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Proofiness – A quick review

January 18, 2014

One of the more interesting Christmas presents I received this year is the book Proofiness: The Dark Arts of Mathematical Deception by Charles Seife my wife bought me. I must admit that I have not read the entire book but I am progressing slowly through it’s eight chapters.

In Chapter One ,Phony Facts, Phony Figures, Seife introduces an idea I had not thought much about, the idea that “there are numbers and there are numbers” (p. 9). Another words there are the “pure” numbers that are the domain of mathematicians and there are those that are meaningless without a unit attached. Indeed he notes that for most of us “A number without a unit is ethereal and abstract” (p.9). He expands this idea by noting that any number that is not pure has a degree of uncertainty attached to it. Following this argument he attests that many things we try to quantify has a large degree of uncertainty in the measurement. Some of the examples include intelligence scores, pain or happyness. I have found his arguments in this regard persuasive as I can relate from my personal experience. After all what I would consider a 9 out of 10 for pain may hardly register for someone else.

Have you ever noticed that there are somethings that sit outside of your awareness but when your attention is drawn to them you can not escape them any more. Proofiness introduced me to the concept of Potemkin Numbers. In short a Potemkin number is a “fabricated statistic.” Examples of Potemkin numbers highlighted in the book include intelligence scores (p. 12), the impact of eye lashes (p. 14), the impact of smoothness or coolness (p. 15) and the number of people at the “Million Man March” (p. 16). At one point he even notes the formula for happiness reported by the BBC in 2003 which seems to be

Happiness = P + (5 x E) + (3 x H) where
P = Personal Characteristics
E = Existence
H = Higher Order Needs (p.65)

I think most would agree the variable of happiness must all take on dubious measurements in themselves to be quantified.

I had never really thought about before but now when I see an advertisement which states “95% cleaner” or “52% clearer” I have to question where the numbers come from. Indeed now-a-days we seem to be bombarded by statistics many of which may or may not be measurable. Now that you have read this I expect that you will also notice these “made up numbers” more often as I have, at least 42% of the time.

In later chapters, Seife discusses the validity of polls and the history of polling which was also very interesting. If you are a math or social studies teacher and have not read about the 2008 election results in Minnesota chapter 5, Electile Dysfunction is for you. Who knew that counting could be so inaccurate.

These examples are just a few of the gems I have found in this book and overall I would consider the book worth the investment of time. After all it will make you 14% smarter.

Number Tricks or Number Patterns

October 18, 2013

I love number patterns. There is a beauty in the way our number system works and the patterns that materialize. Sometimes the opportunity presents itself to not only investigate a number pattern but to also have some fun.

When the students entered class today I was sitting quietly at a desk and had a copy of The Proof of Fermat’s Last Theorem on the board. I told the students that we were going to have a look at the 140 page proof and that while it was a long proof I thought it was good if they could see how important that it is to “show your work.” Well this did not really interest a Grade 6 class and frankly I was glad they did not call my bluff as I am not sure I could have explained most of the proof.

Some of the students noticed that I had the app The Amazing Mind Reader on my iPad. They asked me what it was and I told them it was a math trick that I was trying to work out. They were more interested in this than in a 200-year-old problem so I suggested that we may have a few minutes to try to figure it out. After all, “I spent my lunch hour working on it.”

If you have not played the app the premise is pretty simple.  The steps to solving the problem are:

Mind Reader Chart

  1. Pick any two digit number
  2. Add the digits together
  3. Subtract this number from the original number
  4. Find your number on a list with some symbols beside them (see the image to the right)
  5. Amazingly the app will pick the symbol that you have on your mind

The first two times we tried the app the students were amazed that the iPad was able to detect their symbol (I could not resist telling them that this was because the spirit of Steve Jobs was in my iPad). After the second try students tried to work how the iPad was able to do it.

Eventually students were able to explain how the iPad “did it.” This is where Fermat’s Last Theorem comes in as we talked about the importance of begin able to clearly articulate your solution. Over the course of the next 10 minutes we were able to refine their explanations so they made sense. In the end this was likely an even more important lesson than the  identification of the patterns which made it easy to identify the symbols.

As a final note, I asked students to determine if they could extend the pattern to three digit numbers? Four digits?

Numeracy Tasks

October 1, 2013

Wikipedia defines numeracy as “the ability to reason and to apply simple numerical concepts.” The website links the definitions of numeracy and mathematical literacy indicating that these terms are interchangeable. Indeed I believe most people would see these terms as one and the same. I think the true differences in opinion is not in the word used but in the definition of “simple numerical concepts.” What is simple to one person is not so simple to another. I think many would agree that it is important to be able to tackle a problem with the mathematics one knows is an important aspect of any definition of numeracy.

It is important to examine the type of problem which would be considered a numeracy problem. Many of the questions in a math textbook I do not consider numeracy problems. They are closed and generally have one solution. Don’t get me wrong I am not passing judgement on these questions. They do serve the purpose they are designed for they just do not go far enough to extend rich problem solving and develop students overall numeracy. They are the tools in the tool box.

To extend and develop numeracy concepts in our math classes we have been working on problems that I would define as mathematically rich problems. As stated by  Simon Fraser University faculty member Peter Liljedahl, I believe the Qualities of a Good Numeracy Task exhibit

  • Low Floor
  • High Ceiling
  • Huge Degrees of Freedom
  • Fixed Constraints and
  • Inherent Ambiguity

I encourage readers to check out Peter’s website for a more thorough definition of these qualities.

As we have completed numeracy tasks over the past couple of years we have found that the tasks that best embody these traits are the most interesting,  challenging and capable of assisting in the development of students overall level numeracy. Students not only think inside the box but also have to extend their thinking outside of the box.

Any teachers who are looking for examples of good numeracy tasks and learn more about rich mathematics problems are encouraged to check out Peter Liljedahl’s site and examine the Enriching Mathematics site. If anyone knows of other good sites for rich problem solving please share in the comments section below.

Squares – Conceptual Understanding of Square Roots

September 30, 2013

Last year I made a post on the Squares and Square Roots which examined developing a conceptual understanding of square roots while taking the class outside. I thought the activity was interesting and told a lot about the prior knowledge of the class.  I wanted to follow-up on this post based on a similar activity we did this year.

As anyone from Victoria knows the rain comes about this time of year. The last few days have not been ones that anyone would like to take their class outside. I did want to replicate the activity we did last year so it needed to do some modification due to the weather. While I would still have preferred to conduct the task outside the modifications made for a better learning experience.

Students were organized in groups of three and asked to create three squares out of string taping the string to their desks. The squares were to be 1 m^2, 2 m^2 and 3 m^2. To complete the task students first needed to determine  the amount of string needed to create the square. Once the length was determined students were to come to me and get their string.

I thought the 1 m^2  would be easy for students to create after last years experience. I was surprised with the amount of time it took some students to complete this square. There were some misconceptions among  students about the difference between perimeter and area. These were relatively easily overcome in group discussions and likely in a  much more effective manner than I could have done.

I correctly guessed that the 2 m^2 square would be harder to make.  Most students tried to use sides 2 metres in length resulting in a 4 m^2 square. Many groups did pick up on their error as the larger square as it was more than double the size of the 1 m^2 square. The key piece of the lesson was to have the two squares overlap so the areas could be compared and have students discover the error on their own. This was an improvement over the prior year and something I plan to continue. As a side note, only one of the groups was able to complete all three squares.

Students enjoyed the activity and even those who pride themselves on being strong computationally had trouble with the task due to some gaps in their conceptual understanding of square roots. To be fair this was an introductory class on square roots so I did not expect everyone to solve the problem easily. However, many had said they understood square roots but I think this understanding was limited to performing the calculations on a calculator. Building the squares lead to a better conceptual understanding of square roots.

Completing this activity underscores my belief that computational fluency while important is no replacement for conceptual understanding. They go hand in hand. To have one without the other is like being blind in one eye. You can see what is in front of you but you can not see the depth of mathematics.

Number Bases

September 29, 2013

Over the past couple of weeks I have had the opportunity to work with some Grade Six students on extensions to their regular mathematics studies. At first my intention was to provide some extension beyond the normal Grade Six learning outcomes. Our first task was to examine converting between different number bases and base ten. I have since come to realize the value of this content in better understanding place value. Students quickly made connections with the importance of using zero as a place holder and indicating no value in a particular digit after working with base 2.

Counting in Base 2

Our lesson started with students being asked to “turn the lights half on.” This results in a few laughs and the realization that there are times when only two conditions exist, such as in base 2, and times when there are multiple conditions possible, such as a dimmer switch or base 10. To an adult with lots of world experience this is obviously something that can not be done. For students they all want to try.

I also found that linking these ideas to how computers work is a good hook for students and increases the general interest level. Any stories about computers in the past are very interesting. My first purchase of 2 MB of RAM for $1000 was a huge topic of conversation.

Our lessons progressed in phases looking at:

  • Exponents
  • Place Value
  • Scientific Notation
  • Binary Numbers and their use in computers
  • Other Bases
  • Extensions (number bases in history and use with computers)

I hope to refine and extend these lessons but this has not happened yet.

I have placed the information from our school noodle within this post. I am happy to share what I have created to date in exchange for any feedback. Many heads are better than one. If anyone would like the html code please see the attached file linked below in “doc” format. The code can be inserted into Moodle or another html editor and will be displayed as a web page.

numberbases

New School Year Resolutions

September 29, 2013

I love the start of a new school year. Unlike January 1st, a new school year really does feel like a reset to me. It is more than a new date on the calendar. For a teachers  and students September marks a fresh beginning and a chance to dot he things that we hoped to complete the year before.

The paragraph written above was completed on September 2nd and it has taken me almost a month to get back to completing my thoughts. This speaks to the speed that events within a school happen. Unlike summer when time is your own the school year is “full on” for both teachers and parents. Academics, sports, clubs and meetings dominate the day. Ask anyone, “How are things going?” and the first sentence is sure to have the word, “busy.” Indeed everyone is.

This brings me to my New School Year Resolutions. I have resolved to focus on the speed of the day and I have resolved to:

1) Take the time to listen (not hear) a story or two told by the students.

2) Take the time to tell a story or two

These are small resolutions and seem simple however, teachers and students should all remember that we are in a people business. Education of students goes beyond the lessons in a textbook and teachers can learn from their students. A month into the school year and these two simple resolutions have made for a wonderful, and busy, month. I can only hope the rest of the year goes as well.

Side Walk Chalk and Mathematics

September 21, 2013

The school year is off to a great start. Students are excited to learn mathematics and I am even more excited to learn from them. It seems that every class I teach I learn even more. What a great profession.

As a proponent of multiple intelligences (http://en.wikipedia.org/wiki/Theory_of_multiple_intelligences) I start each year by asking students and parents to complete a survey to determine the strengths and weaknesses of the class. This year the class profile showed a preference for working in the bodily-kinesthetic domain so I wanted a chance to play to this strength while building visual-spatial intelligence which was identified as an area needing some support. Sidewalk chalk seemed to be an answer.

We have started the year in both Mathematics 7 and 8 examining geometric concepts. In the past I have used sidewalk chalk as a way to expand the size of the whiteboards in the classroom and take advantage of the great fall weather.

This past week we were examining transformations and I numbered paving stones around the school. After a brief review of translations, rotations and reflections students were asked to identify the single translation that would move one paving stone to the position of a second one. Students had fun examining these transformations and were able to assist each other in making sense of the mathematics.

The best part was that we were able to discuss the importance of having a point of reference when talking about direction. This was not something I had thought about as it is obvious on Cartesian Plane. When standing around the bricks everyone wanted to use their “view” to refer to direction. As a result, “up” for one student was “down” for another or “left” for a third. Soon students realized the value of using the same reference point. As one student said, “that is why we use right and up as positive.” This was a lesson I did not expect to come out of the lesson.

After school I was pleased to hear some students outside of the classroom discussing transformations with their parents. As a math teacher it is great to hear students explaining their thinking to their prarents. I was easily able to assess the learning of these students and give them credit for their work without them even knowing it.

We have used sidewalk chalk for other learning opportunities over the past few years and have always had success. Some other uses include geometric constructions, geometric proofs, squares and square roots and calculating surface area to name a few. I am sure there are other ways to use chalk and I would love to hear ideas that other may have.