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# Week 10 Number sense and scale sense (March 19-25)

A record number of participants named number sense and measuring as themes important to them, as you can see from the mind map: http://p2pu.org/en/groups/ed218-developing-mathematics-the-early-years/sign-up/

1. Find or design a good hands-on measurement task that depends on counting, adding or subtracting units. Briefly explain qualities that make it a strong learning task.
2. Find or design a good hands-on measurement task that depends on scaling, folding, splitting, stretching and other actions that are NOT about counting, adding or subtracting units. What operations correspond to your task, in the formal math language? Again, explain why you like the task.
3. Curricula of some countries (such as US or China) emphasize counting tasks more, and curricula of other countries (such as Eastern Europe or Singapore) emphasize scaling tasks more. This is not new: for example, Ancient Egyptians were more into counting and Ancient Greeks more into scaling. Needless to say, there are Math Wars about these choices in the current math ed circles. What is your take on the two approaches to the number sense?

Here is a cute example - an episode of the 1970s Russian animated mini-series, hugely popular among preschoolers. Scroll to the measurement parts:

• 3:00 Monkey: (Folding) You are exactly two of your halves long, Constrictor. Or four of the halves of your halves"
Constrictor: You can't measure me in halves! For I am whole!
• 6:15 Parrot: You can measure yourself in parrots. However many parrots fit in you, that's how long you are.
Constrictor: But I don't want to swallow that many parrots.
Parrots: Why, don't swallow anyone. And one parrot is quite enough. Me! (Walks along Constrictor, singing a Dr.Seuss-like, math nonsense song: "One-two, left-right, five times five, it's very easy" etc.) Your length is 38 parrots and one extra parrot wing.
• 7:00 Monkey: What else can you use to measure?
Parrot: Anything!
Monkey: (Counting, by doing cartwheels along Constrictor) Five monkeys.
Elephant: (Counting, by wrapping Constrictor around him) One, two. Two elephants, I am sorry to say.
Constrictor : Hehehe... I am much longer in parrots!

The last phrase became a saying for making fun of people who use their own, favorable criteria.

• 1. My first task about measurement I adapted to food and measurement. This "dinner time" activity would involve students working in groups or individually to prepare a meal on time. Given times on how long each food will take to cook and prepare they will have to figure out at what time each food needs to start preparation in order to be served at the right time. I believe this task lends itself to other math activities as well such as measuring and converting depending on the age level and readiness. Students could also extend this activity to actually preparing foods or relate foods to other cultures, therefore connecting to other subject areas.

2. My second task I found on the internet abotu scaling. It involves creating a superhero and then scaling it to a poster board. I think students would have a fun time interacting with each other to create and design this superhero. Again this activity could be collaborative or individual, but would probably take a few classes to complete. As far as lesson planning goes I believe it is important to connect lessons and subject area and that is what makes a lesson strong. Therefore I think this task relates to art obviously and also could relate to literature if the class is reading about heros and what their idea of one is.

3. Growing up in the US I obviously prefer counting. Until I read this post I had no idea that there was such a controversy over counting and scaling nor that different countries prefered different methods. I always thought that sclaing was more of an art task and was actually an art project of mine. But through this class I am realizing more and more how relate art and math are. In the end though I am a counting person and would probably focus on counting more.

• 1. A hands on activity that I thought of actually gets the students to use their whole bodies to do math. Start off by showing the kids an example. Call two students to stand up in the front of the class to the right. Call three more students and have them stand up to the left of the classroom. Make life size addition, subtraction, and equal signs for other students to hold up. If you were doing addition have a student hold the addition sign in between the groups of students in front of the class and the students holding the equal sign will stand up at the far left. Have the students sitting down solve the problem as a group and send the answer (the amount of students) to the problem up to the front of the room. You would do the same thing for subtraction problems. Let students take turns calling up two different numbers to solve. I like this task because it gets kids to be active within the classroom and work as a team. Everybody gets a chance to participate and solve the problem. It's a creative way of doing math problems.

2. A good hands on measurement task is making various geometric shapes with play-doh. Have the students mold the play-doh into the shapes (it can be flat or 3D). Print out different geometric shapes or bring in actual shapes for the students to reference from. You can also have pictures of a house, animals, or any other object you think would be fun for kids to make. This will allow them to learn how to make things symmetrical on both sides and making sure all sides are even. I like this activity because it's a great way to have students see what it means to have something be symmetrically even while having fun. They're not just looking at an object and talking about it. They get to actually make it and explain the process of doing it. Play-doh will always be popular amongst kids.

3. I believe both approaches are really good to teach to students. We shouldn't teach one more than the other, it should be evenly taught. Both come across in our everyday lives so why not teach both in class?

•

1.       One activity using measurement and counting would be baking cookies. Children would split into groups of four and be given the ingredients needed to make the cookies. They will follow the recipe and measure the proper amount of flour, sugar, etc. in measuring cups and teaspoons. They will have to keep track of how many cups of flour they put in and count aloud with group members. It is a fun way for children to learn about measurement and to see how much math is used while cooking. It is something practical that they will have to learn for everyday use.

2.       Students draw a picture with points at certain coordinates and then multiply all the coordinates by two to double the size of the picture. This would teach them scaling. They can then multiply either the x coordinate or the y coordinate only which teach them about stretching. This teaches the children how to change pictures mathematically.

3.       I also prefer counting over scaling but it is what I am used to. However, I think scaling is more applicable to real world situations. It is a faster way to figure things out and simplifies the process. Counting to me is basically scaling in small increments. If children are taught with emphasis on scaling, they will be better equipped to handle problems that they face in real life.

• 1.)

I think walking the students around the outside of the building to measure and count things would be great for the first measuring task that does involve numbers.  It would probably also be a good idea to do an initial walk-around, ahead of time, alone, just to figure out what would be easily accessible, and the right size for measuring.  In preparation, you could pre-measure some of the things you will be asking the students to measure, (but not in order to limit them).  If the students have the urge to measure a plant or something that's not on the itinerary, they should feel completely free to do so. When you return to the classroom everyone could discuss measurements and compare the lengths of what they measured outside to objects found in the classroom. Is the front step of the building shorter or longer than the teacher's desk etc.  If it's longer how much longer?  Everyone could estimate the difference in length between the two, and then subtract in order to calculate the actual difference. Since this activity is engaging and kinesthetic, it will be sure to be memorable, and capture the student's interest. It is also what I would consider an open-ended activity which lends itself to many variations and many answers.

WARNING: Do not use yardsticks for this activity because the male children will be tempted to engage in battle. Cloth measuring tapes would be preferable.

2.)

The second measuring activity I came up with, involving no numbers, is a sort of experiment with car design.  Designing and actually building rubberband-powered cars would be a bit too involved for younger children, so I would pre-make several unique cars, or have them made in varying designs.  I would bring them to class so that we could race them on a track.  The students would then identify how different aspects of design affect speed and distance.  There would be no need to actually measure anything in this activity.  If the cars were all being raced on a track, simultaneously, you would clearly see which car traveled the farthest distance, and which traveled the least etc.  In this activity you could also explore other variables, such as how stretching the rubberband more tightly will affect the distance and the speed of the cars.  This activity is hands-on and lends itself to experimentation and discovery.  Children learn so much more when they can make a discovery on their own, so this activity would be very relevant.

• 1. One measurement task could be clock-based math. Students could create a schedule for the classroom and consider how many hours, minutes, seconds, etc. and between each of the daily activities. They could see that there are two hours between activity X and Y, there are 5000 seconds provided for activity Q, etc. This would help the students' knowledge of clocks and how to read them, while providing an opportunity to convert numbers. This would be a strong activity because the students would be connected to what they're learning, and could see the practical applications in the classroom. Plus, the kids could work together to form schedules, making it easy to utilize other areas of the curriculum.

2. Another hands-on activity that involves scaling could be having children select their favorite skyscraper or building and then making a scale model of it. We could decide as a class what a good scale would be (10 feet = 1 inch, etc.) and then use that as the standard for all of the projects. The kids could use a variety of materials to build their models and would learn how to scale things down. There would be a strong emphasis on ratios, and figuring out how to convert into smaller units. This activity would be fascinating for the kids, and would be very number-focused.

3. I would agree with Sandy that I prefer using counting more. I think that both have their place in the math world, and that each have core elements that students must learn. Yet, I feel like counting is easier to comprehend for myself, making it easier for me to teacher. I like things in a very structured manner, so counting is more applicable in my life. All I have to do is look at my Google calender, filled with classes and work hours and whatnot, to know that I base my life far more off counting!

• 1. Find or design a good hands-on measurement task that depends on counting, adding or subtracting units. Briefly explain qualities that make it a strong learning task.

An easy hands-on measurement task using counting would be to have the students count the number of tiles on the floor.  We could first measure the size of each block. We could measure the length of the room.  I could then ask the students how many blocks are needed to go from wall to wall.  Students could then count the blocks to verify their answer.  This is a strong learning task because it involves a number of steps.  There is more than one measurement needed, and the students will need to factor the answer.  They will then prove their answer in a kinesthetically.  This exercise can be done individually, as a buddy, or as a team.

2. Find or design a good hands-on measurement task that depends on scaling, folding, splitting, stretching and other actions that are NOT about counting, adding or subtracting units. What operations correspond to your task, in the formal math language? Again, explain why you like the task.

I wasn’t sure about this, but I found this exercise that I think works great to teach fractions without counting.  Students will be able to see the fractions which will lead to a better understanding of what a fraction is.

Introduction to fractions

Students are given various lengths of paper strips or pieces of paper streamers. Ask the students to fold their paper strips into halves and ask a question such as: "How do you know you have folded your strip into halves?" Ask students to compare their half strips with those of other students. Students are then shown other students' attempts to show one half of a rectangle (Figure 1).

* Which of these students have successfully shaded their rectangles to shows one half? (Some students will not recognise that Mike's rectangle is showing one half as they think the left hand side is one half and the right hand side is two halves.)

* Why is Jackson's half different to Mike's half?

* Why do you think Jen has shaded her rectangle how she has?

Comparison of half of a square

Students are handed two squares of paper and asked to fold each square in half. Once students have folded one square in half, ask them to fold the other in a different way.

* Which half is larger: the triangle or the rectangle or are they both the same?

* How do you know?

* Prove it. (Show me.)

Students enjoy proving that the triangular half is the same size as the rectangular half.

Folding paper strips

Students are given a paper strip that is 20 cm long and asked to fold it into two equal pieces. Discussion includes questions such as:

* How many parts are there?

* How many folds are there?

* What do we call each part?

* Show me one half of the paper strip. Show me a different half.

* How many halves are there in a whole?

Students are then asked to fold their halves of paper strip in half. Before opening their paper they are asked:

How many parts will there be?

* How many folds will there be?

* What do we call each part?

* Show me one quarter of the paper strip. Show me a different quarter.

* Show me two quarters. What is another name for two quarters?

* Which is larger: one half or one quarter? How do you know?

Students are then asked to use their paper folding to show: three quarters, four quarters, one half and one whole. After folding their paper streamer in eighths students will be asked questions that involve equivalence, showing fractions that are larger and smaller than given fractions, and questions such as: Show me a fraction that is larger than one eighth but smaller than one half?

If using paper folding for the first time then just fold halves, quarters and eighths. If students have used paper folding before another paper strip will be folded into thirds, sixths, ninths and similar questions asked as for the halves, quarters and eighths. Strips can then be folded into fifths and tenths. Students should be challenged to fold a paper strip into sevenths (Pern, 2011).

3. Curricula of some countries (such as US or China) emphasize counting tasks more, and curricula of other countries (such as Eastern Europe or Singapore) emphasize scaling tasks more. This is not new: for example, Ancient Egyptians were more into counting and Ancient Greeks more into scaling. Needless to say, there are Math Wars about these choices in the current math ed circles. What is your take on the two approaches to the number sense?

To me, counting seems more easily understood.  Perhaps that is simply because it is what I am used to here in the US, but I have trouble with spatial relations, and being able to judge based on my own visual perception would be very difficult for me.  I like the concreteness of numbers.  For me, it is more exact, and is seems to offer me a more reliable answer.

Pern, C. A. (2011, Winter). Using paper folding, fraction walls, and number lines to develop understanding of fractions for students from years 5-8. Retrieved March 2012, from Resource library: http://findarticles.com/p/articles/mi_7030/is_4_63/ai_n28463804/pg_2/?tag=content;col1