of mathematics and beauty

I love math

I really do. I think it represents a profound aspect of humanity that is both beautiful and surreal.

What I love even more is seeing that same excitement I feel reflected in my students, and also pushing them to proofs, because this, ultimately is what mathematics is about.

I had two students jump out of their chairs today when they discovered that the number 35 can be written as

17 + 18 = 35

and

2 + 3 + 4 + 5 + 6 + 7 + 8 = 35

Students were also excited by the fact that their peers hypothesis were either totally correct, or that there were some exceptions to the rule. (such as triples of consecutive sums add up only to an odd number.


Students today worked on a continuing problem from Friday. Here was the challenge:





One of the major conjectures students came up with today was the following:

An odd number plus an even number is always an odd number.

I asked them if they would bet 2 million dollars that this hypothesis was true. Some decided to renege on the answer, while others were more insistent. The next step in this is me asking them to prove it.


Here is the formal proof below for my mathematically inclined readers.

Conjecture: An odd number + an even number = odd number

Let k represent some natural number in the natural numbers set.
Let m represent some natural number in the natural number set.
Let n represent some natural number in the natural number set.

Therefore, we want to show the following:

(2k + 1) + 2m = 2n + 1

Proof:

By commutativity, 2k + 1 + 2m = 2k + 2m + 1 = 2 (k + m) + 1

Let n = k + m, such that n is a natural number in the natural numbers set.

Therefore, 2(k + m) + 1 = 2n + 1

By definition, 2n + 1 is an odd number. proof complete.

Thus, an odd number + an even number is ALWAYS an odd number.

Beautiful, elegant, and simple no?

In any case, this is where I want my students to head towards; the idea that there is truth and elegance in this world.

I strongly recommend this book:





It has really rich and insightful problems that can cater to a wide range of mathematical curricula.

I have found that updating grades is a lot easier when listening to this song.

today....

first brief glimpse of what my students deal with on the outside.

it ain't pretty.

and it makes me sad.

tomorrow I start advocating for them, one day at a time....

undermining bridges

Today would rate as a B - or translated into a 'meh'

Activity went okay. I did all the good utepy things you're supposed to when doing a group building activity:

- job roles for students
- clear expectations
- modeling positive and negative behavior
- clear worksheet with directions
- materials ready in advance to be passed out
- job helpers to pass out materials
- and it goes on....

however, this was all undermined by something very foolish and obvious. I did not explain that the job roles they were being given were IN ADDITION to the group discussion they were supposed to be having...

sigh*

well at least I've learned my lesson...and now will refine that idea for tomorrow.

I bravely look towards tomorrow with sniffles and a head cold.

plans have been made
photocopies need to be hurriedly rushed in the morning.
charts need to go up.
greetings, addresses, and the daily tumble begins.

triumphs and tedium

I am very exhausted this week, and have one more day to go still tomorrow.

It has been a week of realizations, battles, and brief moments of getting it right.

Too tired to type deeply, so I'll speak with pictures.

Here are some of the 'positives' for the week:


Group D's Hopes and Dreams. I need to put up Group E's and F's, but I am waiting for the imminent arrival of the SmartBoard and its subsequent placement before I tape up 60 more of these...


My very simple but effective rules for the classroom. To make it actually concrete as opposed to abstract we did the following activity below:



Students, in their study teams, received a packet of 'anecdotes' of behaviors I had seen last week, both positive and negative. They then arranged them according to category. This became quite powerful as there were some negative stories that fit into all three rules of being broken. Students realized that their actions affected more than themselves. (Of course, not everyone got this, but it was still a powerful exercise nonetheless)

This was a time-heavy activity to make with all the materials. I made envelopes of about 20-30 stories for group, 20 groups in total. At least two hours of prep time, but it was worth it, I feel.


We've started bar graphs and our graphing unit this week. Middle school girls apparently really like Chris Brown....so let's put him into mathematics eh?

I've also been purposely making mistakes on my graphs, so that I can ask students during share out what problems they are noticing about my graph. This is leading up to an activity where they look at different graphs to find errors....still working out the details on it however.....



Yay! I am using my math word wall, and used it to jump start Math Simon Says today. Tomorrow, they begin formally defining the words for themselves. Math Simon Says went well, despite some disruption and behavioral problems.

Okay....bed time

Sara

realizations for today

1. girls really like to chat. you might as well capitalize on it and do something constructive. Today, they did an activity in which they had to put together anecdotes with my classroom rules. Even though many groups were talking about a lot of different things, they definitely understood the idea at the end that if one of us chooses to be disruptive, it effects the whole community.

2. I feel a lot less scared, but definitely still overwhelmed. one step at a time. I am beginning to believe that I can actually do this teaching thing....


3. I am realizing that my transitions are terrible. What can I do to help fix this problem with a bunch of very loud girls? The problem is, is that it's hard for me to get them back together after a group activity. I use the hand raise signal, but I feel like there's already a rate of decay happening with its efficacy. Should I use music? a bell? lights? Janet's head tapping strategy? Or do I just stick with the hand raise. Help....


That is all.

Till tomorrow,

Sara