Using Appropriate Contexts to Teach Mathematics

Have you ever been in a math classroom where a student asks the teacher, "why do we need to learn this?" or "How is this going to help us in real life?" Often times these questions is asked by students when an abstract topic is being taught. Crafting questions and presenting difficult/abstract topics with contexts that connect the material to real life events could help engage students who are questioning why they should learn certain topics.

A great example was done in our class this week where we had to figure out how many classrooms a billion empty shopping bags would fit into. To help solve the problem we were given measuring tape, about 20 shopping bags and a container. Our group decided that we would fill the container given with as many shopping bags possible, then measure the dimensions (length, width and height) of both the container and the classroom. Once we found how many bags could fit within the small container we compared the volume of the container containing the bags to the volume of the classroom. A ratio was then done to show how many bags would fit in 1 classroom. Since 1 billion bags would not fit into just one classroom we found how many more classrooms would be required. It was found that about 225 classrooms would be required to fit 1 billion shopping bags. 

This question was interesting because it addressed a variety of topics such as:

• Measurement
• Calculation of Volume
• Environmental implications
• Ratios

Figure retrieved from: http://images.teamsugar.com/files/users/6/61259/33_2007/Carrier_bags.jpg

Students are able to use what they have learned within the classroom to solve problems such as these that provide a real-world context.

With that being said, the development of these types of questions may not always be easy. A question such as the one shown above requires the students within your class to be quite familiar with a variety of topics. Some classrooms may struggle with the complexity of problems such as the one above and need more guidance. Because every classroom is different, it is important to develop questions with context that can be solved by the students within your classroom based on the knowledge of your students assessed by the teacher. If you are looking for more real-world type math problems an excellent website to check out is http://www.realworldmath.org/ which has a variety of lessons for teachers to help teach certain topics with questions that give real-world contexts.

Figure retrieved from: http://www.realworldmath.org

Not all context questions require an activity to be performed though. There are many other ways in which teachers can accommodate context into their classroom. The first way is through a "hook" at the beginning of a lesson, A hook is used to generate student interest on a certain topic and can be done in different ways such as an activity, a video, a discussion, etc. Providing a context within that hook allows students to see why it is important to learn the topic that they are about to cover and also may increase classroom engagement.

Overall, providing a meaningful context to mathematical problems can help improve student engagement within the classroom but the way that context is taught should be based on the needs of your students.

What do you think about the importance of context within our classrooms?

Effective Uses of Technology Within the Classroom

New technologies are always emerging to make our lives easier. Many new technologies and gadgets such as Smartboards and graphing calculators have even entered our classrooms to improve the learning environment and classroom engagement. The use of technology within classrooms is also outlined within the Ontario Mathematics curriculum documents on page 8 it reads that,

"The prevalence in today’s society and classrooms of sophisticated yet easy-to-use calculators and computer software accounts in part for the inclusion of certain concepts and skills in this curriculum. The curriculum has been designed to integrate appropriate technologies into the learning and doing of mathematics, while equipping students with the manipulation skills necessary to understand other aspects of the mathematics that they are learning, to solve meaningful problems, and to continue to learn mathematics with success in the future. Technology is not used to replace skill acquisition; rather, it is treated as a learning tool that helps students explore concepts. Technology is required when its use represents either the only way or the most effective way to achieve an expectation."

 Note the final line highlighted in red font. Technology is actually required to be used when it is the best way to achieve a specific expectation within the classroom. For example, in an academic grade 9 mathematics class, many student's in the class may be struggling at understanding how the slope and y-intercept of a linear line can be used to generate an equation for that line. Platforms such as Desmos, an online tool used for graphing, can be used to engage students in graphing linear lines and also allows them to explore a variety of different linear lines by just simply inputting the equation of the line into the program.

Figure: Using Desmos to graph various linear lines.

The tool is easy and very quick for students to use. It will allow them to easily graph multiple lines and see how manipulating the equation changes the slope and y-intercept of the lines. 

Smartboards are another popular new technology present within many classrooms. Smartboards can be used to create interactive lessons that can help increase classroom engagement. The lessons can be created so that students can come up to the board and interact with various objects to solve problems, much like using manipulatives for factoring. 

Overall, technology should be used to help benefit the learning environment for your students. I believe it shouldn't be used to teach every expectation but should be used when it is the most effective way of teaching a certain topic.

Diagnostic Assessment

The Summer has just ended and the student's are coming back to school. Some students are excited and eager to learn while others are wishing the Summer would have never ended. A variety of different mindsets are entering the classroom and as the teacher it is your role to make sure that a positive learning environment is present for every student. One of the difficult tasks at the beginning of the school year for teachers, besides motivating students after a long holiday, is assessing the strengths of each student effectively. One of the ways this can be done is through diagnostic assessment, which is not always an easy task.

According to the document Growing Success, diagnostic assessment occurs before instruction begins so teacher's can determine student's readiness to learn new knowledge and skills, as well as obtain information about their interests and learning preferences. 

I remember back in high school each year when I returned back from Summer in my math class we would write a quiz to give feedback to the teacher about what we remembered from math last year. The issue with this type of diagnostic assessment is that sometimes students will forget certain topics if they haven't taken math in over a year (for example, last math they took was first semester of the previous year and they currently have math second semester of the current year) and they'll only need a reminder to refresh their brain on how to do certain topics. I believe a more effective method for diagnostic assessment would be to break students into random groups to allow them to work on questions together. This will allow the students to meet some new classmates and refresh their learning in a peer friendly environment. The teacher can walk around the classroom and assess the conversations being had about the math questions and assist groups in they are having troubles with certain problems. 

Figure retrieved from: http://media.oregonlive.com/portland_impact/photo/roosefreshenglishjpg-e31ff160773d042d.jpg

Diagnostic assessment will help teachers get a better understanding of the strengths and weaknesses of their students so that they can prepare their lessons and assessment tools accordingly as the school year progresses. 

Open Questions and Differentiation

One of the many goals in mathematics is to help students develop their critical thinking and problem solving skills. Open questioning is a method that can be used by teachers to engage a students inquiry. According to the asking effective questions document from the Ontario Ministry of Education, open questions are effective in supporting learning because they encourage a variety of approaches and responses. Open questions also intrinsically allow for differentiation, as the student responses to the questions will reveal their individual understanding and knowledge of the topic the question is based from. What are the differences between an open question and a traditional question though?

To use arithmetic as an example, a traditional question would be something along the lines of:

What does 5 x 6 = ? 

Students can do their procedural multiplication technique or use a calculator to easily find that the answer is 30 and then move on to the same type of questions using the same procedures over and over.

An example of an open question using this same topic would be something along the lines of:

Using multiplication, find two or more numbers that can multiply to 30.

This question has various answers and requires students to think about how multiplication works and could also even challenge their algebra and division skills as well. An example of some solutions to this open question could include:

1) 5 x 6 = 30
2) 2 x 5 x 3 = 30
3) -15 x -2 = 30

Notice the various complexity of the answers shown here. Answer 1) uses very basic multiplication to find the solution but still shows understanding of how to use multiplication. Answer 2) is a little more complex requiring students to think a little bit about their algebra skills. Lastly, answer 3) not only shows correct multiplication but also shows that two negative numbers multiply together to make a positive number, combining two different topics into one to solve the answer showing critical thinking.

Open questions can even be brought to the high school level and used for more complex problems. For example:

Create a linear function that passes through the y-intercept at the point (0,4). 
Students can come up with a variety of answers:

Figure: Different linear function solutions to an open question.

For this type of question students need to understand that the y-intercept component of their equation will not change, but the slope can be any number. This allows students to explore linear functions and see how manipulating the equation effects the graph of each function.

Overall I believe open questions are an excellent tool for testing student inquiry while also providing differentiating learning which provides great feedback for the teacher on how well students are understanding certain topics.

The Use of Manipulatives Within the Classroom

Manipulatives are excellent tools that can be used within the classroom to help students understand abstract mathematical processes such as factoring. What are manipulatives though? Manipulatives are any type of object that a student can touch or move to help them learn a type of mathematical concept.

As mentioned earlier, one of the concepts that manipulatives can be used for is factoring. Factoring is a common mathematical process used to break down polynomials. When I was first introduced to factoring I was taught the many steps at how to factor a polynomial. For example, lets use:

To factor this equation a student would have to:

1) Find two numbers that multiply together to equal 9 but also add together to equal 6.

2) After figuring out that 3 and 3 both add together to 6 and multiply together to 9 the equation would have to be rewritten to:

3) The equation can then be broken apart where the student must know exponent rules and how the exponent rules work with variables as well:

4) Lastly the equation can be rewritten as a binomial:

After all of those steps this is the final outcome. For some students this can be easy if you follow the procedure and get your correct factors for step 1, but this is not simple for all students. As an alternative to the stepwise method of finding common factors, manipulatives can be used to acquire the same answer except visually. 

Some manipulatives that can be used to help with factoring are algebra tiles. In our class this week we used the 3 different sized algebra tiles to represent the variables and the two different coloured tiles to represent positive and negative.

 

Figure: Three different algebra tiles used as manipulatives to help with factoring equations.

Now lets try to factor that same equation using these algebra tiles as manipulatives. In order to do this properly the tiles need to be arranged within the box that they are stored in to help with the organization of terms.

 

Figure: Equation represented using algebra tiles.

The algebra tiles are arranged in a square (or sometimes rectangle) to represent the equation that is being factored. Now finding the factors for this equation is very simple once represented in this fashion! To find the factors you only use the "x" tiles and the "1" tiles along the borders of the box. 

Figure: Equation factored using algebra tiles.

The "x" tiles are placed adjacent to the xsquared tile and the number tiles are placed adjacent to the "x" tiles that are within the box. At the top border you have 1 "x" tile plus 3 "1" tiles representing x+3 and the same on the other border. These two equations are your factors for the overall equation!

A benefit to using manipulatives for topics such as factoring is that it allows students to apply the knowledge they learn in a fun way instead of being lectured by the teacher and just doing homework problems over and over again. I think this would be a much better way to help teach factoring to students!

"Perspective"

One of my fears as a future mathematics teacher is presenting my knowledge in a student friendly language that will help them best understand the content being taught. The reason why this is challenging is due to what I would call "mathematical perspective."

What is this "mathematical perspective" that I speak of? Well, in class this week we did an exercise that involved three rounds of a Tug of War between various creatures. The goal of the exercise was to figure out who would win the final Tug of War based on the outcome of the first two. 

Figure retrieved from: http://www.fotosearch.com/illustration/tug-war.html

The first Tug of War was a tie between 4 frogs and 5 fairy godmothers. The second Tug of War was also a tie between a 1 dragon on one side and 2 fairy god mothers and 1 frog on the other side. The final round had 1 dragon and 3 fairy godmothers on one side and 4 frogs on the other side, but the outcome was unknown. 

Like many mathematics problems, there is only one solution to this Tug of War but, there are multiple ways one could reach that solution (the dragon and fairy godmothers won). These multiple routes to the solution are based on your "mathematical perspective" or how you personally would solve a mathematical problem. 

, 

When our class of nine students was given this problem there were about three different methods to solve this problem, all giving the correct answer. In a high school classroom, this same outcome will likely occur where many different students will have various perspectives at solving mathematics problems. This activity showed us how many students even with the same academic background can look at mathematical problems from different perspectives.

As a future mathematics teacher, I would like to become more familiar with methods that allow me to cater to the various different mathematical perspectives that will be present within my classroom. One of the most interesting and frightening things to me when becoming a teacher is how every student learns differently, but I would like to do the best I can to help each student achieve their full potential.

Ryan

Introduction

My name is Ryan Alt and I am currently in my fifth year at Brock University in the I/S stream of Concurrent Education. My first teachable is chemistry and my second teachable is mathematics. 

I have always had an interest in mathematics and how mathematics can be used to describe so many different things in the universe. While growing up and going through the school system, mathematics was not always my strongest subject though. Along the way I hit a few road blocks, but those road blocks did not stop me from learning and enjoying mathematics. My goal as a future educator is to use my passion for mathematics to help students avoid the road blocks that I encountered and also help them gain a better understanding of the subject. 

Future posts on this blog will be reflections used to engage my learning and create new ideas with my colleagues at Brock University who are also aiming to become future mathematics teachers. This year I hope to learn how to effectively communicate my passion for mathematics through creative lessons and activities to give my students the best learning experience possible. 

This will be an exciting experience to share with everyone. I hope you enjoyed reading.

Ryan