The Real World Presents: Math

As a student I never realized the true depth involved in teaching. We started with the fundamentals: identifying numbers and letters, using them in a sentence, and so on. However that part just sort of… happened. Then, we get into school. We learned simple math and the next year it was a bit more complicated and the next more complicated still but to students, these new concepts and new levels of difficulty are independent from one another. Boy, were we wrong….

Each year the curriculum builds on the required expectations of the year before. Therefore in order to continue our mathematical education we must have a concrete knowledge of the concepts being taught. As teachers, we must ensure we have done our best to cover every expectation and ensure that students have a deep understanding before sending them forward because the moment they miss something is the moment their grasp on any future concepts starts to unravel.

How many times have we all heard the sentence? ‘When will I ever use that in real life?’ or ‘who actually needs math anyway?’

© 2016 TheThingsWeSay

There are aspects of math that can be considered easy or self-explanatory and therefore can be given less attention than desired in the elementary grades. One of which is integers. Which is problematic because it really is an aspect of math that is used everyday (have you ever checked the temperature?!)

As teacher candidates, we have the ability to read a textbook and sort through information to make sense of a simple mathematical topic such as integers. But for elementary students, it is important that all of the expectations outlined in the curriculum are given the time and energy they require. It is also important that we demonstrate why integers are both important to our continued mathematical education as well as our general day-to-day life!

‘An integer is a whole valued, signed number. It can be positive or negative but can’t be between wholes.’

I myself said, ‘then isn’t an integer just a number?’ But no, there is so much more to know about integers that needs to be understood in order to grasp more difficult concepts taught later on in math. 

The first thing to note is for such a seemingly simple concept is how difficult it can be for students to grasp the addition and subtraction of positive and negative numbers. When I was a student, the word integers always came with a strict set of guidelines that were supposed to prevent the confusions associated with the addition and subtraction of integers. So naturally, I’ve forgotten every rule I’ve ever learned.

Today, a fellow teacher candidate’s presentation on integers was able to very simply clear up my lifelong struggle with understanding equations involving integers while simultaneously proving Pat’s point on the importance of using real life examples in our teaching.

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Adam explained integer addition and subtraction by describing the ups and downs of the average day. He demonstrated how the number of positive and negative experiences you have can be added together and how the number of each impacts your overall mood.

After being taught in this way, I found it significantly easier to sort through the activities related to integers found on CLIPS (Critical Learning Instructional Paths Support). CLIPS is an extremely helpful resource when teaching math because it not only provides instructional activities on the different math units but it also provides mini ‘lessons’ or explanations to help simplify the topics. CLIPS is definitely a resource I would share with my own class in the future! Of course I would only invite the use of this resource once I had used Adam’s technique and ensured students understanding by teaching using real life examples!

Who knew there was so much to learn about teaching fractions!?

This week we had our second lesson on teaching fractions; we talked about how we could represent this idea in a ratio….

We’ve had 2 lessons on 1 unit making the ratio 2:1

We looked at a problem about the classic recess game, ‘red light, green light’. The problem provided the ground gained by each player; this information was represented through fractions. Most of the fractions had different denominators. We were expected to put these fractions in order. The important thing to note is that we were not told what sorting method to use or what order the fractions were to be in (ie. Biggest to smallest, etc.). The openness to this problem gave us the option to choose the order we wanted to use to sort our fractions, making it less restricting and easier to begin.

We then discussed our ideas as a class. Having us first solve the problems on our own ensured we had a personal understanding of the problem; having us discuss our ideas as a class provided us with ideas of other ways we could have approached the question. Some of the orders people chose were:

•       Biggest denominators first (biggest gap between numerator and denominator)
•       Smallest gap between numerator and denominator
•       Unit fractions first
•       Change all to percentages

I originally chose to start with the fraction that had the smallest gap between the numerator and denominator, thinking this was the easiest solution. However, after our group discussion I changed my mind, deciding that changing all the fractions to percentages would make it more visually appealing and much easier to sort by size. It was really helpful to go through the problem to come up with my own solution before being able to hear other peoples strategies. It allowed me to compare my strategy with those used by other students and thus come up with a final strategy that lead me to the solution in the simplest way. 

It seems as though the most important thing we learn each week is to always use open problems to provide an opportunity for each student to grasp the concepts at hand.

A good problem has a ‘wide base and a high ceiling’. It is open enough that it allows every student to start, regardless of their learning level, but it is also extendable, meaning you can add new portions for students who may finish quicker than others and still want a challenge.  A good problem has many solutions, therefore allowing each student to comprehend in their own way but then further extend their understanding through group work and group discussion.

A second activity we worked on today was discussing fractions using Tangrams. We read a children’s story about a porcelain square that shattered into 7 pieces and then had to try and remake the square using our own tangram manipulatives. A smart board demonstration assisted in showing how tangram pieces could be used to display fractions of the square.  This activity would appeal to auditory, visual and kinesthetic learners by explaining the idea of fractions through the book and then allowing students to manipulate the tangrams themselves.

We were given an allotted amount of time to make our attempt at creating the square, but not everyone was able to solve the problem before we had to move on. A separate lesson learned from this activity was to show the importance of celebrating hard work as much as we celebrate ‘success’. Teachers need to encourage their students for working hard and persevering through difficult problems in order to propel their motivation. It is important to let students know that success is attained through diligence and they should be proud of themselves for sticking with it whether they find a solution or not.    

A Fraction as a Comparison as a Pizza as a Percent

This week in class we talked about fractions.                                            
We discussed the many ways of thinking about fractions and some different resources that can be used to teach fractions in order to adhere to different styles of learning and levels of comprehension.
           When I hear fractions, my immediate thought is ‘a part of a whole’ and the example I think of is a large pizza with one or more slices missing.

© CLASSWISH 2008-2016

No, that is not incorrect but it is also only a small part of understanding fractions. There are so many ways to describe a fraction…

A fraction as one thing compared to another

This explanation is not one I would think of when discussing fractions. However, after this week’s lesson it is probably one of the first I would teach my students.

          Why? Over the last few weeks we have discussed the importance of bringing real world scenarios and ideas into our lessons to encourage a deeper understanding. We are always making comparisons in real life.

Looking back at the pizza we can understand it as parts of a whole:

One part of the whole pizza is missing, the whole pizza has six slices so therefore there is 5/6’s of the pizza left.

Or we can look at it as a comparison:

If I’ve taken one slice of the pizza then there are five slices left for my friends. I am comparing my one to the other five in the box. If we add these up to create a total we have once again created fractions.

By demonstrating fractions as comparisons we are still teaching the ‘part of a whole’ definition but using the same steps one might go through with any day-to-day situation. The ‘fractions as comparisons’ definition might be one I would use to introduce fractions to my class or as review before delving deeper into the more complex aspects of the unit.

This concept was presented to us in class this week using Hershey’s Fraction Book. Hershey’s Fraction book is a children’s novel that uses the 12 portions of a Hershey’s classic milk chocolate bar to demonstrate fractions; one of the ways it does this is through comparisons. 

The book asks you to break the chocolate bar into pieces and compare how many you have in each group to create fractions (eg. If you break off 5 squares of chocolate you have 7 left, therefore providing you with one group of 5 out of 12 chocolates and one group of 7 out of 12 chocolates making your fractions 5/12 and 7/12). Children's books can be great resources for teaching math to students because they often break concepts down into very simple forms making them easier to grasp.  I would definitely use this book in my own classroom because it explains the big ideas in a very understandable way and uses an engaging topic (any excuse to use chocolate in the classroom is good with me!)

Once fractions have been introduced, we can provide phrases that show the connections between different arrangements of numbers:

              

A fraction as a ratio as a decimal as a percent

Just as we provide students with manipulative's as well as pencil and paper examples to help them grasp a concept, we can provide students with different ways of presenting fractions. Some students may really like the idea of comparisons and will prefer to look at fractions as a ratio and some students may find it easier to understand a fraction when it is converted to percent form. It is important we, as teachers, demonstrate that all of these arrangements are equal and as long as we can explain how to properly convert them students can choose the option that works best for them. Although this definition may be more applicable to students with a greater understanding, I chose to focus on it because I liked that it provided options. One of the things about math I struggled with most as a student was the idea that there was only ‘one way’. I think that if we show students they have options it will make math far more appealing and understandable to students of all learning styles and abilities.

Ensuring students have a strong grasp on fractions is very important as they are not only a vital part of basic math but they are used all the time in every day life. The ‘Exploring Fractions’ video is a great resource to show students to help them understand how often they will use fractions throughout their own lives.