy presents for her sisters. How much can she spend on her sisters?" Answer: 72-m.

The third type of question asked each student to make a double-operation formula out of only numbers. Example: "Sue made 72 dollars by washing cars to buy holiday presents. She decided to spend 32 dollars on a present for her mom and then use the remainder to buy presents for each of her 4 sisters. She will spend the same amount on each sister. How much can she spend on each sister?" Answer: (72-32)/4.

The fourth type was the most challenging. Each student was asked to make a double-operation formula using both numbers and variables. Example: "Sue made 72 dollars by washing cars to buy holiday presents. She decided to spend 'm' dollars on a present for her mom and the remainder to buy presents for each of her 4 sisters. She will spend the same amount on each sister. How much can she spend on each sister?" Answer: (72-m)/4.

The actual questions were varied, such that any given test paper only had one question about Sue washing cars, but the test questions were all equivalent to the examples listed above and in the paper.

The Results
The lowest-performing students were able to do the simple arithmetic formulas (Type 1). The next step was to to the more complicated arithmetic expressions (Type 3). Once they are good at that, they start to understand the single-operator variable problems (Type 2). All students in the study who could correctly answer the Type 2 questions also correctly answered the arithmetic ones. The last step to being able to master algebra is the double-operator variable problems (Type 4).

Most people assume that it is the language used in word problems that makes algebra so hard; given the design of this experiment, the researchers have very conclusively shown that language comprehension is not the issue. After all, the language used in the Type 2 and 4 problems was exactly the same, and yet one was significantly more difficult than the other.

The researchers ran a separate experiment to see what is needed by students to make that final leap into fully grasping algebra. Since it seemed like students have trouble combining multiple steps together when there are variables involved, they trained some students for an hour on the following types of problems, dubbed symbolic substitution problems: "Let X=72-m. Let B=X/4. Write a new expression for B that composes these two steps." Answer: (72-m)/4. Student performance on algebra word problems increased after just one hour of practicing these sorts of problems.

What this shows is that it is not only the language of word problems that needs to be understood, but the language of abstract mathematical concepts needs to be understood, as well. Not only do students have to understand what the problem says, but how to create the right abstract representation, which are two very different things!

Here's a way of thinking about it: Lots of people say that if you can read, you can cook. My husband says that this baloney, because he can read, but he can't cook--cookbooks are written in a foreign language, as far as he in concerned! All of those verbs mean something very specific, and involve skills that he generally does not have; everything is also written in shorthand. He says it's like this: If you can read a cookbook and perform all of the required actions, then you can cook!

How Waldorf Education Teaches Algebra
Arithmetic is introduced in the first grade, and its form takes on both of the first two types of algebra skills, both inside and outside of oral word problems. For instance, they try to figure out what can be added to 7 to equal 12 (a type 2 problem). This early work with numbers and variables skips over the work with complex number problems; however, this is gradually introduced the next couple of years with number adventures, where the children do a lot of operations to a given starting number in their heads. While they may not be writing down these concepts, they are busy internalizing them.

The final step is introduced slowly over the 6th-8th grade years (particularly in 8th grade), as the children work with formulae that they have themselves discovered, the manipulative rules of algebra (many of which they discovered but did not name back in first grade, like 1+3=4 and 3+1=4 (the commutative property)), polynomials (which can be designed into symbolic substitution lessons), and finally, algebra word problems.

The early, deeply-ingrained numeracy lessons of the Waldorf curriculum lay an excellent foundation for introducing the abstract concepts of algebra; based on the study and my inquiries with a Waldorf teacher, the hurdles that the students need to jump over are slowly laid out in a logical way, making 8th grade algebra a fun, playful challenge, instead of drudgery.

The Importance of Creating a Social Community Within a Classroom

Today's review comes from an article found in The Proceedings of the Twentieth Annual Conference of the Cognitive Science Society, University of Wisconsin-Madison, 1998, edited by Morton Ann Gernsbacher and Sharon J. Derry:

Students' Sense of Community in Constructivist/Collaborative Learning Environments, by Helen V. Bateman, Susan R. Goldman, J. R. Newbrough and John D. Bransford.

Background
I will start with a quote that nicely describes what the researchers are referring to when they say that a classroom or school fosters a sense of community, as this is important to the whole article: "McMillan and Chavis (1986) proposed four criteria for defining a sense of community: a) Membership--a feeling of belonging and acceptance, of sharing a sense of personal relatedness. Personal investment and boundaries are important elements of membership. b) Influence--a sense of mattering, of making a difference to a group, and of the group mattering to its members. Influence is bi-directional. c) Integration and fulfillment of needs--a feeling that the needs of the individual will be met by community, as well as a feeling that the needs of the community can be met by the individual. d) Shared emotional connection--an emotional bond that gradually builds as members of a community share events that require high investment of time, energy and effort."

Students in the inner-city tend to not have a full-fledged social community to support them at home; hence, the special school program being studied here is trying to make up for the lack of healthy social instruction found outside of school. A handful of studies done prior to this one indicate that strong group membership in a school can improve academic performance and decrease the incidence of drugs and gang-joining. This study is trying to delve further into this line of research.

The Study
Inner-city students at standard middle school classrooms were compared with those in experimental classrooms in a program called "Schools for Thought." At the beginning of the 6th grade year, all of the students were given a battery of tests and were found to be equivalent in academic achievement in reading, math and science.

The regular classrooms were teacher-led, standard fare. The Schools for Thought classrooms were quite different: The teachers gave the students complex, real-world problems to solve and some guidance, but each student was required to become an expert on some small facet of the subject area, which they then taught to the others and used as necessary. The students did independent research to become experts, and a self-assessment feedback system was put into place. All of the students had to give it their best for the class as a whole to solve the problems assigned by the teacher.

At the end of the school year, another battery of tests was given to the students. The Schools for Thought students had a significantly higher sense of community and of cooperative skills as measured on self-assessment tests. The Schools for Thought students had significantly lower levels of fighting based on reports from the students on their own behavior and on their classmates' behavior. The Schools for Thought students performed significantly better on the testing battery that was used to determine their ability to use collaborative thinking skills to solve social problems.

Significance for Waldorf Education
Except for those Waldorf schools located in third world countries, the students in Waldorf schools generally are not from such a deprived environment as the inner-city kids in this study were. However, that being said, there is much to learn from this study. Waldorf schools use many community-building tricks starting from the very first week of first grade as the class is guided into a healthy, cooperative community. The parents are also led into a community of support for the children of the class (an advantage not available for the Schools for Thought participants). At no point in 1st-8th grade are the children given as much personal leverage in the design of the lesson plan as the Schools for Thought kids, but given the fact that the community-building has been going on for years in a Waldorf school, this does not seem to be a problem at all. The Waldorf high school kids are given something that approximates this amount of control. Cooperation and collaboration are hallmarks of the Waldorf curriculum, and are a very important part of a healthy relationship with the world at large and with fostering a lifelong love of learning.

It would be quite interesting to see testing done on Waldorf students using the same social-talent metrics used in this study.

I really think that the world might be a much better place if Waldorf schools were opened in inner-cities! Imagine the good that could be done if the children were taken into a classroom where they felt like they belonged and could constructively contribute from day one, not just in middle school. The kids were able to build a community within their classroom in a single year, at a point in their school career in which a lot of them are ready to give up (or have already given up) and the benefits were already being seen by the close of the school year.

Humans are social creatures; we crave instruction in this arena. Any curriculum that encourages and supports social learning, cooperation, and community-building is going to provide a valuable tool for its students for years to come.

Definition: Lateralization

The brain is divided into two hemispheres; for many tasks, either one side or the other is in charge. So when neuroscientists say things like, "Linguistic function is lateralized in the left hemisphere," they are basically saying, "The talent of language happens on the left side."

Thus far, it is pretty straightforward. However, there are some nuances that are worth mentioning here.

The Motor Cortex
This section of the brain is special: first of all, part of it resides on each side of the brain (i.e. there's a bit of it in each of the two hemispheres), which does not happen with most other functions. It is what controls our movements. The fascinating bit is that the left motor cortex controls the muscles in the right side of the body, and visa versa.

How does that happen? It comes down to brain development before birth. In the early fetus, there is a brain attached to what will be the spine in a nice, straight line. Then, as the brain folds on itself (making all of those curvy, wiggly lumps), it flips over. It does a somersault. Pretty wild, eh? The result? The left side of the brain is attached to the muscles on the right side of the body (and the right to the left muscles). So, when you wiggle your left thumb, you are using a particular set of neurons in the right motor cortex.

The senses of the skin (such as touch, temperature, pain, etc) do this as well.

Right-Handedness Versus Left-Handedness
All claims of lateralization are always made based upon right-handed subjects, because they are all wired the same way. So, the statement above about language being on the left only applies to righties. Another typical left-brain item is logical reasoning. Many studies have shown that emotional reasoning and music are both housed in the right brain. Again, these things only apply to righties.

And what about the lefties? I'll have to do this more thoroughly later (I need to find an article to review that covers all of the details), but here is the basic gist: a simple majority of lefties (about 60%) are exactly the reverse of righties (so language and logic are in the right brain while music and emotions are off in the left); about 10% are built like righties; and the remaining lefties do not show lateralization at all! They have language and logic in the left and the right; they have emotions and music in the left and the right.

This is why brain scientists are not interested in left-handed subjects when doing studies to determine where in the brain certain functions are located. There are some researchers who are trying to figure out what exactly is going on in lefties' heads, but they could not get started until much of the brain had been mapped out in righties: this makes sense, since they could not possibly be able to understand anything without a base-line map.

Lefties' motor cortices work exactly like those of righties: the right motor cortex makes a person move their left thumb.

It is not too hard to understand why it is that forcing a lefty into being a righty can cause permanent brain damage!

In sum
Lateralization at its most basic is about which bits of thinking are located in which places of the brain. However, the generalizations only apply to right-handed people. Also, certain functions happen on both sides, but each side of the brain controls the opposite side of the physical body.

Definition: Neural Plasticity

I have been trying for two weeks to come up with a simple, straightforward definition of "neural plasticity" with a nice example, and I have been completely stumped! Hopefully, I will be clear enough!

To understand "neural plasticity", you have to go back to the original definition of "plastic". I always think of some hard substance; but really, to make a hard, plastic item, they pour into a mold a liquid that is able to harden into any shape. This is the key piece: to be plastic is to be infinitely shape-able.

What does it mean for your brain to be plastic? Basically, there is a bit of wiggle room within brain function. For instance, an area that is not being used for anything can be taken over by neighboring areas. Also, there may be one ideal part of the brain for learning a particular type of knowledge, but other areas can step up to bat in a pinch: I will definitely be discussing this when we get to teaching reading.

Usually, each section of brain is prepared to engage in a particular type of activity. The ability to switch from one function to another is remarkable and is quite handy for those who suffer from brain injuries. However, there is a limit to this type of self-repair.

Exactly how plastic the brain is is an active area of research; the concept figures prominently in many research studies.

Definition: Critical Window of Development

The quick explanation: A critical window is the period of time during development that the brain is open to a particular type of experience to result in a particular talent, and after the window closes, this talent can no longer be learned.

Boy, that's a mouthful!

Here's a pretty straightforward example from vision: the condition called strabismus (lazy eye). With this condition, the muscles that move the eyes around do not act in a coordinated way; one eye lolls about while the other is looking at the object that the person is interested in. There are ways of fixing this problem. However, if it is not fixed by the age of 4, then there is an aftereffect: the person is never able to see in 3-D. They can take guesses based on relative size and looking at shadows, but they cannot take the information from each eye and integrate it into a single picture with depth, the way most people can. I have a friend whose strabismus wasn't fixed until too late, and she cannot drive at dusk and dawn because the hazy light removes all of her depth cues. She can't tell if things on the road are 10 feet versus 50 feet away, which can be a major safety issue.

Once the critical window for depth perception has shut, there is no way that a person can learn this vital skill. There are no surgeries or therapies that can make the brain use the pictures from both eyes at the same time to figure out how far various objects are. Try spending 10 minutes walking around with one eye closed. You can estimate some distances (like my friend can, using shadows and sizes) but not all of them. Try it just after sunset to really disorient yourself!

Most people have 3-D vision, because most people have the experience of having their eyeballs pointing in the same direction before their 4th birthday. The only way that we know that there is a window at all is that there are some people who do not follow the regular pattern and have lifelong debilities as a result.

It can be tricky figuring out when developmental windows open and close. After years of research, however, it is a given that they do exist. What makes a brain become receptive to the input (the window opens)? What makes it no longer receptive (the window closes)? That is still an open debate; for numerous windows, however, it looks like the rush of hormones at puberty are the trigger. Many windows close between 12 and 18 years of age. Of course, several others open up at this time, too, so I'm not saying puberty is bad per se, but that the hormones are a needed part of proper brain development, acting almost like a timer.

You can try to teach someone something until you are blue in the face, but if it is before their window has opened up, or after it has closed, then they will not learn it. Or they might learn it a little bit, using the wrong part of the brain, but will never be good at it.

Not everything has a critical window. Bicycle-riding, like all talents that rely on man-made tools, does not have a window: once your sense of balance is pretty good, you could learn. Or you could wait 20 years, and you could still learn! I suppose that an elderly person who has lost their sense of balance and has never before tried to learn how to ride a bike would have trouble, but that is not because a specific "critical window for bicycle riding" has closed.

So, a quick recap: The potential for learning something becomes available inside the brain, the person has some time (usually measured in years) in which to learn it, and then a change happens inside the brain that prevents a person who has not yet had the right experiences from learning it. All critical windows open at a specific time and shut at a specific time during the course of development.

Learning about Fractions

Today's review comes from a chapter in a book in which a series of studies about fraction-learning is used as an argument in favor of the author's view of learning. The focus of this post will be on the fraction studies themselves.

Epigenetic Foundations of Knowledge Structures: Initial and Transcendent Constructions
by Rochel Gelman, UCLA
In S. Carey and R. Gelman, (Eds.). The epigenesis of mind: Essays on biology and cognition Hillsdale, NJ: Erlbaum Associates. (pp. 293-322). 1991

Internationally, fractions and numbers with decimal points are a big sticking point with students, even into high school. Kids don't really grasp deep down what these sorts of numbers are, so they just memorize a bunch of methods to cope with them, that end up not working all the time.

The group of studies related in this chapter all involved testing children who had been learning about fractions and non-whole numbers with a standard textbook/workbook series that first introduced the most basic concepts in kindergarten, and added more lessons each year thereafter. Gelman was an author on all of them.

The first study tested children in kindergarten through second grade. They failed miserably on the researchers' tests. While they had mostly memorized facts such as that "1/2" was pronounced "one-half", the understanding went no deeper.

The second study tested kids in grades 2 through 8. The children's scores were divided between the regular students and the ones enrolled in the gifted classes. All of the children did poorly in all of the tests in the 2nd and 3rd grades. The first difference between the groups appeared in the 4th and 5th grade scores. There were 5 types of questions asked. For each question type, about half of the regular kids got the answer right, while at least three-quarters of the gifted kids got the answer right. The 7th grade gifted students all got perfect scores, while the regular students still did not reach that point by 8th grade. The children were also asked to explain their reasoning behind their answers for certain questions. All of the gifted children gave better explanations (even though the youngest kids still got lots of answers wrong), which indicates that they were all somewhere along the path of developing a sense about what these strange numbers really mean.

The third study looked at a new way of introducing the concepts of fractions. The researchers worked with kindergarteners and first graders. They worked with some of the children on some tasks using a number line and shapes (circles or squares that were whole, in halves, thirds, etc.). They also related the shapes to the fraction names. All of the children were then tested using a bigger number line as well as with other tests. In general, the children with the lessons did better than those who did not work first with the researchers, but they did best when the names of the fractions were not used. In other words, they could deal with the shapes (and the ideas that they represented) but not with the names that had already been drilled into them at school!

The conclusion to this chapter was that kids come into school convinced that numbers are only things that can be counted. This is enough for a young child to know, but not enough for a true understanding of the larger world of mathematical reasoning. Something needs to happen in the child's understanding before the wide world of math can be pried open by the child. Unfortunately, the way that fractions are generally introduced can prevent the child from including it as a part of the number stream. The author suggests immersing children in number experiences, and allowing them to practice using part-whole concepts, before going on to teach the mechanics of using fractions.

How Waldorf Schools Teach Fractions
There is a big to-do among Waldorf teachers about the "9-year-old change", which my oldest daughter is now entering into. The children become more grounded in their bodies and more aware of themselves as individuals. They begin to understand part-whole relations because they can finally feel, deep down, what being whole is. This is why fractions are not introduced until 4th grade, a point at which the entire class has finished the change. Then, the children are deliberately led to the fractional concepts by manipulating blocks, drawing items and partial items, woodworking, and other hands-on activities. Only after they have spent several months working with the unspoken concepts are they given names and written labels. And the whole class knows what these numbers are and how they fit into the counting numbers by the end of the year.

By waiting until the kids are ready, and then immersing them in appropriate learning situations, all before even thinking about introducing labels (both written and spoken), the Waldorf curriculum actually goes well beyond the recommendations of Gelman and her colleagues on the ideal way to teach fractions.

Introduction

Why I am starting this blog and what are my initial plans for the site? Enquiring minds want to know!

I send my kids to a Waldorf school, and I am going to be reviewing journal articles from a variety of fields (mostly neuroscience) that seem relevant to this educational model. Am I biased? I hope not! However, you need to understand this about me: I chose a Waldorf education for my children because I already knew that the vast majority of the brain research supports it. If I do manage to find a study that questions Waldorf methods, I will still review it.

We recently had an all-school parent meeting, focused on satisfaction and retention. The point that I brought up was that it can be very alienating to keep hearing the curriculum explained in terms of "because Steiner said so". I happen to think that the guy was a genius, and way ahead of his time, but the fact remains that when a parent asks "why?" and the answer is that some guy who has been dead for 80 years said so, it actually gets interpretted this way: "Steiner said so and no one has come up with a valid justification ever since." Which isn't true! After this meeting, a number of people urged me to write an article (or a series) for the Waldorf parent magazine, Renewal. The thing is, with 4 kids aged 8 and under, I don't have 40 hours to sit undisturbed and hammer something out.

Hence, the blog format. I am hoping to review 1 or 2 articles per week; after 6 months or so, I ought to have something of sufficient depth and breadth to work with. In the meantime, if you all could tell me how clear my writing is, ask me questions that are most pertinent to you, and suggest articles to review, I would greatly appreciate it! All of this will help make my eventual article(s) of the highest possible quality.

At first, I will be reviewing papers that are lying around my house. I graduated from MIT nearly 10 years ago, and then spent 2 years in grad school, so the freshest papers that I have around here are at least 8 years old. But this is not a problem! Cutting-edge research may play a small part in this enterprise, but bodies of work that span decades will probably end up being more useful, so the age of the materials is nothing to worry about. And then, after exhausting my in-house collection, I will go seeking other papers; I suspect that a lot of them will be recent.

So, that is my modus operandi. I look forward to working with you to make this site as useful as possible!