Lesson Play: A hypothetical scenario

The following is our lesson pay for the Minds On component of our lesson plan.

Teacher: Good morning students! How is everyone doing this morning? Brandon you watch the baseball game last night?

Student: Brandon → yeah kinda.

Teacher: Anyway, Has anyone ever noticed that certain things have connections? What two things can you name that are connected? Like hair colour, genetics, age and hair colour etc?  … Yes Emily?

Student: Emily → ummm so the older you get the grayer your hair?

Teacher: Yes exactly Emily! Those are two connections. Anyone else? Marcus?

Student: Marcus → My brother and I have both have brown hair. And both my parents have brown hair as well? So is that a connection??

Teacher:  Yeah that is a connection. Since both your parents have dark hair you are more likely to be born with dark hair. You will learn this in biology more, about recessive and dominant genes. But that is most likely the case. But there are simpler examples with two things that have a connection. Say we do a pop quiz right now. The people who study their notes daily, do their homework and are paying attention in class would probably do better than the people who did not do those things. Do you guys agree?

Student: John → Yeah that makes sense. So the quiz and the student are the connection?

Teacher: Yeah basically. So the more prepared you are for the quiz the better you will do. So it’s not necessarily the student and the quiz, we are looking at how well a student prepares for a quiz and the results of the quiz. So let’s say Gerald studies every night for 2 hours, reviewing his notes, while Brandon has a nap everyday after school. So pretending Brandon isn’t a mathematician, he is more likely to do worse on the quiz than Gerald.

Teacher: What is the importance of knowing connections between two things? Like being prepared for a quiz and doing good on a test? Carol do you have any ideas?

Student: Carol → ummmmm. I guess then people know for next time that the more you study the better you will do on a quiz.

Student: Jermaine → (Interrupts abruptly) Sir i feel this is self explanatory. Like everyone knows that the more you study the better you’ll do. It’s common sense.

Teacher: Thank you Carol. And Jermain you are right as well. In this specific example it kind of is self explanatory. But how do you know what you are saying is true? It might be common sense to you, but how would you prove this?

Student: Jermaine → What do you mean? Its pretty obvious?

Teacher: Well we need to convince the whole class that what you are saying is true. How can we do this? How can you showcase this?

Student: Jermaine→ Oh, umm, I guess I would see who studies  and who doesn’t.

Teacher: Yeah so we would take a “sample” from our class in order to gather data. So say we have all the data we need, we have the exact hours of study time per student and the marks they receive on the quiz, but what would we do with this now? If we want to find out if there is a connection with study time and success on quizzes, how would we do this?

Student: John→ We can make a graph, and see if there is an increase in mark when students study more. I think if we had a picture of the data it would make more sense to more people.

Teacher: Exactly! Jermaine do you see what I mean now? If it doesn’t make sense to everyone, graphing the relationship between two variables can make it easier for everyone to understand a connection between two variables such as quiz marks and study time.

Student: Jermaine→ (nods in agreement)

Technology & Games in the Classroom: Adapting Learning to the Modern Era

https://elearningindustry.com/byod-training-professionals-need-know

Hello everyone and welcome back to my blog! This week I would like to talk about something that has always caught my interest, technology and gaming in the classroom. Considering we are all twenty first century learners and are going to be teaching in the "digital age", the topic of technology in the classroom is extremely important. School boards have already adapted fairly quickly from altering their policies of no cellphones and technology allowed from when I was in school, to an open door policy of BYOD, bring your own device, which is an extreme turn around of previous policies.


It's great that school boards are advocating students to use tools that they consistently operate on a daily basis, however, it is important to keep students engaged and on track now that they have more freedom in the classroom. Depending on internet that has firewalls and blocks social media and other websites is not reliable, as my experiences volunteering in a high school quickly taught me that students just download free apps such as "Free VPN", which changes the users internet settings temporarily and allows users to get past the firewalls. Furthermore a recent article by the telegraph describes how humans have a shorter attention span thanks to the technological age, however, are able to multi task more. The article reveals how the average human attention span has fallen from 12 seconds in the year 2000 to about 8 seconds now, reinforcing the stronger need to keep students engaged in the classroom. How might teachers keep students engaged in the classroom you might ask? Its simple... by allowing students and kids do activities they like to do. Playing games.

http://www.home-speech-home.com/speech-therapy-activities-using-hedbanz.html

This week in my education class, we had the pleasure of playing several games that incorporated mathematics. The first game we played was HedBanz. HedBanz is a simple game that works around answering yes or no questions. Students are given a "headband" with a mathematical problem on it. Without looking at the problem on the card, they have to place it on their head and partner up with someone else in the class. Asking yes or no questions to their partner, they have to figure out what the problem was. In order to play this game, the students need to have a good understanding of mathematical relationships and processes, in order to get to the answer on their head. For instance in our class all the questions related to quadratic equations, and so students were asking if it was written in factored form, what degree the equation was in, if there was a vertical compression or not, and much more. In this light this is a great game for a formative assessment that keeps the students engaged. To get a better understanding of the game, watch the video from the famous Quentin Tarantino and Brad Pitt movie, Inglorious Bastards. 

http://www.askmen.com/entertainment/galleries/top-50-all-time-tarantino-characters-16.html

Although this game required students to have a lot of previous knowledge about the subject, another interactive game students can learn that is not as dependent on previous knowledge is on the mathematical app/website Desmos. Desmos is an interactive graphing calculator that students can use to graph simple and complex graphs and get a visual understanding of the subject matter. Along side Desmos is the game "Polygraph". Polygraph works similarly to the game "Guess Who", where students select a graph among 16 options and their partner has to ask yes or no questions in order to eliminate the options he or she deems unfit.

https://mathcoachblog.com/2014/12/21/class-opener-day-74-polygraphs/

This makes it a little easier for the students compared to the HedBanz game because it gives an idea of the questions students should ask rather than completely coming up with questions out of thin air. Looking at the picture on the right, you can see students have a selection of graphs that their opponent has chosen. In both games the teacher can assess student knowledge by analyzing the questions they ask. The game works by trying to get to the answer in the fewest amount of questions. In our class, the quickest solution took a student body of future math educators four questions to get to the answer.

In the 21st century, we need to cater to our students' needs. This mean incorporating games into the classroom and keeping up to date with what keeps students interested. I look forward to using the above games and much more into the classroom in the future, and for people who do not see the benefit of incorporating games into the classroom, I strongly encourage them to see the benefits of gaming in an educational context like James Paul Gee advocates below.

Learning Where Your Students Stand: Diagnostic Assessment

Last week I discussed differentiated instruction and why it is a crucial element in every classroom. Every student is unique and therefore learns in different ways in comparison to other classmates and the teacher. It is important that teachers address each individual student's needs, and develop a lesson in a way that allows each person a fair opportunity to grasp the content. Unfortunately, this is not always the case as teachers are restricted in time and may not be able to cater to all thirty people in their classroom before moving on to the next lesson. As a result, students who are are doing well in class continuously advance while the students who are struggling to keep up continue to decline. Eventually, there is a gap between the students in the former and latter group that continues to grow and grow the older they get. 

www.ontariomathedresources.ca

As teachers, our duty is to be concsious that there is a learning gap present in all classrooms, and actively work to close the gap. In order to realize where the gap is, teachers need to conduct a diagnostic assessment to understand student's prior knowledge. One avenue which teachers can use to address what areas students need help with, is provided by the government of Ontario and referred to as "gap closing". Gap closing resources are designed for students who need additional support in mathematics. For each topic in the resource, there is a diagnostic and a set of intervention materials that will help teachers uncover typical problems students have with a specific topic. The resource can be found by clicking here, and there are nine modules that cover a wide arrange of mathematical topics. 

After reviewing the student booklet and the teacher resource associated with each module, I can clearly see how this is a diagnostic assessment. It covers various areas throughout the grade levels that would showcase where students are struggling. Although this is great, I do not see where the actual "gap closing" would take place. All the questions in the booklet followed a written test format. In my head this seemed quite ironic because this is probably the number one reason why the students are falling behind. We learn in teacher education how all students are different and learn differently, so I was under the assumption that therefore the diagnostic assessment would allow different avenues for students to demonstrate their knowledge. 

Instead, the booklet looked like a written test or just a series of worksheets no different than how they probably learnt. There was evidence of some sort of open questions, but even as a future math educator I could see how these tasks would be difficult for students to complete. When I first heard of gap closing, it was with a colleague of mine who discussed how she used this booklet at her job with the Hamilton-Wentworth District School Board as a "Tutor in the Classroom". She expressed how her duty was to use these booklets in order to close the gap, but it did not seem to work with many of the students she was working with.

So then how do we close the gap? Well although there may not be a clear-cut answer, the best path to take is to provide different avenues for students to showcase their math knowledge. Instead of getting students just to write out the answers or respond to a series of worksheets in a booklet, students can have oral conversations with teachers, construct mind maps and other informal techniques that still engage the student and allow teachers to assess what they have learnt. If teachers are expected to instruct students in different ways, then teachers need to accept various different forms of student work so that there is equity in the classroom.                                                                                                        

Most students have anxiety when they feel like they are being evaluated and tested. Considering this is a diagnostic assessment and the teacher is just trying to figure out where the student stands before the lesson takes place, the teacher should consider using informal techniques in assessing student knowledge. There are mathematical games students can play, quizzes students can take such as Kahoot, completing interactive mathematical website activites such as Khan Academy, and many more options that teachers can use. 

Research by John Holloway showcases that teachers who use assessment that supports learning and provides useful information to both teachers and students, in addition to technology that influences the mathematics taught, are the biggest factors in improving equity in the classroom and aid in closing the gap. Using the same sort of formal testing and written evaluations to close the gap is not an efficient method in providing an equitable mathematics classroom. It is my hope that in the near future there will be other forms of assessing students prior knowledge, and if you are aware of other methods please leave a comment so I can enhance my future classroom and help students who are in need. It is my goal to create an equitable classroom and close the gap as much as I can. 

References:

Picture 1: ontariomathedresource.ca 

Picture 2: metrokids.com

Picture 3: http://elementary-math-and-science-differentiation.wikia.com/wiki/Lesson_Plan_Outline_for_Differentiation

Holloway, John H. "Research Link/ Closing the Minority Achievement Gap in Math." Improving Achievement in Math and Science 51, no. 5 (February 2004): 84-86

Differentiated Instruction in the Math Classroom

Hello everyone, and welcome back to my blog! 

This week I would like to talk about a very important concept in any classroom, differentiated instruction. Now when I first heard the term differentiated instruction, I thought it had to be very specific and cater to each individual child's learning needs. While this is part of the solution, this is not entirely true. When I entered teachers college and first began discussing differentiated instruction in my cohort, our instructor explained how differentiated instruction is not changing the lesson for each specific student, but catering to every student, so it could be as simple as talking slowly and clearly, enlarging the fonts, and other small adjustments. A better way to understand differentiated instruction is provided by edugains.ca by Our Lady of Lourdes Catholic Secondary School. They explain, 

"Differentiated instruction is not individualized instruction; it involves considering and selecting from a variety of instructional approaches and making frequent use of flexible, short-term groups to address a variety of learner needs and preferences." 

In relation to this definition, a great instructional approach that teachers can use in their classroom to identify the variety of learner needs and preferences is the traffic light system. A detailed video about the traffic light system in the math classroom is demonstrated in the St. Mary's SS video. 

The traffic light system is extremely beneficial in the classroom because it allows the students to illustrate how comfortable they are with the lesson, and therefore lets the teacher know who needs further assistance as well as who is comfortable with the lesson that was just taught. This allows the teacher to better manage their classroom because the students who have learned the lesson can do other activities, while the teacher can form short-term groups and further assist the students who identify themselves as yellow and red with a different method of instruction. 

Another important factor of differentiated instruction is providing student choice. One method teachers can use in the classroom that gives student choice is by implementing open-tasked questions. There are two types of open-tasked questions. The first is open-ended, which means there are a variety of strategies a student can use and multiple correct answers. The second type is open-routed, which means that there are multiple strategies a student can use, but only one correct answer. Both are important in differentiated instruction because open tasks are inclusive of all students' mathematical thinking and its relative to students' zone of proximal development. This means that no matter what stage the learner is at in the lesson, they are working in their own comfort zone and the teacher can assess their mathematical reasoning and big ideas which are part of the mathematical processes. 

Retrieved from: The Literacy and Numeracy Secretariat;
Capacity Building Series

In the end what is most important is the fact that all students receive and equitable treatment in the classroom. This means that teachers need to address the need's of all the students in the classroom and use techniques that will help all the students in the class, not just the majority of the students. It is the teacher's duty to make sure that every student gets a fair chance to learn, and this is why differentiated instruction is instrumental in the classroom and something all teachers need to take into consideration.

References:

Knowing and Responding to Learners in Mathematics.2015. Booklet

Capacity Building Series. The Literacy and Numeracy Secretariat. September 2008.  

Instrumental and Relational Understanding

This week in my teaching journey, my University math class uncovered two different perspectives of math that I never thought about before. We looked at an article by Richard Skemp that assessed math in the twentieth century, however, what he concluded in his paper is still extremely relevant today. In his article, Skemp addresses the idea of two different types of understanding, instrumental and relational.

https://alearningplace.com.au/2-ways-to-teach-and-learn-maths-2/

Now if you are like me, you may be familiar with instrumental understanding, but not that accustomed to relational. Instrumental understanding means a child knows a specific rule or procedure and has the ability to use it. In other words, the student knows how to follow instructions and specific steps to get to an answer. Skemp insists instrumental understanding are "...rules without reasons" (90). Take dividing fractions for exmaple, in order to get the right answers, students blindly follow steps not knowing why they are doing it. 

https://buildingmathematicians.wordpress.com/tag/nixthetricks/

https://alearningplace.com.au/2-ways-to-teach-and-learn-maths-2/

 Relational understanding is when the student knows exactly why they are doing those steps. In essence, instrumental understanding is part of relational understanding, however, the key fact here is that the student knows why they are doing it. This is important because relational understanding allows students to tackle questions that are out of the ordinary or that look unfamiliar. Relational understanding helps facilitate mathematical reasoning. With instrumental understanding, students are just trying to get to the answer, and if the teacher asks a question that does not quite fit the rules they have learnt, the student will get it wrong. (Skemp 90).

A notable analogy that Skemp uses to understand the difference between instrumental and relational understanding is taking a walk in a park. Imagine you are in a park, and you want to get from point A to point B. You learn from someone else a certain path to take and you get to the destination fairly quickly. Eventually, you add more points to your locations you want to visit and you know how to get there. Step off any of the known paths, and you are quickly lost and can even develop a fear of losing your way (in this case math anxiety). You never really develop an overall understanding of the park, and you may not know about other connections between points that might be quicker. This is instrumental understanding. 

Now instead of walking around the park through specific paths, you get to wander all over. For some parts of the park you are guided, for others, you walk around aimlessly. In time you get an overall picture of the park and this allows you to figure out a shorter path than you are accustomed to, or how one point in the park is related to the other. If someone showed you a short-cut, you would understand why it worked and why it was faster than the path you took before. Now you wouldn't be afraid of walking off the path, because even if you did, you can find your way back easily. This is relational understanding. 

http://www.gocomics.com/calvinandhobbes/2011/03/09

The above comic highlights the importance of relational understanding. Teachers need to allow students to 'wander around the park aimlessly' in order to tackle math anxiety and better prepare their students to tackle diverse problems. The real question is how do we as teachers facilitate relational understanding in our students, and how do we assess if they understand a situation relationally or instrumentally? Furthermore, for students who do not plan on having math as a primary factor in their future career, would instrumental understanding be better suited for them? Maybe for the future educators and policy makers should analyze the high school math streams and possibly advocate relational understanding and instrumental understanding based on their future careers. 

Skemp, Richard. "Relational Understanding and Instrumental Understanding." Mathematics Teaching in Middle School 12, no. 2, 88-95.

Debunking the "Math Person"

In my Introductory post I brought up a famous quote by John Dewey, "If we teach today's students as we taught yesterday's, we rob them of tomorrow".  One of the most common misconceptions about mathematics, which teachers seem to believe, is the notion of a "math" person.

Retrieved from: https://www.youtube.com/watch?v=Ukt4A5GCfQU

As Professor Jo Boaler from Stanford University explains in the video above, no one is born with a "math" brain but we all have the potential to grow and change our brain all the time. This notion is very important because from my experiences tutoring high school students in math, one of the most common statements I get from pupils is the fact that they are not a "math person" and therefore need help in math. This statement speaks to me because as we have just learnt, there is no such thing as a math person. When the students say they are not a math person, they are setting themselves up for failure. They are already creating an excuse for failure right from the beginning because according to them it is not in their nature to be a successful math person. In the education field, we call this type of thinking a fixed mindset. 

Retrieved from: http://carriekepple.com/2015/04/24/growth-
mindset-vs-fixed-mindset-which-do-you-have/

In order to fix this misconstrued idea of a "math person", teachers need to teach students the difference between a growth and a fixed mindset. If the teacher can get the students to change they way they think from a fixed mindset to a growth mindset, not only will the student be successful in math, but they will also be able apply this problem-solving in real world situations. In addition, by getting students to start thinking with a growth mindset, the teachers are getting students to use mathematical processes that are part of the Ontario curriculum such as problem-solving and reasoning. 

Several techniques teachers can use in the classroom to change the mindsets of their students from a fixed mindset to a growth mindset can be found on mindsetonline.com. Furthermore, a study conducted by Mercer and Ryan (2010) explains how teachers play a big role in fostering a growth mindset in students. The way teachers provide feedback is crucial due to the fact that it may promote a fixed mindset if the teacher focusses on grades, or it could promote a growth mindset if the teacher pays special attention to the learning process, belief about developing the student's ability through hard work, and commenting about the learner's efforts during the feedback (p. 442). 

In the end by moving away from the idea of a "math person", or in general one particular subject persons, teachers are enabling their students to broaden their abilities, while at the same time increasing confidence in the classroom. The teacher is providing a positive learning environment and more importantly, creating an atmosphere in which all students can thrive and enjoy math, rather than being disengaged and feeling discouraged.

References: 

Mercer, S., & Ryan, S. (2010). A minsdet for EFL: learners' beliefs about the role of natural talent. ELT Journal, (64)4, 436-444.

Welcome To My Blog!

Hello everyone and welcome to my blog!

Before I begin my educational posts and reflections about math, I thought it would be better suited if you all got to know a little bit about me. My name is Bevan Fernandes and I am currently a teacher candidate at Brock University. I hope to become a secondary school teacher with teachables in Math and History, which most people find odd considering the fact that they are not complimentary in the conventional school system. However, I feel like there are many ways that you can connect the strands between the two subjects, for instance, the process you take to tackle new problems in both subjects. It is my goal to get students to find an appreciation for both math and history, two subjects that most students traditionally find disengaging.

Retrieved From: https://editorial.rottentomatoes.com/article/get-
down-with-the-get-down-15-things-to-know-about-netflixs-newest-drama/

Outside of my aspirations to become a teacher, I enjoy playing various sports such as soccer and volleyball, and I have a growing interest in music and musical instruments. Lately, I have been more and more interested in different types of music such as jazz, 90s hiphop and music throughout the 60's to the present day, which has allowed me to connect with multiple students on different occasions during my volunteer experiences. This idea influenced the title of my blog, since it not only brings the idea of the basic necessities in mathematics, but also the popular culture reference to music and Djing, a growing industry in the twenty-first century and among highschool students.

The purpose of this blog is so I can discuss and improve key insights in the mathematics field, hopefully collaborating with my peers on future posts in order to find better methods and techniques in teaching math in the future. A notable quote comes from John Dewey who says:

"If we teach today's students as we taught yesterday's, we rob them of tomorrow".

Retrieved From:http://www.nigeriancuriosity.com/2009/06/chaos-
status-quo-and-nations-future.html

My goal is to challenge the traditional and conventional ways of teaching math, and more importantly reflect on how I can be an advocate for change and reform in the classroom. I feel that the quote by Dewey encompasses this idea as it hints towards the notion that we as educators need to take heed to the constant developments in education. The best way to do this is to reflect on the current practices in math and work together as a community of educators, in order to change the traditional system so that we can create a more inclusive learning environment that helps all students achieve their full potential.