Education ∪ Math ∪ Technology

# Tag: learning(page 1 of 2)

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.

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.

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.

I recently came across Frank Noschese and John Burk‘s collection of posts on Pseudoteaching. In Frank and John’s words:

What is pseudoteaching? This term was inspired by Dan Meyer’s pseudocontext, which sought to find examples of textbook problems that on the surface seemed to be about real world problems and situations, but actually were about make believe contexts that had little connection to the real world, other than the photographs that framed the problems. After reading many of Dan’s pseudocontext posts, John Burk and I had the idea of pseudoteaching [PT] which we have defined as:

Pseudoteaching is something you realize you’re doing after you’ve attempted a lesson which from the outset looks like it should result in student learning, but upon further reflection, you realize that the very lesson itself was flawed and involved minimal learning.

We hope that though discussion, we’ll be able to clarify and refine this definition even further. The key idea of pseudoteaching is that it looks like good teaching. In class, students feel like they are learning, and any observer who saw a teacher in the middle of pseudteaching would feel like he’s watching a great lesson. The only problem is, very little learning is taking place. We hope pseudoteaching will become a valuable lens for critically examining our own teaching, and that the idea will spread to other teachers as well.

How does this apply to presenters?

Most presenters will tell you that their primary job is to teach new ideas, but it’s not. The primary job of a presenter is to be asked to present again (because if you don’t get to present again, you don’t get to share your message). The secondary job of a presenter is to teach. If the primary job of a presenter were actually to teach, they would use methods of presenting that might actually result in learning, rather than just entertainment.

Here are some quick checks you can use to tell if the presenter you have is actually teaching.

• Do they use formative assessment during their workshop, and then modify their workshop in response to the results?
• Do they ask questions that provoke thinking… and then expect everyone to respond to those questions?
• Do they give opportunities for teachers to collaborate and discuss during their session?
• Do they attempt to uncover and address incomplete models1 related to what they are teaching?
• Do they check to see if teachers can apply what they have learned?

If not, you are probably listening to an edutainer.

1. My colleague, Scott Bruss, introduced me to the idea of using the phrase incomplete model instead of misconception. A misconception implies that something can easily be addressed by presenting the conception. An incomplete model suggests that in order to help a learner develop a complete model, you need to know what model they are currently using.

I’m reading Dylan Wiliam’s "Embedded Formative Assessment" book (which I highly recommend) and this paragraph jumped out at me:

Mathematical notation has been developing since the introduction of writing and has largely grown organically with new notation added as it is needed. In fact, if a mathematical concept is developed in different cultures, it is entirely likely that each culture will develop its own mathematical notation to describe the concept, and these mathematical notations inevitably end up competing with each other, sometimes for centuries.

This observation by Dylan Wiliam suggests to me that difficulties in mathematics for some students are almost certainly related to the notation that we use to represent it (especially in classrooms where mathematics is largely presented to students in completed form, rather than being constructed with students), and that people who end up good at math in school may be good at being able to switch meaning based on context.

Can you think of any other examples of mathematical notation which are potentially inconsistent with other mathematical notation? I’ll add one to get the list going:  which is clearly inconsistent with algebraic notation, and potentially with fractions too.

My youngest son recently learned how to walk. He’s certainly not an expert by any means, but he can now toddle around for 10 to 12 steps at a time and not fall down. I noticed something strange about his walking though – he makes no effort to avoid obstacles in his way.

This is strange because I remember when he first learned to crawl a few months ago, he had the same problem, but he fairly quickly learned how to crawl around the obstacles on the floor and therefore avoid them.

There are two possibilities – he knows about the obstacles and chooses not to avoid them because he wants to focus on keeping his balance, or he doesn’t notice the obstacles at all. The first possibility seems unlikely to me since he normally hesitates in his walking when he seems uncertain and he does not even look at the obstacles at the floor. The second possibility seems more likely to me.

How is it that he can not notice obstacles while walking that he avoids easily while crawling? My theory is that there are three factors at play – first that his perspective is different while walking than while crawling and that he is not able to adjust his internal model of "looking for an avoiding obstacles" because of this. Another is that he does think that he should worry about obstacles while walking because it does not occur to him that the obstacles he successfully avoided while crawling could be obstacles while walking. Finally, walking probably takes my son quite a bit of cognitive effort. He has to spend his time thinking about balancing, and he is unable to think about all of the other things that could happen while he is doing this. Walking is not something for which he has automaticity. All of these reasons are results of a change in context between crawling and walking.

The space is the same, the obstacles are the same, but the context is different, and this makes the tasks different enough that even what seem like simple obstacles to us are currently challenging for my son.

We need to consider this issue for our students as well. How often do we see them unable to solve problems in an even slightly different context that they seemed completely capable of solving before?

"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.

Derek Muller sent me this link to a very popular video animation that attempts to explain fundamental forces in nature. You can watch it for yourself below.

The video uses analogy and some cute animations to attempt to explain how forces in nature come from difference between measurements of those forces in different parts of the university. For example look at the screen-shot taken from the video shown below.

If you look at this picture, does it accurately represent the statement given by the narrator? It seems to me that if you are going to use a visual to explain a concept, it should be clear from the visual what you mean. Visuals should support your explanation, and if your analogy strays too far from the concept you are trying to explain, your visuals do more harm than good. What was the first thing you thought of when you looked at this visual? I bet it wasn’t "Measurement by itself is meaningless, but as surprising as it sounds, that meaningless is exactly what causes the fundamental forces in nature" which is what the narrator says at this moment.

Here’s another screen-shot.

This visual says two things. The first is not stated by the narrator, but is suggested by the equation shown, specifically that what we are going to look at next is very complicated. The second is suggested by the crossing out of the word Quantum. In this case, the visual definitely describes what the narrator is going to do in the rest of the video – ignore quantum effects on the four fundamental forces. The bad news here is that ignoring quantum effects means that whatever follows is going to be out of date by 100 years of science, and not necessarily a very good representation of the apparent strangeness of the universe. In other words, what follows is a bad model that one will probably not understand.

Now let’s see what happens next.

My question here is, what of the previous 1 minute and 30 seconds do you remember? I’m going to suggest that you probably do not remember much. This new model is so vastly different than the old model the narrator starts with (and that previous model was not well explained, as you may recall) that the transference of the introductory model to this new model is not likely to happen. If you happen to be an expert in the area of the fundamental forces of nature, you may not notice this effect, since the earlier model is (maybe) describing something you already understand, and have already internalized. If you are not an expert, I very much doubt that a 1 minute explanation is going to make you one.

Further, if you look at this section, you may notice the model for currency transaction (which looks a lot like a function machine, an analogy mathematics teachers often use to explain functions to students) in the middle of the currency exchange. The currency portion of this implicit analogy probably makes sense, but the symbol in the middle may be lost on a lot of people, particularly since the narrator doesn’t take the time to explain what this symbol means.

Now this is where the narrator makes a huge assumption. He assumes that people have been able to make a somewhat over-arching generalization from his single example. He says, "Hopefully now you can see why measuring things differently in different places inevitably gives rise to a long range interaction, mediated by a particle." I doubt that anyone would be able to make that generalization without a fair amount of expertise in long-range interactions themselves.

It is a form of cognitive bias to assume that because an analogy makes sense to you, that it will make to other people. Analogies are useful as a sense making activity when the analogy describes a shared experience between two people, and very few people have an experience of currency exchange (surprisingly, only a small percentage of any population travels to other countries). In other words, using an analogy that people lack experience with is unlikely to lead to further understanding of a more complicated phenomena.

This particular video, when I watched it, had over 156, 000 views, and over 5000 likes, which suggests to me that one cannot take the popularity of a video and use it to gauge the effectiveness of the learning from the video. I recommend reading the comments on the video. You will see more than a few people who are confused by the video, or who add messages which  are essentially unrelated from the video itself. The most popular discussion point I saw, in the 100 or so comments I read, was that this "minutephysics" video was in fact longer than a minute.

My complaints while directed at this one video are generalizable. Analogies used in videos should be related to common experiences, where possible. Visuals matter – using visuals which are confusing, or even wrong, not only distracts from the intended objectives of the videos, it can introduce other possible misconceptions. Avoiding people’s misconceptions in the videos, and attempting to present clear explanations means that people will, in general, incorporate the new information into their existing schema, leaving their current misconceptions intact.

This comment on the video essentially summarizes my main point (notice how many people agree with it).

One of the first times my son rode his bicycle.

I went bicycling with my five year old son yesterday with his new bicycle from his Birthday. We haven’t had much of a chance to bicycle recently together, so my son hadn’t actually ridden his bicycle since the summertime. As a result, he had really forgotten a lot about what he learned over the summer about riding bicycles, and was really struggling to even get his bicycle going.

As the minutes wore on, he became more and more frustrated, although I was encouraged by his willingness to fall down and then get right back up over and over again. I did give him some advice and encouragement during this time, but since I knew he was capable of riding his bicycle, I didn’t want to be too helpful.

I realized that he had forgotten what it felt like to ride his bicycle. He was on an unfamiliar bike that was just a wee bit too large for him, and just couldn’t seem to get it together.

So I asked him to pause for a second, and get off his bicycle. I knelt, and we were face to face, and I asked him if he remembered riding his bicycle before. He said he couldn’t and his face fell a bit. I asked him if he remembered riding it down Auntie Juniper’s driveway (which is where he first learned how to ride his bicycle), and his face lit up while he nodded vigorously. I asked him to close his eyes, and imagine himself riding down Auntie Juniper’s driveway. He closed his eyes and I reminded him of how much practice he had put in, how much fun he had during the summer, and how good he had gotten at riding his bicycle. We spent about two minutes remembering together the feeling of the first time he rode a bicycle.

Right after that, he got back on his bicycle and started to ride it. He only fell down one more time while riding his bicycle, and even managed to ride it all the way around the park twice without stopping (he was pretty proud of this accomplishment). He went from unable to get his bicycle going more than a couple of feet to being as capable as he was during the summer after all of his practice.

This incident reminded me of a few things about learning:

• What you know how to do is tied to your emotions. It is not enough to simply know things, you have to have some feelings attached to those things for them to be useful. When my son lacked confidence, he wasn’t able to ride his bicycle. When he regained his confidence, and remembered the joy he felt riding his bicycle during the summer, all of his knowledge about how to ride a bicycle came back to him.

• Focused and contextualized practice are important in learning. You can’t really get better at riding a bicycle by talking about riding a bicycle, you have to do it. My son spent many hours riding his bicycle in order to become better at it.

Although I see the obvious value in learning in practicing, I want to re-iterate how important it is that this practice be in a meaningful context. I often see comments on stories about mathematics education, for example, where the people talk about this or that cashier who was unable to make change without a calculator and how this points to an obviously sorry state of mathematics education. The question I want to ask in response to the often repeated story of the cashier is, when did they practice making change? They practiced arithmetic repeatedly in schools, no doubt, but how many schools have students play the part of cashiers and make change for pretend customers? How often is the skill of arithmetic practiced in context?

• Practice should be part of a shared experience, and should have a positive emotion attached to it. If my son had learned how to ride a bicycle on his own, I wouldn’t have been able to help him remember his previous experience. If he had spent his entire time practicing in frustration or in anger during the summer, I doubt he would have remembered how to ride his bicycle yesterday.

When my son woke up this morning, he asked me to make him waffles. Having never made waffles before, I was going to refuse, but then I decided to take a chance, and just learn how to make waffles. Most of my adventures in the kitchen in the past 6 years have happened with the help of my wife, but I really need to spend more time cooking by myself, like I did when I was a bachelor. Both my wife and I agree that more balance needs to happen between us in terms of who makes meals (although she’s pretty happy with me doing all of the dishes…).

I looked up a recipe for waffles online and decided to make sure I had all of the ingredients. With my son’s help, we looked through the kitchen and found all of the ingredients for the waffles, except we only had 1 egg, and the recipe called for two. We also didn’t have enough vegetable oil, so I had to do a couple of substitutions.

Together, my son and I measured out the ingredients for the waffles and put them into a bowl and mixed them all up. I then pulled out the waffle maker, and figured out how it worked, with my son’s help. It certainly makes waffles easy to make!

Unfortunately, I didn’t know how much of the wafflie mix to put into the waffle cooker. I decided to take a guess and glopped some mix into the cooker. As you can see, this didn’t work out so well.

The mixture overflowed from the waffle cooker, and onto the counter. Oops! I’d put too much in! After some experimentation, and more messes, I figured out how much was the right amount to cook.

The big moment came, when I actually got to try my waffles for the first time.

My wife and son agree with me, my waffles were yummy! I was pretty pleased with myself, and although I realized afterward that making waffles is really not all that difficult, I still felt a sense of accomplishment.

As I ate my waffles, I thought about how this experience should translate to student learning.

• I picked a project which was meaningful to me.
• I created a plan to complete my project.
• I followed through on my plan, which required me to trouble-shoot, revise my plan, and clean up after my mistakes.
• I enjoyed and shared the fruits of my labours at the end of the project.
• I learned a skill I can almost certainly use later.
• I took a risk and met the challenge successfully, while overcoming some obstacles in my way.

While it’s clear to me that not every learning experience can be as successful, or as self-directed as my waffle-making experience, it’s also clear to me that too few experiences of children in schools mirror my experience at home. We spend a lot of time directing the lives of students, and I’d like to see more schools with structures in place that allowed students to be in charge of at least some of their learning.

Update: There has been some recent research that suggests that while the timeliness of feedback is one aspect of good feedback, it may not be the most critical aspects of feedback. Awful feedback given immediately is much less useful than carefully constructed feedback given later.

Abstract

A review of the literature on the role of feedback in learning shows that student feedback is critical to student learning.  Although different studies emphasis immediacy in feedback to different degrees, all of the studies reviewed agree that timeliness in feedback is important.

The Role of Immediacy of Feedback in Student Learning

Without feedback of any kind, we would not learn at all, period.  We would end up doomed to repeat the same mistakes over and over again, as the fable of Sisyphus (Camus, A. & O’Brien, 1975) demonstrates.  As teachers then, one of our primary roles for our students is to provide opportunities for feedback, preferably in different forms.  Examining the literature on student feedback, we can see that this claim is supported.

According to Nicol and Macfarlane (2006, p7), there are seven principles of good feedback practice.  Good feedback:

1. helps clarify what good performance is (goals, criteria, expected standards);
2. facilitates the development of self-assessment (reflection) in learning.
3. delivers high quality information to students about their learning;
4. encourages teacher and peer dialogue around learning;
5. encourages positive motivational beliefs and self-esteem;
6. provides opportunities to close the gap between current and desired performance;
7. provides information to teachers that can be used to help shape the teaching.

When Nicol and Macfarlane (2006, p9) clarify these expectations, they indicate that “high quality information” about student learning means “that feedback is provided in a timely manner (close to the act of learning production), that it focuses not just on strengths and weaknesses.”  Quality feedback includes a provision that the feedback is provided close to when the students are learning the material.

Chickering and Gamson (1987, p2) also have seven principles of good practice in practice for education.  They indicate that good practice in undergraduate education:

1. Encourages Student-Faculty Contact
2. Encourages Cooperation
3. Encourages Active Learning
4. Gives Prompt Feedback [emphasis mine]
6. Communicates High Expectations
7. Respects Diverse Talents and Ways of Learning

Note that here, Chickering and Gamson have indicated that feedback needs to be prompt to be included in their list of good practice for undergraduate education.  It is fair to assume that good educational practices at an undergraduate level of schooling are also good practices at any level of schooling.

Learners themselves have an understanding of the importance of feedback in learning.  According to a study done on the expectations of students as to levels of support provided by the educational service provider, Choy, McNickle, and Clayton (2009, p8), found that the services found most highly regarded were:

1. clear statements of what I [the learner] was expected to learn
2. helpful feedback from teachers [emphasis mine]
3. requirements for assessment
4. communication with teachers using a variety of ways, for example, email,
5. online chat, face to face
6. timely feedback from teachers [emphasis mine]

Note that feedback from the teachers was listed twice with the qualifiers of helpful and timely.  Clearly the students in this study felt that feedback was important enough to mention twice.

McTighe and O’Connor (2005, p5) reiterate from Wiggins (1998) that “To serve learning, feedback must meet four criteria: It must be timely [emphasis mine], specific, understandable to the receiver, and formed to allow for self-adjustment on the student’s part.”  They have only four requirements for feedback, and the first of these they list is how timely the feedback must be.

One could argue that timely feedback is most critical in student learning.  “[T]imely, detailed feedback provided as near in time as possible to the performance of the assessed behavior is most [emphasis mine] effective in providing motivation and in shaping behavior and mental constructs” (Anderson 2008). Students need the feedback for learning to happen near to the event of learning, according to Anderson (2008), in order to learn effectively, which is what he means by “providing … mental constructs.”

If we view the analogy of learning a physical act, we can see how obvious it is that timely feedback is important.  Although feedback from the learning of sport, or even the act of walking is not necessarily directed by teacher, the very world around us provides us with feedback.  If we fail to walk properly, we fall down!  Kick the ball with your toe, and it is sure to go over the goal.  We learn physical actions very quickly because we receive lots of timely feedback about everyone of our actions.  The only physical actions which are difficult to learn for some people, assuming capability of performing the action, are the ones where the feedback is delayed.

It is clear that any informed educational practice should take into account how feedback will be provided to the students.  Feedback needs to be timely and relevant to the learner’s needs in order to be effective.  Educators must therefore provide assessment opportunities for students with timely and relevant feedback built into the assessments or these assessments are limited in value.

References

Anderson, T. (2008). “Teaching in an Online Learning Context.” In: Anderson, T. & Elloumi, F. Theory and Practice of Online Learning. Athabasca University.

Camus, A. & O’Brien, J., (1975). The myth of Sisyphus, published by Penguin books

Chickering, A. & Gamson, Z., (1987) Seven principles for good practice in undergraduate education, AAHE bulletin, 39, 3-7

Choy, S.; McNickle, C. & Clayton, B., (2009). Learner expectations and experiences. Student views of support in online learning, National Centre for Vocational Education Research

Higgins, R.; Hartley, P. & Skelton, A., (2002). The conscientious consumer: reconsidering the role of assessment feedback in student learning, Studies in Higher Education, Routledge, 27, 53-64

McTighe, J. & O’Connor, K., (2005), Seven practices for effective learning, Educational Leadership, Association for Supervision and Curriculum Development. 63, 10-17

Nicol, D. & Macfarlane-Dick, D., (2006). Formative assessment and self-regulated learning: A model and seven principles of good feedback practice, Studies in Higher Education, Routledge, 31, 199-218

Wiggins, G., (1998). Educative Assessment. Designing Assessments To Inform and Improve Student Performance. Jossey-Bass Publishers