The relationship between area and perimeter
Area, the two-dimensional space inside a region, and perimeter, the distance around a region, are continually a source of confusion for students. This is in part because students may simply be given formulas to use and not understand the concept of area and perimeter. The two activities below are useful to help students understand area and perimeter.

Activity #1
Give students a loop of non-stretching string that is exactly 24 centimeters in circumference. The task is to decide what different-sized rectangles can be made with a perimeter of 24 cm. Students can be given 1 cm grid paper to place their string on. Each different rectangle can be recorded by students with the area calculated.  Students can also do this activity with just the grid paper by being asked to find rectangles with perimeters of 24 cm.

Activity #2
Provide students with centimeter grid paper. The task is to see how many rectangles can be made with an area of 36 cm squared. Each new rectangle should be recorded by sketching the outline and the dimensions on grid paper. For each rectangle, students should determine and record the perimeter as well. 

​An escape room is a game in which teams solve multiple puzzles using clues, hints, and strategy in order to figure out how to escape from a locked room. It has been found that the use of puzzles and gamification in mathematics increases students’ participation and engagement. The prevalence of escape room businesses has increased in recent years. Escape rooms used in the classroom can provide an enjoyable and memorable challenge for students as they work together in teams. Incorporating escape rooms is one way to engage students, encourage productive struggle, and foster teamwork.

In designing and classroom testing escape rooms I have developed  important principles for preparing escape rooms to be used in the mathematics classroom. These include a unifying theme and a brief backstory, structures to help students persevere in problem solving, and a compelling twist.  The backstory provides information on the context of the problem and what students must do to finish the escape room ​(Stohlmann, 2020).

​Students that participated in the escape rooms have enjoyed the mathematical work situated in a fun challenge and they were able to persevere in problem solving by demonstrating many characteristics associated with a growth mindset. Students commented that the time went by quickly as they stayed focused on completing the escape room. As groups worked on the problems they shared their mathematical thinking and developed their knowledge (Stohlmann, 2020).

​Students also felt a sense of pride in the work and effort that they put forth. For example, a student at the conclusion of one of the escape rooms commented, “I feel so accomplished!” I have also written a book based on the Michael's Movie Moves escape room with example student responses that highlight a variety of strategies that students can use to solve ratio and proportional thinking problems. Included in the book is useful information for teachers and students on proportional thinking strategies including scale factors, tables, unit rates, tape or strip diagrams, double number lines, pictures, equations, and graphs (Stohlmann, 2019). An example from the book is below.  

References:
Stohlmann, M., & Kim, Y.R. (In press). Game-based learning: Robotics and escape rooms. The Australian Mathematics Education Journal.
Stohlmann, M. (2020). Escape room math: Luna’s lines. Mathematics Teacher: Learning and Teaching PK-12, 113(5), 383-389.
Stohlmann, M. (2019). Escape room: Michael’s movie moves. Seattle, WA: KDP.

Technology can allow for new tasks or improved tasks that help students make connections between representations and preserve in problem solving; as well as enables teachers to elicit and use evidence of student thinking in new ways. Recent technology has the potential to enable students to work on higher-demand tasks as delineated by Smith & Stein (1998).

​These tasks make use of multiple representations and focus on mathematical concepts, processes, or relationships. Technology can aid in students being able to make connections between representations. This is done through analysis of real life videos and pictures as well as explorations through dynamic geometric constructions. Students are able to quickly explore different ideas and receive feedback on the results of their actions. Productive struggle has the potential to be enhanced as students progress through technology based activities and receive immediate feedback. Scaffolding can also be incorporated through important questions that focus students’ thinking and also through the incorporation of making other students’ thinking visible to all students. This allows teachers and students to elicit and use evidence of student thinking. When students are able to easily view others’ ideas it can lead to richer discussion and understanding. Mathematical knowledge is then not viewed as solely residing in the teacher but as a shared collaborative knowledge building.
 

Reference:
Stohlmann, M., & Acquah, A. (2020). New directions for technology integration in K-12 mathematics. The International Journal for Technology in Mathematics Education, 27(2), 99-112.