Friday Two Cents: A Problem Solving Model

In it there are four levels to achieve your goal to solve a problem.  The great thing is that you can use this problem-solving model not just for mathematical problems but also other problems in you may face in your life.  Here is my break down of the model…

A couple of weeks ago I wrote several posts about resolutions and making plans to help you beat the winter blues.  I have always thought that planning out things helps to solve the problems that I face on a daily basis, yet I have wanted something tangible that I can see in black and white to help me explain it to others.  I did find such a piece when I took a mathematics additional qualification course a couple of years ago.  It was in the Ontario curriculum, Grades 1–8: Mathematics (revised) document of all places. It is on page 13, figure 1: Problem Solving Model.  

Understand the Problem (the exploratory stage)

This stage should be self-evident.  I cannot tell you how many times I would have students come up to me and say, “I don’t get it.”  Then I would ask, “Did you read the question?”  Most of them would say ‘Ahhh no.’  ‘READ THE QUESTION’ I would say and then look in the question for the pieces you need to answer it.  Others I would rephrase the question to emphasize the important information they may need.  Basically what is the question asking of you: what is the problem?  The best advise I say to people is talk to someone about the problem so you can see it for different angles, especially after you read it several times.  

Make A Plan

Is there another situation that you may have seen a similar problem?  I tell students don’t try and reinvent the wheel.  Or in other words don’t start from scratch look at another situations where you solved a similar problem and try and rework it for this situation.  In essence, “Make a Plan”, think of a strategy you used before and use that plan.  Tweak the plan to fit your needs.  

Carry Out the Plan

Put you plan into motion.  Draw, write, use objects to help you visualize the plan and then implement it. Use different tools to make you plan work, monitor it and make adjustments when needed.  If you planned for something and you don’t need it, don’t use it. Why waste time and energy when you do not need it. 

Look Back at the Solution

Check you results, go back to the question to make sure that it actually answers it.  Does it make sense?  You have to go through the process again from the beginning to refine you answer or correct any mistakes you may have seen.  Could you get the same result another way, perhaps and easier way? This way you can use that revised plan in the future.  

I created this visual to help myself and others try and visualize how to solve a problem.  Maybe it will inspire other teachers and students to think more about how any problem can be solved so long as you have the tools and desire to make the effort in trying.  Remember what Napoleon Hill once said …

‘Effort only fully releases its reward after a person refuses to quit.’ Napoleon Hill

A Problem Solving Model

Ontario Ministry of Education. (2005). The Ontario curriculum, Grades 1–8: Mathematics (revised). Toronto: Queen’s Printer for Ontario. http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf

Friday Two Cents: A New Year With A New Experience

 

With the New Year I begin a new adventure.   I have been fortunate to been given the chance to be the teacher in a grade 1 class. It is a tremendous opportunity to teach a full grade one class and to put my years of education and experience into practice. I was lucky to teach in a kindergarten class at the end of last year’s school year and yet now I get another chance to teach many of the same students again in grade 1. I have been with them for the first week and it has been tiring but very enjoyable and enlightening.

The interesting thing is that I worked with a grade five class for about a week and a half before the Christmas break and now I go from a class of 10/11 year olds to one of 6/7 year olds. To say they are very different grades would be stating the obvious but they all have their own rewards. These little people in grade one are so full of wonder and excitement to learn that it has become a bit infectious. They have been eager to learn and I in turn am eager to teach the lessons with these students.

We work on subjects that I enjoy such as art but their enthusiasm for the core subjects like math and literacy is amazing. One literacy activity we did is a “Roll and Read” sight word game. Each student gets to roll the dice and whatever number it lands on they must say the words under that number as quickly as they can. I introduced it on Monday and everyday since they are eager to play the game.

Roll and Read Sight Game

Another activity I introduced is a subitizing activity. Subitizing is the ability to recognize the number of a group of items at a glance and without counting. I used bingo dabbers to make dots on a card and we practice this just before the math lessons. The real nice thing is that I noticed the students practicing on one-another during their indoor recess time.

Subtilizing Card game

It is only the bringing and I have no idea how long I will be in this class, but one thing is for sure, I am going to enjoy this experience with the students. Already I have noticed that they are learning new things but on the other side so am I.  

Friday Two Cents: Two Cents For A New Year

Last week was the final week at school before the winter break and I was fortunate to work in a grade 5 class for the entire week. I have supplied in that class before and the students all know me and I enjoy their company and quirky personalities. The teacher left her plans for me to follow but she also left me room to do a bit of what I’d like to do with them. She knows I have a visual art background so art of course was one of them, yet I also have a fondness for math.

The lessons for the week were around measurement and one unit was on money. I did the lessons with the students but one thing came to my attention with a question from a student. She asked about why do we have to know about pennies when we do not use them anymore. I told her that this was an excellent question and she was one of many people, teachers, administrators and parents included who were asking the same question.  I told her; even though we do not use them we still need to understand the math behind calculating making change and other necessary interactions when it comes to money. They mentioned the rounding up or down depending on the value. For example if the cents end with a 1, 2, 6 or 7 we round down to the nearest 5 or 10 but if the cents end with a 3, 4, 8 or 9 we round-up.

They got that concept but a question came up, “Does the rounding benefit the consumer or the retailer?” I was curious and then I heard about a study done by an undergrad on this very subject. She found that the rounding is benefiting the grocery stories in the range of over $3 billion Canadian or about $157 per store a year. A spokesperson for the Retail Counsel of Canada disagrees with the finds but it does make me wonder. Therefore I made it a point to look at my purchases a little closely for this issue and I did notice that the bills of many favoured the retailer more than myself, the consumer.

‘Penny rounding adds $3 million a year to grocery stores’ bottom line, study says’

I noticed this a lot with small purchases like coffee or when I went out for a bit to eat. Yet I also noticed this when I bought groceries and during Christmas shopping. I for one do not like using the debit cards when making purchases I prefer cash. Yet this has opened my eyes a bit and I have been changing the way I shop. Most recently I no longer use cash when buying coffee at Tim Hortons, I use a gift card that I reload, with cash. This way if the bill is $5.88 I pay $5.88 not $5.90. On large purchases I use my credit cards but I think I need to get a cash card for the stores that I frequent so I can pay the exact amount without rounding.

Upon reflection this is exactly what many people want, the beginning steps towards a cashless society. I know that I am not on the for-front of this but I enjoy using cash. It helps me to budget what I spend and how much. I also like the feel of it in my hand. Every time I feel those coins in my hand I would imagine what someone a century, two centuries even a millennia or two ago felt like when making a purchase. I once found a bronze Roman coin dating back to about 300 AD and as I held it in my hand I wondered, who else held this coin? A slave, a blacksmith, a farmer or even a Roman Legionnaire who knows but just thinking it made me realise that now I was part of this coin’s history.   Money, coins in particular have been around since the beginning of civilization, it is a part of who we are. It is a way for us to remember where we came from and be apart of history instead of just observing it.

My two cents and my new years goal is that I will probably go cashless on many things, just to make sure that the system benefits me more than the retailers. However in my heart I will probably still use the age-old traditions to help me with this goal, if only to remind myself and imagine a simpler time.  

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Friday Two Cents: Discovering More Through Play

This week and next I will be working in a kindergarten class where the teacher has asked me to present a lesson centred on math. Math has always been my favourite subject second only to art and when given the opportunity, I raised to the challenge.

It is the beginning if the year and I know that these students are new to counting, especially in French. So I decided to begin by seeing how much do they know when it comes to counting. I came up with several activities to help them and the teacher assess where the students are in their math skills.

I began with some activities with the students using 10 frames, rekenreks, printing the numbers and a bingo game with snap cubes. I noticed a few things that made me see that no matter where I go and with whom I work with, children learn a lot though play. Several of the activities are centred on a play base system. Yes they are doing their work by learning to count in French and they are printing the numbers but to get to those numbers they have to use a spinner like you would use in a game. And the more I and the teacher played / participated in the activity, the more the students, 1) appeared to enjoy it, 2) hopefully learn something from the experience and 3) we learned where each student is in their counting.  

Every time I enter a classroom I always have this quote from Plato in the back of my mind…

 

‘You can discover more about a person in an hour of play than in a year of conversation.’ Plato

 

These activities I did with the students are a clear indication that Plato’s words ring true.

 

Number printing and counting

Paper clip and pencil spinner

Spinner & 10 Frames

Rekenrek counting

Rekenrek counting

Subitizing bingo sheet

Snap Cubes from counting

 

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Friday Two Cents: I Enjoy Mathematics

 

This past week was the first week of my summer vacation, yet instead of going away somewhere or doing some work around the house I decided to improve my teaching knowledge. I decided to take an additional qualification course (AQ) in mathematics for the primary and junior grades. Many people asked my why I would take such a course because I am more of creative person who enjoys art. Other said “YUCH!! I hate math.” or “I was never good at math.” Many people had a negative experience with math in school.

Yet truth be-told, I have always enjoyed math since an early age and it seems to come natural to me. Yet I have discovered that knowing how to do math and teaching it is a lot different. I wanted to increase my knowledge in the subject matter to better help the students understand math and perhaps enjoy it as I have.

What are numbers?

The course has been very eye-opening yet there was one notion that made me pause in refection. As part of the course I read a chapter from a book entitled, ‘Adding It Up: Helping Children Learn Mathematics’. In it you are made to try to understand what are numbers. The professor then posed a question… ‘What statement best captures your understanding of “number”?’ – Number are ideas, nouns, adjectives or something else?

When I was first asked the question my first thought was that numbers were quantitative / measurable units. Yet after the first lecture and reading the chapter I am starting to see that numbers are a lot more that I gave them credit.

Numbers are what we make them to be. They can be apples, cars, money, people or even time. Without the context of what the number is, it is just a digit, a visual image representing something but we are not sure what. When you say 5 what is five? Not until you say five cars does the number have value. I never thought of this concept but it makes sense.

Numbers are ideas, nouns and adjectives. They are all of these and more. It puts into perspective the idea of giving the students equations. The concept of 1+2=3 is good if you are doing high-end mathematics, yet we are talking about students in the primary and junior grades, children from ages 3 – 12 years.

This reminds me of when I took calculus in high school and university. It is all about numbers being part of an equation that needed to be solved. Yet I remember using many similar equations in physics when doing formulas. We knew that each variable represented something (f = force) and we would plug-in numbers to calculate the values. Even back then there was value to the numbers like mass (m) or speed (m/s) yet I never though of that when I would be teaching math.

This has definitely changed my thinking around how I present and instruct math. But it has also helped me to know what I can say to parents about, what people are calling the new way of teaching math. I can help them understand that numbers are ideas, nouns and adjectives. That the numbers are only part of the lesson, that we are helping the students to connect the number to their own life and thereby allow them to carry on this knowledge into their every day life.

This has only been a beginning of my journey to understand and help others understand math. There is a lot more to learn and with time, practice and perseverance I know I can improve my knowledge of how to teach math to the students. Perhaps even change society’s negative preconception on math.

Comic Strips: Out of Date Technology

 

Last week we sent out the newsletter and the latest instalment of, The Craziest Things along with it. This month’s comic continues the battle of when to use technology and when not to. Or more importantly, can the new generation of students do the work without it.

This scene has been played out to me, not word for word but very close to this scenario. For my practicum placements the one subject I always taught was math. I find it easy to understand and whenever anyone had difficultly teaching it they would always come to me. I too had some difficulty in the beginning because I understand the subject at a high level but how was I to bring it down to the students’ level of understanding. I found a happy medium by when to use technology and when not to. Using a pencil and paper instead of a calculator that sort of thing.

I came up with the comic and the students seemed to like the outcome I hope you too will enjoy, The Craziest Things; Comic Strips: Out of Date Technology.

 

Friday Two Cents: Are Boys Better At Math And Science Than Girls?

 

Currently I am finishing my Early Childhood Education (ECE) degree and I discovered something many should be made aware of.  I had to look into the an ago old stereotype that says that boys are better that girls when it comes to math and science.  Is this a myth? Well I found something that will make you think twice when dealing with girls when it comes to math & science.

The Myths that boys are better at math and science are just that, Myth.  Girls are as capable as boys to do well in these subjects, however I have discovered there are studies that have shown the difference comes from the teacher teaching the subject.   

If the female teacher teaching the subject has an anxiety about the subject, then this anxiety will be passed on to the same-sex students.  A study shows that students at the early years are more likely to model behavior from the same-sex teacher, including their anxiety about math.  This could come from the perception that women/girls are not good at math because it is socially acceptable (Beilock, et al., 2010).  

In the study, at the beginning of the school year, students who were taught by female teachers found no difference between the boys and girls when it came to math achievement.  However, at the end of the school year the boys continued to have no difference in achievement but the girls did.  It found that the higher the level of anxiety from the teacher, the more likely the girls would believe that their abilities should fall under the stereotypes of acceptable norms.  As a result, the more the teacher believed that they were bad at math, then this would be reflected in the girls math performance (Beilock, et al., 2010).    

Calling on my own experience this stereotype is indeed present.  Within my own school, there is a homework club and the children can do their homework and ask for assistant.  On several occasions the female teachers would redirect the math questions to me saying, “Mr. Paul is better at math than me.”  Even though I am good at math and science the teachers would already put down their own math ability.  Furthermore, many children would not even ask for help on a math or science question from a female teacher but would come straight to me with it.  Is the stereotype present that men are better than women in math and science in the mind of the child.  Unfortunately, yes it is.  

A further indication is within my classmates at the university.  We are currently starting a course on statistics, and a large portion of the course is math.  I, like many other in the class, got the textbook and looked through it before hand.  I found that many of the equations are similar to what I learnt in Physics and Calculus classes some years ago.  I have no reservations about it even though I know it will be a bit hard relearning the equations.  However, if you could see the rest of my class, their anxiety about this class is through the roof. 

Many are panicking about it and are already asking me things such as, “Paul you have to sit with me to help me.” Or “Paul can I call you to help me with the math.” I reassure them that this will all be alright and they can do the math just fine.  But you know what their response is?  “No Paul, you’re a guy.  You’re better at math than I am.”  Wow, there is some truth to that study after all.  Maybe their teachers had anxiety about math and it was passed on.  And now it might be passed on to the students they may teach.  Fascinating. 

In the end, it comes down to what you believe and how we present these beliefs to the children. Good and bad, both conscious and unconscious.   As teachers and professionals, we all must be aware of how we model behavior.  But all of you remember one thing.  We adults, whether teacher or parent, what we do or say nonchalantly, may have ramifications on the next generation.   

 

Reference

Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2010). Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences, 107(5), 1860-1863. Retrieved from: http://www.pnas.org/content/107/5/1860.full#cited-by

 

Related articles

• Math Stereotypes (educatoral.com)