Sunday, 23 October 2016

Who doesn't love a good math game?

Hi everyone!

Welcome to another math post! This week in lecture, my classmates and I got the opportunity to play with some interactive math games- both online and offline.

The first game that we played is called Hedbanz. This is a classic game that people play at family events, board game nights with friends, and office parties (I imagine)! However, my professor altered it to help students practice their knowledge of the different forms of quadratic equations.

At the beginning of class, my professor handed out the headbands to every student in the classroom. Each headband had a quadratic equation on it, written in factor form, vertex form, and standard form. We were not allowed to look at the equation on our own headband. The goal of the game was to pair up with a partner, and ask each other "yes" or "no" questions, in order to figure out the equation on our forehead.

My best friend (K) and I immediately paired together to tackle this challenge. We put on our headbands and quickly wrote down some questions before we approached each other:

1. Am I in standard form? No
2. Am I in factored form? Yes
3. Do I have a coefficient in front of my brackets? No
4. Do I have more than one odd root? No
5. Do I have an odd root? No
6. Do I have two even roots? Yes
7. Do I have one positive, and one negative root? Yes
8. Are my roots the same digit? No
9. The only thing left to guess were the digits of our roots.

Results: K won the match, but we both had a great time and shared a few laughs during the game!

Here is a cute picture of the two of us afterwards! (Please forgive my awful selfie skills):


Why did my teacher let us play Hedbanz instead of handing out a worksheet on the different forms of quadratic equations?

Simple. It all comes down to three words that ALL teachers love: engagement, interactivity, and creativity.

My professor could have easily given us a worksheet, so that we could practice identifying the different forms of equations, but it would not have been nearly as successful! Even though the Hedbanz game was not a typical mathematics game, it gave students the opportunity to use math terminology in their questions (roots, standard, factored, vertex form, coefficient, etc.), and there was no pressure to arrive at a quick solution. Each pair could ask as many or as little questions as they needed to figure out their equation, and they were able to create questions that would help them best understand the different aspects of their equation.

This game is a great way to assess how well the students can use prior knowledge, problem solving and reasoning to figure out their headband equation. This game could be used as a diagnostic assessment before starting a unit on quadratic equations, or it could be used as a formative assessment before a major assignment or quiz. As the teacher observes the game unfold, he/she could make notes on areas of strengths and improvement for the class.

How can we use this game in our classroom? 

Hedbanz is versatile game that can be used to practice/review a number of math concepts in a fun and engaging way for students! We used it to test our knowledge of quadratic equations, but it can also be used to test students knowledge/application/thinking/communication skills in regards to prime factors,  graphs of functions, exponents, factoring, and more! The possibilities are endless for this game. This is also a great for ELL (English Language Learners) because it gives them an opportunity to practice asking questions, using math terminology, and demonstrating their knowledge by applying it in a low-risk setting that removes some of the anxiety and pressure from participating in a larger group setting.

I hope that you all enjoy this wonderful game, and have a great week ahead!

Dayna

Friday, 14 October 2016

Which one doesn't belong?

Hi everyone!

So today I thought we could play a math game! It's called "Which one doesn't belong?"
Objective:

There are four shapes/numbers/letters/graphs shown below; find a reason why each of the four (numbers) does not belong with the other three.

Rules: 

1. Have fun!
2. Be creative!
3. Try to think outside of the box.
4. Your thinking does not always have to be related to math!
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Take a moment, and try to come up with your own guesses/rules for why 9 does not belong with 16, 25, and 43.

My thoughts: 

The digits that make up 16, 25 and 43 all add up to 7. (16=1+6=7; 25=2+5=7; 43=4+3=7). The digits of 9 do not equal 7, so it does not belong. 

My mom's thoughts:

9 is the only single digit number! 

Both answers are right because they both find a rule that groups 16, 25, and 43 together such that 9 does not belong in the group. Both answers are mathematical because they use math terminology such as "add", "number", "single", "digit(s)". And both answers discuss the different properties of numbers.

In my math teaching course, we were introduced to a website called "Which one doesn't belong?" (www.wodb.ca). It was created by a woman named Mary Bourassa who accepts entries, tests them to see if they fit the criteria, and then posts them on her website for people to solve.

I love this activity because of it's flexibility. This activity presents a puzzle for students to solve, using their prior knowledge and experiences to help them find patterns, create categories/groups, and form rules that each number must follow in order to be part of the group. It is also OPEN to possibility. As long as a student sees a pattern that can be proven, their answer is correct! This opens up the playing field for all students because there is no "right answer" or set formula that has to be followed to arrive at an answer. This game is about being creative, open-minded, and thinking outside of the box.

I tested this game out with my mom because I wanted to see if we would get any different answers by examining the problem from different perspectives. My mom used to be a banker, so she is very good with numbers, but she doesn't study math as a hobby. This offered a new perspective because she thought in terms of the operations (addition, subtraction, multiplication, etc.) between numbers, while I tried to find patterns between the digits of a number. We tried three different "Which one doesn't belong?" puzzles, and I found that sometimes we came up with the same rule or category for a group, while other times, we had completely different responses! I think this would be a great activity to do with a class as a warm up exercise because it allows the teacher to perform a quick diagnostic assessment to see where students are at in their learning, but it is also a great opportunity for students to learn from each other and see that a puzzle can have multiple solutions.

As an expansion, I thought that it might be a fun challenge for students to try to make their own "Which one doesn't belong?" puzzle, and either share it with their classmates to solve, or e-mail it to Mary Bourassa. If it is solvable and Ms. Bourassa is able to test it successfully, then it may be published on her website for other people from around the world to solve. How cool would that be?

That's it for now! I hope everyone has a great weekend!

Image result for math jokes

Dayna

Tuesday, 4 October 2016

Open Questions = Open Access for all Students

Hi everyone!

In my Math Teaching course, we learned about the power of using open ended questions in the classroom.


Often times, students have a fear of math simply because they are worried that they will not be able to find the correct answer. Many students worry that they will be put on the spot in front of their teacher and peers, and they will be forced to admit that they "do not know the answer" or worse, attempt to provide a solution and realize it's the wrong answer. This fear of public humiliation can cause math anxiety that continues to build the longer that students feel like they have nothing meaningful to contribute to their class.

By starting a lesson with an open-ended question, all students are able to participate in the class discussion because there is "no right answer". As my teacher put it, "If I went around to each group, I know that every student in that group would have something to tell me about the picture/pattern/question." It would be amazing to design a question that can capture the attention of all of my students, so that they can all feel like they are a part of the discussion. By using open-ended questions at the beginning of a lesson, it gives students an opportunity to get into the flow of the class, gets them excited and engaged, and it eases some of their anxiety when they are able to play an active role in the classroom.

A great example of an open-ended math question is estimation. Estimation is a skill that students can develop to make accurate guesses that will help them determine if their solutions are realistic or completely off-base. Estimation can also be used in the real world. By helping our students develop their estimation skills, we are helping them determine if they have enough money to pay for their items when they are standing in the check-out lane; we are helping them determine if their furniture will fit into their college dorm room; we are helping them estimate how long it will take to walk home or drive somewhere; and we are helping them plan events when they need to estimate how much pizza needs to be ordered for a large number of people. This is a great tool to foster because it brings the real world into the math classroom, and it brings math into the real world.

We can use estimation as an open-ended question because every student has their own opinion and internal guesses; therefore, every student can participate in the lesson without fearing that their answers will be rejected. Estimations are accurate guesses; they do not always have to be exact, so this may ease some of the math anxiety associated with getting the correct answer.


A great resource for any teacher that would like to include estimation in their lessons, is Estimation180. It is a website that provides pictures and videos that students can watch/examine to make predictions; once they have all decided on an accurate guess, the teacher can show the answer image or video and students can compare their answers with the actual answer. The teacher can then have a follow-up discussion, where students can determine why their answer may have been a bit off. Here is a video example from Estimation180 that helps students begin to think about adding fractions:

For more fun ways to use estimation in your classroom, check out Estimation180.org. 


Thanks for reading everyone, and have a wonderful day!

Dayna


Yay for Manipulatives!

Hi everyone!

Today, I wanted to introduce the topic of manipulatives, and address some of the misconceptions that we may have around this subject. 



What are manipulatives? 

Manipulatives are tools that students can use to help them approach and solve math problems. They are concrete items that students can move and explore. Even though math is the "universal" language, sometimes it is difficult to understand or decipher, so students can use manipulatives to represent math in a new way that makes sense to them. 

What are some of the benefits of manipulatives? 
  • Manipulatives decrease math anxiety because it offers students a concrete way to interact with abstract concepts 
  • They are fun to explore! Math doesn't always need to be represented on paper!
  • They can be used in all grade levels and streams. Yes, that's right; they can be used in both ACADEMIC and APPLIED. Manipulatives do not just apply to one particular stream or age group, since they can be used in Grade 12 and College/University. 
  • They provide both a visual and kinesthetic approach to mathematics, so they are a great way to offer differentiated instruction to all learning types. 
  • In order to save money and provide more manipulatives for your students to work with, you can make your own! 
    • I was able to make my own algebra tiles by gluing a piece of red and blue construction paper together, and cutting it into different shapes (long, thin rectangles represent the x term; and small squares represent the constant term 1). This cost me a total of 60 cents, since I only used 4 sheets of construction paper.    
Construction paper Algebra Tiles

Algebra tiles can be used for a number of different algebraic concepts, such as factoring, adding and subtracting expressions, multiplying and dividing expressions, and completing the square! (Who knew!) I focused on using algebra tiles to collect like terms, which I have included in a video below. I hope this helps for anyone who is new to using algebra tiles! 





Are there any problems with manipulatives?

Personally, I don't see a problem with manipulatives because they allow students to learn at their own pace and discover new ways of representing and understanding mathematics. 
However, there tends to be a problem when students are unable to use them during a test, when they previously had access to them during lessons and homework sessions. I understand that teachers want to see if students are able to demonstrate their work using the abstract procedures shown in class, but how are students able to display this level of abstract thinking, when they need a concrete tool to help them understand? Tests already place a heavy amount of pressure on students, especially when these students have math anxiety. 
Taking away their manipulatives during a test, might be the equivalent of throwing them into an ocean without a life jacket. No wonder students have a growing fear of math. 

Some students will use their manipulatives during class in order to help them understand an abstract concept, but will no longer need them for the test because they have grasped the abstractness. However, others will depend on their manipulatives to help them through a stressful assessment; we should not be taking that away from them. I believe that a student should be able to use any strategy or tool to arrive at an answer, and they should not be penalized based on their chosen strategy. As long as students are able to express their answers and explain the procedure they used, then they have effectively demonstrated their learning. 

My final critique on manipulatives is that some teachers lack the training to use them in the classroom. Often times, the Ministry deems that manipulatives should be implemented in the classroom (which is great!), but they do not provide teachers with the proper training to effectively incorporate them in their lessons. This lack of training may cause confusion for students, frustration for teachers, and the benefits of manipulatives to not be reaped. I think that an easy fix to this potential problem would be to provide teachers with PD days that would help them learn more about the manipulatives, and provide examples and suggestions on how to use them in their lessons. 

Overall, manipulatives are a great tool for teacher's to have in their toolbox. We just need to know how to incorporate them in our lessons in order for them to be effective!

Thank you all for reading, and I hope you have a great day!


Dayna