Let’s Congress

We explored a unique way of fostering student’s thinking by exploring the technique known as a math congress. The congress was a combination of last week’s kitten problem and then using the solutions in what is called a math congress (developed by Dr. Cathy Fosnot). I really liked this method as a congress allows for students to be engaged in the math problem, and in trying to figure out “a” solution, which is not necessarily the “only” solution. We began with the kitten problem of comparing 12 cans for $15 at Bob’s store versus 20 cans for $23 at Maria’s store. Then we began with thinking and reflecting on our own to figure out a solution, followed by sharing with our elbow partners to figure out a solution. As a group of four, we had to collectively decide on one way of solving a problem. This allowed for each of us to collaborate by sharing our ideas, thoughts and perspectives. This great technique used in classrooms is known as, “think, pair, share.” These solutions were then saved to be used in the class to conduct a class discussion the following week.

Once each group had a solution, this is when the math congress came to life. Each group joins another group to form a mini congress, where two students facilitate and guide the discussion. Each student uses their respective solutions to share their groups thinking to the rest of the congress. This goes on till all the solutions at the congress are understood and the thinking of each group has been made visible. The teacher also carefully picks which groups should work together to display various samples of thinking and problem solving, allowing for a rich understanding from various perspectives. One solution that really stood out to me was one that was presented in another class. Students developed a strategy to assign a dollar value to each can and compare which deal is better using both pictures and numbers. Although I did not think in this way, I really like their strategy of figuring out a unit price to compare. Overall it was great to see all the brilliant strategies that can be discovered if math focused more on the depth than breadth of a student’s understanding.  Once our thoughts, ideas and perspectives were shared, we did a gallery walk to see all the other solutions. Once again, it was incredibly fascinating to see all the various ways of presenting the solution to just one problem.

Finding unit price of each can at each store
(Khalid, N © 2016)

Numerate vs. Innumerate:

In the following video Dr. Cathy Fosnot discusses the kitten problem and shares solutions that students in a math class discovered and presented. At the very end of the video, she explains that numeracy is when an individual looks to numbers first and comes up with an, “elegant, efficient strategy given those numbers.” While, “someone who is innumerate uses the same strategy for all problems no matter what the numbers are.”  I have to agree with her, although like I have mentioned previously, it is hard to think differently when various algorithms have been drilled into our heads. These new techniques make it apparent the need for change in the way we approach and teach mathematics. 

Cathy Fosnot - What is numeracy? from LearnTeachLead (1) on Vimeo.

Fun with Fractions

This week we continued exploring other methods of engaging students when teaching fractions.

Red Light, Green Light:

A stimulating technique is to use games to teach fractions. This activity is played by having the teacher at the finish line and a group of students at the starting line. It is green light when the teacher at the starting line is looking away from the students and red light when the teacher faces the students. If any of the players are caught moving while teacher is facing them they have to return back to the starting line. At the end of the game, each player is a certain distance from the teacher. They now have to communicate in fractions, describing the distance they are from the teacher. I found this game fun and engaging and I could see this being used at my placement.  The teacher at my placement has allotted ten minute energizers where she allows the students to play an outdoor activity. Depending on the weather this activity could work really well. Since this grade 6 class is taught in a portable it allows for the students to easily enter and exit the class without wasting much time.  

When describing the distance, students would be encouraged to use a variety of descriptive terminology to explain how far from the teacher each of them are. This is known as using soft math terms such as; close to, far from and about half-way, just to list a few. When students are comfortable with this terminology, they can be introduced to fractions such as 1/2, 1/4, 8/9, and 2/47. Students can then be given a number line to see if they understand where each fraction would be placed. They can also work in groups to share ideas of why certain fractions fit in certain places on the number line. This allows students to communicate and collaborate with other students and gets them comfortable with using fractions and math terminology.

Mr. Tan’s Tile:

Another engaging way to develop a student’s understanding of fractions is through the use of stories. Many students enjoy listening to stories especially the junior grade. Incorporating stories into instruction captivates the attention of the students and helps in processing information.  A story can be constructed to get students to help Mr. Tan put all the broken pieces back together to create a complete tile. Students can relate each of the pieces to represent a fraction or a portion of the entire tile. I think this is a great introduction to fractions and proportions. Students can also have fun by using the tiles to create various other images.

I created a house with a chimney (Khalid, N © 2016)

Each tile of the tangram representing a fraction
of the whole tile (Khalid, N © 2016)

Making Fractions Meaningful:

Dividing fractions same
as multiplying (Khalid, N © 2016)

This week in class we learned the meaningful way of dividing and it was as simple as using the same technique of multiplying fractions. The numerators are divided and then the denominators to give an answer. If the denominators cannot be divided evenly, we can find a common denominator. This basic way of dividing fractions is easier and similar to multiplying fractions. When dividing fractions, I never really understood why we used the technique of inverting and multiplying when we could skip one step and just divide numerator by numerator and denominator by denominator. The cool thing about this method is that since the fractions are divided to give a smaller quotient, reducing this answer is much easier. Whereas the invert and multiply gave us larger equivalent quotients making it hard to reduce the fractions and adding multiple steps when dividing two fractions.

Overall this was an insightful class that opened up many possibilities to making learning and teaching math engaging, fun and most of all meaningful. Here are some great websites students can explore to make their experience of learning fractions even more exciting and fun:

http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.NUMB&ID2=AB.MATH.JR.NUMB.FRA&lesson=html/object_interactives/fractions/use_it.html

http://www.mathplayground.com/index_fractions.html

Identity Crisis

There is no escaping the realm of fractions! We are bound to encounter fractions in every aspect of our life. Instead of avoiding fractions, it is best that we as teachers are comfortable in this area of math. In class this week we tackled the number sense strand of math and specifically explored the identity of fractions.

We started the class with picking our favourite fraction, and I chose 1/2. I was thinking of picking another complicated fraction but decided to stick with this easy and more common fraction not knowing what adventures lay ahead. The fraction was then represented using a variety of manipulatives. I then had to add the fraction my elbow partner had, and we both happened to pick 1/2.  This made it easy as our sum added to a whole or 1. I appreciated the class discussion that later took place where I got to observe why certain individuals chose their particular fraction. Some chose mixed fractions (which I had not even considered), while others chose numbers like 5/8 just because it was a complex fraction. Fractions come in all forms and are quite a diverse form of numbers just like the diverse range of learners in a classroom. Fractions are found in percentage, as whole numbers, mixed numbers, proper and improper fractions. Getting students comfortable with the various identities of fractions and their special trademarks will allow them to easily recognize their various forms.

Different ways to represent 1/2 with reference to yellow block
representing a whole (Khalid, N © 2016)

With the abundance of manipulatives to represent fractions, students can explore in ways that allow them to engage in their learning. Possibly every child’s favourite and mine off-course, chocolate! We were generously offered a bar of chocolate per table of four and were asked to cut the Hershey’s chocolate bar into 12 pieces. We could then follow along with the book that was being read to us by the teacher.The book represented all the fractions that dealt with 12 parts of a whole. This book makes the listener, not only engaged in the story but also take an active role in representing all the various fractions with the chocolate in front of them. Who knew fractions can be so delicious! However, if I chose not to use chocolate in my class, I could also use an inexpensive manipulative such as egg cartons or clocks printed out on paper. I personally did not know there were so many ways of representing fractions and was pleasantly surprised!

Chocolate used as a manipulative (Khalid, N © 2016)

All the activities were great, the colourful plates used to represent fractions really stood out to me. Children like nice bright colours and things they can touch and feel. What was interesting was that musical rhythms were incorporated in this activity to show that fractions happily exist in the world of music as well. Musically attuned children can easily engage in this activity, and with a great introduction, the whole class can be engaged in this activity. The deck of cards was used as a competitive game to see which student got the highest fraction. However at my placement I saw the deck of cards used differently. Students had to generate the highest number only with the cards dealt to each player. It was great to see a variety of different ways to use a deck of cards. I walked away from this class with a hope that with all the wonderful tools and techniques available for our use, the journey to teaching fractions will be stimulating and an engaging one.

Differentiated Instruction

This week in class we continued learning the different strategies in which we can add numbers. As well as learned various techniques of subtracting and multiplying. The purpose of having multiple ways of doing the same problem is to allow students to explore various paths of arriving at one or few solutions. This goes back to the purpose of differentiated instruction, since we have a range of learners in class, it only makes sense that as teacher we have multiple ways of arriving at a solution.

So what is the purpose of alternative algorithms?

When learning is done instrumentally, a student does not know why they arrived at that solution. As mentioned above, everyone learns in different ways and we should compute a variety of methods in finding a solution. Differentiated instruction creates a safety net that students fall back onto when they feel like they are struggling. This form of instruction also gives our teaching a creative touch, broadening the way we think, and at the same time respecting the various learners in the classroom. Having this method of instruction allows for students to be engaged in class while helping them relate to the problem.

An example of differentiated instruction is “Skip Counting.” This allows students to visualize their adding and subtracting problems by drawing them out a number line. The students break the numbers down in chunks that are easier for them to add or subtract by skipping in small and easy chunks on a number line. However, ground rules need to be established when teaching this method; the number line increases to the right and decreases to the left. As long as students are aware of what they are doing, the bumps illustrated on the number line are up to the student as to whether they correlate proportionately to the amount skipped.

Skip count to add demonstrated in class (Khalid, N © 2016)

We continued to explore open math problem investigating the different ways we could achieve an area of 8 m². This math problem further demonstrated that a good math problem allows students to relate, thus captivating their interest. It gives everyone a chance to start on the problem no matter the learning levels. Another benefit allows for collaborative group work, observing different points of view and reasoning. When students are engaged, they involve themselves and ask supporting questions to understand the problem better. Student engagement and interaction results in reflection, making it easy for students to recollect not only the ‘how,’ but also the ‘why’.

Activity demonstrated by Fellow teacher
candidate: Using missing numbers from 1 - 9
 to solve math equations (Khalid, N © 2016)

Finally we explored math activities demonstrated by fellow teacher candidates. One of which was to use dice to perform various numerical expressions. While at my teaching practicum I saw a similar activity being implemented by my associate teacher. The activity involved students rolling a dice 5 times and recording the numbers that appeared on the dice onto each of the five blanks, with a decimal between the last two blanks. This process was repeated nine additional times. Next a random draw was made to multiply or divide each of the ten trails in blanks by 10 or 100. This activity tested their ability to multiply, divide and round numerical values. Overall as teachers we need to realize that when students are focused and engaged learning manifests itself.