The Good, The Bad, and the Questions

              Before I started this class, and my journey towards being an effective math teacher, I would have told you that all math questions are inherently bad, mainly because they are math questions! Like I have stated throughout my last blog posts, I am learning so much about math, and how to teach math that the subject is slowly morphing into something that I no longer fear, rather something that I want to embrace. At first thought, I could say what makes a math question good, is whether or not it is an open question, because that is the kind of questions we have been focusing over the past few weeks. This is somewhat true, as open questions embrace collaboration and the use of students’ imagination to work through the questions. But through further instruction, we have learned about other key points as to what makes a math question good, and what makes a math question bad.

                Obviously clarity in how you are asking and presenting a question is the first step towards creating good math questions, or what is known as the use of “Soft language”. Educators like myself should focus on creating questions with what we can call a “High Ceiling”, which means structuring questions in a way that students that are learning at a higher level are able to continue their learning past what the question is initially asking, questions that allow all learners to get the most out it. Good math questions also have what can be dubbed as a “Wide Base”, another term that embraces the learning of all students, focusing on asking questions that enable all students to get started.  Good math questions should also be relevant to the students, like sticking to themes or interests the students have which promotes engagement. Good math questions normally involve an activity and or manipulatives as well as embrace student collaboration.

                One key aspect that good math questions have is being structured in a way that embraces different ideas and methods in order to solve the problem. An activity and problem that we worked through in class is an unbelievable example of a math question that embraces different methods to solve the problem. The problem, titled “Joel’s Kitten Problem” asked us to determine out of 2 stores what had the better deal for kitten food, one store selling 12 cans for $15 or the other selling 20 cans for $23. We were asked to find the better deal without using division to simply find the unit price. Even within our group, we had differing ideas on how to do it. What made this question great was that everyone could start it (Wide Base) and that each group basically came up with a different method to determine the better deal. The picture below, Method #1,  displays how a group determined that the way both stores sold their kitten food enabled them to buy 60 cans of food. By doing that, they could compare the two stores prices for 60 cans and determine the better deal.

Method #1

 Another method that a group used was to solve the problem by visualizing each can from either store being a standard and easy to work with value. After each can had the same value, they started to divide the remaining money into each can, essentially discovering a unit price and revealing what is the better deal.

Method #2

             Both groups shown in this post discovered that the better deal was at the store selling 20 cans foe $23.  Joel’s Kitten problem exemplifies how an educator can create good math questions. Answering these questions with the use of manipulatives would also be a good idea, also illustrating how this is a good question. Educators have a lot to consider when developing lesson plans, but it is important o remember the key aspects for what make questions good or not, because the quality of the question has a direct relationship to how much every students takes away from the exercise.

A Rational Reflection About My Ratio-based Presentation

                This past week had been one of the most interesting weeks I have had since enrolling in teachers college, specifically in regards to my “becoming a math teacher” journey. I had the task of creating a 8-10 minute lesson for my colleagues in my math class, on the topic of ratio’s. I must admit that when I began the process of putting together my lesson and presentation, I was very nervous. There is defiantly something different when you compare getting in front of colleagues to getting in front of students when conducting a lesson. I have never been shy while in front of a crowd, but because I was getting ready to essentially teach them a subject that I have never been that successful with, I was slightly nervous.

                It was not until I laid out how exactly I was going to teach ratio is when I became comfortable with what I was doing. What helped most was relating to what we have been working on a weekly basis, that is, learning how to teach math in a completely different way compared to how I was taught. With this idea being in my mind, I decided to gear my activity towards Grade 6 students and structure it in a way that would emulate that grades general introduction to ratios by focusing on proportional geometry through the lens of ratios. I also made the commitment to include simple, real-life scenarios while I introduced the topic, scenarios that if those kids were actually going to be in the class, would be very relate-able for them. I decided that my lesson would start with a short presentation that introduced not only the definition of ratios, but  I wanted to include two examples that simply explained what a ratio is as well as how they can apply to comparing two similar geometric shapes. The other half of my activity was based on a worksheet I developed (which can be seen in the image below) that consisted of 5 questions that asked students to explore ratios by comparing similar shapes.

An Image displaying my worksheet for my presentation 

                I can honestly say that I values this assignment greatly for many reasons. Primarily, I now believe that by making sure I am prepared, I can confidently say that I will be able to teach more than just a 10 minute math lesson. This was actually the first presentation I have had to make this year, so not only was it refreshing to get it out of the way, based on the fact that it was in my math class has made me even more confident and excited to be able to teach lessons in the future. This assignment also introduced me to lesson planning, albeit it was on a small scale, I value the experience greatly. While putting together my assignment, I noticed that both the curriculum documents and our text “Making Math Meaningful to Canadian Students, K-8”, obviously included what topics should be specifically introduced but also identified key mistakes and misconceptions students have when dealing with Ratio, percent and proportion. These key items the focused on helped e improve my lesson but also made me think that in addition to knowing the best way’s to teach students math, educators must also be aware of common mistakes and misconceptions students have in order to avoid them and strategize how to help students who are showing signs of struggling.

                My presentation went very smoothly, and I enjoyed my time in front of a class while teaching math. That is a sentence that I would have never thought would come out of my math just 2 short months ago. This week I made great steps towards becoming confident in my ability to teach math in the future. I look forward on continuing my education when it comes to math, but also telling you all about what I will be learning in the future as well!

Thanks for reading!

Often Less than 1, But it Doesn’t Fracture the Importance of Teaching Fractions

It is safe to say that fractions may be often over-looked. Early in their lives, students grasp the simple aspects of fractions; but as educator, we must illustrate to them that they have many functions. Quickly think about how many uses fractions have in our lives. They can represent ratios, measurements, area, volume and mass. With so many uses and applications, introducing students to fractions should be considered as one of the core operations that elementary students need in order to further their math learning later in life.

                Our textbook, Making Math Meaningful to Canadian Students , K-8 written by Marian Small,  does an unbelievable job at illustrating the various meanings fractions can present to students. It brings up the fact that eventually students must pull together the various “meanings of fractions” that a fractions are not just a number with a numerator and denominator, that they can represent part of a whole set, compare things ad be part of a continuous set.

                Students often start with learning very simple fractions, very often known as “unit fractions” that have a 1 as the denominator and any other number as the denominator. An example of a unit fracture can be seen in the image below. Unit fractions are always less than one. Fractions, like many aspects of math, have specific divisions and traits that stay constant no matter how complicated they may seem. These traits or definitions also include Proper fractions, characterized by still equaling less than 1, but have a number more than one at the numerator, e.g. 4/9 . There are also Improper Fractions, where the numerator I more than the denominator e.g.  8/6  or have a whole number and a proper fraction, e.g.  2 4/9 . Getting a grasp on the simple definitions and trait about fractions is the first step or students. It is the educator’s responsibility to build upon this, highlighting that fractions have many uses and exists everywhere. Like most math operations, more understanding about fractions is obtained with the use of manipulatives. 

JudeGRolfe., (February 13, 2013). A Unique, Unit Fraction. Retrieved from http://bit.ly/2cWKwHm

                  One of the most common manipulatives used when teaching how fractions can work is egg cartons. Not only are students already prone to saying “half a dozen”, students can also be shown how often fractions are used during activities like baking and cooking. Manipulatives such as egg cartons, and Pattern Blocks (depicted in the picture below), can be used by students in order to visualize how fractions and change and are able to b compared and altered. Physically touching the objects allows students of all ages to develop deeper understanding.

Jimmie., (August 1, 2009). Pattern Block Math Manipulatives. Retrieved from http://bit.ly/2dvvGye

Fractions can be simple, but they also can be very hard for students to grasp. Most of the errors that occur when dealing with fractions is when students start to add, subtract and multiply fractions. The use of manipulatives should not be limited to the younger grades as they are still useful when it comes to the more advanced operations of fraction operations. The list of what fractions can represent is almost endless, which is why teachers of math should see their lesson plans involving fractions as crucial classes because it can have a profound effect on their students ability to learn the topics that follows.