Thursday, July 18, 2013

Fraction War Task

A while ago I wrote a post on embedding atomic skills into tasks so that the basic skills are developed and used as tools of mathematics rather than the ultimate goal of mathematics. I try to develop tasks that follow this framework. I want the student to choose a pathway of thought that enables them to use basic skills, but doesn't focus entirely on them.

Recently, I was reading Young Children Reinvent Arithmetic: Implications of Piaget's Theory by Constance Kamii and came across one of her games that she plays with first graders in her game-driven curriculum.

The game was called double war. Each child got two piles. Each turn consisted of both players turning over a card from each of their piles for a total of two cards each. They would then decide who had the highest sum. This game--although powerful for first graders--holds little value for my high school students. I got to thinking how I could capture the spirit of the game as well as a relevance for the grade nine and ten range.

I wanted to stick with the "war" theme. I also wanted students to decide whether numbers were larger or smaller. I also thought I would target fractions because they present my students with perennial difficulty. The bulk of the task would be completed in pairs. 

I strategically chose a series of numbers between one and ten. The goal of the task is for the students to create an "army" of fractions (so to speak). The army needs to consist of the largest ten fractions possible. (they are given 20 numbers total). That goal is designed to create discourse within and between groups. 

What makes an army the largest? The sum of the entire army is the largest possible? Maybe two armies are compared on a fraction-by-fraction basis? Each army could send out their champion and they could battle.

To further increase discourse and challenge, I added two conditions to the army creation:

  1. No two fractions can be the same
  2. You can not create a fraction that will reduce
The task is delivered to the pairs in three stages. Each stage has an accompanying handout. For convenience, I have linked this post to each handout as a Google Doc. 

Stage 1 - Entry
Each group gets a copy of the Playing Cards. The handout includes two copies of the twenty numbers. In stage one, I want students to begin to discuss how they will approach the problem. Nothing is formally recorded, but I am sure to stop by and pick the brains of each pair. Even in stage one I am hoping for the discussion to bleed between groups. 

Stage 2 - Formalization
I give each group a copy of the How To... handout where they are required to talk through their strategy to create the strongest set of ten fractions. I really encourage students to work with the restraints that they are given. Mathematics is about sense making around a set of parameters. Here, students should use the fact that no two fractions can be identical as proof; relatively prime numbers should become a logical truth in their arsenal. This stage usually lasts fairly long. I am sure to gather main ideas from groups as they write. These carefully selected and sequenced ideas will become the basis of a class discussion. 

Stage 3 - Extension
After a class discussion, I give each pair an Army to Beat handout. If your class is short, this may end up being a take home assignment. It would be a great one for students to pull out after dinner with parents. Here, students are asked to design an army that "beats" a carefully pre-designed army. Again, their interpretation of "beat" is important. Some may match up fractions mano-a-mano while others will justify based on a global sum. Both hold important mathematics.

The goal of the task is to get students thinking about how fractions work and what they mean. I want them to get a deep understanding of what happens to fractions as their numerators get smaller or their denominators get larger (or vice versa). I want them to realize that fractions are a composition of numbers and they have a spot on the number line like other numbers. All of these deeper, more robust learnings are bolstered by the atomic skills of building, comparing, and adding fractions. Hopefully these things are all captured within Kamii's framework of autonomy provided through self-directed games in math class.


Saturday, July 6, 2013

Spinner Data Task

The difference between what should happen and what does happen is a difficult distinction for students. They are so used to finding exact answers in the back of textbooks, that differing experimental results create an sense of uneasiness. At an early age (Grade 9 in my province) we begin to introduce students to the ideas of sampling and experimental probability. 

The topic is usually approached with a project or survey of schoolmates. The results are then tallied and then used to create "probabilities" of various things such as favourite sports team, food, or colour. I love the philosophy behind the project approach; student initiative and autonomy is a powerful thing. I, however, don't like that the experiment involves humans. Here's why...

Students see humans as unpredictable, but often view mathematical concepts as very predictable. Can we blame them? For years they have been calculating, to remarkable precision, answers neatly coded in the back of text books. Calculating experimental probabilities on unpredictable subjects allows students to connect probability (a naturally unpredictable thing) exclusively to (seemingly) unpredictable events. 

I want my students to encounter something counter intuitive. Hopefully, they can construct a sense of stochastic thinking through this cognitive conflict. 

Enter something very predictable... something so mathematical... the coloured spinner. 

Hand each group a Spinner Data Task: Spinner Charts handout.

I begin by establishing the predictability of the device. I ensure the students that each section of the spinner is exactly one-fifth of the total area. We also talk about ties. What should happen if it lands on the divider between two sections? Because this is digital, I explain that ties are not possible because of certain rounding. In a nutshell, the spinner is completely fair. 

Have the students predict the results of ten spins. What is the logical conclusion? Is there any dissension? This a great time to to play on predictability. 

"So you're telling me that it MUST land on each segment exactly twice?"

Make the students verbalize uncertainty; add the language of probability to their discourse. 

Have them predict the results for 100 spins. Are they more willing to see variance because the number of trials went up? Is anyone more certain that the spread will be uniform? You will get interesting arguments form both sides:

"There are more spins, so more chance of the spinner landing on the same piece over and over."
"There are more spins, so there are more chances for them to even up as you go."

Both excellent conversations.

I show them the result for 200 spins with a 6 second Vine video. This provides a quick reveal and an immediate feedback loop. It also allows teachers show the technology even if they don't have access to individual technology for each group. The Vine is pausable at any point. Have a discussion with your students about the result.

Is it fair? Is the program rigged? Where is the breaking point? At what point would you no longer believe the computer simulation? A spread of 10? 20? 30?

The second spinner on the handout shows the section labeled "1" as twice as big as the other four. I would go through the same process with the students as before--but with a growing awareness of uncertainty. 

The next spinner has section "1" worth three times sections "3", "4", and "5". Section "2" is double that of "3", "4", and "5". Repeat process. Have students predict the results and graph on the accompanying charts that mirror the Vine videos. 

The last spinner has section "1" is worth four times sections "3" and "4" while sections "2" and "5" are double the size of sections "3" and "4". 

After the discussion around the proportions wanes, I want to reverse the process to see if students can get a feel for the "population" when given the data from a "sample". I give each group a copy of a handout with a set of data. They are asked to draw the spinner that was used to collect the data. 

I provide them with a protractor and allow them to use their calculator. Students will probably assume the data was generated with "nice fractions". Some groups will take the percentages and try to match them with the closest unit fraction. The nice corollary of this is they try and add all their fractions to one. Entertaining...

It is important that students write out their logic and record the sizes of their regions. (as fractions or percents). When each group has attempted a solution, I digitize a couple and run a simulation to see how close they were. I recommend getting familiar with the spinner website before class. 

The task develops the notion of probability within a seemingly predictable context. It encourages good math discourse, lays groundwork for bias in real world sampling, and has students conjecturing, arguing, and creating all at once. It works on basic ratio skills as an added bonus.

Common Core State Standards Addressed

Saskatchewan Curriculum: Math 9 Curricular Outcomes Addressed

Using a predictable context to study sampling, data, and experimental probability creates a deeper appreciation for the mathematics of the unknown. The assault on their common sense creates an ecology of rich mathematical discourse as students learn that not all of mathematics can be summarized neatly in an answer key.


Tuesday, July 2, 2013

Ambiguous Case Vines

We all live in a consumer's world, and we do an amazing job at acting entitled. These two factors have culminated in the invention of Vine--an app used to create six second, looping video clips.

Yet another way in which students can create, share, and network around media. Unfortunately, I feel like my students don't often have an attention span longer than a Vine video.

In my opinion, Vine has two qualities that make it an interesting tool for math education.

  1. Stop motion video capabilities allows students to quickly see what might otherwise be acted out or verbally described by the teacher. (How many times have we used our forearm to represent slope?)
  2. Rapid looping means that students can watch multiple times. This should allow for pinpoint tutoring. (If it is possible to pinpoint a student's exact misunderstanding.)
The digital classroom is expanding rapidly. Agents of the flipped classroom are posting lecture videos in order to use class time for tasks that focus on deep understanding. The Khan Academy was set up around the idea that lectures can be available instantly to anyone who seeks them out. 

Various math teachers have developed things like 3 Acts, #WCYDWT, #dailyDesmos, and MakeoverMonday in an attempt to harness the powers that technology have and the unique connection it enjoys with today's youth. 

Vine presents a different animal all together--a sort of ADHD math tutorial. They cannot (and should not) be used as instructional pieces, but rather as illustrative aids. Students can pause whenever they need to, restart in a matter of seconds, or focus on a particular portion of the loop. 

These first few are really just an exploration of mine into this new software. Vines could be used to quickly introduce estimation tasks, show graph shifting, or step-by-step equation solving. Some applications open doors for deeper thinking, and others tie the bow on a topic or procedure. 

My jury is definitely out on Vine's educational significance, but I'll try anything once.

I targeted a topic that students had trouble with last year; the topic had to include some kind of spatial movement. I settled on the ambiguous case of the Sine Law because the students who could visualize the situation scored significantly higher than those who just plugged blindly into a formula. 

My textbook tries it's best to summarize the cases. It uses dashes and colour as best as it can. Paper just can't capture the movement necessary to build an effective visualization. Compare the four cases on paper, and then via Vine. 

I like the questions that the book asks, but don't think that paper has the ability to make them obvious to the students. The movement allows students to see why the number of solutions can be changed by the lengths of the sides. Vine also allows students to see which sides and angles must stay constant, and which can vary. 

On the whole, it is an interesting way to develop spatial intelligence in students. Vine could be a powerful visualization tool, or another passing fad courtesy of the rapidly expanding interweb.