Inspired by Lillian Ho's article on using origami with adult ESL learners (see references below), I decided to build a lesson for high school ESL students around the

hopping frog model.

The basic activity went like this:

1. Students were shown how to fold the model without being given any verbal instructions.

2. Students were given a version of the instructions with all written instructions removed.

3. In groups of 3 or 4, students were asked to provide written instructions on chart paper to go along with the diagrams.

4. The written instructions were shared by placing the chart papers up around the class, and students were asked to identify the important words that were used in the instructions.

5. As a whole class, we folded the frog again, noting the mathematical ways we could describe each step.

If you try an activity along these lines, I expect that should be split over two or three sessions. We did step 1 at the end of another lesson, steps 2 and 3 on a second day, and steps 4 and 5 on a third day.

In thinking about the sort of descriptions that the students should be guided towards, it's helpful to note some of the observations provided in an article by Koichi Tateishi (see references): (1) the written words are not a replacement for the diagrams, but should be thought of as complementary, and (2) we should avoid technical origami terms (mountain fold, squash fold, etc.).

When one group of students uncovered a useful word, it would get written up on the board for all groups to share, so as we went we developed a list of helpful words. With reference to the steps on the

instruction page, some words that the students used included or discussed at each step were:

**step 1**: paper, card, square, rectangle, fold, half, long, short, side, middle, open, line;

**step 2**: diagonal, top, side, edge;

**step 3**: do again, repeat, other side;

**step 4**: turn over, flip, corners, intersection, lines crossing;

**step 5**: push, pinch, squash, together, peak;

**step 6**: tip, up;

**step 7**: center, in, inwards;

**step 8**: flaps, outside, inside, arms;

**step 9**: bottom, nose, legs;

**step 10**: down, knees;

These words were used in the context of providing instructions and directions - a very important type of speech act that students are routinely confronted with.

In the appendix to her article on the benefits of origami lessons for middle school students, Norma Boakes (see references below) provides a great sample mathematical dialog for the jumping frog model (step 5 of our lesson plan). Like Boakes suggests, in our mathematical discussion we talked about rectangles, squares, quadrilaterals, right angles, bisectors, 45 degree angles, triangles, pentagons, right-triangles, parallel and perpendicular lines. Each (now very familiar) step of the frog construction providing a concrete model for the concept we were talking about.

Through this lesson, we learned and reviewed a lot of everyday language around the giving and receiving of instructions that involve everyday spacial terms, and were also able to then apply a mathematical lens to deepen our understanding of what we were doing.

**References**
Boakes, N.J. (2009). "The Impact of Origami-Mathematics Lessons on Achievement and Spacial Ability of Middle-School Students." In Lang, R.J. (Ed)

*Origami 4: Fourth International Meeting of Origami Science, Mathematics and Education*. Natick, MA: A K Peters Ltd.

Ho, L.Y. (2002). "Origami and the Adult ESL Learner." In Hull, T. (Ed.)

*Origami 3: Third International Meeting of Origami Science, Mathematics and Education*. Natick, MA: A K Peters Ltd.

Tateishi, K. (2009). "Redundancy of Verbal Instructions in Origami Diagrams." In Lang, R.J. (Ed)

*Origami 4: Fourth International Meeting of Origami Science, Mathematics and Education*. Natick, MA: A K Peters Ltd.