Education ∪ Math ∪ Technology

Tag: education reform (page 1 of 2)

Six things about math education which do not work

There are six things (at least!) about mathematics education which do not work:

  1. pacing for coverage of curriculum rather than focusing on effective student learning,
  2. fear that if students take more than five seconds to solve a problem, they will give up,
  3. teachers spending more time talking than students get to spend thinking,
  4. teachers working in isolation to plan lessons, units, and understand their students,
  5. students being forced to work in isolation from their peers as potential resources,
  6. and an obsession with procedural fluency over conceptual understanding.

The objective of my current work is (collaboratively with the rest of the members of my team at New Visions) to develop tools for teachers that will help address as many of these issues as we can. These tools will be used collaboratively with teachers to look at student work and try to address the question, "What were these students probably thinking?" and "How can I help this student further their understanding of mathematics?"

Education reform ineptness

"If you are absolutely no good at something at all, then you lack exactly the skills that you need to know that you are absolutely no good at it."

John Cleese

Perhaps this is part of the reason that so many education reforms that are attempted fail so badly? Could it be that at least some of the people involved in education reform are just so completely inept that they are not even able to judge their own performance at all?

The good news is, if you know you are bad at something, then at least you are not completely inept, because if you were, you might think you were good at it. The bad news is, people who are completely inept are unable to judge or recognize their ineptness, which may make them push ahead in ways which are harmful to the rest of us.

Many ways of learning how to ride a bike

When I learned how to ride a bicycle, I practiced with training wheels first because my parents thought that it would be too difficult for me to learn how to balance myself, steer, and pedal all at the same time. I eventually learned how to ride a bike without training wheels but it was challenging for me. 

With my son, we started him on a like-a-bike which would let him practice balancing on a bicycle (which some parents argued with us was the most difficult part of learning to ride a bike) before having to learn how to pedal. We then later gave him a bicycle with training wheels so he could practice pedalling separately from balancing himself but honestly, we didn’t find it helped much. When he finally had a standard bicycle, he still needed to learn how to pedal while keeping himself balanced.

I’m not sure either method is best. It probably depends on the kid which technique we should use. Maybe we should have just started our son on a regular bicycle.


So now look at mathematics education. Our goal is to have students think like mathematicians, and to know enough mathematics to be able to use it in their thinking. Is it entirely necessary that all of them learn it the same way? Can we find different ways to engage different students in the act of learning how to be a mathematician?

Children are not railroad trains

"Timetables! We act as if children were railroad trains running on a schedule. The railroad man figures that if his train is going to get to Chicago at a certain time, then it must arrive on time at every stop along the route. If it is ten minutes late getting into a station, he begins to worry. In the same way, we say that if children are going to know so much when they go to college, then they have to know this much at the end of this grade, and that at the end of that grade. If a child doesn’t arrive at one of these intermediate stations when we think he should, we instantly assume that he is going to be late at the finish. But children are not railroad trains. They don’t learn at an even rate. They learn in spurts, and the more interested they are in what they are learning, the faster these spurts are likely to be." ~ John Holt, How Children Learn (1984), p155

John has certainly identified the problem, the question is, how would we build our system differently?

A lot of people have identified this problem, but I have seen less solutions to it than people expressing their outrage at it. It is certainly true, we do treat children like railroad trains, and expect far too much regularity in how they learn.

Further, our education system has become more like an accelerating railroad train in which each year children are expected to be able to do more sooner. Algebra in 8th grade. Reading in kindergarten. Essays in 5th grade. Why do we feel the need to keep up with the Joneses?

Designing a new system will be tremendously difficult. We have an enormous amount of cultural inertia in our current system. It is a difficult problem! How can we take a system wherein we fund students to attend school at a ratio of one teacher for every 20 children (on average) and find ways for each of these children to learn everything we feel is important in order for them to become adults?

Here are some suggestions, which are by no means exhaustive.

  1. Trim the list to that which is really important.
  2. Cultivate a desire to learn more, and the ability to learn for oneself.


Why people often do not accept the research

Via the @BCAMT email list-serve:

"[T]here is an interesting (and disturbing) literature on situations in which information does not change prior biases or decisions. The word I have seen is ‘motivated reasoning’.

Interestingly, I ran into a problem of ‘motivated reasoning’ with a class of future teachers. The question is: when would research about the teaching and learning of mathematics change their classroom practices. A common response to articles, given some practice in critiquing research, was:\

– if I agree with the conclusion, the article was reliable;
– if I disagree with the conclusion, then here are x reasons why the article was not reliable and I should not change my practices!" 

Dr. Walter Whiteley

Dr. Whiteley works with pre-service teachers, and would like me to point out that they are still in the middle of articulating their own personal theories of how learning and education work, thus they lack experience in schools from the other side of the desk. It is therefore possible that this is an issue isolated to pre-service teachers.

On the other hand, I have seen people vehemently defending a position that has no merit simply because they are unwilling (or unable) to see that the evidence is mounted against them. I have also noticed many times that months later, this person has changed their perspective, sometimes claiming that the opposite to what they had previously believed was their belief the whole time, so maybe that argument influences their thinking later, and they are more willing to change on their own.

It takes enormous strength of will to remind ourselves of our cognitive biases, and act against our instinct to defend our mistakes. I can’t say I’ve succeeded at this all that much. Does anyone?


Imagine something different

See this piece of paper?

Piece of lined paper
(Image credit: D Sharon Pruitt)


Throw it away.

Imagine the limitations of the piece of paper shown above do not influence how you share the record of learning your students have done, with their parents, and the wider community.

Now remember the history of grading, which started with one William Farish (in Western culture – Chinese culture has been apparently giving grades to students for many centuries for the purpose of sorting their children into social classes.). William Farish (re)invented grades as a way to increase the number of students he could "teach’ for the purposes of lining his pockets (at the time, more students meant more money).

What would you do differently to share your student’s evidence of learning, if the limitations of the paper above did not exist, and if your purpose was neither to sort students into social classes or line your pockets by being able to teach more students? 

An Unfamiliar Revolution in Learning

This video, shared via the Good blog is a must watch. Find six and a half minutes to watch this video, and ask yourself what changes would be necessary in your school to make it more like this one.


The work that this school does on teaching empathy, and understanding what it feels like to be another person, is an incredibly valuable life-skill. The abstract reasoning that one gains as one learns empathy has to have side-benefits for academic reasoning as well. If I know what it feels like to be you, and what you likely feel like, I may be able to better make predictions about other types of objects in the world as well.

I particularly like the five habits of mind the school has used for their conceptual framework:

  • Evidence – How do you know?
  • Conjecture – What if things were different?
  • Connections – What does it remind you of?
  • Relevance – Is it important? Does it matter?
  • Viewpoint – What would someone else say? How would someone else feel?


Our words are not enough: It’s time for action

I’m fortunate to work in a school which gets it. We do a lot of the stuff that people on #edchat are describing as innovative, particularly in the area of student leadership and assessment policy. I feel respected every day, and my opinions and thoughts have a real impact on the direction our school goes. I know this is not true for many teachers though, and I hear it through the discussions we have on Twitter. It seems most teachers work in places where they have very little influence on school policy.

We discuss a lot of stuff on Twitter, but given the number of people meeting, and our individual influence, I have often wondered how powerful we could be as an organizing force. I’ve often found #edchat to be a great starter of ideas, but the ideas seem to go nowhere and we often talk in circles without seeing any change. Sometimes #edchat feels like a gigantic echo chamber where we all pretty agree with each other, and find the best ways to share our agreement in 140 characters. 

I don’t want to change #edchat, but I would like to see space for organizing group action we can take, and it seems to me that Twitter would be a valuable tool for doing this. I’ve proposed the #edaction hashtag, where educators can post ideas for action we can take, and then we can meet to decide on actions (and brainstorm future actions) we will take for the week. 

Here are some suggestions of relatively easy actions we can all take:

  1. Talk to a neighbour education and your vision for what it should look like. Listen to their opinion. If you disagree, discuss core beliefs and find out what you agree on.
  2. Write a letter to a traditional print newspaper. We might be firm adopters of the digital world, but many people with influence do not read blogs & follow Twitter. We need to spread our message to a different audience.
  3. Share ideas about education we have in #edchat with your colleagues at school. You don’t have to become an evangelist for #edchat, but we need to see what our non-Twitter educator colleagues think and ensure that when innovative ideas come out of #edchat that a larger community hears them. We need critical evaluation of our ideas as well, and our colleagues are a great source of criticism, and improvement of our thoughts.

Please post other ideas you have for #edaction in Twitter using the #edaction hashtag. Our objective? Take action to transform education.

School Bells Interfere With Learning

I hate being interrupted in the middle of a good learning session with my students. It has happened hundreds of times in my career because of an archaic device we use in schools known as a clock. The clock itself isn’t evil, but the way we use it in schools has serious ramifications on how our students learn.

First, because we partition students into neat packages called subjects, they are implicitly taught that learning is something we do in compartments. If you try and introduce a little bit of another subject in your subject, students object, saying "This isn’t English, Mr. Wees. Teach us Mathematics." (I’ve actually had students tell me that). Where in the real world is learning sectioned off like this? Mathematics use English (and other languages) when they explain their discoveries to other people. Biologists use geography to decide where to start their research. All of what we learn is interconnected, and more of these connections need to made obvious to the students. This is not easy to do in a school with nine 45 minute separate blocks.

Next, we tell students to stop working on a particular project when the time is up. We enforce time limits on learning! While I’ll grant that real life has deadlines and limits, it very rare indeed that someone has to complete a task "within the next 15 minutes because class will be over" (I’ve said this in my classroom, so many times I can’t keep track). Maybe you have to finish something by a particular day, or by the end of today, but you are in charge of how much you work on the subject, and not the clock. It is ridiculous the number of times I’ve seen students actively engaged in learning and have it wrecked because the end of class came. Worse, I’ve filled the last 10 minutes of a class with a meaningless activity just to ensure that I use every minute I’ve got.

We also assume that each subject area needs the same amount of time each week, and try to make sure that everyone gets their equal share of the carefully apportioned time for courses. In our school I teach IB Mathematical Studies, which requires at least 150 hours of in class instructional time. My school has carefully arranged for about 160 hours, just in case I lose some to field trips, student illness, snow days, and other time sinks. Oh right. Field trips, those banes of our teaching existence which make it so hard to plan. It’s not like any REAL learning happens during field trips anyway.

Clocks are part of the systems world of a school but they have come to rule our life world. We have let ourselves become subject to fixed schedules, daily routine, and the drudgery of a factory-like system. I’m not saying that we can do without the clocks, but maybe we need to find ways for our system to be more flexible, to allow the learning to extend when necessary, and even send off kids early for another opportunity to learn, when their lesson with us is done. Maybe we should even rethink how we schedule kids, and consider other instructional models. There are schools where there are no bells, no classes like what you would see in a traditional school, just kids (and adults) learning.

I used to be a master at memorization

When I first started my career I struggled. A lot. My first job was in the School for Legal Studies which when I joined it was a relatively small high school by New York standards. I had three classes each day, two of which were double period classes. If you’ve ever watched Michelle Fiefer’s "Dangerous Minds" you’ll understand what my classes were like. It took me 3 months before I actually got one of my class’ attention.

I had one lesson which worked really well during my first semester. It was suggested to me by an Assistant Principal for Math. Basically, I started a class singing the quadratic formula song. Instantly the class went quiet. One student asked me to sing it again, so I did. By the third time I was singing it, some students were joining in. By the fifth time, only the quietest and shyest of kids weren’t singing with me. After the singing I managed to hold their attention for 20 minutes of examples of how to actually use the quadratic formula to solve equations.

For three weeks every time my students came to class, they sang the quadratic formula song when they entered. I’m still in touch with some of the students from this class and all of them remember that we sang a quadratic song although most of them don’t remember all of the words.

Over the next three years, I learned a bunch of tricks to help students memorize the bits and broken pieces that represented the NY State Math curriculum. Together my students and I sang songs to remember formulas, used hand signals to remember the relationship between an implication and its inverse, converse, and contrapositive, and deciphered calculations of algebraic groups to look for transposes and inverses. None of it made any sense to the students, it didn’t have to, they could memorize it.

I didn’t use flash cards or other tricks to help my students memorize these math facts. I used every other trick I could think of. I became a master of memorization. My students did reasonably well on their exams each year compared to their peers in other classes but I never felt like I achieved more than mediocre success because my pass rates really never exceeded 60% overall.

I regret that I did this. I wish I had more guts back then and had been willing to slow down and instead of trying to race through a bunch of disconnected concepts that I pulled out the ones which were most relevant in these students’ lives. I also wish that I had discovered my constructivist methods of teaching earlier in my career.

This actually isn’t the whole story; this is what I regret most from those early years. I also remember another side to this story which was that of an educator who endlessly experimented with different techniques to help his kids understand math.

I remember staying after school with students building model water slides so we could experiment with time-distance graphs. I remember bringing in pictures of buildings in my students’ neighbourhood so my classes could figure out the equation of the lines in the pictures. I remember buying a class set of long tape measures and protractors so we could go outside and calculate the height of the gigantic block which passed for a school in NYC. I remember being a good educator.

I do regret the endless drills and worksheets I passed out to my students. I am also eternally grateful that I found another way, a better way, and no longer rely on cheap parlor tricks in my teaching.