Education ∪ Math ∪ Technology

Month: September 2013 (page 1 of 1)

Welcome to the no fun school

When my wife and I moved our family to NYC from Vancouver, we knew that we would have to get used to many changes, we just did not know how large some of those those changes would be.

We started our planning in January because we knew that finding an affordable place to live and a school for our son was going to be challenging. We created a map to overlay the information about schools we were gathering with information about rental prices. Very quickly we noticed a few trends.

Rental prices in New York appear to be at least partially influenced by whether or not the local zoned school is considered good by the community. We found ourselves, over and over again, finding a school that appeared from the outside to be good, only to find that the cost of rent in the community surrounding the school was out of our price range.

We also noticed something else disturbing when we started searching for schools. Every school review website we looked at included the same type of statistics at the top of their review; what percentage of kids were eligible for the school lunch, what the average number kids being successful on the state math and reading tests, and what the racial distribution of the school was. 

We found the sharing of these statistics to be kind of bizarre in a city whose education system claims to support reducing inequity in education. If you wanted to greatly increase racial segregation in schools, I think of no better way than to loudly announce which schools "those black and latino kids go to." If you wanted to create the misconception that schooling is primarily about getting high test scores, a great way to start would be to highlight those test scores as much as possible for each school.

We even looked into some private schools but either found them to be completely unaffordable or making dubious claims. One Montessori school my wife visited told her over the phone that "they stop using the Montessori materials after kindergarten." When she visited the school, she found their classrooms were filled with wall to wall workbooks and the desks were carefully arranged in rows. We were reminded of the saying "Buyer beware!"

We were also really surprised by the lack of options for alternative types of elementary schools in the NYC public system. For a system that advertises itself as a "school system of choice" we were shocked to find out that there were no Montessori, no Waldorf, no International Baccalaureate, no Reggio Emilio, and no Democratic schools in the public system, notably all of which exist as public schools in British Columbia (there is one Montessori charter school in the Bronx that opened recently). There is not even, as far as we know, a way to search for NYC public schools by educational philosophy. We did find a few progressive schools but they were either open by lottery or located in some of the most expensive areas of NYC. Choice in NYC, it seems, is limited to the rich.

We eventually settled on our local community public school, because they advertise that they are exempt from the mandated city-wide curriculum because of good performance, they have a strong arts and music program, and it is conveniently located two blocks away from a house owned by my wife’s family.

We moved to NYC at the end of June, rented a suite in the family owned house, and spent the summer anxiously waiting to see if we had made the right decision.

About a week before school began, my wife went down to the school to register our son for school. Actually, she went down three times, each time being told that school registration wasn’t open yet, and that she should come back tomorrow. On the third day, she finally found someone to help her fill out the paperwork required to register my son, which included a report card from my son’s previous school that would end up never being forwarded to his teacher, despite assurances to the contrary.

The first day of school was a bit disorienting for us. The sidewalks outside the school were jam packed with kids and their parents, and there were various adults trying to shepherd the crowd around and keep them off the street. There was no inspiring first day of school speeches, no friendly faces welcoming our son into the school, just loud voices telling us to "stay off the street and to leave as soon as our child is safely inside the school." We waited until our son was safely inside the school, and then we left, slightly bewildered by the experience.

"This is the no fun school," my son told my wife a week later, "I don’t like it." He told us he had done nothing but worksheets by the end of the first week of school.

I took the morning off from work, and my wife and I went together to the school’s Meet the Teacher event. Before we met with my son’s teacher, the Principal of his School talked about the school year. "We don’t have any classroom aides this year, unfortunately," she started, "We had to let them go because we couldn’t afford them anymore." During her speech, the challenges with the school budget came up a couple of times, and the need to prepare students for the 3rd grade tests came up three times. It is probably worth noting that the group of parents she was talking to all have children in the school in 2nd grade. After our time spent with our son’s classroom teacher, we both left with a lot more questions about the school.

There is no morning recess. There is a lunch recess but the kids were not allowed outside for the first two weeks of the year because there were no aides available to supervise them. The kids aren’t allowed to run outside during lunch recess because the courtyard is too crowded. My son only has gym once a week (and apparently some children do not get gym at all) during which all he has done so far is walk around the gym. The dance teacher was sick for the first two weeks of school but fortunately they were in this past week. He only has science and social studies for two short periods each a week; the rest of his time is spend on reading, writing, and arithmetic.

My son has homework for the first time, lots of it. He is supposed to read for at least 15 minutes a day, which is never a problem for him, and write down 5 sentences based on some spelling words he needs to know, and do a math worksheet. He typically finishes the math worksheet in a minute, but he has been struggling with the writing, which is easily ten times as much writing per day than what he was expected to do last year.

We met with the teacher recently to voice some of our concerns. I won’t share exactly what was said in this meeting, except to say that my wife and I both left feeling like my son’s teacher is a caring and thoughtful person who is deeply committed to her students, and feels greatly constrained by the mandates she is given by her school district.

A couple of years ago, a friend of mine told me that my concern about what type of school I wanted my son to attend was overblown. He told me, "The most important factor in determining how well your kids do well in school is if you have books on your shelves at home. Do you have books? Well, then your kids will do fine no matter what school they go to."

While I understand my friend’s argument, I think he and I have different goals for our children’s schools. I’m not so concerned that my son does well at school, which is largely a function of how well he does compared to other children (and I’m not really interested in that kind of comparison), I’m concerned that he will learn messages about what it means to learn and to be human that are very different than what I believe. What messages about the value of learning will he learn in a school that keeps talking about the tests the kids will have to take? Is my son learning to be an autonomous learner? What about the value of living a healthy and balanced life?

A couple of days ago my son told my wife, "It’s okay mommy. I’m getting used to this kind of school." 

I know son, and that is what scares me. 



Update: The server this blog is on had its hard drive crash recently, which means that 12 comments on this post have vanished forever. I’m sorry about that. Thank you for everyone’s support!

Experimenting with random walks

Root 2 in visual form

Based on this image created by Matt Henderson, I decided to write something for myself that would explore other possible random walks, although mine are generated in a slightly different way than what Matt did.

Each of these images (which will only appear in Chrome, Safari or Firefox, sorry people still using Internet Explorer) is a visual representation of the first 100, 000 digits of the number where each digit of the number corresponds to a different rotation. They may take a while to calculate, depending on the speed of your computer.

Note that for the last "random walk", I increased the scale significantly over the other random walks so that it made the pattern more obvious.

Ways to use technology in math class

Here are some ways you can use technology in your math class which are more interesting and innovative than using an interactive white board or having students watch instructional videos. Note that these ideas are all examples of potential student uses of technology.

Students could:


Any other suggestions of ways students can use technology in order to improve their mathematical reasoning?

There are no aha moments

If we understand learning to be the developing of neural connections in the brain, then necessarily there cannot be true aha moments (or more accurately, every moment is an aha moment).

Lets suppose that a child has a (flawed) model of how something works. Each time they are presented with information, they build new connections between neurons in their brain, while also occasionally (usually while they sleep) removing connections that are not used. Over time, this gradually results in a child having competing models for understanding how something works.

At some point, the model that works best is the one that becomes used, and the neural connections that represent the unused model are eventually severed, and we might say that the child is exclusively using the new model.

At no point does a child suddenly flip from only using one model exclusively to using an entirely new model because this would require making many, many different neural connections simultaneously for the new model to function. This sudden flipping is what is commonly called an "aha" moment, and it doesn’t exist. Learning is not the sudden acquisition of new models of understanding the world, but the gradual shifting between competing models, none of which probably completely describe the world.

Aside: No one has a perfect model for understanding the world, because that requires a complete set of all possible "true" states of the world.

Different possible stimuli of the same basic information could lead to different models being used. For example, suppose I asked students to solve 4 + 5 verbally as compared to representing this symbolically in writing. It could be that students use one of their competing models to answer the verbal form of a question, and a different model to answer the written form, and in some cases arrive at different results. It could even be true that different people asking the same question results in different models being used!

When my son was in grade one, I participated in a student led conference in which he laboriously demonstrated using a regrouping method how 3 + 9 = 11. In the context of his classroom and with a written question, he answered the question with one model. I did nothing to offer feedback on his model at the time. 10 minutes later, we were driving in the car, and we played a number puzzle game where we took turns saying numbers and trying to figure out how to get that number using arithmetic operations. I said 12, and my son responded with 1 + 11 is 12, 2 + 10 is 12, 3 + 9 is 12, and so on. My son used a different model, and arrived at a different result.

If this theory is correct (and to be clear, it is just a theory), then it has implications for instruction. The first implication is that in order to help students develop models, we need to introduce all them to both different representations of ideas (ie. representations of other people’s models for understanding), from different people (ie. teachers, students, and parents), and different modes of processing that information (verbal, written, symbolically, manipulatives, etc…). It also suggests that we should not assume that because a child can respond in a way that suggests they have a solid model of understanding once, in one context, that this means that they actually have such a model. It also means that what children know how to do, or do not know how to do, is unlikely to be successfully captured by a system that assumes binary understanding of concepts (concepts are not known or unknown, we have models which seem to work in some contexts, and may not in others).

Note: It is probably worth noting that my use of the word model is a simplification of the set of neural connections we use in our to process and store information, and is almost certainly an incredible simplification of those processes.

How do you help develop mathematical curiosity?

A recent New York Times article talks about how to fall in love with math. Related to this issue is how to develop mathematical curiousity in your students as a math teacher. In no particular order, these are some of my suggestions.

  1. Build a strong positive relationship with your students. They will follow you farther into the unknown if they believe in you and trust you.
  2. Give students opportunities to explore mathematical ideas for themselves without a predefined goal, except that which your students may define themselves.
  3. Introduce your students to mathematical mysteries; such as the fact that there are as many fractions as whole numbers, but too many decimal numbers to count, or that there are shapes with infinitely long perimeters, but finite areas.
  4. Ask questions you cannot answer.
  5. Give problems which are easy to state but which either have no solution or the solution is not yet known to anyone.
  6. Be mathematically curious yourself and demonstrate this curiosity to your students. Do mathematics yourself and make the mathematics you discover public.
  7. Let your students do most of the thinking in your class. Too often we do important thinking for students, and if one is not thinking, one cannot be engaged (obedience without thinking is compliance, not engagement).
  8. Open the black box of problem solving and give students problem solving heuristics they can use themselves.
  9. Don’t grade everything. Leave as much as you can as activities which are worth doing because they are interesting, not because someone will judge their performance.
  10. Develop learning mathematics as a social activity. As the African proverb goes, "If you want to go fast, go alone. If you want to go far, go together." Let students explore things on their own, or in small groups, but do some mathematics together in ways which respect every student’s contribution.
  11. Help students learn some of the history of mathematics and the social contexts under which it was developed.
  12. Teach mathematics as a narrative, rather than as a series of disconnected facts. Too often children experience learning as a series of sitcoms. Math should be more like an epic journey.

The Teaching Gap

In the fall of 1994, after several months of watching tapes, the project staff met to present some preliminary impressions and interpretations. We invited distinguished researchers and educators from Germany, Japan, and the United States to attend, and we listened intently to what they had to say. We were ready for a fresh perspective. It came late on the last day of the meeting. One of the participants, a professor of mathematics education, had been relatively silent throughout the day. We asked him if he had any observations he would like to share.

"Actually," he began, "I believe I can summarize the main differences among the teaching styles of the three countries." Everyone perked up at this, and here is what he had to say: "In Japanese lessons, there is the mathematics on one hand, and the students on the other. The students engage with the mathematics, and the teacher mediates the relationship between the two. In Germany, there is the mathematics as well, but the teacher owns the mathematics and parcels it out to students as he sees fit, giving facts and explanations at just the right time. In U.S. lessons, there are the students and there is the teacher. I have trouble finding the mathematics. I just see interactions between students and teachers." (James Stigler and James Hiebert, The Teaching Gap, 1999, p25-26)

The Teaching Gap is a synthesis of the research done during the TIMSS video study. This study was an effort to randomly sample classrooms from Japan, Germany, and the United States, and videotape randomly selected lessons from those classrooms. From those lessons, another randomly subset of lessons was chosen, and carefully analyzed and coded by researchers. The objective of the study was to attempt to determine if there are cultural differences in how mathematics is taught in different countries.

The conclusion of the researchers is that although the classrooms look very similar in many ways, there are vast cultural differences in how mathematics is taught, on average, in the different countries. They also concluded that these differences far outshadow the relatively-minor-by-comparison differences between individual teachers in each of the countries. In other words, the difference in teaching methods between a typical Japanese teacher and a typical U.S. teacher are greater than the average differences in teaching between randomly selected U.S. teachers.

The quote above does not quite accurately capture the U.S. classrooms. The researchers concluded that there is mathematics being taught in U.S. classrooms, but that the mathematics content of a typical U.S. classroom is less than that of the other two countries.

A couple of pages later in the book, there is a table comparing typical lessons for German, U.S., and Japanese classrooms. One footnote was particularly interesting to me. In the U.S. classroom, it is "[t]ypical for teacher to intervene at first sign of confusion or struggle," and in the Japanese classroom, it is "[t]ypical for students to struggle with task [sic] before teacher intervenes." (The Teaching Gap, 1999, p30).

Japan currently ranks 5th in the world in an international comparison of mathematics achievement worldwide. The U.S. currently ranks 9th in the world using the same comparison.

Is this data (from nearly 20 years ago) still relevant? If it is, should U.S. teachers adopt the style of Japanese lesson plans? How much do other factors in culture from outside the classroom influence these findings?

Balance puzzle

Screen shot of the balance puzzle

I created a new puzzle, based on something I played with when I was at the New York Hall of Science. The idea of the puzzle is that given three numbers, you are trying to balance both sides of the equation, with the constraint that different positions on each side of the equation are worth different amounts. At this stage, this is a prototype, so the numbers are chosen randomly, but the eventual plan is to build this balance puzzle into a series of challenges, each of which will probably have multiple solutions.

One hope is that a puzzle like this helps bridge the gap between arithmetic and algebra, by providing an activity which uses arithmetic operations but embeds algebraic reasoning. I also hope that this app helps reinforce that the equal sign (and the inequality symbols) are not indicators that one should do an operation, but rather that they represent a relationship between the two sides of the equation.

Check out the puzzle here:

I’d love feedback before I move onto making this puzzle more robust.

What is developmentally appropriate?

My son programming
(This is my son learning how to program in Turtle Art, at age 4)

As educators, we give work to students based on what we consider to be developmentally appropriate, and what we feel they have the capacity to learn, which then helps them develop exactly in the ways we expect. Isn’t this a bit of cyclical reasoning?

Shouldn’t we challenge our students to find out what the limits are to what they can do, rather than assigning them limits before we have sufficient evidence of what those limits might be? I’m not saying that the children we teach do not have limits, only that we do not know what those limits are without attempting to stretch them.