Fun with Fractions

OnlineMathLearning.com. (2015). Frayer Model of a fraction
 [Online Image].Retrieved from
http://www.onlinemathlearning.com/fraction-grade4.html

            Last week in class, we discovered a progression of fun activities we can do with our students. Whether they are learning about fractions for the first time, they have grasped a good understanding of fractions or they are looking for a challenge, many activities are available to engage students. The activity I presented was for beginners and it provided students the opportunity to develop their spatial reasoning skills by seeing the size different fractions visually and comparing them. A more advanced activity had students break a shape into continually smaller, but equal pieces. And another activity encouraged students to convert fractions into decimals by providing students with both fraction and decimal form, and then matching the correct ones with each other.

            As a class, we further developed our understanding of fractions through use of the Frayer model (right). The Frayer model is a visual diagram of four equal sections with the topic of interest in the middle where all the sections connect. Each of the four sections is filled with a definition of the topic, characteristics of the topic as well as examples and non-examples of the topic. In class, our topic of interest was proper fractions, but I tried to encompass all fractions in my Frayer model (as seen below). 

          I find fractions are difficult for students to understand because fractions consist of two numbers. Students are so used to associating numbers with just a quantity. However, we know that the "top number", the numerator, is the number that expresses the quantity and that the "bottom number", the denominator, expresses the name of the quantity. Or more simply, how much and of what. I think this is a very crucial component to understanding fractions. Then, when working with fractions, it will be easier to understand why we need to have the same name for fractions when adding or subtracting them, and why we only add the numerators, once this common name has been found.

      I also found an excellent online resource for helping students understand and learn about fractions. Khan Academy is an online website that offers free world-class education to anyone, anywhere. Its specializes in teaching mathematics, but also offers help in history, art and grammar. For 4th grade fractions, it provides activities for common denominators, equivalent fractions, and comparing fractions. Once the learner completes these activities, they win stars. Once they have earned all their stars, they are eligible to complete a unit test and win the grand trophy. It is engaging, interesting and educational!

There's a fine line between a numerator and a denominator. Only a fraction of you will get that joke.

The Importance of Counting Principles

Welcome folks!

This week was filled with fun activities presented by fellow peers and critiquing the activity based on its levels of appropriateness and effectiveness for the intended grade. We also talked about some counting principles that are essential for building a solid mathematical foundation. And this is what I will further analyze in this post.

In class, we learned that there are seven counting strategies or principles: stable order, order irrelevance, conservation, abstraction, one-to-one correspondence, cardinality, and movement is magnitude.  Curious about why these principles are so important I dug a little deeper and found that these strategies form the basic sense of number and quantity, also known as the core of mathematics. Without mastery of these principles at a young age, students struggle more and more as they progress in mathematics classes without having the opportunity to grasp these strategies. These principles appear in various forms under the Number Sense and Numeration strand of the Ontario curriculum from grades 1-8, yet instead of focusing on mastering these strategies, students are expected to also learn elements in the Measurement, Geometry and Spatial Sense, Patterning and Algebra as well as Data Management and Probability strands (Ontario Ministry). Yes, all of these other strands include these counting principles, but students in grade one who are still learning about these principles are expected to learn other mathematical elements at the same time. Shouldn't we focus on building a strong math foundation for our students before we introduce all of these complexities?

Kyle Pearce, a K-8 Mathematics Consultant in the Greater Essex County District School Board uncovered another critical counting principle: unitizing. In short, "unitizing involves taking a set of items and counting by equal groups" (Pearce). Unitizing explains how our base-ten number system works and is vital for understanding place value, fractions, unit rates and other big ideas connected to proportional reasoning (Pearce). As a math instructor, I have seen a lot of older students who struggle with fractions and proportional thinking because they are not masters of the unitizing principle.

Pearce, K. (2017, January 21). "Counting Principle: Unitizing" 

[Online Image]. Retrieved from http://bit.ly/2xlXz11. 

I firmly believe that one of the reasons a lot of students are so disinterested with mathematics is because they are not given enough time to become masters of these vital strategies. Mathematics is a progressive field of study, but we cannot expect our students to progress well if we don't give the needed attention to develop some crucial skills. When a house is being built, contractors make sure the foundation is solid before they build on top of it to support the rest of the house. Let's do the same with our students! Let's build and ensure they have a solid mathematical foundation that will support them throughout their mathematical life.

Why does a calculator make a great friend? Because you can always "count" on it!

References

Ontario Ministry of Education. (2005). The Ontario Curriculum Grades 1-8: Mathematics [Program of Study]. Retrieved from http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf

Pearce, K. (2017, January 21). Tap Into Teen Minds. Counting Principles - Counting and Cardinality. [Blog Post]. Retrieved from https://tapintoteenminds.com/counting-principles-counting-and-cardinality/

Growth vs Fixed Mindset

Welcome back!

This week in class, we discussed a lot about growth mindset versus fixed mindset and the importance of having a growth mindset.

First off, I would like to point out some differences between having a growth versus fixed mindset. When one has a growth mindset she/he:

- embraces challenge,

- realizes mistakes are okay to make,

- realizes she/he improves by having gone through the experience,

-believes that her/his ability can grow and she/he can become better (Small, pp. 5).

Lu, J. (2017, May 18). Growth vs Fixed Mindset
[Online Image]. Retrieved from http://bit.ly/2f5BsaA

Compared to a fixed mindset, which is when one:

-shies away from challenges in fear of failing.

-relates mistakes to being stupid

-is quick to blame others for his/her failures

-is embarrassed by his/her failures (i.e. shying away from a problem because he/she got it wrong last time).

In my opinion, it is important to have a growth mindset as both a teacher and a student. On one hand, teachers must have a growth mindset in order to foster one in their students. As teachers, we want the best for our students and having a growth mindset sets the stage for students to be able to tackle any problem and handle it constructively. On the other hand, it is important for students to have a growth mindset because we know we are always learning no matter our age, exemplified by finding different ways to solve one problem.

This mindset applies not only to mathematics, but also to all subject areas of teaching. As teachers, we need to ensure our students feel safe to make mistakes. We can do that by building on students' mistakes to steer them in the right direction to understanding, instead of scolding them for getting it wrong. We can create a safe and inviting classroom environment where every student is aware of his or her own voice and feels heard. Most importantly, we can continuously encourage our students to try and not give up, no matter how hard or impossible it may seem.

You have to be 'odd' to be number one! 

References

Small, M. (2017). Making Math Meaningful to Canadian Students K-8, (3rd ed.). Nelson Education.