Fractions

Dr. Tim Pelton subbed for our class today and talked about fractions with us. After hearing his lecture, all I can say is wow! I have never understood fractions this fully before and I’m excited to teach fractions using this method with my future classes. I’m going to use his explanation of adding fractions for this post, but he also talked about subtracting, multiplying, and dividing fractions.

When I was taught fractions in school, teachers just presented the standard numerical representation with no manipulative or diagrams. They taught us rules to memorize for each operation, and we were expected to perform these formulas in exactly the manner we were taught without asking questions. Dr. Pelton pointed out that asking questions is not only okay, it is essential for our students to really understand the math we are teaching them. If they don’t ask questions, they will never really learn.

So, to teach adding fractions Dr. Pelton suggested first coming up with a story for the fractions to make it a real world instead of an abstract problem: something like “I have 3/7 of a pizza and you have 1/4 of a pizza, how much pizza do we have in total? Having a story also makes it easy to draw out what those fractions actually mean, which is what he did next. Then, you can lead the students to realize that you need to be comparing the same thing to add them, so that means changing the denominator. Dr. Pelton changed the above fractions to 12/28 and 7/28 and drew that out on the board to help us visualize what was happening. Having all of these illustrations made it really easy to see that the answer was 19/28. I am really excited to use this method with my students!

Finally, I really appreciated how Dr. Pelton acknowledged that traditional algorithms and traditional terminology are important. These algorithms and words have meaning and have been developed by mathematicians over the years. They are generally accepted as the best and simplest ways to perform operations and describe equations. As math teachers, it is not our job to come up with new terminology and formulas – how is my first year university mathematics knowledge supposed to compete with someone that has a PhD in math? It’s also wrong for us to throw students into solving a problem and allow them to figure it out however they want. How will they move on to higher math if they don’t know the standard addition algorithm? Rather, it is our job to help our students understand the “why” of the formulas and terms developed by mathematicians. If we can guide them through creating these formulas and understand why they work instead of telling them to memorize or just throwing them into solving a problem with no algorithms to use as tools, then they will really understand the mathematics we are teaching and if they choose be able to move on to higher math, possibly creating algorithms and terms for future students to learn.

Scaffolding Measurement

I really liked how both of Liedtke’s books explained how to scaffold measurement concepts. As I think about teaching math, it is often hard for me to break the overall concepts found in the IRP into manageable steps. I definitely plan on using this method in my classroom in the future.

Liedtke describes a particular pattern for teaching measurement that I will describe here. I will stick to linear measurement for my analysis but the concept works for any type of measurement. First, he suggests beginning with real-world, non-standard, inconsistent units of measurement such as hands, feet, or steps. This shows children that they can measure objects in their environment using whatever methods are available. I see real value in starting here because it allows children to see measurement as a real world activity rather than measuring lines on a piece of paper. I could see challenging the children to count how many steps it took them to get to school (perhaps using a pedometer) or to measure how many feet it is from their desk to their cubby. Liedtke then says to have children compare their measurements and come to understand that measuring by steps or hands or feet is ineffective because everyone’s hands and feet are different sizes and everyone’s steps are different lengths.

Next, Liedtke suggests introducing a non-standard but consistent unit of measurement such as a paper clip. This keeps the measurement activities within the real world but gives a unit of measurement that is the same for every student. Now, students could measure things like the size of their lunchbox or their height (hopefully choosing a unit bigger than paperclips for the latter!) and compare results between classmates. After doing this students should be able to understand that having a standard unit of measurement is important so that people can compare with each other, even with people around the world. How cool would it be for students in my class to compare their height in notebooks to another class’s height in notebooks?

Finally, Liedtke suggests introducing standard units of measurement such as centimetres and metres and helping students to understand that these standard units allow people all over the world to compare distances. At this point, perhaps I could extend the class comparison of heights by having that class measure something like the distance around their school and having my students do the same.

When I was in elementary school, I was only introduced to standard units of measurement. While I did eventually understand how to measure, I remember it being hard to understand at first. Hopefully by scaffolding my teaching of measurement as Liedtke suggests I will be able to make this process much less confusing for my students.

Liedtke, W. W. (2010). Chapter 8: Measurement. In Making Mathematics Meaningful For Students in the Primary Grades. (pp. 129-153). Victoria, Canada: Trafford.

Liedtke, W. W. (2010). Chapter 6: Measurement and Measurement Sense. In Making Mathematics Meaningful for Students in the Intermediate Grades. (pp. 145-163). Victoria, Canada: Trafford.

Notes on a Triangle

Today in class we discussed the video “Notes on a Triangle” by René Jodoin and the National Film Board of Canada. I found myself mesmerized by this video. Who knew you could create so many shapes from a single triangle? As Jennifer mentioned in one of our readings for next week, I think this video could be cool to show to many different grades, and I’m sure each age would get something different out of it. We discussed how this video would be great for teaching transformations and slides, different types of triangles, and even other 2-D shapes that can be formed from triangles.

After discussing all of the different concepts contained in this one video in class, I thought a great way to bring the learning into a more personal space for my students would be to have them create their own version of the video.  I would use the Stop Motion Studio app and have students draw or create triangles and other 2-D shapes out of paper. They could then play with the different translations, rotations, and reflections they can do with the shapes they have chosen and sequence them into a pattern like Jodoin does in the video. This would be an activity to do with older grades (perhaps grade 5 and up) as it would require quite a lot of patience and concentration, and translations do not appear in the BC IRP until grade 5. I’m excited about building this idea into a lesson plan and I hope I get to use it in a class someday!

Geometry Manipulatives

When I was in elementary school, geometry was often taught by memorizing the names and calculating surface area, volume, perimeter and area of different 2-D and 3-D objects. These objects were always presented to us as drawings on a sheet of paper or in a textbook. It was rare to see actual manipulatives used in our geometry lessons, and I think that is something that was lacking in my own education.

We discussed many different ways to teach geometry in class today, and I think these ideas can be built into a logical progression. It makes sense to move from 3-D to 2-D objects (like we discussed in class), because 3-D objects are part of the real world and students encounter them outside of the classroom. It also makes sense to begin with real world objects, such as a toblerone box or a bouncy ball, and to introduce abstract geometric solids after students understand that geometry is a part of everyday life. So, a good geometry unit can spend some time working with real world objects, then move to 3-D solids, then move into 2-D geometry with manipulatives.

I also think it is important to use manipulatives all the way through geometry, just as we would to teach place value or division. I found it almost impossible to remember the names of all the geometric solids when looking at then drawn on a sheet of paper, and I’m sure it would have been easier for me to remember if I had had more of an opportunity to work with physical objects.

Finally, I agree with our Making Math Meaningful textbooks when they say that geometry is an important part of the curriculum and should not be relegated to the last few weeks of the year, if there is time. Developing spatial sense is as important as knowing basic addiction facts and so geometry should receive some focus in the curriculum.

Liedtke, W. W. (2010). Chapter 7: Geometry and Developing Spatial Sense. In Making Mathematics Meaningful For Students in the Primary Grades. (pp. 111-128). Victoria, Canada: Trafford.

Liedtke, W. W. (2010). Chapter 7: Geometry and Spatial Sense. In Making Mathematics Meaningful for Students in the Intermediate Grades. (pp. 164-178). Victoria, Canada: Trafford.

Strategies and Algorithms for Division

Today in class we talked about different algorithms that can be used for division. These include the traditional algorithm:

Repeated subtraction:

Changing the problem to something you know:

And the area model:

I can see the value in all of these models and I see how they could be used by different types of learners to understand division. For example, someone who has great spatial awareness might love the area model, and someone who is great at remembering a step by step procedure might love the traditional algorithm. However, all of these algorithms seem very cumbersome. No matter which model you choose, it involves remembering at least four or five steps and in some cases many more. The steps also always have to be performed in a certain order. Some models have more flexibility than others but in every case doing the last step first will result in the wrong answer. I wish an algorithm existed for division that was less cumbersome. Personally, I always turn to a calculator when I need to divide now since it takes far to long to work through any one of these algorithms in my head. I am worried that my own students will come across this frustration when I teach them these algorithms, so hopefully I am able to come up with a solution for them!