Talking with Children: Shape Centers

I have been spending considerable effort looking for situations to "mathematize" in my daily interactions with students. Sadly, upper-level students are so mark and answer focused that they spend little time wondering about emerging problems with me.

This is not the case with my 8-year old friend.

While he was inventing his word problem, he stumbled upon the idea of a middle. Specifically, he told me that four train tracks met in the middle. I quickly asked him if three tracks could also meet in a middle. He responded with an annoyed, "of course".

A problem was born.

I drew a circle and asked him to find the center. He traced his way around the circumference and made some chords across the shape. When he was satisfied, he placed a dark dot in the interior of the center and smiled. "There".

I asked him how he knew he was right. He told me that he found the point that was the same distance to each edge. (At this point I thought about asking him how many edges a circle had, but I decided to stay the course.)

I drew a square.

We repeated the process. He used the vertices and tried to find a point that was equidistant. When I asked him if he could use the sides, he discovered that it yielded the same point. He was a little weirded out, but we pressed on. 

I drew a rectangle.

This time his strategy changed. He couldn't seem to find a way to be the same distance from each side. He placed a point and tried to measure by "pinching" the space with his thumb and index finger. Finally, he erased all his points and traced the perimeter of the shape. He then continued to trace another rectangle inside that one. And another, and another. As he worked, it was obvious that his brain was miles ahead of his hand because the lines got sloppier and sloppier. 

When he was done, his rectangle looked like this: (I asked to keep his work)

He proudly explained that tracing his way into the middle of the shape provided a perfect center. I was fascinated by this process, and asked him to apply it to his square.

Interesting. I drew a triangle, and he hesitated. I could tell that he thought triangles had no center. Nonetheless, he pressed on:

I was about to add a shape when he stopped.

"Wait"

He went on to explain that a center had to be a dot, and a dot had to be a circle. Using his method, he was creating smaller and smaller rectangles, squares, and triangles. He went on to show me that the dot in the middle of his triangle is just another triangle if "you put it under a microscope". In fact, he stated that there could never be a center to any shape because the dot would be round and the shape would not be. It could therefore not be the same distance from each side. 

I was impressed with his ability to iterate into the infinite. Not only was he constructing geometric knowledge, he was now struggling with the basic structures and how they related to infinity. (Infinity has been his favourite topic for years). 

I asked him if a circle has a center.

He waited and responded... "No".

He explained. Each circle could have a "dot" as a center, but that dot is just a smaller circle, and must have a "smaller dot" as it's center. Each dot can always get smaller "until it is only a molecule thick".

And if that is the case, "then there can't be a middle for anything". He slumped his shoulders with the weight of this reality. 

Here we have a young boy struggling with his personal construction of logico-mathemaitics as juxtaposed to the notation that is meant to describe it. He has been introduced to the formality of "points", "edges", and "centers" without ever having the opportunity to construct a deep understanding of what a center is. He is left at a crossroads between his own construction of reality and what the school worksheet tells him.

It is a sad predicament--one created by the burdens of time, curriculum, and the dreaded "test".

NatBanting

Talking with Children: Word Problems

My wife and I spend a lot of time with friends who have three young children. I spend most of that time engaged in a combination of trampoline dodge ball and mathematical discourse. I have begun to document the snippets of conversation on the "Talking with Children" page, but decided that larger ideas warrant their own post.

The middle child is most willing to think mathematically. During one of our conversations, he decided to turn the tables. What resulted is a wonderful look into a child's perception of what "mathematics" does. 

Him:  Maybe you can answer my question?
Me:   Sure. What is it?
Him:  Ummm... (literally scratches head)

I could tell that he was reaching for straws, but just as I was going to suggest some possible pathways, he began detailing his problem. What followed was a hodge-podge of textbook drivel. Roads met at points, trains travelled at speeds, the vehicles took neatly-timed breaks, and the whole thing finsished with a simple question:

Will the trains crash?

After the initial digestion of the problem, I asked him to draw it out so I could get a visual. Despite major changes in the question, he managed to draw this diagram and detail the problem:

His "textbook ready" diagram

Ok... um. There are 4 trains. One of them is 3km long. Another is 2km long. Another is 5km long. Another is 1km long. The 3km train goes 2mph. The 2km train goes 3mph. The 5km train goes 3mph and the 5km train goes 3mph.

All the tracks are five miles long. The 5km train stops at a stoplight for one minute. The 3km train stops for 30 seconds. Will the trains crash? 

 

There are so many awesome things going on here. First, we must keep in mind that the child's natural curiosity has been polluted by school mathematics. He constantly shows desire to make sense of his world through mathematics, but when required to produce a question, he can only think of manufactured, one-size-fits-all textbook problems. He hasn't even covered the mathematics of rates but knows that, when doing "math", it is important to list every detail neatly and orderly. He knew that each track needed a length and each train needed a speed. He even managed to mix up his units--for good measure. (Pun intended).

Second, the child makes the problem more difficult by adding more details. When I asked him if he could make the problem easier, he told me I could take away one of the trains. To him, more details always equals more difficulty. I asked him to make the problem more difficult, he paused and then told me that an asteroid hits Train 1.

Third, he assumes that there is no real basis for the questions his teachers ask him. He knows that they include real life objects (such as trains, stoplights, and asteroids), but has no inkling that mathematics is used to model and describe real life. He also has a sense of mathematical neatness. The entire situation is tied up with one, nice question: Will the trains crash?

We have to be constantly aware of what are students are perceiving as important details. We have built an educational culture where all the details are readily available to those who are willing (or are literate enough) to search for them within drawn out paragraphs of textbook banter. Students need to be presented mathematical situations where they are required to make decisions, gather resources, and apply strategies. I'm worried that school is not offering opportunities for students to reason and model mathematically. What would your students create if asked to write a math question?

NatBanting

Vedic Maths: Lipstick on a Pig

I was alerted to this video by a pre-service teacher that helps in my room every week. Before this post makes any sense, you should watch the video below. Try to watch the whole thing--I found that task very difficult.

As I watched, I found myself becoming increasingly annoyed with the topic. The presenter claims that the problem with current  mathematics is the algorithms that we teach. Ironically, he couples this solution with a boring lecture complete with lack luster audience polling, inadequate wait times, and dry humour. I imagine that a lot of what is wrong with math education can be pulled from his very presentation.

While I was watching, I was reminded of two phrases that have become part of my every day worldview.

The first is a quote from President Obama during his first run for election. (Yes, even us Canadians follow some American politics.) Although education only makes a cameo in the speech, the message rings true for change in any capacity. Obama accuses Republican promises as identical policies with different names and goes on to inform the audience that "you can put lipstick on a pig." (retrieved from YouTube http://www.youtube.com/watch?v=58FVeYjHpM8 )

To truly create a measurable gain in holistic numeracy across the world, educators need to undercut the surface gimmicks and stop putting makeup on pigs.

The second quote comes from a small but potent book on mathematics education written by Paul Lockhart called A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. This book is a staple on many educators' shelves. 

While musing on the tendency of school to get caught up in notation, he remarks that "it's like rearranging the deck chairs on the Titanic!" (p. 37). Education so often becomes a battle between frivolous things; in this case, it has become a war over which mindless--and preset--algorithms produce the best math scores.

With all of that said, there are three reasons why I feel that Vidic Maths is not revolutionary for math education:

It's spearheaded in a corporate manner

People with business degrees trying to push products in education scare me. It seems--from this presentation--that those behind the "system" of Vidic Maths are out to distribute their algorithms for profit. What scares me further is the fact that he bothered to mention the legal struggles he went through to gain the right to mathematics. Mathematics belongs to no one; You cannot take a patent out on deep understanding. He has attempted to take base-10 manipulation and claim that he invented its patterns and caveats. I chalk them up to nothing more than the hidden and beautiful patterns that emerge throughout mathematics. 

The presentation spends very little time on meaning

I put myself in a student's shoes while he flashes the slide with the number theory logic behind the trick. The slide was shown for under two seconds. He does the same thing with the other "trick". The focus of Vidic Maths is not understanding. In fact, the focus is on the exact opposite. The collection of tricks is designed to be memorized and correctly implemented. The whole time, comments like, "easy, isn't it?"(2:21) push instruction forward. The description of the video calls the method the "High Speed Vidic Maths" method. Same empty understanding, just faster. (Even that is debatable).

The method feigns numerical flexibility

This one stings me the most. People will look at this guy and think that he has a great understanding of how numbers work and interact, when really he is parroting. All that Vidic Maths accomplishes is the replacement of a long, yet robust, algorithm with a series of shorter, less versatile ones. Actually, it eliminates what little mathematical sense we did have in the old multiplication algorithm. Now we have many more mindless manipulations to complete. Also notice that the presenter still uses the language of the "inefficient" system. He speaks of "carrying" (2:13) in his example. A topic that is born from the understanding of the traditional algorithm. 

At the six minute mark, he mocks a student's method of multiplying eight by seven. Although the student miscounted their circles, they show an understanding of what multiplication is by drawing out sets of circles and then counting. Such learning is much more valuable, applicable, and transferable to a society that is in a mathematical crisis. Vidic Maths doesn't teach the beauty of our base-10 system, it exploits it and renders it unthinkably rigid. 

I believe that basic math skills need to be built up through the deep understanding of numerical flexibility. In other words, I would love students to understand why the processes they use to arrive at answers work. Vidic Maths creates more problems than solutions. Instead of focusing on the mechanisms behind the methods, they promote the rote memorization of more facts. Unfortunately, these rules work in very limited capacities. (Multiplying two numbers close to a power of ten, or multiplying two digit numbers by eleven, etc.)

Unless taught with a focus on understanding the mechanisms, Vidic Maths is reduced to yet another way we can teach students to follow directions in stead of creating directions. 

NatBanting

Note: My full impression of the system comes from this video. I can envision a system of education which uses understanding of these methods as a launchpad into deep understanding. Such a system is not portrayed in any shape, way, or form in the video. 

Rubricized: Thoughts Provoked by Skemp

This week I had the privilege of chatting with other math educators about an article written by Richard R. Skemp in 1976. We have formed a sort of ad hoc reading group built around reading classic and contemporary pieces of mathematics education research and discussing their application to our daily crafts. The inaugural meeting (so to speak) consisted of Raymond Johnson (@MathEdnet), Chris Robinson (@absvalteaching), Nik Doran (@nik_d_maths), Joshua Fisher (suspiciously un-twitterable), and myself(@NatBanting).

The full conversation--facilitated through Google Hangouts--can be viewed on Raymond Johnson's blog here.

It must first be said that I loved the article; it would be a great starting point for any math educator to begin to wrestle with the distinctions obnoxiously present in the field of math education. It also holds timeless value for those more experienced in the digestion of such literature. Full reference can be found at the end of this post.

Moving forward...

In a nutshell, Skemp proposes that there are two types of understanding when it comes to the field of mathematics. The first--relational understanding--describes the process of knowing what to do and why you are doing it. The second--instrumental understanding--describes the process of applying rules to arrive at answers. It must be said right that these two understandings are not mutually exclusive. In fact, most of our conversation revolved around their classification, necessity, and interrelation.

The group--following Skemp's lead--addresses the issue of assessment in light of relational understanding. How can we begin to assess deep understanding of mathematics, especially understanding that is not our own. The methods of portfolios and interviews were suggested, but dismissed as regular assessment pieces due to the extensive time burden. 

Here is my thought on the difficulty of assessing relational understanding:

Relational understanding is difficult to assess because teachers use an instrumental approach in all assessment.

Allow me to briefly explain...

A lot of  teachers match instrumental--or algorithmic--assessments to their instrumental teaching. That seems like the correct thing to do. Recent work in mathematics education suggest that including lessons that build relational understanding has certain benefits for student learning. The only problem is that teachers need to be able to spot and reward that learning. 

Let's say you are working through a task where a student has generalized a problem, worked effectively with others, deduced possible pathways toward a solution, checked the reasonableness of their solution, and even posed further problems or abstracted universal truths. In order to judge their mathematical progress, you naturally start looking for key indicators of success. 

Things like:

• Did they use a diagram?
• Did they consider all possible cases?
• Did they link the problem to a previous one?
• Did they check their answer?
• Did they persevere throughout the process?

Teachers are using an algorithm to assess work that is designed to empower students to see the larger picture. I believe that, over the years, teachers develop their own set of rules that govern assessment and then apply them instrumentally to assess students. (Now this is a very young belief, and I am open for counter-beliefs). 

I also think that within this relational-instrumental struggle exists the birth of the infallible (**cough**) assessment tool known as the almighty rubric. All rubrics do is attempt to instrumentally describe relational skills, yet they are upheld as the ideal way to achieve relational assessment. As teachers, our entire assessment framework has been rubricized; everything must fit into a nice box in a nice grid.

This is the portion where I should suggest a solution, but I am far from it. I have only just begin to realize that even my observations during otherwise "relational" tasks are of an instrumental nature. Algorithms are a part of mathematics, and assessing them is--by its very nature--algorithmic. I am now wondering what a non-algorithmic assessment looks like.

NatBanting

Reference:

Skemp, R. R. (1976/2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88–95. Originally published in Mathematics Teaching. Retrieved from http://www.jstor.org/stable/41182357

Fair Dice Task

The recent curriculum renewal has placed a (well-deserved) heightened emphasis on counting, set theory, and probability. Just under a half of a Grade 12 "Foundations of Mathematics" course now covers the three topics. This is a huge improvement from the token, disjointed topics strewn around the last courses. It allows teachers to set a different tone--a tone of curiosity that seems inherent in probability. 

I came across the idea of Grime Dice (named and pioneered by Dr. James Grime @jamesgrime) late last year after I knew I was to be teaching probability this winter. I knew right away this was a great task to get students tinkering with probability before defining its inter-workings theoretically. A great description of their function can be found on the PlusMath website written by Dr. Grime himself. They are available for purchase from MathGear.co.uk. 

Basically, the dice have been altered to compete at different strengths against varying dice. A very interesting phenomenon all to itself. I wanted to find a way to capture the simplicity of a two-dice game, couple it with the quirkiness of non-transitivity, and wrap it all up in a task filled with student initiative. My result follows.

I began by purchasing five novelty dice from a dollar store. I re-worked the numbers on the sides to match Grime Dice A, C, and E. (see previously linked article). I didn't make dice for every group, but the task could be extended quite neatly into an experimental probability lab. 

To begin the class, I introduced the class to the altered dice by showing them a graphic. 

I then handed out a sheet with three dice charts on it.

Dice Chart between Grime Dice "A" and "C"

Students were then asked to answer the question:

"Which dice is the strongest dice? why?"

I circulated and probed further as groups finished the relatively easy task. Soon students began to see the circular nature of the dice. As a class, I explained how this could be exploited by choosing last in a game. We also talked about the probabilities themselves. Students had a natural grasp on odds and probability and offered great explanations for their value. We briefly touched on possible values for probabilities, and the idea of a complement. 

Enter Phase Two.

I attempt to build in student authorship into tasks. Student thought flourishes when they are not only problem-solvers, but also problem-creators. The second portion of the lesson was designed to get students creating mathematics. 

I posed the following problem while handing out a fresh set of three (blank) dice charts:

"Can you design three unique dice that all have equal chance of beating one another, but none are numbered the standard one through six?"

After a few minutes, obvious solutions began to emerge. (These are Grade 12s). When the ripple effect came full class, we held a discussion to highlight the logic, and I re-framed the question. 

"Can you design the dice only using the numbers one through five?"

Students worked hard altering their dice; I could see the tenants of probability working as they switched numbers to alter the respective chances. The conversation of ties came up. Are they allowed? How do they effect probability? Can you end with an odd amount of ties?  I didn't make a hard rule, but allowed the class to self-govern. 

In the end, I got some extremely creative solutions. Underlying it all, students were introduced to the mysterious nature of probability. The task began to introduce some of the lexicon that would be used for the unit. Words like favourable, total, outcome, complement, probability, and odds were established. Students saw (and defined) the process of creating basic probabilities, and compared their values in fractional and decimal forms. The task accomplished everything I wanted it to.

A well-designed task takes the notion of "motivational set" and weaves it throughout the lesson and unit. It uses a broad scenario and student authorship to get students actively working with the topics. What initially seemed like a neat activity now serves as an effective anchor for an entire unit.

NatBanting

Relation Stations

This semester I desperately wanted to improve how I taught linear relations to Grade 9 students. I had tried some interesting activities in the past, but lost patience and ended up drilling them with notation and algorithms. I wanted to find a way to show the students that equations were just explanations of patterns. I began compiling different linear patterns and dug in for the long haul. 

I stumbled upon a collection of abandoned, square tiles and decided to use them to put students in the center of the pattern making.

I began the lesson by dividing the class into groups of three. Each group was given a handful of tiles and a sheet with three blank T-Charts on it. To begin the activity, I placed the following pattern on the board:

Students were asked to model the pattern for the next three stages and record how many blocks it took to build each instance. As they were filling out their charts, I went around to each group and asked them to explain the pattern. Most began by saying, "you just add four more each time". I kept prying at this explanation. I asked them to be more specific.

"You start with a center, and add one on each arm"

"Every time you make the next step, you need to add a tile to each arm"

"The arms have to be as long as the step you are on"

I left after I was sure the groups had made the connection between the four added tiles and increase of four in the table.

Two more patterns were shown, and I repeated the process. 

After the third time, students started to be very concise in their wording. They began to tell me what I wanted to hear. It was after the third round that I asked:

"What do all of these patterns have in common?"

They quickly decided that each pattern started with a base and then added a constant amount each time. I (happily) wrote that observation on the board, and handed out a sheet 11"x17" blank paper to each group. I wanted students to create linear relations with the two newly-defined parts before I formalized their existence as the constant and rate of change. 

I explained to the class that each group had three minutes to create their own pattern like the three we had seen so far. The only rules were:

1. You must use your blocks to model the first three stages of your pattern.
2. You cannot copy or repeat a pattern we have already seen.
3. Your pattern must have a starting point and a constant change.

I circulated and heard some great conversations about how they were going to create their patterns. Students would name portions of the pattern "top", "bottom", or "middle" and then determine how much it was going to expand each time. This guaranteed that each stage would have a constant change. A wide variety of patterns emerged at a wide variety of skill levels.

My students have used stations before, so they were very familiar with the concept of circulating and completing a T-Chart for each pattern. They had twenty minutes to copy down the first three stages of the pattern, fill out the T-Chart, and decide what the constant change was from every station. They handed in their work at the end of class.

The nice part about the activity was that I could hand back their work the next day when we worked on building equations from the relations. Students had seven great examples to practice building relations. The paper was large enough that the group could easily share. I noticed that more students understood the make-up of equations once they had built their own pattern. They had experienced (and deduced) the existence of a solid piece around which a constant change was occurring. 

We went on to discuss other examples of linear relations. Ticket sales at a concert, a taxi cab fare, and Pokemon attack strength. Students got so good at finding the rate of change that filling in the charts was a breeze. It provided a great bridge to graphing and using the relation as an input/output machine. 

It's a very scary thing for a teacher to give up control. Last year, I stood in front of the class and delivered a very good (Read: Twenty-five minute) explanation about how to fill out charts based on patterns. When it came to introducing equations, they got lost and I barreled forward. There is something intangible that occurs when you provide a student with a well-structured task. I have seen the benefits countless times, but still get anxious before I loosen my grip on the class.

This activity now provides an anchor for the learning. I can always refer to the two critical parts of a relation and students will have experienced them first-hand and on their terms. 

NatBanting

YouTube Relations

My goal this semester was to continue to improve my use of formative assessment (largely through the use of whiteboarding) and expand the role of Project-Based Learning in my classroom. Up to this point, I have developed a wide-scale PBL framework for an applied stream of math we have in the province called Workplace and Apprenticeship Math. Those specific topics lend themselves very well to the methodology; they are a natural fit for PBL. I am still looking for ways to branch the intangibles from PBL into a more abstract strand of mathematics--one that includes relations, exponents, functions, trig, etc. 

I decided to build a small project for the end of each unit of study in my Grade 9 class. I chose this goal because:

1. I currently have a half semester of Grade 9s, so I'm only responsible for four topics.
2. The classes will switch at the midway point. This gives me opportunity to make little adjustments to improve the projects quicker. 

In the midst this goal, I have been asked to sit on an informal committee to review how technology can be embraced in our building. I am no technology expert, but am always willing to try something with upside. 

History lesson over--let's move onto the task.

My students will complete this mini-project at the end of the unit on Linear Relations. They are not given any time in class to complete the project, but half a day is devoted to me introducing the students to the necessary technology. (YouTube, Paint, and LinoIt.com). 

Students are to use the miracle of self-publishing to find an appropriate video on YouTube. The video must be longer than ten seconds but shorter than thirty. They will then choose variables, create a constant scale, and graph the relation portrayed on the screen. My guess is the grading will involve plenty of sports bloopers and "Fail" compilations. 

I initially was going to get the students just to write out the video's URL and hand in a hard copy of the graph, but then decided that the entire class would benefit from seeing the relations. I needed a place where the graph and video could be fully functional in one place. I decided to use LinoIt.com.

LinoIt is an excellent place for online collaboration. I have used it as a simple class website in the past. Parents could access the URL and see homework assignments, student work, links to school sites, and class announcements. The boards can be private or public. My suggestion is to make the board public (anyone can post) but not to broadcast the URL. That way, students do not need to make an account. This speeds up the posting process. 

The students download a standard blank graph (directly from the LinoIt board) and are asked to graph their relation accurately. When they are done, they can easily post the video, the graph, and their name to the board. All the instructions are given to them in a handout. The handout as well as graph template can be downloaded here. 

When completed, each student's homework will look something like this:

The tools are connective and intuitive. The project is simple and creative. It opens up an opportunity for students to begin to see the world through numerate eyes. After the project is complete, I will make the URL public through my wiki page. 

The assessment is three tiered:

• Students assess themselves based on accuracy, challenge, and creativity.
• Students rate peer's graphs based on the same three criteria. (As we show them to the class).
• I will assess each graph on the previous three plus a completion grade. 

My hope is that small, unit-ending projects such as this will begin to include some of the intangible benefits that PBL has brought to my other classes. I want to leave ample opportunity for student choice, autonomy, and innovation. Adding technology only opens those three avenues further. 

NatBanting