Chinese Puzzle Problem

"how many guests are there?" said the official.
"I don't know," said the cook, "but every two shared a dish of rice, every 3 shared a bowl of broth,
and every 4 a dish of meat."

There were 65 dishes in all
How many guests?

Lets start off letting x = number of guests.

We know that there are three different dishes - rice, broth, and meat.  And we know that the total amount of rice dishes, broth bowls, and meat dishes is equal to 65.

Looking at the first statement, "every two shared a dish of rice," we can deduce that every guest has half a dish of rice.  So x/2 = number of rice dishes.

From the next two statements we can deduce that
x/3 = number of broth bowls and
x/4 = number of meat dishes.

So now we can get an equation using the fact that there are 65 dishes in total.
number of rice dishes + number of broth bowls + number of meat dishes = 65
or
x/2 + x/3 + x/4 = 65

From there we can solve for x, which is the number of guests.

6x/12 + 4x/12 + 3x/12 = 65 (Finding a lowest common denominator)
13x/12 = 65 (adding like terms)
x = (65 * 12)/13
x = 60

We can also solve this problem by guessing and checking

Lets guess that there are 120 guests.  That means that there are 60 dishes of rice, 40 bowls of broth, 30 dishes of meat.

60 + 40 + 30 = 130 which does not equal 65.

But we can notice that 130 is double 65.  so our guess was off by a multiplication of 2.
therefore we know that there are 60 guests.

I chose 120 because the numbers work out nicely, but this should work for other numbers with some rounding.



Extensions:  Does it make a difference that this is a Chinese puzzle?  Is culture important in math?

I don't think in this specific problem it makes a difference that is a Chinese puzzle.  The terminology used was irrelevant in my opinion.  It could have been french fries, burgers, and milkshakes.  

But I do believe that culture is important in math.  I remember in highschool we were doing a section on probability and the teacher did an example of a football game.  It wasn't much of a problem to our class as everyone knew what football was and how it was played, but if there was some European students in the class they might not have fully understood the problem.

Introducing Algebra Into the classroom video

My initial reaction when the video started was..... seriously?

But soon after I realized what the teacher was trying to accomplish and he accomplished it quite effectively.  One of the parts that stuck out to me was when he introduced the negative numbers.  Tapping on the board from right to left, he went from 3,2,1,0,-1,-2,-3.  The students seemed to comprehend that.  However, shortly after that he started at -6 and then tapped the stick to the right.  Some students mumbled out -7, and some students mumbled out -5.  It is easy to see why students would get confused because it's hard for them to comprehend that -7 isn't actually greater than -6.  That moment, I think, really cemented what a number line was and how it worked.

It was interesting to me because he introduced so many mathematical concepts(number line, negative numbers, preliminary algebra) with only using the ability to count, and barely speaking at all.  Pretty much any student of any level could comprehend what was being taught.

Hewitt : Arbitrary and Necessary

I quite enjoyed Hewitt's article as it brought up some new perspectives on why and how we teach mathematics.  In particular, the idea of received wisdom caught my attention and it was explained clearly in the following example,

"If a teacher stated that the angles inside a triangle add up to half a turn rather than offering a task for students to become aware of this, then students are left with having to accept what the teacher says as true.  In this case, it becomes just another 'fact' to be memorised.  I call this received wisdom."  

This particular example stood out to me because I would have to believe that most mathematics is taught this way, or at least it was during my highschool career.  Providing this 'received wisdom' or arbitrary content simply just provokes a battle of who can memorize the best.  As Hewitt explains, there are arbitrary facts that we simply cannot avoid the need for pure memorization, such as the x before y explanation, or names of certain shapes, however when necessary facts are explained in an arbitrary way, it becomes a problem because then we aren't teaching mathematics at all.  While I tend to agree with what Hewitt was saying, I question that it's much easier said then done.  It is easy for me to see how a teacher can assist in memory, but it is hard for me see how a teacher can educate awareness.  It seems to me that in educating awareness, the teacher is barely doing any educating at all, rather they are just facilitating or directing?  The students are simply figuring things out for themselves?  This concept confused me a bit.  

I would like to see more examples of what Hewitt is doing to get a better understanding of the differences between teaching arbitrarily and teaching awareness.  The idea of received wisdom is definitely something I will keep in mind and I hope to find instances in my lessons where I can stray away from it when it is not needed.  

Teacher Change and development via research

After reading Robinson's article one of the ideas that struck me was videotaping yourself teaching as a means of self analysis.  Your perception of the teacher you want to be versus the teacher who you are is something that a video can display.  It is important to receive feedback from external sources, but it is equally, if not more important to receive feedback internally.  I also enjoyed the section on incorporating group activities into mathematics lessons as it seems to be a common emphasis throughout all of our education courses.  It is good to see some first hand experience and verification of this being extremely beneficial.

There are obvious problems that come with group work such as certain students contributing to much or to little, not being respective of other students opinions, of conversation begins to go off topic.  Robinson seems to tackle a lot of these problems by designating each student a responsibility within a group and getting students to complete a synthesis evaluation.  One question I would ask Robinson would be, are there other major problems you experience by incorporating group activities?  If so, how do you manage them?

How will I continue to develop as a teacher after the education program?  

One way I will continue to develop is by having an online presence.  I think the internet, social media, and technology are all great tools and resources for students and teachers.  It's a great way to bounce ideas off other teachers, find, use and tweak creative lesson plans, as well as it allows for an easier learning experience as you don't have to go through all the same hardships that other teachers already went through.  I have yet to attend any major conferences but I think going to these events, not only just the math specific conferences but other disciplines as well will be informative and help me keep up with the current pedagogical trends.

Also, I will try and get feedback from my students on a consistent basis.  Students are the reason why we teach so their input is the most important.  Being willing to change and manipulate lessons based on who I am teaching is something I will always try to do.

Mathematics education for democratic citezenship

After reading the Simmt article I agree with the fact that everything in today's society is quantified and numerical so it's important that our mathematics education prepares students for this and that it is also crucial to acknowledge and incorporate the fact that mathematics can be a tool to provide the necessary civic skills required in our society.  I think the main challenge and problem with mathematics is the fact that we are over concerned with correctness.  It's very authoritative in a sense.  The way we test, mark, and teach mathematics is for the most part all based on getting the right answer.  This creates an individually competitive scene which isn't necessarily a bad thing as competitiveness is a form of motivation(I think math needs students who are more motivated).  But it becomes a problem when we start isolating ourselves.  As teachers, and students, we should be promoting collaboration, communication, and conversation amongst each other, much like Simmt points out as one of her arguments for citizen education in mathematics. As Jacob Brownowski says, 

"It is important that students bring a certain ragamuffin barefoot irreverence to their studies; they are not here to worship what is known, but to question it." 

Students should be willing to challenge and confront ideas rather then just adhering to what the teacher said because he/she is in authority.  Seth Godin says in a similar fashion,  

"You have been brainwashed by school and by the system into believing that your job is to do your job and follow instructions."  

It's important that students learn both skills in my opinion.  Being able to effectively listen and execute what someone else is saying is a life skill that is important in the work force.  But at the same time students should be able to formulate their own opinions and reasoning.  One of the main ways I will teach for democracy is by trying to provide a less isolated environment by getting students to work together, sharing their knowledge with other students via discussion groups, mini presentations, or group research projects.  However, from personal experience, I know that group work can often be problematic as it is hard to delegate who does what and it often turns out to be a one man job.  I also think that technology can be used to teach for democracy.  Via technology we can post notes, videos, links to other resources etc.  This can allow students to learn at their own pace rather then having to struggle to keep up in class.  Also, through the use of blogs or other social networking it is easier for certain individuals to speak their thoughts and is simply better for overall communication.  I know from personal experience in highschool, raising your hand to say something was often very daunting.  Now we can sit behind our computer screens and think about what we really want to say and then ask.  I think technology provides so many opportunities but the problem is that not every student has the latest gadget or device.          

In Mathematics there are plenty of different methods to solve the same problem so we should be encouraging this, not limiting it.  It's important to get collaborative input from all students perspectives. Mathematics shouldn't be based on who is the best "memorizer" but I feel like this is exactly what is happening in Secondary Math.  It's a big contest to see who can follow and memorize the steps correctly.       

"My contention is that creativity now is as important in education as literacy, and we should treat it with the same status.” - Sir Ken Robinson

It is important that we acknowledge creativity rather then linear memorization.  However, the problem is that instrumental teaching is often easier and less time consuming.  But we shouldn't be content with the easy way out. 

Ill end this post with an interesting quote I found from another article speaking on democratic education

 "in the end, we universally teach mathematics in schools in order to educate students; we do not universally educate students in order to teach them mathematics" (For the Learning of Mathematics , Vol. 17, No. 3 (Nov., 1997), pp. 11-16).  

Family Relations Problem

Brothers and sisters Have I none, but that man's father is my fathers son
Assuming the speaker is male, what is his relationship to "that man"

This problem shouts at me to make a diagram.

First lets see who is in this problem.  We have the speaker(me or I), the speakers father, that man, and that man's father.

So lets draw those out

The problem asks for the relationship between the speaker(me) and That man.
Now lets take a look at the second statement, but that man's father is my fathers son

so that man's father = my fathers son = ME

The first statement also says the speaker has no brothers and sisters so we can conclude that "That man" is my son.

If students are struggling:
         -give them a better understanding of how family tree's work
         -drawing diagrams helps a lot

Extensions: Can you create a paradoxical or difficult statement using double,triple, quadruple negatives
one you could use with your students?

1.I found a funny clip of the big bang theory using quadruple negatives!

http://www.youtube.com/watch?v=NAjFgVM0y5o

2."I don't owe nobody on my team nothing"

3. 

A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie.

You meet two inhabitants: Zed and Alice. Zed tells you, `I am a knight or Alice is a knave.' Alice tells you, `Of Zed and I, exactly one is a knight.'

Can you determine who is a knight and who is a knave?

Microteaching reflection

My initial thoughts and reactions after microteaching were mainly positive.  Myself and my peers agreed that for the most part I covered all aspects of the lesson plan, though sometimes not thoroughly.  Personally I had troubles with creating a hook and having a good way of summarizing/concluding.  These are important as they are the first and last impressions of your lesson so it is something I will continue to get better at.

Some of the feedback I received on the strengths of the lesson included
"Payed attention and offered feedback to every individual student"
"Simple and focused and did not try to overload on information"

These are two things that I completely agree with.  Having a decent amount of coaching experience I know the importance of giving feedback on an individual level and offering different types of feedback(verbal and visual).  Also, keeping it simple and not overloading students/athletes with too much information is critical in my opinion.  You need time to develop each individual part of the puzzle before putting it together.

Some of the feedback I received on areas of improvement were
"more eye contact"
"demonstrated a bump with a ball"
"more obvious hook and summary"

I always thought I did a good job of making eye contact with everyone but clearly I was wrong!  I will definitely make more of an effort to make eye contact with each individual consistently.

I was supposed to demo the bumping technique with a ball but I completely forgot too! I did it without a ball but forgot to do it with a ball.  Maybe for next time I should take a quick glance at my lesson plan here and there just as a reminder.

And yes I completely agree with a more obvious hook and summary, I had already noted it as one of my weaknesses and I will continue to develop this aspect of my lesson planning.

One thing I noted was the inclusion of extra activities.  To my surprise, the athletes were quite good at executing the proper technique and had successfully done this faster then expected.  I had a couple other progressions to add in to the drill, which I did, but again they were quite good at adapting to the variations to my surprise.  So maybe for next time I should make sure to have a fair amount of variations and other drills I can do if they group is excelling.

Great learning experience overall and the assessments were a valuable tool.