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).

Ask questions such as:

* 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.

Questions to ask students include:

* 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