Education ∪ Math ∪ Technology

Month: June 2013 (page 1 of 1)

[Bad math] Polling

Poor polling practice

I hope no one takes the results of this poll by the Conservative party very seriously. Before they even collect results, I know that those results will have significant bias.

  1. They promoted their own political party in large bold font just before their survey question. This is equivalent to saying "Chocolate is AWESOME!! Do you like chocolate, yes or no?"
  2. They are relying on people to take the time to respond to the survey themselves, which suggests that they will end up with a small group of people who are die-hard opponents of the Conservative party, and probably a larger group of Conservative party supporters. Who else would bother voting in this poll?

Note: If you were to substitute any other political party into this poll or a similarly run one, it would be just as invalid.

Lego and Minecraft in Math

This article originally appeared in my school’s monthly magazine, and was co-written with Chelsea Todd, a 4th grade teacher at my school. Note that the lesson idea presented below is Chelsea’s idea.


The 4th grade math classes were recently completely engaged with a challenging but slightly unusual mathematical task. Students had been studying area and perimeter with the guiding question, “How do area and perimeter help us to construct things?”. For their summative project, students designed and constructed a model building design, starting first with the architectural blueprints and completing their project with the structure. They had the choice to build their world virtually using Minecraft or to use Lego blocks. During the two classes, it looked like every student was engaged in the task for the entire class time, and in fact sometimes their concentration on their work during the task made it difficult to interrupt them to ask questions!

In the first class, students designed blueprints on graph paper, being sure to include accurate labels of the values and calculations of each shape’s area and perimeter. They spent the entire time busy at work, trying to create complicated designs, knowing that they would be asked to recreate these designs the following day. They used their knowledge of area and perimeter to accurately label their work, often re-checking their values to ensure that their blueprints accurately captured their intentions.

In the second class, students recreated their blueprints, either using Minecraft or Lego blocks. They brought in their own computers if they were using Minecraft, and at one point we noticed that there were seven different kinds of computers in use. If they had Lego at home, the students brought that in as well, and extra Lego and building boards were provided for all other students.

The students had to use their blueprints and their models to calculate the area and perimeter of their designs, with some students also calculating the total volume, and explain their solutions. This task therefore assessed the students’ ability to create and read a blueprint, calculate perimeter and area, and communicate their understanding of mathematics. It also required them to demonstrate their visualization skills to transfer their 2-D plans to a 3-D structure, an area of mathematics that often falls second to calculations.

During the lesson, students were observed problem-solving using mathematics and applying mathematics skills they had previously learned. The students had to continue to develop their numeracy skills while at the same time starting some pre-algebraic reasoning, as they tried to get the scale and dimensions of their blueprint to match their constructed diagrams.

The context of the problem was realistic, so that students should have seen that mathematics is something that one can use in one’s life. The context helped students draw connections that they may have otherwise not understood; for example, the relationship between multiplying two numbers and the area of a rectangle. We hope this will keep students, and their teachers, inspired about mathematics!

Probably the best sign that this is a project worth doing actually came from the fifth grade students. When they heard that this year’s fourth grade class was breaking out the Lego, they said, “Oh, I remember doing that project. That was fun!” The fifth graders did a project that involved practicing skills, learning mathematics in context, and they not only remember doing it, they found it fun. That sounds like a success to me.


Lego and Minecraft 1 Lego and Minecraft 1 Lego and Minecraft 1
Lego and Minecraft 1 Lego and Minecraft 1 Lego and Minecraft 1
Lego and Minecraft 1 Lego and Minecraft 1 Lego and Minecraft 1
Lego and Minecraft 1 Lego and Minecraft 1 Lego and Minecraft 1
Lego and Minecraft 1 Lego and Minecraft 1 Lego and Minecraft 1
Lego and Minecraft 1 Lego and Minecraft 1 Lego and Minecraft 1
Lego and Minecraft 1 Lego and Minecraft 1 Lego and Minecraft 1
Lego and Minecraft 1 Lego and Minecraft 1




I will miss you

Yesterday was my last day working with students as a classroom teacher, at least for the foreseeable future. Next Tuesday is my last day of work in my current school, and on Wednesday I will be flying to New York to start a new job.

I will miss hearing piano music drifting into my office as I work.
I will miss people asking me questions about all sorts of things.
I will miss the sound of students working and playing.
I will miss wandering around our quirky but lovable buildings.
I will miss standing on the street corner to make sure kids are safe.
I will miss our kids.
I will miss calling a group of people who are in no way “mine”, “my” kids.
I will miss discussing the big (and small) ideas with students.
I will miss random treats in the staff room.
I will miss grateful letters from parents and students.
I will miss grad pranks and graduation dinners.
I will miss taking kids on field trips to see exciting things.
I will miss heart-felt discussions about life with a group of people that has hardly lived any of their own life yet.
I will miss seeing an aha moment when someone gets it.
I will miss watching a group of kids grow and mature.
I will miss the bustle of a class working on a project that has meaning to them.
I will miss being surprised by what students can do.
I will miss my friends.

DIY Catapults!

Fork catapult - side view

One of my jobs at my current school is work on mathematics enrichment projects with three 4th grade students. For the past couple of weeks, in between their camps, my camp, and other end of year events, we have been working on looking into a mathematics modelling task, specifically, a fork catapult. The 4th grade boys and I completed this project over a series of 3 lessons with me, and 2 more lessons working on it independently between our sessions.

I built the fork catapult fairly quickly for the boys, mostly because we lacked a lot of time to work on this particular project, and partially because I wanted them to focus on the collection of data, not specifically the design of their catapult (As it turned out, the boys found time to create three more of their own designs anyway).

I set up the catapult, starting opening up the clip on the clipboard, and right away one of the boys asked, "I wonder how far it will go?"

Fork catapult - side view

So, we collected data. What we did was open up the clip part of the clipboard so that it was 1cm, 1.5cm, 2cm, 2.5cm, and 3cm open, and shot a mini-whiteboard eraser by releasing the clip, 10 times for each position of the clip. We then measured the distance (as accurately as we could) the eraser flew across the room. The initial prediction from the boys was that the wider the clip was open, the farther the eraser would fly, which the boys continued to believe, even once we had actually collected all of the data.

I showed the boys how to plot 1 or 2 points, and then I then asked them to carefully graph the rest of the data. This is what they produced (the red dots were added later).

Graph of results

Once the boys had their data graphed (do you see the small error in the graph above?), we looked at together to see what it meant. From the graph, the boys decided that it was clear that their initial hypothesis (they didn’t use this language, I did) was incorrect and that there was a maximum distance that the eraser could be launched. One of the boys had the insight that the problem was that when the clip was "open too wide" that too much of the energy went into throwing the clip up, and not enough went into catapulting the clip forward.

The boys also reasoned that the eraser would be shot the furthest when the clip was between 2cm and 2.5cm open. Our next step was to test their new hypothesis. As you can see from the red dots graphed above, their hypothesis was probably right, although in order for the boys to see that, we first had a discussion about finding average values from the graph.

At the end of the second class, we started cleaning up, and the boys asked me to photocopy their graph and data, and let them take home their other catapult designs. One of them said to the other boys, "Let’s keep working on this next year!"

There are a lot of ways one could go with this project, but one thing I really liked about it was that we created a mathematical investigation from some pretty basic supplies, that the data that results has enough experimental error that it makes it more obvious that collecting many trials is useful, and that the graph was clear enough that students could read the results from the graph fairly easily.

Bias in assessment

Every form of assessment of learning has bias. This bias may be hidden, or it may be quite obvious. As Cathy O’Neil points out, assessment is a proxy for what we want to measure – learning. We cannot measure the building of connections between neurons that is happening in the brain directly (or even potentially understand what that growth even means) so we use a proxy in the form of an assessment of the externally visible signs of learning.

One bias therefore is our tendency to forget that we are not actually measuring learning, we are measuring a proxy for learning.

Another bias is the language we use to do the assessment, whatever form that language takes. Language is necessarily a construct of our minds and no matter the appearance, it has deeply personal meaning based on our own experiences. As such, our use of language to communicate an assessment to a student contains an inherent bias based on our understanding of the language we have used.

The medium of the assessment also introduces bias. An assessment done on paper on pencil is limited to what can be collected in this form. As Dan Meyer and Dave Major are showing, digital assessments have different affordances (and different biases) that can change how students can interact with the assessment. Students who share their understanding verbally may have a very different explanation than if they write down their explanation.

There is further bias in the assessment based on our interpretation of the results of the assessment. If one knows the names (or race) of the people doing the assessment, is one sometimes more or less lenient? Look at this video of people assessing whether or not someone is stealing a bicycle. Do they exhibit any bias? Is it possible that teachers may experience similar bias (hopefully to a lesser degree!) when examining student work? How much of a difference to the student’s marks does it make if the work is messy or neat?

There are no doubt other biases in our assessments that I have not mentioned.

To combat bias, we must be first aware that it exists, and next that we should look at our assessments and ask ourselves, what bias is likely to exist in this form of this assessment? Can I address this bias? If not, can I use a second assessment in a different form and compare results?

Access to computers

President Obama recently unveiled a plan to have broadband Internet access in every school across the United States by 2018. There’s only one huge problem with that plan; according to the US government’s own research, as of 2006, there was only one computer for every four students, and many of those computers are old. Outfitting the rest of the students in the United States with a computer, and upgrading the existing ones to be useful, will come with a hefty price tag.

Include with the plan (linked above) is this statement: 

In addition to connecting America’s students, ConnectED harnesses the ingenuity of the American private sector get new technologies into students’ hands and support digital learning content.

I read this as, "We will use public money to buy computers for students via private companies" and very likely, those private companies will make enormous profits, given the size of the US education market.

Here’s a more creative solution: develop open-source hardware for schools, like the Raspberry Pi. Not only will the costs be lower in the long run (since the US government can then mass-produce the hardware for schools at cost), it will create jobs within the United States, and allow for innovation in the field through end-user adaptation.

There are a couple of arguments against this idea.

First, one thing that brings down the price of computers is production in mass scale. To this I say, the number of students in the US school system is more than sufficient to allow economy of scale to bring down prices to reasonable levels.

The second argument is that competition between different manufacturers of computers reduces prices, which to some extend is true. However, technology companies also artificially increase their profits in a variety of ways, including delaying new features for their computers to force turn-over of their devices when they introduce these features, and continuing to build their hardware for planned obselescence rather durability and life-span.

While I think that there are tremendous benefits to technology in schools, I also think that schools should use public money wisely. The United States certainly has the technical capability of developing high-quality, durable, open-source hardware. The question is, why aren’t they using it?

What did you learn in school today?

I’m grateful that I work in a school where I do not believe that any of these (updated) lyrics by Pikku Myy apply. Via the Blue Skunk blog.


What did you learn in school today, dear little boy of mine?
What did you learn in school today, dear little boy of mine?
I learned that I must pass a test
To sort the learners from the rest
That winners win and losers lose
And TAKS test scores is how they choose
And that’s what I learned in school today
That’s what I learned in school

What did you learn in school today, dear little girl of mine?
What did you learn in school today, dear little girl of mine?
It matters what my parents earn
I’ll get better grades with cash to burn
If I don’t speak English I can’t be smart
And no more music and no more art
And that’s what I learned in school today
That’s what I learned in school

What did you learn in school today, dear little boy of mine?
What did you learn in school today, dear little boy of mine?
Teachers fill my empty mind
So that I won’t be left behind
I’m learning how to play the game
And all right answers look the same
And that’s what I learned in school today
That’s what I learned in school

What did you learn in school today, dear little girl of mine?
What did you learn in school today, dear little girl of mine?
Learning’s just a job I do
From seven thirty til half-past two
And all my interests have to wait
‘Til I drop out or graduate
And that’s what I learned in school today
That’s what I learned in school


If they apply in your school, what are you going to do about it?

Things I did not learn in teacher college

The best leaders are the best learners


These are some things I wish I learned in my teacher training:

  • The goal is not classroom management, the goal is effective student learning. It may be that a well-behaved class is an excellent environment for learning, but the means by which you end up with that well-behaved class matter.
  • Most of your early lessons are going to be awful. Remember the ones that aren’t and build on your successes.
  • Connect with other educators as much as you can. You are each other’s best lines of support.
  • You are always in charge of your professional development. Any experiences which other people require you to do are training, not professional development.
  • Never stop learning. You should continue to explore your own subject area, because a teacher who is inspired by what they teach is more able to inspire others. Always take time to learn more about teaching, because what you learned in college is only part of the story.


Note: It may be that this advice was given to me during my time in teacher college, but I didn’t learn it then, and I sure wish I did.

How first aid training is like mathematics education

(Image credit: drewleavy)

I talked to someone recently about first aid training, and they expressed their frustration at how ineffective first aid training usually is.

Unfortunately, according to my friend, many people who teach first aid actually have very little practical experience using first aid. As a result, the agencies that are responsible for first aid certification give their instructors "idiot-proof, deadly boring, text-filled presentations" to use in their training so that every first aid course is at least minimally useful. According to my friend, instructors are expressly prohibited from using their real life experience and telling stories of themselves actually using first aid, and also from explaining the reasons behind the protocols used in first aid. Further, most people taking first aid training have little to no interest in first aid themselves, they are almost certainly required to take a first aid course as part of retaining certification in their line of employment.

It occurred to me that this situation is remarkably similar to the position that we find ourselves in mathematics education, at least in the k to 12 level. Mathematics teachers often lack experience either making mathematical discoveries, or even applying mathematics as might an engineer or physicist. Consequently, curriculum is packaged in such a way so that nearly anyone can follow the text and make sure the students get at least a minimally effective mathematics background. We aren’t prohibited from using stories to relay our experience in mathematics, but we often have pretty frustrating limitations on what mathematics we can teach, and at the end of their k-12 program, how our students will be assessed. Finally, almost none of our students is really interested in mathematics; most of them are in our courses because they are required to be there.

Fortunately, first aid training is occasionally successful. My friend suggested that about 80% of the time, people who have first aid training are more useful in an emergency situation than people without the training. At my school, a 6th grade student, learned the Heimlich maneuver during a recent first aid training, and then used it to save the life of his mother two weeks later (An aside: according to my friend, the correct response when someone is choking, after encouraging them to cough, is 5 sharp blows to their back with the heel of your hand, right in the middle of their shoulder blades, followed by abdominal thrusts, then CPR if they fall unconscious). Not teaching first aid is obviously not an option, even when the programs are usually limited in effectiveness, but I wonder, what would happen if we let people who were highly experienced in using first aid have a bit more flexibility in how it was taught?

In the same way, our mathematics education is not a complete disaster. Many people go on from their subpar mathematics education to be able to use mathematics in a meaningful way, and some of those people even make new mathematical discoveries. However, the surest proof I have that our system is less than adequate is the enormous number of people I have met who will happily admit that they were terrible at mathematics, hated it, and now never use it.

The question that I think defines my career as a mathematics educator is, what can we do about this issue? It is of limited use to complain about a problem, especially one as well discussed as mathematics education, without proposing some sort of solution.

What if mathematics educators who had actually used mathematics to solve problems, or had developed new areas of mathematics, had more freedom in how they were able to help their students learn mathematics? If someone has significant experience in topology, number theory, mathematical modelling or any other area of mathematics, why not let them teach this area of mathematics to their students, perhaps leaving out some other area of our curriculum so that they would have time to do so. What if we taught mathematics in such a way that our primary goal was to inspire our students into further study of mathematics?