What did I learn from this course?

- Everyone can do math! Yes, this is true. So many times do I see students think that they can not do math, or they are just no good at math. At one of my observation days, I was upset to see that one grade five student, just sat there until I would give her the answer. She had the preconceived notion that math was difficult, and did not make sense. Students should be taught that math does indeed make sense, and everyone can do math.

- As a teacher, I need to learn to shut up!! There is no value in just telling a student an answer. Students need time to attempt a problem. Students need to be active learners, not passive learners. Students need to be involved to learn and understand math.

- “ I don’t have the answer”- Mary Cameron. Students are taught that there is only one answer, and the teacher knows it. But what happens if there are more than one answer, or more than one way of getting the answer? As a teacher, problems should be posed that have multiple entry points, and multiple answers.

- Math should be presented in a realistic way to students. Problem solving activities gives meaning to math, and makes it more realistic to students, opposed to the traditional math worksheet. As a teacher we are asked so many times: “why are we doing this?”, and like the book Math Curse by Jon Scieszka states , “you can look at anything as a math problem”. This is very true, and students should be given realistic things in math to solve, thus making it more meaningful to students. Bringing me to my next point...

- There are tons of great books that can be used to teach math. Books are not only for language arts, and in class I was introduced to many books that would be very helpful; The Best of Times by Greg Tang, The Number Devil by Hans Magnus Enzensberger and the Math Curse (found on my blog) . Literature in math is a great parallel to teaching math. It introduces math in a fun interesting way.


- The answer is not important, it is the logic, and the process, it is about students understanding NOT memorizing.

- Give students the option of using manipulatives. Manipulatives are great in exploring abstract concept. Straws, or popsicle sticks are great for learning any math concepts. In class we explored how these can be used to aid in subtraction or addition. Along with 5 frames, 10 frames, arrays, etc. And how can I forget Fraction Kits. I love this idea, and see the many benefits in not only using them to solve problems, but the many benefits in getting students to actually create one. The making of the fraction kit, may be more beneficial to students, than using them. Math should be hands on, and meaningful, which can be achieved through the use of manipulatives!

- Give students many chances to interact with each other. Students can work together to solve problems, and socially construct their knowledge. Students can learn a lot from each other.


- Reflecting on math? Yes this is possible and should be practiced and implemented in every lesson. Ask students how they got to a particular answer, and what things did they consider when solving a question. Writing in words how they solved a problem is a great way to get students to reflect on how they do something. Students should be given a chance to share how they got an answer, this way students can rethink their problem and make sure it all makes sense to them, and it even may help some other students in the understanding of a particular problem. Reflection is very important in math, and allows students to have an in depth look into their thinking. By students reflecting, they will be able to remember longer, and have a deeper and better understanding. This also reinforces the idea that math is not about a memorizing of facts, but more about understanding.

Teaching Episode (Reflection)

I believe my teaching episode went good. There are many things that I liked about my lesson, and a few things that I didn’t like and would like to do differently. It seemed that all students enjoyed the first activity. Students were asked to estimate how much a certain amount of water is, and then check to see if they made the correct prediction. This helped with students to build in their number sense, and help build the concept of capacity. By asking students what strategies they were using in estimating, they got to reflect upon what they were doing, and become aware of the process they are using.When students were predicting the different amounts, it was interesting to see the different strategies they used when predicting. Sometimes students used the reasoning that they knew how much was in the full bottle, and other times they referred to their previous measurements. I liked this activity because it was very hands on, and students can learn visually. It also allowed them to work in groups, so they can construct their knowledge socially. It was so great to see how excited students were when they predicted the correct measurement. Best of all, this activity captured the students attention, I was not struggling to keep them on task. By the student’s reaction, and the minimum noise in the class, I knew students were enjoying this activity, and learning the concept of capacity. In my class, I will be using this activity in the future.

The second activity, did not go over so smoothly. When students were asked to design their own question, it was apparent that the students did not want to do this. How come? Well, it could have been the fact that the class was over, and we were going over time. OR, students was not ready to design their own problem. Why? The problem we asked students to do was abstract, and may have been too abstract. If students were used to doing abstract things that have more than one answer, they may have been able to complete the problem. As teachers, we should have scaffolded students to a higher level. I do not think that students were very clear on the concept of capacity and 1ml = 1cm3. Students should have been given a number of different problems before the problem they had more than one answer. If students are not comfortable with this concept, they will not be able to do higher order questions, such as the one that we asked our fellow classmates to do. It is much easier to answer questions with one answer, but if the question involves more than one answer, it can get a bit tricky. The beauty about letting students design their own question is they can make it as challenging as need be. They can differentiate it to their own ability levels. The greater the understanding of a concept, the more in depth the students can design their questions.

In conclusion, I like that my lesson allowed students to work in groups to socially construct their knowledge. I also liked that students were given manipulative to visually explore the concept of capacity. Students really seemed to enjoy the activity. The activities chosen were evidently motivating and intriguing for students. The second activity did not work as well, I think further scaffolding should have been done for students to grasp the capacity concept. Also one thing that my professor pointed out, is in the lesson I should not say, you can get manipulative’s if you need them. Instead, manipulative’s should have been given and tell students they may use them. By using the word “need”, it makes students feel that when using them, they are not as smart as those that are not using them. Also, thanks to my fellow students, I have learned many different ways in grouping students. In my lesson, I simply asked students to get in groups of 4 or 5, this did not work to well. It would have been better to assign them to groups. I really liked the idea of using different ways to represent a number, to get students into groups (AGAIn, thanks to me fellow classmates). For example, they had six written, the number six, and symbols to represent six. Everyone that represent the number six was in one group. In older grades (this lesson was for grade 6), students can be in groups by matching all fractions that equal eachother. For example; 2/3, 4/6, etc. All in all, I would do this lesson again with some areas to improve on.

Teaching Episode (Lesson Plan)

Ed3940: Peer Teaching Episode
Date: April 2nd, 2009
Group: Robin Skinner, Sarah Rowsell, & Kirsten Watters

Grade: 6
Unit: Measurements
Lesson Topic: Capacity
Outcomes:
Students will be expected to:
- D1 -> use the relationship among particular SI units to compare objects.
- D3 ->demonstrate an understanding of the relationship between capacity and volume
- D6 -> solve measurement problems involving length, capacity, area, volume, mass and time.

Our lesson will be dealing with capacity and enhancing the students’ estimation skills. The student will be working hands on with manipulatives and real life materials to discover capacity. To start off we will recap what the students have “previously learned” (that 1cm³=1ml). The teacher may have a few different sized bottles with the same amount of water in them to show that capacity is not always how high the liquid goes in a container. To be sure that the students fully understand this concept they will be using 2L pop bottles, water, funnels, rubber bands, and capacity cards to practice estimating the different water levels on the pop bottle.

Students will work in groups of four and be sure that each student has a turn to give their opinion on where the water level will be. After the students have a chance to work with some small capacities and some larger ones the teacher will move on to a more difficult concept in which the students will apply this knowledge/practice.Throughout this activity the teacher will circulate the classroom to see how well the students understand before moving on. A checklist may be used here. The reason why a checklist was chosen is because it is an easy and effective way of assessing students while making the teacher available to the students at the same time. The teacher is able to guide the students through the lesson while still having the students assessed. The teacher will have the checklist on hand, and available on one page. This way, it is easy to check off names and components while guiding through the lesson.

Once the teacher has determined that the students understand how to apply capacity and different measurements the next activity can be introduced. To introduce this activity a word problem will be given on the overhead. After thinking with their group about the problem and discussing the answer students will create their own problems, using the manipulatives provided. This allows for diversity because each student/group can create a problem that suits their own ability level. The teacher may provide some struggling learners with extra hints to get them started, or an advanced learner may want to provide the assistance to give them extra practice and reinforce their learning.

Required Materials:
· (6) 2L pop bottles· Funnels
· Water supply for each group
· Extra water in case of spillage
· Paper towels· 1cm3 blocks
· Elastics· Capacity Cards
· (6) Containers to represent pools

The benefits of making your own fraction kit!!

In class we made a fraction kit. This is something that I would definitely use in my classroom. My professor asked us to create whole, 1/2, 1/4, 1/8, 1/3, 1/6, 1/12, then 1/24. Each fraction would be on a different color of paper. Making fraction kits are great for students to understand the abstract concept of fractions. For example; when making ¼, I made ½, and then halfed both of my halves. Similarily with 1/8. To make 1/8, I halfed my paper, then halfed one half, and then halfed between each line. It sounds complicated, but it really was not that bad. By students creating their own fraction kit, they can see the relationship of each fraction to each other. The benefits of the fraction kit does not end there. Students can use their newly made fraction kits has manipulative to figure out problems. Also, with their fraction kits they can see four ¼ makes one whole, as well as the various other fractions that are equal to one another. For example, we were asked to create 6 1/8's. There was more than one answer to this problem. Some were; 8 1/24's, 9 1/12's and 1/2 + 1/4. This fraction kit proves to be beneficial in aiding students concepts of how different fractions relate to eachother, they can visibly see which ones are bigger, and even must think about other fractions when cutting for a particular fraction.

Making Sense of Math

When asked in class to count grains of sand I immediately thought, there is no way I would be able to solve this problem. I soon realized that it was not the answer itself that the professor was concerned with, but the process. This was the first time that I was forced to think that it is not the answer that is important, but it is the logic you use and the process. In class I really enjoyed the idea of doing math problems with more than one answer or more than one way to get the answer. I will definitely be using this method in my own math class. It was interesting to see the different ways in coming up with the answer ‘five’. The possibilities were endless.

This class caused me to immediately think of an observation day in a grade three classroom. While in the class, I was excited to see actual problem solvers taken place in the classroom. When doing the problem solvers the students had to write their strategy in solving the math problems, their workings and their answers. I really liked the idea of this, unfortunately I did not agree with how the teacher taught the mathematics. I felt she never allowed the students enough time to solve the problems themselves, and I didn’t feel her explanation of the problems were clear and felt many students were being left behind and not really understanding what was going on. This was a substitute teacher, and I felt she was more concerned with getting the work completed rather than an actual understanding going on with the students. She also never accepted different ways in solving a problem. A student solved a problem with subtraction and she said it was the wrong way because he should have used addition. When correcting the problem solvers of individual students, I could tell the students did not understand how they came to the answers and comments were made on their sheets that subtraction can be used but it was the wrong way to get the answer.

This teacher appeared to view mathematics as using numbers to get an answer. The product was what was important and if you practice them enough you will understand. Process and meaning appeared to be neglected .Also given the teacher was a supply teacher, time became an issue of having all tasks completed. The substitute teacher didn’t have the time to follow the student to see where their thinking may have been going.

Math Curse

Math Curse by Jon Scieszka and Lane Smith illustrates how you can look at almost everything has a math problem. When students are able to solve real life problems then math would become more meaningful to students. There are many activities you can do with this book. I can see myself using it as a read aloud book in my classroom, solving the many problems introduced in the book through whole class instruction. After being introduced to this book in the classroom, I went to 'Chapters' to purchase it. Unfortunately it was not in stock. I am still in my search to owning my own copy of this book, and using it as a resource in my classroom.

What is Mathematics?

Reuben Hersh’s Talk was ‘beautiful’. I have never thought about math as a philisophical discipline, yet Hersh described it in a way that just made sense. I really liked how he summed it up into one great sentence: The humanistic philosophy brings mathematics down to earth, makes it accessible psychologically, and increases the likelihood that someone can learn it, because it's just one of the things that people do. This is a matter of opinion; there's no data, no tests." As a future teacher, this sentence alone puts a new perspective on how to effectively teach mathematics. His talk helped me sort out some of my confusions and clarify my own personal feelings/experience with mathematics. It helped me to see why I enjoyed problem solving and why I felt inadequate with High School mathematics.

The big ideas about math is that the concepts need to be meaningful and that teaching occurs through social interaction and hands on learning. Students need to be involved and not passive learners. One strategy I liked was where the teacher read the student's body language to measure engagement and altered his delivery accordingly. Also, the idea of the teacher waiting (or shutting-up) to get the students involved was a very effective strategy.

Although Hersh’s Talk caused me to think differently about mathematics he also created more questions for me. I need to know the expected outcomes at each grade level and how can I as a teacher make this meaningful and help the students see how this fits into their lives. I also need to find ways to keep students interested and motivated. More importantly, I belive that I must stay interested and motivated in order to ensure that my students see that mathematics is important but can also be fun.