Expanding Your Curriculum Knowledge Beyond High School

Secondary school teachers are often well versed in understanding the components of the grade 9-12 mathematics curriculum's and how to develop lessons that address each of the curriculum requirements. One component that is often overlooked by high school teachers, especially those teaching grade 9 mathematics classes, is delving into the elementary mathematics curriculum. This gives teachers an understanding of where our students are coming from and an idea of the knowledge that they have likely already obtained.

It is important for teachers to understand the knowledge basis from which our students are coming from. This allows us to develop effective lessons that will scaffold on the previous knowledge that the students have already obtained in elementary level mathematics.

During assessment it could also be noted that students may be struggling with a certain concept. This could be due to the lack of prerequisite knowledge in a certain topic from elementary which is hindering students progress in learning new material. Although not part of the high school curriculum it is your role as a teacher to ensure each student is achieving their individual successes, which may require you giving a brief lesson on elementary level topics to refresh student's minds on certain topics.

In our class this week we were given curriculum expectations for grades 7-9 and tried to develop continuum's that would link concept ideas across each grade. This proved to be a challenging task as not all of the topics made direct correlations, especially when going from grade 8 to 9. This showed how big the gap was between content in grade 9 and and elementary, which is why it is important that teachers teaching that grade should become familiar with the elementary level content.

Continnum chart linking curriculum expectations from grades 7-9.

There is also a variety of resources available on Edugains that link curriculum expectations across the grades.

First Teaching Block

For my first teaching block I was teaching the light and geometric optics unit of a grade 10 academic science class. During the practicum I learned many new skills and strategies that could be applied to any type of classroom. In this post I will talk about some of the experiences I had while on my teaching block as well as some of the challenges and difficulties that I had to overcome.

This practicum was the first time I had taught in a classroom setting. I had some tutoring and coaching experience but had never run a classroom before. One of the first challenges I encountered while teaching was pacing. Due to being in a university style setting for the past 4 years, my pacing was a little bit too fast at the beginning of block but as I gained more experience the pacing became much better and was definitely one of my biggest improvements.

Being organized was extremely important during the first block. I learned many strategies that I would use in a future classroom to help keep everything organized within the classroom. One of the key organizational strategies that my teacher used was a binder that was always within the classroom that contained extra notes/worksheets. It was the students responsibility if they missed a lesson or multiple lessons to pick-up the notes/worksheets they are missing from the binder which was organized by date of the notes and divided by each class as well. This made it much easier for putting student's who had missed days back on track.

Having an academic class, classroom management and behaviour was not a huge issue for me. Some students would be a little rowdy at times but I found that proximity during lessons and changing pace often would quiet them down.

Although I wasn't teaching a mathematics course there was still some math involved in the optics unit. The mirror equation shown above was one of the equations we would use in the class to find the distance of an image produced in a mirror. It was interesting to see how all of the students had different math capabilities within the class. I broke down each question step by step and included the class when solving problems to help cater to all of the different leveled math learners within the class. We also did a lab which had a component that applied the equation which the students found useful!

Image retrieved from: http://previews.123rf.com/images/alexraths/alexraths1209/alexraths120900001/15134543-Teacher-with-a-group-of-high-school-students-in-classroom-Stock-Photo.jpg

Overall I found my first block to be very successful. I learned many useful skills that I could use in any future classroom and gained the valuable experience of teaching in a real classroom. Can't wait until my next practicum!

Online Session Reflections

Online Session 1

In the first part of the online session we looked at Five Practices for facilitating mathematical discussions around cognitively demanding tasks within the classroom. As a beginning teacher I found that some of these tasks may be a bit challenging for me such as anticipating likely student responses. Because we are all different learners there are students in my class that will likely think of different ways to solve mathematical problems I present that I did not think about. This is perfectly acceptable but as a beginning teacher I may find it more difficult to anticipate all of the student responses to questions due to my lack of experience as a teacher. With more experience though and getting to know my students better it will make it easier to anticipate individual student's responses to questions.

I find the Five Practices beneficial for the classroom because they provide a student-centered approach to learning that creates a more dynamic and engaging classroom. For example, one of the practices is selecting students to present their mathematical responses. This can be beneficial because it shows student's how their peers solve mathematical problems and in an ideal classroom where there is no judging other students you can also occasionally select students with a common error in their formulation of their solution to the problem which can generate discussion as well as enlighten other students who had made the same mistake.

The second part of the first online module challenged us to think outside the box by solving problems using different approaches that students may use. This exercise was useful because it challenged me to use different approaches to solving a problem then what I would normally be used to which is good preparation for when I will have to anticipate student's responses to math problems while teaching.

Online Session 2

Formative assessment is important tool used to determine the level of understanding students have on certain topics/concepts/ideas. There are different ways that you can approach using formative assessment but there are five key universal steps that can be used to help achieve effective formative assessment. One of these steps is sharing learning goals with the students. This allows the students to have an idea of what they should learn at the end of a lesson so that they can asked themselves "did I achieve the learning goal today?" Another very important step is providing feedback that moves learning forward. Feedback can be provided in written, oral or through a demonstration and should force students to engage cognitively about their own work.

Written feedback to a student should provide a question to the student rather than just a "correct" or "incorrect." You should provide the student with a reasoning for why their answer was good or incorrect that allows them to think about other options or methods. For example in a substitution/elimination problem that a student solves using substitution only you may write down as feedback "great answer! can you think of another method that you can use to solve the problem?"

The teacher should also guide student's using feedback rather than providing them with the correct answer right away. This allows students to think back to the problem rather than just seeing the correct answer and moving on.


Another important thing with feedback that I believe is very important is providing enough time for students to read and think about their feedback. You can provide students with feedback on an assignment or test and tell them to read the feedback and they may or may not do so. Providing time in class to look at the feedback makes it more likely that student's will look at their feedback and cognitively engage in it.

Overall I believe feedback is important tool for all students whether they are struggling or excelling at a topic it can be beneficial for both those types of students if used effectively and this module was a great way of thinking about feedback and how it can be used in a math classroom effectively.

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.