Previously...

On our Mathlicious Part One, I have introduced my two main teaching pedagogy, which are: seeing every child as a “whole child”, meaning paying attention to their emotional and cognitive development, and learning with a child through a step by step thinking process that I define as “logic.”  The fundamental of thinking step by step in an organized and thoughtful way enabling a child to reach an understanding and not stop at knowing, which will lead the child to be able to analyze and critically thought of a concept.

I came across one beautiful mathematical quote; I don’t know about you, but I find quotes are particularly useful to boost up my mental and emotional stage when a reality seems to be so disarrayed or seemingly too far away to cope with. This one is a quote that has motivated me in thinking and imagining mathematics...it only makes sense to introduce mathematics through a creative imagination of a teacher who takes the time and courage to understand each student. Creativity and understanding are key parts to create the “delicious” taste of mathematics.

“ Many who have had the opportunity of knowing any more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination.” ( Sofia Kovalevskaya)

Problem Solving...

it is the fun of knowing what to do and not merely getting the result. In my class, the right answer is merely a result and not a goal of learning mathematics; the doing of mathematics is the goal of learning mathematics.

It is Fall and it is a bit chilly...

What could we connect Fall season with? The very thing comes to mind is APPLES...I love apples and well...I love food in general...Let us do a problem with Apples.

Problem One:

My Mother and I are planning to bake 4 apple pies. Each apple pie will need 2 apples. When we were about to start, we realized that we only had 3 apples on the kitchen counter. What do you think we should do? Please read each of the possible answers below and circle the one that you think make the most sense! ( There is the time when more than one possible answer is “possible”  and there might be a time when there is no possible answer is provided:) ).

1. We should not bake any apple pie since we don’t have enough apples.
2. We should plan to go to apple picking.
3. We should think of how many apples we need, then we go to the supermarket and get some.
4. We should make one apple pie instead of four apple pies. 

Now, do you think we should count the number of apples that we would need to make 4 apple pies? If you think so, how are you going to do it?

1. You would add the number 4 and the number 2 together.
2. You would add the number 2 four times.
3. You would add the number 4 four times.
4. You would add the number 4 two times.

The next step, you know that we have 3 apples at home, and you know how many apples we will need altogether to bake 4 apple pies, then what do you do?

1. I should subtract (take away) 3 from 4.
2. I should subtract (take away) 4 from 3.
3. I should subtract (take away) the total number of apples we need from 3.
4. I should subtract (take away) 3 from the total number of apples that we need. 

Note:

My intention on the problem above is not just to solve the problem, but more importantly, it is to think of a problem. When you spend the time with your child doing it, I would highly recommend for you to use lots of open and ended questions, be patient and give sometimes for your child to think through, provide any kind of hands-on manipulative that might help your child to understand the problem, and if the right answer does not come, instead of saying, “No, that is not the right answer”, say, “Well, that is how you think of it; can I share with you how I think of it?”-- this kind of approach provides an alternative, rather than merely negates the child’s answer, which is equivalent with cutting his or her thinking process off. What we want ought not to be “cutting off the thinking process”, rather ought to be “expanding the thinking process, introducing possibilities” that leads to a better thinking process.

Problem Two: ( Shapes and Creativity and Imagination)...

In this problem, we are concentrating on three shapes: circle, triangle, and square. In relations with Fall session, ask your child what comes to mind when we are speaking about “Fall”...well, these are some possibilities:

1. Pumpkins....it could use circle, triangle and square... ( triangle and square are for the stems)
2. Scarecrows...we could definitely draw scarecrows using all of those three shapes.
3. Turkey---yummy and Thanksgiving...yeap, we can use all of those three shapes.
4. Craving pumpkin faces could use the help from triangles, circles, and squares. 
5. Leaves....my favorite colorful leaves...we can use those three shapes to create beautiful leaves...

Note:

This particular activity is not just for the youngsters. At times, we misunderstand that this kind of “math” is not math...it is just a “play” for the youngsters. I beg to differ! This kind of activity is a mathematical activity; the idea of putting an abstract concept such as shapes and relating it with what is around us requires a logical thinking. The older students could provide more details on their imagination, and yes, we might be able to add a bit more complicated problems to the activity, but, do not let them miss out the ability to “KEEP CALM AND IMAGINE AND CREATE.” When a student is allowed to imagine in the beginning, it is like a warm-up before marathon; the mind is opening up and more often than not, the thinking is expanding and the fun is happening. 

What could we possibly think for the older students in relations with shapes?

Well, if you are ready to introduce measurement, provide a certain measurement on the squares and triangles, and encourage them to learn using rulers.

You could also introduce the idea of area and perimeters through the activity. For example, 

Each side of a square is 3 inches. so the perimeter of the square will be: 

4 times the length of each side, and if the student is not comfortable with multiplications, I would encourage him/her to use additions to solve the problem. 

Note: 

To learn the formula is not my goal in teaching; it is to understand the definition of a parameter through this formula is my goal. For beginners, I withdraw from using an unreasonable measurement; in other words, my students will be able to use a ruler to actually draw a square that represents the length of each side-- seeing the real representation of a certain length and the actual size of the square. I won’t use 145 inches for each side since it is way too big for a paper and for beginners learners, this won’t make any sense since no connection is truly seen. 

We could also introduce fractions using these three shapes and I could go on. The possibilities are multiple. See what your child is ready for, and adjust accordingly. 

Last Problem...

My students, especially the youngsters, love to say the word “infinity.” On other day, one of them who is 5 years old asked me if infinity is a number and what it would look like other than the symbol itself. From that question, which is seemingly simple, and understanding his readiness to learn ( I’ve known him for 2 years now since he has been taking my class since he was 3), I told him that infinity is not a number; it is a symbol. However, mathematically, we can talk about infinity in two different sets of categories: countable and uncountable sets. In the countable sets, there is a possibility of us to have an infinite countable numbers, and in this case, counting is possible. In the uncountable set of numbers, we can’t really count since the name itself is “uncountable.” We explored on different things around us that could be considered countable and uncountable, such as sugar will be uncountable and the total number of watermelon seeds will be countable infinite set of numbers-- and I told him that we use the word “infinite” is mainly because most of us do not have enough patient to “really” count the number of seeds, hence I introduce an alternative concept, “estimation”, to provide him with a better understanding. 

Is my explanation somewhat simplistic? Yes, it is; however, it reaches a goal that both of my student and I are looking for: thinking and understanding a somewhat abstract mathematical concept. At times, or often times I shall say, SIMPLE does it. I believe that one understands a concept thoroughly when he or she can share the knowledge at it most simplistic way. 

On this note...

I am leaving you with a quote, and I hope that you are keeping this quote in mind whenever you are spending time with your child, when you are trying to understand your child...this is from one of my favorite fiction authors,Haruki Murakami, 

““Adults constantly raise the bar on smart children, precisely because they're able to handle it. The children get overwhelmed by the tasks in front of them and gradually lose the sort of openness and sense of accomplishment they innately have. When they're treated like that, children start to crawl inside a shell and keep everything inside. It takes a lot of time and effort to get them to open up again. Kids' hearts are malleable, but once they gel it's hard to get them back the way they were.”

Many of parents come to me and say that “my child lacks of confidence”, and many of my students, I do feel and know, do lack of confidence. Please keep in mind that confidence is not innate; it is learned, hence it could be unlearned. What we want is a child who thinks outside the box, and what a child needs to think outside the box is the bravery of possibilities, the bravery of falling and getting up, and often times, we are, as adults, forget to be that role model...the role model of bravery of possibilities, the bravery of falling and getting up by as simple as not providing the chance and the opportunity for a child to experience the falling, to experience the uncertainty, and to experience mistakes. I do want to encourage parents to explore than the sentence “ To be with my child”, often times, is just it is...TO BE...it does not mean not to provide structures or guidance, rather it means to adjust the structure and guidance according to  the careful observation on and understanding of your child when your child is just “BE”...

Until my next part...

Dr. Chandra Budi

Unconventional and Resourceful Learning

(URL Tutoring and Consulting, LLC)

E : itworks@urltutoring.org