Education ∪ Math ∪ Technology

Month: January 2013 (page 2 of 2)

Teaching division algorithms

I’m assisting our 4th grade teachers in finding resources for their upcoming unit on division, and I’m hoping to avoid either of these situations from arising (see the videos below). Actually, these videos could be great hooks to see if students can figure out what mistakes the people in them are making…


A key problem in both of these videos is that these (fictional) people are making key mistakes in their use of a standard algorithm which they clearly do not understand, nor do they seem able judge how accurate their solutions are. Now, my suspicion is that students will not make such egregious mistakes, but they may make more subtle mistakes based on the same principles.

The key components of any division instruction should help students by:

  • emphasizing understanding what they are doing when they use a division algorithm,
  • providing tools for students to estimate their answer,
  • encouraging them to use their estimation skills to check the reasonableness of their answer,
  • suggesting alternate techniques to double check the accuracy of their answer,
  • giving time to see the connection between different division algorithms,
  • embedding the use of division in a problem solving context.

If I were going to teach division, (after confirming that students understand multiplication very well) I might start with a problem about equal sharing for students to do that every student can access. Next I might gradually increase the size of the numbers in the problem until students start to see that their "naive" strategies may work, but are very inefficient. One obvious problem I can foresee happening here is that some of the students in many grade four classrooms already know a division algorithm, and may be reluctant to come up with their own strategies as a result.

Next, I would compare different strategies that students use in a whole group setting. We would spend some class time discussing the strategies and talk about the benefits and drawbacks of each. We might also explore some other alternate strategies not yet shared, perhaps using a gigsaw technique where each group explores a specific strategy, perhaps suggested by a video.

One interesting activity for students might be to watch the videos above and try and figure what mistake the people in the video are making when they calculate.

Here are some division strategies that your students may not have seen before:


I’m also looking for projects and/or puzzles which require the use of division as a strategy for solving the problem/puzzle.


An aside: I am not convinced that these algorithms have much value for our students. More time spent understanding what exactly what division is would be more useful, I think, for our students in the long run, and then just give them a calculator. However, that is not my decision to make, it is up to the BC Ministry of Education to decide.

Social interactions and learning

A conversation on learning
(Image credit: Vandy CFT)

One of the trends with technology today is that it is beginning to redefine the means through which we are social. Prior to the invention of writing, social learning meant discussing ideas with someone in person, the invention of writing allowed social interactions to span geographic and chronological barriers. The trend throughout history has generally been to expand the distance between participants in a conversation and reduce the time delay in a response until today we can potentially interact with people on the other side of the planet via video link with almost no delay.

But what does it mean, "to be social"? If I leave a comment on a five year old blog post amongst hundreds of other comments, how much have I really interacted with anyone? If I see someone else’s response in a learning management system, is this social interaction? Technology has begun to make fuzzy what used to be a clear definition of social interaction. Can we come to a definition of social interaction which clears up some of this fuzziness? Maybe the exact definition of social interaction doesn’t matter?

There is plenty of evidence that learning works better in a social context. BF Skinner’s teaching machines and IPI, both attempts to automate learning, are failures without social interaction. The key affordable of social interactions in learning is timely, contextual feedback on the schema the learner is attempting to build. Without this feedback, it is too easy for our brains, designed to find concrete patterns more easily than abstract patterns, to construct an alternate set of schema that might satisfy all of the information we have, but which is too inflexible to add new information. It is my experience that we tend to more complicated, convulsed explanations of reality without feedback, than less complex but more abstract explanations.

This affordance of a social interaction will help us define what the minimum amount of social interaction should be in a robust learning environment. If the interactions between participants of a learning space lack timeliness and context, then they are unlikely to be useful as a social tool for learning. It may be that they are one-way timely (where you write or say something in a video, and when I consume it years later it acts as useful feedback to me in that moment) but these interactions lack the fore-knowledge of the non-interactive party of what understandings the interacting party has. A one-way interaction cannot give personalized feedback based on the interaction.

If social interactions are so critical to learning, why is it so many learning experiences are designed solely around content? Given the ability of technology to connect people across vast distances, why is so little of this technology available in the learning environments of today’s online courses which include, at best, primitive forum discussions? The primary difference between the MOOCs offered by Stanford, MIT, and others, and the more organically designed cMOOCs (like #etmooc and #oldsmooc) is that the first set of MOOCs treat social interactions as a side-line to the main show, whereas in the cMOOCs, the social interactions ARE the show.

Raising mathematicians

I read a recent article about the importance of early number talk with children and was pleased that this issue was being brought up. The article shares research on a few of the stark differences in how parents talk with their children about numbers. For example, parents tend to talk to their daughters about half as much about numbers as their sons. Parents also range in how much they use number words around their children from about a dozen times a week, to as much as 1800 times per week.

However, I felt that the list of suggestions the article had at the bottom was incomplete. The article’s author essentially makes suggestions which I feel will only help children develop an instrumental understanding of mathematics, as opposed to a more useful, interesting, relational understanding.

Here are some more things my wife and I do with our children from a very young age to help them develop a deeper understanding of numbers.

  • We play games with our children that involve numbers. We roll dice, we play cards, we solve puzzles together, and we play hopscotch. Through these games, my children gain an understanding of the relationship between numbers and actions we take in the games themselves. We ask questions like "how many ways can you get a 10?" My son recently answered that question with this sequence of answers: 20 – 10, 30 – 20, 40 – 30, 50 – 40, etc…
  • We talk about how we use numbers in our day to day life. We talk about fractions of food (that are physically present in front of us), and talk about the relationships between these different fractions. We cook, do our finances, and share as many uses of numbers as we can with our children.
  • We look for patterns in numbers. We play with relationships between different numbers. We celebrate discoveries our children make. For example, my eldest son noticed that he only needed to remember the very last digit in a large number to figure out if a number was odd or even. Now, he delights in asking people to give him gigantic numbers like 30,938,309,830,983 and being able to tell right away if the number is odd or even. As a mathematician father, I try hard to balance between giving my sons space to come up with their own discoveries, and expanding where their explorations might go.
  • I also try very hard to remember that when my son makes a mistake, with further experiences, it is likely that he will discover these mistakes later for himself. For example, my son was convinced for a long time that numbers went 90, 100, 110, 200. He did not understand place value well enough, but over time, and with further exposure to our use of numbers, this misconception of his has disappeared. He now has a very good understanding of place value up to 1000, although he still does not understand larger numbers (he will say things like 233 hundred thousand, 293 million, 389 billion, and 43), but I am confident that as he continues to be exposed to numbers in his day to day experience, he will understand these numbers better.
  • We balance our discussion of patterns in numbers with other types of patterns. We look for patterns in art work, in sidewalks, in tiled floors, and wherever else they may form. We create patterns ourselves in our art work and notice where they came from. We doodle, we draw shapes, we watch clouds, and we look at maps, all of which help my son develop a sense of shape and space.
  • We ask for evidence from our sons about why they think something is true, whether or not what they are saying is accurate or not. When they have discovered something, and provided solid evidence for why they think it is true, we celebrate it. We might ask questions like, "How did you discover that?" or "Wow, that’s neat. Does it work all the time?"
  • We give our children plenty of creative time to explore the world through art work, Lego, blocks, reading, playing games, and other self-exploration activities.
  • We see the development of our children’s numeracy as a process, rather than a race. I have no interest in accelerating my sons through the elementary school curriculum, instead I focus more on providing opportunities for enrichment.

The key to all of these activities is that we view numbers and quantities as ways of exploring, and we nurture our children’s sense of wonder about the world.

Andragogy vs pedagogy

Andragogy is a theory of learning as learning applies to adults rather than children (pedagogy). According to Malcolm Knowles, there are 6 key components of adult education.

  1. Adults need to know the reason for learning something (Need to Know)
  2. Experience (including error) provides the basis for learning activities (Foundation).
  3. Adults need to be responsible for their decisions on education; involvement in the planning and evaluation of their instruction (Self-concept).
  4. Adults are most interested in learning subjects having immediate relevance to their work and/or personal lives (Readiness).
  5. Adult learning is problem-centered rather than content-oriented (Orientation).
  6. Adults respond better to internal versus external motivators (Motivation).

I fail to see how these six things are not also true for children.

The objects that adults produce as part of their learning should be different than the objects children produce. Adults don’t need to create posters (although this may still be a valuable learning experience depending on the context) at the same rate that children do during their learning.

Adults have some different external concerns (children, job, home, etc…) than do children that sometimes interfere with their ability to learn in a classroom setting, but these concerns are just different than the concerns of children, they are no less important to the learner.

Adults come to their learning with more experiences than children, and this may make any unlearning (if necessary) more challenging for them, but the fundamental process through which they learn should be significantly similar to the process children go through.

The primary difference I see between adult learning and children learning is how much power they are granted during their learning.

Measurement by Paul Lockhart

You may remember Paul Lockhart as the author of a Mathematician’s Lament. I’m currently reading his newest book, Measurement. I’m halfway through it and reading it every chance I get. Here’s my favourite quote from the book so far:

"All of the events — past, present, and future — of our whole ridiculous universe are writ on this one four-dimensional canvas, and we are but the tiniest brush strokes." Paul Lockhart

Of course, what Paul says is true. Here’s a great mathematical investigation: Given a football sized canvas, how large a brush stroke would all of the aggregate movement of the human species require, assuming the canvas represents the entire universe.

If you are a math teacher, or just want to understand what people find fascinating about mathematics, I recommend reading Paul’s book.