Week 5

The latest concept explored in Math 251 would be the "base" concept. So to explain, when we add, and the sum of two placeholders is greater than 10, we carry the one to the next placeholder to be added there. Its something drilled into our heads at a young age and becomes second nature later in life. The issue with these "second nature" equations is that when trying to explain why we do this to children, our only answer is "because thats how you do it." So to better understand why, and how, this concept occurs, we experimented with different bases.

For example, base of four

  231     Why is this? Because when two numbers of a placeholder reached 4, we had to carry one
+121     and add from there, just like we do with ten! So, in the 2nd placeholder, since 3+2=5, we had
_____   lower the one leftover, and carry what would be the base four to the next placeholder.
  1012

Changing the rules to better understand your own understanding of a concept, will heighten your ability to be a better teacher in that concept. Its like seeing things from many perspectives to understand it better as a whole! Which is a great way to teach.

Week 4

Addition. A simple enough concept that we are taught at a young age, though the way we are taught doesn't completely allow us to understand what addition means. The coming together of multiple things to create a new sum, a sum in which can be discovered in so many ways!
This week in class we explored some of those ways.

Traditional:
This form is the method we were taught from day one of formal education, the stacking of the two numbers like so, not much explanation needed.

http://everydaymath.concordnhschools.net/modules/cms/pages.phtml?sessionid=df8d994395a2e08afb0842c63b74e11e&sessionid=&pageid=158683

Partial Sums:
This form is not farm from the traditional form, but it paints a clearer picture of what it is the traditional concept entails. By separately adding each placeholder, for example 34+98 would be 30+90 and 4+8.

http://teacher.sheboyganfalls.k12.wi.us/staff/laschwab/EveryDayMath.htm

Decomposing:
Another more creative visual of addition involves using a number-line to show the growth involved in addition. Similar to the regular number-line we worked on earlier in this year, though you are able to start at one of the numbers involved in the equation.

This example goes a step further to also implement compensation to make the jump easier.

http://www.kcptech.com/dynamicnumber/elementary_number_properties.html

Compensation:
This form of addition makes numbers that are hard to add without the traditional form and "carrying" much more manageable, because as a teacher of young math students the key to most problems are finding a "friendly" or "nice" number.
For example, if we were to be adding 34+28, not many of us would be able to solve this without counting. If we were to round the 28 to a "nicer" number like 30, we would have 34+30. Which you can easily do in your head, you come to 34+30=64 then COMPENSATE for the numbers you rounded to by subtracting the 2 (64-2) coming to the actual answer 62.

http://mindfull.wordpress.com/tag/compensation-addition/

Give and Take:
Similar to the compensation concept, give and take turns difficult numbers into nice numbers, but instead of rounding on its own, you give and take numbers from the numbers in the problem. Therefore you don't have to compensate in the end, because you never added or subtracted numbers that weren't there to begin with!

http://mathcoachscorner.blogspot.com/2012/10/more-mental-math-strategies.html

Addition can be as complicated or as simple as you want! Here are a few ways to do either.

Week 3

In our third week of Math 251 we continued, and elaborated, on the topic of Venn Diagrams. Instead of using simply the intersect, union and compliment sentences, we applied these diagrams to real life situations.
These are a great tool when discussing percentages in a survey of people. For our example we used winter sports, the survey included the following categories:
a. people who snowboarded
b. people who skied
c. people who snowshoed
d. people who snowboarded and skied
e. people who skied and snowshoed
f. people who snowshoed and snowboarded
g. and people who did all three.

They would fit within the Venn diagram like so:

Figuring out how to fit all of the numbers into each category taking into account that theoretically someone who falls under the "likes snowboarding" category could also "like skiing" and could potentially "like all three" is a great mathematical challenge that we were able to address. 

Primarily the concept of starting at the center amount and using that to work your way to the outside numbers.

For a more detailed walk through of a Venn diagram word problem I've provided a video:

http://www.youtube.com/watch?v=MassxXy8iko

Week 2

This week in Math 231, we introduced the concept of Venn Diagrams. Something seemingly outdated and for children, in fact takes a whole new light in this class. The concepts of unions and intersections of groupings create a much more complex concept than once believed.

The concept of a Venn Diagram originates with two groups and their populations and intersection of populations.


For example; people who like country music and people who like pop music are in two separate circles, and those who like both both would fit in the center segment.


The next concept of the Venn Diagram was isolating certain populations of this diagram. For example, to show exclusively the population that like pop AND country music. Written as A intersect B.


Another idea, was exhibiting the entire population, this is called a union and is written A U B


An idea in which you isolate everything but a certain aspect has a line over the subject. So A with a line over it is everything but A


So to combine these events one could draw everything BUT the INTERSECT of A and B and would look like this.


There are many things you can do with a Venn Diagram and I think its a great visual way for kids to learn.

Week 1

Subtraction can be a struggle for all people, from elementary aged students to the teachers that are attempting to instruct them. An effective way for many people to learn is visual aid, we worked with one such device in class this week called a number line. With this tool, we discovered a creative and helpful way to visualize the act of adding and subtracting, and proved that there are many different ways to come to the same answer.

One example we used was 65-48, my initial number line looked like the following:

17__(-3)__20___(-5)___25________________(-40)___________________65
<-------------------------------<-----------------------------<---------------------------------<

The arrow indicated the direction we are working, because though you would start with 65 when subtracting 48, the visual line still must read from left to right. So in this example the numbers -40, -5, and -3 all add up to -48 which is the number we're subtracting. Being able to break up the number into clearly visual parts is extremely helpful for new math learners and a great tool for teachers.

A clearer example of this method is shown here:



With a new tool under my elementary education belt, until text time!

photo cred: http://learnzillion.com/lessons/1583-solve-subtraction-problems-using-a-number-line