Patterns & Five Practices to Model

After having read "Orchestrating Discussion" by M.S. Smith, E.K. Hughes, R.A. Engle, and M.K. Stein, and really enjoying the idea of the five practices mentioned, I've decided to practice them for myself. Currently I'm not teaching in a classroom so all of the practices can not be applied, but I thought thinking about the practices and trying to apply them as far as I could in a given situation would be good practice and perhaps revealing. So I went and found a problem related to patterns on which to try some of them out(Sticky Triangles from NRICH: http://nrich.maths.org/88).

The five practices to model from the article are as follows:

1. Anticipate Student Responses:  As a teacher before introducing the activity in class, do it yourself, consider and push yourself to come up with multiple ways in which it could be solved and that your students may solve it. You may consider misconceptions students might have and how that could influence their work. This allows you to be better prepared to address answers students give.
2. Monitor Students Work & Engagement: Walk around the classroom, observe what students are saying and writing. Notice when students are nearly finished. Ask students about what they're doing and question them to think deeper or consider different aspects.
3. Select Particular Student Work to Share: Having been monitoring their work, you have a good idea of what each student is doing and pick certain students to share based on ideas you want the class to discuss and consider and highlight perhaps different techniques used in solving.
4. Specific Order to Student Sharing: Depending on what you want to highlight or address first, have the order students share in match this. Perhaps to help the flow of the discussion or to clear up misconceptions first.
5. Connect Student Responses to Another & to Mathematical Ideas: What's the take away of everything? What should students be learning from this? What can they learn from another?

So 2-4 I won't be able to do, but I'll be applying and practicing 1 and 5 in relation to the problem of the Sticky Triangles. The problem shows triangles composed out of match sticks like the pictures below:

The problem asks students to explore the pattern that is occurring by finding how many matchsticks are in each, how to make the next triangle, and how to know many matchsticks to create any size triangle. 

Anticipating Student Responses:

• All students will likely find the number of match sticks in the 3 triangles given: 3, 9, 18.
• Finding triangles of at the next sizes, students would likely draw to figure out or if they notice the pattern quickly would calculate.
• For triangles of any size, students then will have to find an equation or way to describe what is happening.
• Might see the center of the second triangle as being the original first triangle. So it's 3 + 2 (3) where the first 3 is the matchsticks in the first triangle, 2 is the number of matchsticks at each extended corner and 3 for the three corners. But then the student would hopefully realize that the small triangle is not in the center of the third. So this does not work
• Might come up with a recursive formula. You are adding a new row of triangles to the bottom of the previous. The triangles added to the bottom are the number of the size of the triangle and each triangles uses 3 matchsticks. So previous plus 3 times the size of the triangle.
• Might look at the numbers, not the pictures, to come up with a formula. From 3 to 9 is plus 6, from 9 to 18 is plus 9.
• Might look at the matchsticks around the edge and then the number in the center. So first is 3+0, second is 6+3, and third is 9+9. For the outside it is 3 times the size of the triangle. The inside is almost 3^s-1 where s is the size. Or again could use recursive.
• Might notice how the number of size 1 triangles in each size are changing.
• Likely some students will come up with formulas that apply to the first and second, but not beyond. And students who do not check to see if their formula can predict further answers.

Connecting Responses:

• From this activity, I would hope students would take away the multiple ways in which a pattern can be described within both words and equations to represent the situation. Students hopefully can learn from others and the different techniques their classmates used to solve the problem. Understand how we can represent a pattern with an equation, and how to use variables in this case (what do they represent, where can we see it). Highlight recursive formulas and how they are represented in an equation.
• The problem also at the end asks students to consider how their findings would change if we were composing squares in a similar manner. I might have students do this after the discussion or take home so that they can apply the ideas we talked about as a group and practice different ways they heard.

After looking at the problem in this way, I feel it really pushes me to consider and see what really can be taken away from a problem. It allowed to think about different ideas, some of which I might not have thought of, and maybe students wouldn't either. It also shows if a problem may or may not have multiple view points, if there would be anything to discuss about the problem. If a problem were fairly simple, using the five practices would reveal this. You likely would not be able to think up many different student responses. It gets you to consider your students and how they might think. I don't have students, but if I did I can see how you'd be able to think back to work they've done and how ideas they have may influence their techniques in the given situation. It also gets you to see what things should be discussed. If the problem is very computational, you can focus on different ways students think through the computations, what order do they do things in, and how can they check their work. You can get at why students are doing what they are doing, because you'll have an idea before hand and can prepare appropriate questions.

The five practices really gets you to focus in on what you are having students do, what are they doing, and why does this matter? You can better plan and be better prepared.

Repeating Decimals

Awhile back as a class we investigated repeating decimals. After some work and discussion, we came to see that when doing the long division of the fraction, if you arrive at the same remainder twice then you have a repeating decimal. For example consider 1/6:

Each reminder will keep being a multiple of 4, which 6 will continue to go into by the same amount each time. Pretty cool.

Then we all went home with the task of investigating repeating decimals further on our own.

What immediately came to my mind to look into was if there were a way to look at the fraction and tell if it would be repeating decimal without doing the long division or typing it into your calculator. My first thought was just that maybe it dealt with the denominator being prime, since I had been focusing on a 1/3 for some time. But this was quickly, like the second after thought, proven wrong. There's a 1/5 and a 1/2, and 1/6 repeats but 6 is not prime. Then I realized I should probably come up with a couple more examples of repeating decimals, so know what the numbers look like. Here's the list I got: 1/3, 1/6, 1/7, 1/9, 1/11, 1/12, 1/13, 1/14, 1/15, 1/17, 1/18.

I also knew that certain multiples of these would become non-repeating (so I would have to look at fractions in simplified form). I noticed from my list that 1/6, 1/9, 1/12, 1/15, and 1/18 related to a 1/3 since multiples of them could be reduced to a third. They are also was of breaking down a 1/3 into smaller pieces, so it makes sense that they would be repeating decimals as well. I then looked at 1/7 and thought to myself that anything with a denominator a multiple of 7 will be a repeating decimal. I then checked: 1/14, 1/21, 1/28. What do you know, it worked. So it would likewise follow for 11, 13, 17, and 19 (the denominators of other repeating decimals I recorded).

I then noticed something all these numbers had in common, their denominators were composed of more than just 5 and 2 (the prime factors of 10). Those composed of only 5 and 2 or that could be simplified to such were not repeating: 1/5, 1/2, 1/4, 1/8, 1/10, 1/20.

I was curious as to why and might explore that later. I had a thought that since the numbers the denominator is compose of go into 10, when doing the long division you would eventually have a point when that value goes into the reminder (a multiple of 10) completely. But I was content with my findings and had fun processing through it: asking different questions, gathering more examples, analyzing what I had, and then being able to make a conclusion and reach an understanding on what I had set out on,

Understanding Division by Fractions

After reading an article by Richard R. Skemp discussing relational vs. instrumental understanding, in class we discussed how we had learned fraction operations (addition, subtraction, multiplication, division). For me and the people in my group, most of us had learned through instrumental understanding. We were taught a rule to memorize and apply in the given situation in order to solve the problem. A "rules without reasons" way of learning.

As we went over the different operations, for all of the first 3, addition, subtraction, and multiplication, we felt we had a good understanding of them. The answers we would get from problems made sense to us and we could explain to someone verbally why. However, when it came to division with fractions, we were a little stumped. Most of us were taught the rule of flipping the denominator and numerator of the fraction you are dividing by and then multiplying that with the other fraction. We knew the rule well and could explain to someone how to solve it. But why was the answer the way it was?

 I thought this was sort of interesting. Why was it so difficult?

I think part of the difficulty comes from the fact that we often don't talk about dividing by fractions. Another part might be that we did not have a good idea of how to visualize what was happening or to describe the process. After looking into division with fractions, these two things coming together really helped.

1). Verbally:
Knowing how to read a division by fractions problem is really important. This helps you to understand what you are solving for. For example, given: 1\2 ÷ 1/4, what is being asked is how many fourths are in a half. This relates right back to division with integers, like 6 ÷ 3 is asking how many 3's are in 6. Once we see that this is what is being asked, we can better represent it with a picture and also understand our answer.

2) Pictorially:
Keeping with the same problem (1\2 ÷ 1/4), it's helpful to begin with drawing what a whole is. After this, we can mark in what a half is and what a fourth of our whole will look like.Then it simply becomes finding out how many fourths will fit into the half.

We see that two will fit, so our answer is two.

3) Making sense of it all:
We can reason to see that the answer we got is correct. Remember, we're finding how many groups of a 1/4 go into a 1/2. We already know that a 1/4 is less than a 1/2, so we should expect our answer to be at least greater than one. Thinking in the same way, we can reason through other division by fraction problems. Like 4 ÷ 5/6 for example. We know that 5/6 is less than one whole, so we should expect our answer to be at least greater than one. We also know that we have four whole pieces, so we should expect for each whole that at least more than one 5/6 will fit into it, so our answer should be at least more than 4.
What if the dividend is smaller than the divisor? Let's look at the example of 1\3 ÷ 1/2. Since 1/2 is larger than 1/3 we should expect our answer to be less than 1, since not even one 1/2 will fit into the 1/3.


Having a better understanding of what fraction division problems are saying allowed me to think through the problem and reason where the answer should be, as well as to better understand how to visualize and model what is happening. Building a relational understanding of fraction division really helps to better understand it and now for me to help explain to others how to do it and incorporate the use of models effectively.

Confusing Contexts

A few weeks back during one of our previous classes, we had a brief discussion on what contexts we would choose to use in teaching fractions. Many of us mentioned food, baking, measuring time and distances, money. These we believed were things students knew about and had back ground knowledge on, and potentially things our students would find interesting or care about - who doesn't like food?

This past week during an observation, I realized more so just how important the context is that we choose to use. I was in a seventh grade classroom. The students were reviewing for their upcoming test the next day over percentages (comparing percents, fractions, and decimals, finding percentage change, what's the percentage of a whole?). I was walking around the classroom, helping students who had questions. Most of the students in the class seemed unafraid to ask questions if they were uncertain about something - which seemed to be a result of the sort of classroom the teacher had cultivated. Anyway, as I was walking around I had a girl ask me over to see if she was solving a problem correctly. The problem gave her the price of a good, the percent of a mark up on the good, and asked for the new price of the good after the mark up. When I asked her what she was thinking, she explained to me how she had solved the previous problem (given the price of a good, and the percentage of discount, find the new price). She asked if she would then solve this one in the same way - after all she was given a price and a percentage and asked to find the new price. So to her the two problems seemed very similar, if not the same. I asked her if she knew what "mark up" meant and she shook her head no. I helped to explain to her the concept and then she was able to solve the problem on her own.

Originally, viewing the problem, it sounded like the previous one. While she may not have known or simply had forgotten, not knowing what a mark up was, she made an assumption that felt logical to her so she could keep going and solve the problem. Luckily she had asked a question, but not all students will. On a test, she might have gotten points off, even though once she understood the situation she knew how to correctly solve it.

Context is important. We want to see what students have learned and understand, instead of whether they know what the term "mark up" means. We may think students know what a mark up is (it uses the word up, so wouldn't students think increasing?), but they may not. Lots of students haven't had jobs yet and are more familiar with ideas of sales and discounts, than the marking up of prices by suppliers. When using contexts, we need to be careful and ensure they are relate-able to students and that students understand them and any vocabulary or ideas that accompany it.

Proportion Sense

Recently in class we began discussing fractions. We began by looking and discussing proportions without the use of numbers. It seemed like it would be simple, but without the use of fractions at our disposal it became difficult.

We were given a picture of two boxes of chocolates(like below) and asked which one was more nutty?

It was difficult to stop yourself from using and comparing fractions - since we already have that knowledge. Our future students, however, will be just developing how to use fractions and relate them. Instead, we had to rely on the picture and ways in which we could manipulate it. What we ended up finding most helpful was the idea of buying enough of each box so you have the same number of chocolates and then comparing how many nuts each one has. An idea  that relates back to finding a common denominator. Once you had the same number of each, making a comparison between the two was logical.

However, after accomplishing this, when faced with another proportional problem we fell into mistakes. We were given a problem comparing mixtures of blue dye with water. Given certain a number of beakers of dye and water, we had to decide which would produce a stronger blue color. It was interesting to see how different we reacted to this problem. Many of us ignored the idea of getting to the same number of beakers and began to cancel out blue dye and water combos. The result was a false belief that the two mixtures shown below would be the same color.

We failed to take into account the ratio of water to blue dye. It was startling to find ourselves making this mistake.

These two activities really helped me to come to understand the importance of proportional sense. My future students will likely be making similar mistakes, but having a discussion on these ideas and working through proportions without numbers will be helpful. Introducing students first to proportions and making sense of the relationships between them will get students asking important questions:

• What quantities are being compared?
• What other factors do I have to consider? 
• What roles do these variables play/what's their relationship?
• How is it affecting the goal(if it's nuttier or more blue)?

Through this, students can begin to develop a sense of how to compare proportions and what to take into account. Then, when dealing with the numbers and fractions they should to able to reason if the answer they arrive at is a logical one or not. They can rely on asking questions to led them to the correct calculations and hopefully dealing with fractions will not seem as daunting.

The Human Number Line

One of my current classes is centered upon teaching middle school math - which looking back I now realize began a decade ago for me. Most of what and how I was taught, I honestly don't remember in much detail. However, I can recall that for the most part I was in my seat and taking notes - likely doodling as well. Maybe once in a while I remember forming groups, but really infrequently. 

During one of our past meetings as a class, we were introduced to the idea of using a human number line - a number line down the center/front of the classroom on which students could walk/stand. Our line was centered at zero and went from -15 to 15. As a class we illustrated by walking various equations and story problems. We played games that involved adding and subtracting of negative integers - trying to beat your opponent by making it to your end first (whether -15 or 15). Even as a college student it was enjoyable to work out how to play the games and interact with another in a different way. 

Instead of having students just sitting and taking notes, students can make connections between action/movement, visuals, and the math algebraically. For example, when given the problem 8-(-5), the student would begin on the number 8 facing the class, then turn toward the negative end of the number line (representing the subtraction) and then walk backwards (representing the negative number) 5 paces. The student would land on 13, the solution to the problem. In the process, students will begin to notice that subtracting a negative (though different) results in the same solution if you were to have added a positive 5. Other students in the classroom not walking out the problem can follow by using a printed out number line paired with an object to move (like a chip or plastic cricket). 

In addition, number lines are useful and helpful later on in students mathematics careers. Allowing them to become familiar with them and their usefulness, just gives students another tool with which to work. Such as later encounters with inequalities and graphing the solutions. 

Doing the same old, same old, is boring. The use of the human number line seems like a good way to break this. Students are up and moving, and shown through multiple modes of representation - connecting ideas and figuring out what works best for them.

All of the Lights

One of my family's favorite Christmas movie is National Lampoon's Christmas Vacation. So since it's getting close and it's that time of the year, I thought I would look into something Christmas related. This weekend, since the weather was nice (no more snow and not too cold!), my family dug out the Christmas lights to put them up. And it got me thinking of that movie and all those lights. How many and how much would it cost to decorate a house as done in the movie - the entire roof and all around the entire sides.

First, I went to find out what sort of lights were used. Based on a picture in a movie, they're larger than the sort I'm used to. So then I searched Christmas lights. I know that there are all sorts of places to buy lights but I choose Home-Depot. The lights cost $8.48/each for a strand of 25 lights at a length of 25 feet. You can however only connect at most 2 sets. But I'm guessing by the looks of it with any lights you wouldn't be able to create that long of a strand. So we're just going to assume that we can attach them all together no matter.

So now how to figure out how many strands are needed. After a little searching, hoping to find the house dimensions (and then go round trying to figure out how many strands), I was able to find that house was decorated with 25,000 lights! That's a ton and no wonder. So divide that by the nice easy 25 lights per strand and you would need 1,000 strands of lights. And based on the website, you'd have to travel to several stores to pick them up - the one near me only has 20 on stock. These lights being bigger than usual, already cost a lot. Multiplying the cost of a strand at $8.48 by the number of strands, it ends up costing $8480.00 dollars. No thank you. I could pay to study abroad instead!

And the length of all those strands, at 25 feet per strand ends up being the same as the number of lights - 25,000 feet.

So then I thought: how tall of a tree would you need to use that many lights? I'm going to use a cone to represent a tree to find this out. Since lights go around the tree, I'm going to look at the surface area of a cone minus the base: pi*r*l. The variable "l" of the cone represents the slant height, which in terms of height equals sqrt(h² + r² ). So our formula becomes: pi*r*sqrt(h² + r² ). Just looking at my own tree, the height is about 3 times the radius of the tree. Using this I can narrow the formula down to having only one variable, height: pi*h/3*sqrt(h² + (h/3)² ). Now to simply the equation a bit:

= pi*h/3*sqrt((4h/3)² )

= pi*h/3*4h/3

= pi*4h²/9

To make things simple, just like the house we'll cover the entire tree with lights! Which I think would hurt to look at up close haha. The width of the light strands is about 2 inches or 1/6 of a foot, so the area of all the light strands is  feet squared. Setting this equal to the surface area and solving:

4166.67 = pi*4h²/9

37500 = pi*4h²

2984.15 = h²

54.63 = h

So the height of the tree is about 55 feet. A little too small to be the tree at Rockefeller Center (69 to 100ft). If the spacing between the lights was increased from nothing to something, the height of the tree would just continue to grow. So again, like I thought, I'm going to pass. I'd rather keep with tradition and continue to put up the fake tree and string a few lights.

A Jumble of Geometry

Recently in class, we began with some talking points - questions related to geometry to spark discussion. One of the questions asked, "From the diagram given below, you can find the measure of angle D":

After a brief look at the diagram, the answer was clear: no. The measure of angle D can not be found because the diagram does not tell have parallel lines - if this was given the answer would be yes! As a follow up, a new diagram was given. Below I will outline my steps in the process of solving for each of the 4 angles presented.

The Process

The first thing I noticed when looking over the above diagram was the 44 deg. angle at the bottom. Since this time the lines are parallel and recalling that opposite side exterior angles are congruent, I knew the value of A is 44 deg. as well. At this point, recalling the vertical angle theorem, I was able to fill in 2 other 44 deg. angles(in blue).

After writing in the values, I noticed the triangles at the top and bottom of the diagram. They now had 2 of there 3 angles filled in. Knowing that the angles in a triangle always add to 180 deg., I was able to find the third angle in each:

180 - (12 + 44) = 124

180 - (44 + 30) = 106

Once those angles were filled, my focus became on B. By the vertical angle theorem would also we 124 deg. And by the same theorem I was able to find another angle of 106.

If you can't tell, the tiny orange dot in the small triangle represents the other 106.

Next, I looked to solving C, because I felt like if I'm going in order thus far I should keep with it. But as I looked at the diagram, D felt like the clearer next step. All I would have to do is find all the angles of the little triangle and I'd be a step away from finding the measure of D. I recalled that angles on the same line add up to 180 deg. This allowed me to find the angle beneath 112 deg, because they are on the same line - so the angle must be 68. By the same reasoning I was able to find another angle of 68 deg.

Now there's a tiny pink dot in the tiny triangle. 

Again, I know all the angles of a triangle add up to 180 degrees. So 180 - (106 + 68) equals the measure of the third angle, which is 6 deg. Then I noticed that the measure of angle D with the angle I just found makes a circle, which has an angle of 360 deg. So using this fact and the found angle, D is 354 deg.

So now I finally had to face C. This one took me a minute. As I looked at the diagram, I kept thinking "how will I find C?" and wishing "if only I could find the measure of the angle that with C results in 360 deg". So how could I find this angle? The shape was irregular and I didn't know anything extra about the lines around C (no parallel lines to be had). I kept thinking, if only I could somehow break the shape in triangles and find other angles that could lead to C. I wished the shape was regular, then maybe I could find the sum of the angles and then find C. As I thought about this I realized that the regularity of the shape was irrelevant. This shape was five sides, so it's a pentagon, and pentagons (all and any) have 540 deg as the sum of their interior angles! 

Since I now have 4 of the 5, finding the fifth became extremely simple: 540 - (124 + 22 + 68 + 52) = 274. To find C, all I had to do was subtract this angle from 360. So C ends up being 86 deg.

Thoughts:

For not having done high school geometry in a while, I felt good about working through this problem. I especially liked the way in which it seems a little daunting (all the lines and angles and shapes) but as you dive in, you realize that each piece of the puzzle isn't too bad. It was also helpful in using several different theorems that I was taught in high school geometry to figure everything out. While the pentagon through me off a little in my search for C, I enjoyed that aspect. It was nice to struggle and face a minor challenge along the way - once reaching the solution, how obvious it now seemed! I think it would be fun to present the problem to high school students, since a lot of the knowledge and understanding required for the puzzle they are in the process of learning. And because it's a process, that one thing leads to the next and has more purpose than simply filling in the blank for one question, students could get a lot out of it - applying the skills they are gaining and diving into critical thinking and problem solving!

Visual Mathematical Laws

Inspired by the following blogpost, I decided to make my own visual representation of mathematical laws. Art being my tied for favorite subject with math, the idea of visual representations sounded not only fun to create but potentially really helpful for visual learners. The visual representations take away the daunting idea or confusion variables bring to many students. If students are confused about variables or just have a general dislike, with the visual representation they are still receiving the same information just in a more understandable way. Along with this, students should be brought to discover or shown how the laws have come to be, since understanding the laws is more important and helpful than just remembering them.

In addition, to me, the visual representation is more interesting. Students see numbers and variables all the time in their math classes - of course! Color, on the other hand, not so much. So maybe students will have a tendency to recall the laws better because of the uniqueness with which they were presented. Plus it's always fun to see math and art come together! - even if in such a small way.

Through the process of making the laws, I became more aware of how some of the laws work. Before I just took it for what I was told, memorizing but not really seeing the connections. The colors helped me to make the connections and clearly see where each piece is coming from. It's difficult to say if I would personally ask my students to create their own - it might take some students a lot of time and they may not see the purpose in it. I would however in going over the laws with a class encourage them to use colors in place of the variables and have a poster of the laws in this way displayed in the classroom.

So for mine, I decided to link it to those seen in the blogpost(which focused on laws of exponents) and visually represent the laws of logarithms :)
They are as follows:

• Logarithm to Exponential
• Canceling Exponentials (2nd and 3rd)
• Product
• Quotient
• Power
• Changing Base
• BONUS!

Working with Algebra Tiles

After using them in class and reading an article advocating for the use of algebra tiles in classrooms, I was left with two main questions.

 The first, deals with completing the square. The article gave a long list of things the tiles could be useful to help teach. Having been tutoring for awhile, I've realized that many students do not know what it is to complete the square or get confused about how. So I wanted to put it to the test and see how the tiles would be able to show the concept of completing the square.

I did a quick Google search to find some problems that require completing the square:

1. x^2 - 4x + 6 = 0
2. -x^2 - 2x - 5 = 0
3. 4x^2 + 4x - 3 =0

Then I completed the square to find the solution, so I could compare when working with the algebra tiles.

1. (x - 2)^2 + 2 = 0
2. -(x + 1)^2 - 4 = 0
3. (2x + 1)^2 -4 = 0

So I began, and at first, I was confused. I collected all the tiles I would need, but somehow could not form a rectangle/square. It simply was not possible.

After a moment, I realized, to complete the square I will have to create a square with the tiles. So first, after placing the x^2 tile, you have to divide the x tiles evenly on either side.

Then I filled in the units to complete the square, with 2 of the six leftover. So when writing the solution you get (-x + 2)^2 + 2 = 0. Which is equivalent to the solution above. As I solved the others, I realized it got easier as I went along. You just have to be careful with the units (that the total tiles is the total units you had in the original equation). Both ways, visual and symbols seem like fine ways to teach it. It seems like it would be helpful for visual/struggling learners and interesting to others. But potentially a waste of time for students who already grasp the idea symbolically, if they must do a lot of work using the tiles.

Here are the visual representations of the other 2 equations:

 

My second question, was if algebra tiles could make realizing if a given binomial can not be factored easier? So again, I Google searched and found two equations that could not be factored and got out my tiles.

1. x^2 + 2x + 5 = 0
2. x^2 + 4x + 1 = 0

After setting up both, the answer is yes! If the equation can not be factored, you will have more or less units than what is necessary to create a rectangle with the tiles. Since students are sometimes asked to factor an equation, or say if it can not be factored, this method seems really helpful. Students can see the solution quickly, whether it can be factored or not. It also allows students to pay attention to the units of the equation when deciding if something can be factored. When they face larger equations, which the tiles would not work well for, students know where to look.

 

So both questions resolved, I recommended the use of Algebra tiles. The algebra tiles help to introduce ideas of factoring (which can feel and seem abstract) in a more concrete way. Students can see, visually, why something works and how to fix the problem. They become more comfortable with the ideas of factoring and gain insight into recognizing when things can and cannot be factored. So hopefully, as a result, factoring will not feel like a daunting task, but an achievable one.

Teaching and Technology

      I have never been the biggest fan of technology. When I was younger my siblings would play video games or computer games, while I only turned the computer on to write stories or play solitaire. As I grew up this stayed about the same. I got my first phone sophomore year of high school and then never really used it until the end of senior year (my friends would call the house because that was the more reliable way to reach me). It's not that I don't appreciate technology or am bad with technology; I can learn things pretty quickly. I've just always had a slight disinterest towards it.


    Thinking about the use of technology in my future classroom, online activities and web-based programs, I leaned toward the side of no. I feel like there is a lot to said for doing things by hand. But the world is changing. While teachers of the past did not have access to such things, I will. Using sites like Desmos and programs such as Geogebra have shown me technology has a lot to offer a math classroom. Students can experiment with graphing equations, how to reflect/translate/transform them. Data collection can be increasingly simplified. There is an abundance of resources for activities and projects. Technology has the ability to help introduce students to new, difficult to grasp ideas, providing a transition into the topic. Students are able to make references to real situations and see the actual motion a graph is depicting.

   So my mind has changed. However, along with this desire to incorporate technology, which can do amazing things, I still have some hesitance. Yes students are seeing ideas in new ways, and yes they can work at their own pace, and yes they get the chance to explore and discover on their own. Yet, if not careful, the message and ideas you are hoping students to see will become lost. Some of the activities - and I know I have not seen them all - were fun and provide a good introduction to a topic; however, they felt semi-easy. I did not feel I was presented with a challenge or something asking me to think deeply and critically. And this is what I think is truly important. I see technology is helpful, but without in-depth discussions to follow up or asking students to create something of their own (to work through the process of solving a complex problem), it is not going to be used at it's full potential. We need to use technology to challenge our students, not just simplify how things can be done.

Eleusis: Problem Solving

The Problem

 
 Given the following Eleusis card set-up, find a rule to describe it. Then, list three more not yet played cards that can follow the set-up.

The Process

  
   My first initial thought when viewing the Eleusis card set-up was that color was not the sole factor of the rule. There were two blacks, two reds, black, red, four black. This did not appear to be any sort of pattern. Next I noticed how color affected cards of the same number being right and wrong. I found these four instances:

• A black 10 could not follow a black 8 but a red 10 could.
• A black 2 could follow a red 3 but a red 2 could not.
• A black 6 could follow a black king (or 13) but a red 6 could not.
• A black 8 could follow a black 7 but a red 8 could not.

   After discovering this, I was able to eliminate adding/subtracting of the cards to be part of the rule. Color would not affect the resulting sum or difference of the two numbers. Getting on the thought of addition and subtracting, I decided to look at the differences between the cards. At the time, I was thinking anything I discover will be helpful - whether I find a pattern or not. As a result I ended up with the following:

   In hindsight, I discovered what I already knew: that the difference between the numbers is irrelevant. Otherwise, if this was untrue any 10 should have been able to follow the 8 because the difference would still be the same. However, I stumbled upon something important. What is the value of the ace? In this rule, was the ace being viewed as 1 or 14, odd or even? To figure this out, I started to propose guesses as to what the rule may be.

• If you go from even to even, you must change suite.

This instance only occurred once. So I ruled it out and reformulated the hypothesis.

• If you go from even to even or odd to odd, you must change suite.

There were however cases not covered by this rule, so this was not it either. For example it could not explain why the 2 of hearts could not follow the 3 of diamonds. While neither of these were the rule, thinking and testing my guesses was helpful. I saw what did not work and began to notice my focus on odd/even and the suite/color of the card. So I made the following chart (assuming the ace is 1):

   This simplification of the information given, brought me to a pattern and a rule to describe the Eleusis set-up

The Solution

  
   The rule for the pictured Eleusis is: An even card must be followed by a red card, and an odd card must be followed by a black card. So based upon my rule I can play the following 3 cards:

Evaluation

 
    Describing my thought process was the easy part of solving this problem. Although, it took an effort to remind myself to record what all I had been thinking. Having stepped back and re-evaluated the problem several times over a couple days and still no answer, I thought to myself: perhaps it is unsolvable. Perhaps the diagram  is too limited to reveal the rule. Unwilling to give up and a strong desire to overcome the problem - it couldn't be that difficult could it? - I continued to search for the solution from where I left off each time. A lot of thinking, as well as various methods, went into solving this one problem. Even when I was not looking at the problem, I tried to think about it. It wasn't so much as arriving at a correct answer. After all, my rule only holds if the ace is 1 and there was no example in the diagram to confirm an even red card can follow an even red card. I kept trying because I enjoyed the complexity of the problem and simply did not want to give up. I wanted to make sense of the problem at hand and I did.

Eleusis: A Card Game

Eleusis is a fun and challenging card game that at it's heart encourages critical thinking and problem solving. Always having been a fan of card games and puzzles, I was excited to learn how to play.

The Rules:

Each Player is dealt 10 cards, expect the dealer (so the game is best played with smaller groups).

The dealer is then left in charge of coming up with a rule that the cards played must follow. The remaining cards are sat in a pile. The dealer will flip the first card over, then the players will take turns placing a card from their hand to test their hypothesis of what the dealer's card rule may be. If the players card does not follow the rule, it is placed beneath the card it could not follow. The player must then draw a card. 

The Object of The Game:

To guess the dealer's rule! After each correctly played card, a player can take a guess as to what the dealer's rule is. Once they get it, they have successfully won the game. From here you can pick a new dealer and the game starts over.

Example:

The first card played in the example is the 5, followed by the correctly played 7. The 10 was played after the 7, however, because it was incorrect it is placed under the 7. Then the 2, followed by the 9 are played after the 7 correctly. In this example the rule is very simple: Red, Black, Red,.....

Helpful Hints:

• Don't make the rule too complex. If your rule is too complex it will restrict a lot of cards from being played. The players will likely be unable to solve your rule and there's no fun in that.
• Don't make the rule too simple. The players will guess the rule quickly and the challenge of the game is lost.
• Play wrong cards. When you think you have the rule, play a card that goes against it. If the dealer says it works then your initial guess isn't quite right yet.
• What's the Ace? The ace can be viewed as high or low, 1 or 14, odd or even.
• Remember the Characteristics. You're searching for a pattern, and in finding this you should remember to consider all the characteristics cards have. Cards are even/odd, black/red, different suites and numbers. Any one of these or a combination can go into the rule.

    While playing Eleusis you get caught up in the game, you forget that you're thinking deeply and critically about each card that is played. Whether a card is right or wrong, you are taking it into consideration. You take multiple steps to try to solve the problem, testing your hypothesis, reformulating it and testing it again. In your head you are adding numbers, subtracting them, analyzing the relationships. Yet, Eleusis is just a card game and you are having fun. After playing the game in MTH 229 (Mathematical Activities for Secondary Teachers) and discussing our thought processes, it does seem like a great game to play in the classroom. Since it is a card game, negative perceptions and connotations of math disappear for the time being. At the same time, students gain and engage in mathematical thinking.

Check out my follow-up blogpost (Eleusis: Problem Solving) where I discuss my thought process in solving the rule for the below Eleusis card game. Try it for yourself and compare results.