Strengths and Weaknesses of "Dividing by Zero"

Strengths:
- The activity was fresh, interesting, and different
- It allows a person to explore their creative side and makes one realize that math does not mean LIMITED by numbers!
- It gives a chance for one to delve and try to understand why dividing by zero is an area that no one want to step into from a different perspective
- It gives a chance for a student to explore in a fun way

Weakness:
- If we choose a word that has a different meaning in a mathematical context, it may make it more confusing for a student.
- It may not be really relevant in mathematics and there is a chance that nothing is learned or gained from it. It's just a "hoop to jump through", leading to a student that is fixed minded
- It could make a student that is not interested in poetry discouraged when they expect to be learning "math"

Interesting:
- I first realized that Math and poetry can bode well in a Harold and Kumar film...

I’m sure that I will always be
A lonely number like root three

The three is all that’s good and right,
Why must my three keep out of sight
Beneath the vicious square root sign,
I wish instead I were a nine

For nine could thwart this evil trick,
with just some quick arithmetic

I know I’ll never see the sun, as 1.7321
Such is my reality, a sad irrationality

When hark! What is this I see,
Another square root of a three

As quietly co-waltzing by,
Together now we multiply
To form a number we prefer,
Rejoicing as an integer

We break free from our mortal bonds
With the wave of magic wands

Our square root signs become unglued
Your love for me has been renewed

- Asides from poetry, I want to try and mix different subjects with mathematics! (Like BIOLOGY. =) )

Divide, Zero, Dividing by Zero

DIVIDE

When one says the word DIVIDE, a series of random words flash through my head. Divide, as in using simple arithmetic with calculators. Divide means math. Divide means fractions. Divide makes one thinks of splitting pie into EQUAL PARTS. Divide also means to share things EQUALLY and fairly. At the same time, the word DIVIDE makes me think of an argument. Arguments between families, arguments due to language barriers, a space and gap between nations causes divides. Is division the opposite of multiplication? Grouping things into groups and finding the number of groups- "divide yourself into groups". Divide and conquer- what does that mean?

ZERO

Zero. Absolute ZERO. Nothingness. A vacuum. Zero = 0 = a nice round number~ A nice round number that has nothing IN it, it's empty inside that circle. Maybe that's why the notation 0 is zero? Zero. Coke zero. What is coke zero? Is it truly possible to eat something with ZERO calories? Is there a catch to this? Zero. Nothing... Does it mean that the number zero has no meaning to it? Do kids understand the concept of ZERO? Of NOTHING? Nothing we've experienced has really had NOTHING in it... except for in a vacuum. The word ZERO. Also a character's name in a Japanese anime. Why was he named as ZERO? What significance did it have? On the number line, zero isn't a positive. Nor is it a negative. It's stuck right in the middle. What is dividing by zero mean? Would there ever be a scenario that you would want to add or subtract by zero? Zero is such a SAD number. Zero is a WHOLE number, but not a natural number. Is it not found in nature? Zero. What a nice round number for a number that is EMPTY~

DIVIDING BY ZERO

You want to split something by nothing?
No! Don't you even dare dream of it.
It's ludicrous, it's blasphemy, it's absolute catastrophe!
Chaos and confusion, that's what it will do.
Dividing by zero- are you absurdly insane?

ONE pie, split among two,
Split it into halves and each get a part.
ONE pie, split it among one,
Then the person gets a whole pie to themselves.
ONE pie, split among ZERO...
How can you split a pie with nobody there?

Thinking about it makes my head pound and hurt.
So don't EVER waltz into that gray area.
Don't dare dream about dividing by zero.
It's just simply utterly impossible.

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Here is the first free write with the word DIVIDE:

Divide, division. Wars- splitting up. What causes a divide? And argument and disagreements. There can also be physical divides- divides like being divided across an ocean. Divide = PIE. How many ways can you split a pie up into different pieces? Divide- the OPPOSITE of multiplication? Like add is "opposite" to minus. Divide... a fraction. Divide, a line with two dots, one on top, one on the bottom. Having a divide is a sad sad thing. A dividor would keep a person very organized! Eight divided by two, how many groups of two can fit into 8? Divide also makes me think of cliques and groups with language barriers.

Here's the second free write with the word ZERO:

Zero. Coke zero. Does that mean there is zero calories? Is it possible to have zero- absolutely NOTHING? Is there any possible way to have nothing? You'll still end up with air... except in a vacuum. Zero. A number in the "middle" of the number line, not a negative, not a positive. Just stuck there. Zero- is it written like 0 because there is a WHOLE in the middle with nothing in it? Zero, such a sad sad number~ All lonely, no one to hear. Nothingness. Zero, why are there main characters in anime where the hero is known as Zero? Is there a special significance around it? Zero is a WHOLE number, but it's not a NATURAL number. Why is it not a natural number- do you not find nothing in nature? Zero is funny number since x^0= 1. And anything times zero is zero. Zero- such a funny word to say, yet, it is a word that makes me feel sad, alone and empty~

Citizenship Education in the Context of School Mathematics

From Elaine Simmt's paper, we take a look at what mathematics education has in correlation to civics educations.

In truth, I originally thought that school mathematics had little relationship with citizenship. Furthermore, through my high school life (up until Grade 10 at least), I had grown up with the understanding that:

1. Mathematics is a "set of facts, skills and processes".
2. Mathematics are facts and fact.
3. Mathematics is either right or wrong.

These are three of the instructional stances and strategies in mathematics that may conflict with citizenship education according to Simmt. I believe that depending on the teacher, the students' insight and use of mathematics could be greatly affected. By teaching in a way where the instructors look for the "right" answer and look for a specific computational skill and algorithm, students ay lose sight of the fact that mathematics is seen all around society. But training our students into looking for information and "stripping from word problems" the facts, word problems are not serving their purpose. They are merely simple math questions in disguise and students lose sight that mathematics and citizenship are integrated with one another- especially in today's world. By asking closed questions, the impression that mathematics is either right or wrong is given to students. Do we as teacher candidates want our students to think close mindedly? By asking questions that are either right or wrong, our students grow to learn not to question the results of mathematics and not think on a higher level of the Bloom's taxonomy.

In order to prevent this from happening, Simmt offers suggestions that promote active and critical participation in society. These include:

1. Variable-entry prompts and investigations: problem posing.
Problem posing is a difficult concept to grasp- however, it is filled with benefits. By giving very open ended questions where the students can construe their own understanding, their knowledge on the individual level becomes more solid. As wll, the students will need to "pose, negotiate, and judge the appropriateness and adequacy of their own and classmates' questions and solutions". By doing so, students not only work on their mathematical skills, but also develop skills in taking an active and critical participation in society that is expected of citizens.

2. The Demand for Explanation
By helping students reazlie that math is not merely about the RIGHT answer but about the "truth" and having them realize that there are multiple truths as long as it is backed up by explanation- their skills in articulating thoughts and assertions are built. By having my students explain their reasoning, I would furthermore be able to teach my students via construetivism. I believe that in this way, we would be able to educate for citizenship with independent and individual thinking.

3. Mathematical Conversations
Through discussing with their peers and teachers, students can perhaps develop the ability to solve concepts and problems through communication. By having interaction with one another, this would help them realize that the skills developed in mathematics is applicable in society as well.

From Simmt's paper, I have realized the importance of not only helping students learn mathematics in terms of "general numeracy" but also as a subject that the power to change society. Hence, though we may not realize it immediately, mathematics plays a great role in citizenship education as it helps us understand society AND develops active and critical individuals.

Pages 33-66 The Art of Problem Posing Reading

I thought that the reading from The Art of Problem Posing was quite interesting and a bit difficult.

Firstly, language was a bit difficult to read through and the details was a bit hard to grapple on. But in general, I really enjoyed the authors use of analyzing how we could approach theorems and "truths".

I can see a few strengths and limitations in the what-if-not approach and using it in a classroom setting. Firstly, by using this technique, it would help students to think on their own and to try to formulate their own thoughts. By using this concept in class and modeling it, the students will be able to perhaps develop this technique and approach the questions in this method. By doing so, they could perhaps make their own discovers, understand concepts much more deeply and gain further insight. Another strength to this method would be that students would not be taking facts at point blank, and coinciding with Brent Davis' lecture, it would promote growth-minded thinking. By questioning what is true, one can figure out and perhaps even make new discovery! As teachers, I believe that using the what-if-not technique and incorporating it into the lesson is very important if it is possible.

One limitation from using this technique is that at the high school level, most students are already familiar with taking theorems as facts. They may not appreciate questioning everything and become more confused in the class setting. As well, due to class and time constraints, using this approach may take much longer than expected and one simple "fact" may take a whole lesson to help students test out their "hypothesis". Furthermore, there is no real end to using what-if-not and if there is poor classroom and time management, this could pretty much go on forever. As well, I believe that the mindset of the students is really important to have them engaged with this approach. Otherwise, they would not take much out of it and only want the "fact" and not want to think too much over the concepts.

But all in all, I believe as teacher candidates, we should try to incorporate this form of problem posing into the classroom. The method is a great way for students to start thinking and questioning and probing into mathematics. By using this technique, some students may find that math isn't as boring as expected and that there is much to discover and much to probe. Furthermore, by using this form of probing questions there are a variety ways of approaching the theorem (for example, with Pythagorean theorem, the students were able to look at it geometrically as well as numerically).

Though the what-if-not approach is an interesting one and it sounds appealing, I would like to test it out in the classroom setting to see if I can apply it in reality.

10 Questions on The Art of Problem Posing

After reading pages 1-32 of "The Art of Problem Posing", I had a few questions that I'd like to post:

1. When should we incorporate these problems in class? Would it be good for a hook/part of the strategy/or is it best to have it in the conclusion?

2. How much time should we spend discussing these questions in class? Will these questions even be beneficial for the students? (Has there been studies done?)

3. What age group should we be posing these students to? At certain ages, would they not just rather have answers than be left with the frustration of not having their questions answered?

4. Should it be up to the teachers to form the problems or should students be problem posing? In either case, how do we get into that "frame of mind"?

5. How would problem posing relate to grades in school? Will they get marked for having "good" questions or making contributions in class? (Otherwise, they might not deem problem posing as something relevant)

6. How do we make sure that the problems as not too broad or too narrow and that we are able to keep on track with the lesson plan?

7. Has this method actually been used in class?

8. Should we have problem posing projects?

9. Is problem posing relevant and important in the eyes of the B.C. Education Ministry?

10. Though there are no such thing as "stupid" questions, are there such things as bad problem posing questions?

From the View of Two Hypothetical Students

Dear Editor,

I'm here to send a complaint on the education system in B.C. In particular, my mathematics teacher.

I have one math teacher, Miss Lo, that I see as dreadfully inefficient in her teaching styles and wish that she would change the way she teaches. Firstly, her teaching style varies too much. Instead of having straight forward notes and examples with homework, she covers a variety of teaching methods. From using group works, projects, and puzzles- the format is a bit too over the place and I wish she would just stick to the classic method of using the good old computer to type up notes. Like really, is it NECESSARY to teach us with so many different styles? It gets a bit too confusing.

Secondly, the quality of homework she gives us is much too difficult! Though she only gives about a page of homework a day, I think it's too hard to understand. Is it really essential for a student who does not plan to become a mathematician to do such abstract and complicated thinking?

Finally, I dislikes the discipline that she invokes in class. The classroom regulations are so rigid and there is no gum chewing allowed in class- we're only allowed to drink water. Like really, who can learn without eating in class?

These are just a few complaints I have with this one teacher and I wish she would switch her style of teaching.

Thank you very much for your time.

Student who dislikes Math Class.

________________________________________________________

Dear Editor,

I am also a student of Miss Lo's class and I would like to write a reply to "Student who hates Math Class" letter.

Firstly, I enjoy the way she incorporates different methods of teaching. It keeps the class entertained and occupied. Furthermore, the reason she does so is because it will allow students of different learning methods to understand the concepts and helps us look at mathematics in different perspectives. Hence, I think that Miss Lo is doing a good job of teaching with different styles and using different forms asides from merely note taking and lecture.

Secondly, though I DO enjoy eating. I respect Miss Lo as a teacher and she needs to set up some rules to make sure the class is tidy. There are times to eat and other times, we should listen and work in class. PLUS, since we're all supposedly busy learning in class and doing group work, there shouldn't be time to eat anyways. We should be eating at lunch.

Finally, I personally enjoy the way Miss Lo gives out her homework. By giving us only two to three challenge homework questions, we are able to try to push ourselves and have a further deeper knowledge of mathematics. Though it is at times very difficult, she provides morning hours for help if you need it, and as well, she does not mind that we work in groups as long as we write out our own homework in our own words after.

So in response, I believe that Miss Lo is the best teacher ever!

Student SMILEY! :D

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My Hopes and Worries:

From what I've written, I hope that I'll be able to teach with various methods and that I'll be able to have the students involved in mathematics whole heartedly. A few worries that I have would be that because I want to teach in so various methods, they may find it confusing. In order to overcome this obstacle, I hope to be very well prepared and give students clear instructions and notes by the end of the lesson. As well, I hope that I don't end up being too strict a teaching. So goal wise, I hope that I'll be able to be entertaining yet educational in class and make it a learning experience for both the students and me. I would like to make math a fun, applicable and yet challenging experience.