The Origami Algorithm And C3 (Common Core Confusion)

When my grandchildren are being tortured by Common Core math homework their mother tries to help them. Sometimes she throws up her hands in frustration and asks me if I can help. Thus began my training in the mysteries of Common Core Math. But before I could help, I first had to figure out one of the great Common Core math mysteries. What is an algorithm? But the answer to that mystery has to wait until you get to the end of this blog.

Working with the grandkids on their homework, I learned the first rule of Common Core math is sequence. You are not supposed to think. You are not supposed to imagine, estimate, or consider options. You are to follow instructions and solve problems in the right order.

This April the grandkids took the sixth grade math test so I decided to look at the questions released by New York State to see what I could learn. I discovered these things. It is a reading test with content specific vocabulary and the prompts contain extraneous information designed to confuse students.

Reading Test: Common Core math, for better or worse, is no longer about calculations; it is about reading. Most of the "released" questions were word problems, which means a student who has difficulty in reading is going to fail math, even if they know how to do the math. This will be especially unfair for English Language Learners.

Vocabulary: Passing the math test means memorizing specific the vocabulary. Word problems on the sixth grade released questions used the words or terms (listed in alphabetical order) approximate, calculate, centimeter, constant, coordinate, corresponding, determine, diagonal, discount, enclose, equation, equivalent, expression, grid, hourly, inequality, kilometer, opposite, percentage, plane, plot points, possibility, properties of operations, pyramid, quotient, rate, ration, rectangular, relationship, represents, trapezoid, units, and vertices. The term function, which I think is a synonym for equation, appears 172 times in the 83-page explanation of the New York State Math Learning Standards but not on the sixth grade math test. I have no idea why.

Extraneous information: The questions often have extraneous information that students must recognize and disregard. For example, according to one question, "The summit of a volcano is 10 kilometers (km) above the ocean floor, as shown below. If the ocean floor has an elevation of -5 kilometers, which statement describes the elevation of sea level and the summit?
A. The elevation of sea level is 0 km and the elevation of the summit is 5 km.
B. The elevation of sea level is 5 km and the elevation of the summit is 5 km.
C. The elevation of sea level is 0 km and the elevation of the summit is 10 km.
D. The elevation of sea level is 5 km and the elevation of the summit is 10 km.

The metric system has nothing to do with this question. Information could just as easily been provided in feet or yards. A student unfamiliar with or unsure about kilometers may be thrown by the question even if they understand the math. The information about the ocean floor is also extraneous and only confuses. The elevation at sea level is 0 km so the elevation of the volcano summit is 10 km.

Whenever you question a Common Core standard or a question on a Common Core test you are always told it was carefully chosen as appropriate using an algorithm, the mysterious Common Core algorithm.

Did you ever bake a cake, found the recipe confusing, or you were short an ingredient and had to jerry-rig a substitute? One way to think of an algorithm is like recipe. As with any recipe, with an algorithm, if the instructions are confusing or the ingredients are inappropriate, the outcome could be a disaster.

Now back to the origami algorithm. I originally developed this activity, creating an origami paper crane, as part of a unit for teaching elementary school students about the impact of nuclear war and the need for peace and middle-level students about Japanese culture. Now I use it to help teachers understand what an algorithm is. It remains a great activity for students and teachers of all ages.

Origami is the Japanese art of paper folding. In Japanese, oru means "to fold" and kami means "paper." Paper folding originally developed in China, probably soon after the invention of paper in the first century AD. In an example of cultural diffusion, paper and paper folding were brought to Japan in the sixth century AD by Buddhist monks. In Japan, paper and paper folding became important aspects of architecture and the Shinto religion. Kami (paper) is a homonym for spirit or god.

While leading a class in paper-folding, I stress four basic rules that are also applicable in Common Core math:

Sequence (steps must be done in order).
Symmetry (what you do to one side you must do to the others).
Concentration (the open side must always be down).
Precision (folds must done carefully and firmly).

Origami Paper Crane

The crane is usually not for beginners but I think it is worth a try. Everyone is not going to be able to create one the first time. It is easier if you work in pairs with one person reading the directions aloud. There are videos online that can help.

  • Start with a square of paper. Usually it is brightly colored or patterned on one side. While you can use any size, I prefer a six-inch square. A larger piece can be awkward to work with and a smaller is just plain difficult for inexperienced or clumsy fingers. 'Top' means the corner or side pointing away from you. 'Bottom' means the corner or side pointing toward you.

  • Place a square of paper on a flat surface with the colorful side underneath. Turn it so that it is a diamond shape.
  • Fold top corner to the bottom corner creating a triangle. Note the bottom side (the side pointing towards you) is always open.
  • Press carefully and firmly along the fold at least three times. You must do this after each fold.
  • Create a smaller triangle by folding the right corner over to the left corner. Press.
  • The next step is tricky. Lift up the flap you just created so your paper forms a right angle. Insert your index finger into the opening of the vertical flap carefully pushing to the back.
  • Gradually open the flap. As you do this, the top point of the flap will start to come down creating a new diamond. Press the new folds.
  • Remember the principle of symmetry. Turn the paper over with the open side remaining down. Create a new vertical flap. Again, insert your index finger into the opening of the vertical flap. Gradually open the flap. As you do this, the top point of the flap will start to come down creating a new diamond. Press the new folds.
  • With the open side down, fold the bottom right side so it touches the middle axis of the diamond. Fold the bottom left side the same way. Flip your figure over and repeat on the other side. The new figure will look like a kite. Press the new folds. Fold the top section of the kite over and press.
  • Another tricky step. You are going to open the "kite" and let the figure form a new shape. If you do it correctly, it will want to do this on its own. You just have to help it. Pull back the top flap. The place where it bends is a key fold. Open the two side flaps. You have returned to the diamond shape. The bottom of the diamond is open. Lift the upper sheet of paper from the bottom of the diamond. Placing a finger from your other hand on the key fold, push the upper sheet all the way back away from you. As you do this, the new sides will start to fold in. Help them. You will now have an elongated diamond shape with an opening in the front.
  • Flip the diamond and repeat the transformation on the other side. Press the new folds. If you have done everything correctly, the bottom of this new figure will have two "legs." To make sure everything is okay, let them walk a little.
  • There are openings on both sides of the diamond. Place one index finger in each opening and bring the upper sheets together. Bring the lower sheets together as well. Lay the figure flat on your surface. It should look a little like a wolf. Press the new folds.
  • With the open side down, fold the bottom right side so it touches the middle axis. Fold the bottom left side the same way. Flip your figure over and repeat on the other side. The new figure will look like a wolf with a narrower face. Press the new folds. Do not despair. You are almost finished.
  • You should have two pointy flaps at the bottom of the figure and two "ears" at the top. Lift one of the pointy flaps up as far as it will go and fold. Fold the top inch of the tip down. Press the new folds. Turn the figure over and lift the other pointy flap up as far as it will go and fold. You are finished folding. Firmly and carefully press the entire figure.
  • We now must open the crane. Let the last two folds spring out. This should leave two folds of paper standing upright. These are your wings. Place one index finger in each opening of the wings and gently pull out and up. A box or body will form in the middle. Keep tugging gently until the body is unfolded. Straighten the head, tail and wings. Tuck the tail up against the box. Your origami paper crane is now complete and you have mastered one of the great Common Core mysteries - the algorithm.
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