# The Reflective Educator

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#### Tag: math (page 3 of 10)

I recently realized that I have a tonne of different math mini-applications that I’ve built over the years, and I will need to take the time to catalog them at some stage (note that some of these will just not run in Internet Explorer). For now, here’s a list of the ones that might be useful, in no particular order:

(Image credit: bterrycompton)

I co-coached my son’s blastball team last year. We spent a lot of time playing blastball, but we also spent some time practicing some of the skills needed to be able to play (while emphasizing how these skills fit into playing the game). One of the skills we practiced was throwing and catching a blastball.

How this worked is that each kid stood one or two steps away from their parent and threw the ball to their parent. If the parent caught it, the parent took a step back, and threw the ball back to their child. If their child caught it, the parent took another step back, and so on. This meant that very quickly parents and children tended to be separated by a distance where they catch the ball about half of the time. This is by itself a good activity that relates to the number line, if you think of each step apart being 1 space apart on the number line.

My son and I went through a particular fun exchange where his objective was to make sure I didn’t catch the ball, requiring me to continue closer to him. As I moved closer to him, I kept indicating where I was like so: "Okay, now I am three steps. Now I am at two steps." Eventually, I ended up zero steps away from him. To continue the joke, he managed to find a way for me not to catch the ball even though we were directly on top of each other. I continued stepping toward him, which meant that we were now facing back to back, with one step in between us. "Okay," I said, "now I am at negative one step." He had lots of questions about negative one step, and continued the game a few more times as I moved into smaller and smaller negative numbers.

While the introduction of the concept of negative numbers is obviously secondary to this activity, it is a way to tie together some fun physical activity with some conceptual understanding of the relationships between different numbers.

Here’s a thought experiment for you (h/t to Dan Meyer for the sports analogy).

Imagine you start learning the game of basketball by learning how to shoot free throws. At no point are you told what the point of shooting free throws is, or how being a good free throw shooter will help you play the game of basketball, or even that there is a game of basketball. Worse, you are occasionally asked to shoot as many free throws you can in a minute, and then judged against your classmates based on your performance.

You are at some point asked to start practicing all of your free throws blindfolded, possibly after unsuccessfully learning how to shoot free throws earlier. If you are lucky, your coach tells you how many free throws you sank, and how many you missed. You finally have some understanding that free throws are important in basketball after years of not having a clue why you were practicing them but no matter how many times you practice, you never seem to get better at free throws.

By way of analogy, this is almost exactly how addition and multiplication facts are taught to students. They spend the earliest years learning addition and multiplication facts with only a superficial explanation of how these facts might be useful later, and most do not learn how addition and multiplication fit into mathematics as a whole and they certainly never get to experience "the game."

In their later grades, their teacher  (although lacking the time to give students feedback on their "basic skills") expects students to work on their higher level mathematics without a calculator or any aid of any kind for their foundational numeracy skills. The premise behind these calculator-less classrooms is that students will likely forget their addition and multiplication facts if they get to use a calculator, and so the use of a calculator is banned. Unfortunately, in most of these classes, very little time is spent reteaching addition and/or multiplication facts, and almost no feedback is given to students as to whether they have even done their addition and/or multiplication correctly, so if you are a student who never understood addition or multiplication in the first place, this further practice without support is unlikely to be useful.

If you are going to ban students from using calculators in your class for basic arithmetic operations, then you must at least take the blindfolds off of your students and help them improve their arithmetic skills. On the other hand, I prefer not to ban tools, but instead find ways that these tools are used productively (and unproductively) and change my teaching to compensate.

Here are two sample programs from a pair of kindergarten classes today (I took screen-shots of their program, and cropped them to fit in this blog).

Program 1:

Program 2:

I started the kindergarteners off the same way I started off third graders last week – they were to program me, and then program their partner. It worked fairly well, as most of the kindies could figure out how to get me to move in a square fairly easily, but an L turned out to be a stumper for a while, and a T was super hard. One could easily have done this entire activity with some adults (or older kids) willing to stand in as computers and be moved around by the kindies.

The idea of this activity is to get students thinking geometrically and systematically – if I want the computer to draw this shape, what do I need to do to get it to work. The key here is that the kindergarteners have to do the thinking, and what they showed me is that they are capable of some fairly advanced logical reasoning when pushed into it a bit. Most of the kindergarteners were able to get the computer to draw a square, or run way off the screen, and nearly all of their programs involved using the repeat function. I really found the students had to think to be able to do this activity, and to trouble-shoot when their programs didn’t work.

I would not advocate this activity replace moving around time, or other drawing time, but if you are stuck at the end of a year with nothing but worksheets to do, this could be an excellent replacement.

I found out through Reddit about a new visual programming language that runs in the browser called Blockly. The system looked pretty good, but wasn’t quite right for my students. Fortunately the Blockly code was fairly easy to figure out, and so I hacked around a bit this weekend, and put together a simplified version of the system for use with my kindergarten students tomorrow. This version allows students to use code to create simple animations. Unfortunately, at this stage the animations cannot be saved.

• It does not run in Internet Explorer. It may not run in a few other browsers as well. I can confirm that it should work in Google Chrome and Firefox, and it probably works in Safari.
• It’s not done yet. I plan on adding more of the advanced functionality, which exists already in the Blockly language, and should be easily implemented.
• I plan on beefing up the forward and turn commands so they are similar in power to what can be done in Logo. One of the chief advantages of Logo is that it is both an easy to understand language, and powerful.
• I plan on implementing save and share features if this system looks reasonably useful.
• I think the graphics for the turn and forward code blocks could be better. For example, they should show an arrow either turning, or going straight.

Do you have any feedback about how I could improve this programming environment?

I recently found this article written by Richard Skemp that Gary Davis (@republicofmath) highlighted on his blog . I recommend reading the whole article. Skemp describes the difference between instrumental and relational understanding, and how the word understanding is used by different people to mean different types of understanding. He also makes the observation that what we call mathematics is in fact taught in two very distinct ways. Skemp uses an analogy to try and explain the difference between relational and instrumental knowledge which I would like to explore.

Imagine you are navigating a park, and you learn from someone else some specific paths to follow in the park. You move back and forth along the paths, and learn how to get from point A to B in the park, and you may even be able to move quickly from point A to B. Eventually, you add more points to your list of locations to which you know how to navigate. Step off any of your known paths though, and you are quickly completely lost, and you might even develop a fear of accidentally losing your way. You never really develop an overall understanding of what the park looks like, and you may even not know about other connections between the points you know. This is instrumental understanding.

Imagine that instead of navigating the park by specific paths shown to you, you get to wander all over the park. For some parts of the park you may be guided, through other parts of the park, you wander aimlessly. In time, you develop an overall picture of the park. You might discover the shortest paths between two points, and you might not, but you would understand the overall structure of the park, and how each point in the park is related to each other point. If someone showed you a short-cut in the park, you’d probably understood why it worked, and why it was faster than your meandering path. You wouldn’t worry about stepping off the path though, since even if you get lost, you’d be able to use your overall understanding to come to a place you know. This is relational understanding.

Here’s Richard Skemp’s description of the analogy.

“The kind of learning which leads to instrumental mathematics consists of the learning of an increasing number of fixed plans, by which pupils can find their way from particular starting points (the data) to required finishing points (the answers to the questions). The plan tells them what to do at each choice point, as in the concrete example. And as in the concrete example, what has to be done next is determined purely by the local situation. (When you see the post office, turn left. When you have cleared brackets, collect like terms.) There is no awareness of the overall relationship between successive stages, and the final goal. And in both cases, the learner is dependent on outside guidance for learning each new ‘way to get there’.

In contrast, learning relational mathematics consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point. (I say ‘in principle’ because of course some of these paths will be much harder to construct than others.) This kind of learning is different in several ways from instrumental learning.” ~ Richard Skemp, Mathematics Teaching, 77, 20–26, (1976)

Instrumental understanding is really useful when you have to know how to do a specific task quickly, and aren’t too concerned about how this task fits into other similar tasks. Relational understanding is useful when you want to explore ideas further, are unconcerned about your destination, and are more concerned with the process.

Unfortunately, our system tends to favour instrumental understanding too much. While it is useful to be able to get from point A to point B quickly, if one is not aware of one’s surroundings, and doesn’t get to enjoy the scenery, it hardly makes the trip worthwhile.

I’m working on a block puzzle game. The objective is to cover the entire puzzle area with blocks of various sizes. So far I’ve got the basic structure up (it will only run in web browsers that support the Canvas HTML element, so Safari, Firefox, Google Chrome, and maybe Opera). Scoring for the game depends on what types of blocks are used (you’ll notice those little 1 by 1 squares are worth no points).

I’m looking for feedback on how to improve the puzzles.

https://davidwees.com/javascript/blockgame/

• Restrict what playing pieces the players can use.
• Randomize the playing pieces to which the players have access.
• Allow more access to different kinds of shapes, such as triangles, pentiminoes, heximinoes, etc…

Update:

Here is some feedback I’ve received as well from other sources.

• Change the images that turn into the blocks into pictures of the blocks. @joshgiesbrecht
• Change from scoring blocks to a par system (like golf) where players get scored on the number of blocks used. @joshgiesbrecht
• Make the point that one gets a higher score from using larger pieces more obvious. @joshgiesbrecht

Annie gives a very short talk that highlights some of the issues in math education, and which I can tie to work various people have done on learning.

Everyone who is trained to become an educator has some fairly strong intuitive sense of what it means to be an educator. They have seen educators work, and they know how to copy the behaviours of the teachers they have seen. Unfortunately, often we want to change teachers behaviours, and so we must address the misconceptions that teachers have about learning head-on.

If you do not address the misconceptions that people have, chances are very good that they will incorporate the new information you present (in almost anyway that you present it) into their existing misconceptions and as a result, not change their behaviours at all. This is a problem that numerous educators have discovered (it seems independently of each other) and one which definitely has implications for teacher education.

Annie’s observation that her teaching college in 1988 was already talking about inquiry based learning, and some pretty serious reforms in mathematics education, and then her description of her beginning practices which were so different, gets at the heart of this issue. She was "taught" that inquiry based mathematics is an effective pedagogy, but she didn’t hear it. She probably did hear it, but she thought that her notion of what inquiry based education meant was the same as what she was doing. She was unable as a beginning teacher to see how different her techniques were than what she was being taught to do.

So if we want to change teacher education, we definitely need to assume that the student teachers coming in have an understanding of what it is to teach, and that much of what they understand is misguided and just plain wrong, and we need to incorporate the wrongness of this approach into our instruction of teachers.

Having spend the last ten years teaching students mathematical notation (while simultaneously teaching the mathematical concepts described by these symbols), I have often reflected on how efficient and amazing it is, and how unfortunately broken it often is.

Some notation shows off some of the power of mathematical thinking (for example, algebra), but some notation has clearly not been designed for clarity. In fact, my suspicion is that much of mathematical notation has been invented to save space.

Of course, a reason why one might one want to save space with mathematical symbols is because paper used to be expensive but I suspect this is not the main reason mathematical symbols are so tightly packed with information. It is also time-consuming to use more clear mathematical notation, and mathematicians love to be concise. In fact, I have often noticed that mathematicians often equate the length of a mathematical proof with its elegance, which over time may have supplied pressure to reduce the notation used to describe these proofs. A few mathematicians have contributed heavily to mathematical notation, most notably Leonard Euler, and these few mathematician’s desire for brevity has defined the notation we use today to communication mathematics.

Look at sigma notation for example. What does the letter sigma from the Greek alphabet have to do with finding sums of things? Absolutely nothing as far as I can tell. According to Dave Radcliffe, Sigma (∑) is short for summa (probably because they start with the same sound), which is the Latin for sum. Euler invented the symbol to use for summation, and we’ve been using it ever since. Essentially, we are using ∑ to mean sum for historical reasons.

The portion of this equation to the left of the leftmost equals sign is summation notation, which I have taught for years. I usually have to spend a class, sometimes two, explaining this specific set of notation. The brevity of the summation notation contributes little to the comprehensibility of this statement. It is essentially equivalent to the following:

Summation (i, 3, 6, i2) = 32 + 42 + 52 + 62 = 86

Unfortunately this notation requires us to memorize the order of the parameters in the summation function, but this is functionally the same as the previous notation, except one more piece of information is given to us; we know we will be doing a sum of some kind without having to memorize the meaning of sigma. With some work, we may be able to improve upon this notation more, and provide even more clarity.

Summation (index: i, start: 3, end: 6, function: i2) = 32 + 42 + 52 + 62 = 86

This notation is somewhat more clear the second option I suggested, since the parameters are defined within the notation. It is significantly longer to write than the original notation (takes up twice as much space) but it has a huge benefit of being significantly clearer. Further, one could imagine that if I were entering this notation into a computer, that the autocomplete function (which is common to code editors) could suggest parameters for me, as well as show me the definition of the parameter as I enter it. Finally, this notation is similar to how we define functions in computer programming (in some languages), and so when we teach mathematical notation, we will also be giving our students some ability to read computer programming code.

This issue about notation is not a trivial concern. The notation used to explain mathematical ideas is often a barrier to some students learning how to communicate mathematical ideas. Quite often students (and sometimes teachers) confuse learning notation for learning mathematics.

Furthermore, notation which is excellent on paper may be somewhat less useful on a computer. I have spent many hours looking for solutions to make adding mathematical symbols to websites more convenient and have discovered that there is no easy way to do this. Every method has drawbacks, and no method is as convenient as adding the same symbols to paper. My conclusion in terms of using mathematical notation with computers is that one of two things (or both) will happen. Computers will develop more touch senstitive interfaces, and software developers will create software that recognize the current mathematical symbols, or we will start to change mathematical notation to be more easily inputted into a computer.

The one huge advantage of our current notation is that it is somewhat universal. Essentially the same notation is used around the world, and by choosing a more amateur friendly notation, we will be creating localized versions of the notation for each language which is obviously problematic. In a computer, this is easily resolved by making the names of mathematical objects translatable so that whomever is viewing a mathematical document can select their language of choice. In print, this is more of an issue, and so we should reluctantly continue to use our existing notation until we have more fully transitioned from our traditional print medium, but the more we use computers to communicate mathematics, the more likely it is that we should fix mathematical notation.

Update:

Here are a couple of critiques of this post:

Yesterday, our learning specialist for science, Ana, read an article about how games are used to help simulate the spread of disease. She suggested that we could turn this into a collaboration between biology and math, and create a game so that students learn some of the principles of the spread of disease (which is a biology topic) from a mathematical perspective.

I created a simulation so we could test what parameters we may want to use in the classroom so that students are most likely to see that the spread of a disease can be modelled effectivelyh, and see the probability of the infection being spread from person doesn’t change the type of mathematical infection curve much. Try the simulation here.

Some assumptions I’ve made with this simulation:

• Individuals once infected, stay infected.
• Each individual has an equal probability of being infected by anyone else in the population.
• The probability of anyone being infected remains constant over time.
• Individuals can be re-infected.

I don’t know if we will end up using this simulation with students, but if we do, I’d like it to be fairly clear  so they can get started using the simulation without much intervention from me.