Math does not have one clear definition. In fact, Farrell and Farmer (1980) argue that mathematics is a verb, as well as a noun. Sarama, et.al (2018) consider mathematics as a noun as they note:

“Mathematics is the study of quantity, structure, shape, and change. It provides a foundation for many aspects of daily life, including for much of science, technology, and engineering. The mathematical sciences include more than numbers and arithmetic—they also deal with such topics as geometrical figures and structures, measurement, and logical argumentation. Mathematicians and children doing math use the practices of mathematics to identify crosscutting patterns and structures and to understand and explain phenomena.” – p.1, CADRE report, by Sarama et al., 2018

Recent efforts in mathematics education, including the introduction of the Common Core State Standards for Mathematics, focus on mathematics as a verb as they call for students to be involved in “doing mathematics” in the classroom. This raises the question: What does it mean to “do mathematics”? Doing mathematics involves giving students opportunities to make sense of mathematics through exploration. It involves engaging students in solving rich problems and asking them to justify and defend their reasoning. For more on how to support students in doing mathematics, see https://www.oerafrica.org/system/files/8230/unit-1-pdf_0.pdf.

We believe that students can be given the opportunity to engage in high-level, quality math lessons when they are integrated with other subject areas, including those of the STEM field. The example projects below were a part of our STEM Collaboration project, but their development began with a math standard and used a supporting science standard, instead of the reverse. The first project, Shape Hunt and Plant Parts, is for first grade and focuses on shapes, with a science emphasis on plant parts. The second project is from fourth grade, with a math emphasis on angles, while the science standards drew from technology and engineering.

First grade- Shape Hunt and Plant Parts

DCI Math and Science

Standards

Math 1.G.A.1 Distinguish between attributes that define a shape (e.g., number of sides and vertices) versus attributes that do not define the shape (e.g., color, orientation, overall size); build and draw two-dimensional shapes to possess defining attributes.

1.G.A.2 Create a composite shape and use the composite shape to make new shapes by using two dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, rectangular prisms, cones, and cylinders).

Science

1.ETS2: Links Among Engineering, Technology, Science, and Society 1) Use appropriate tools (magnifying glass, basic balance scale) to make observations and answer testable scientific questions.

1.LS1.1 Recognize the structure of plants (roots, stems, leaves, flowers and fruits)

Authors Alissa Lange, Amie Craven, & Jamie Price

5Es

Engage. Students will begin this unit in a whole group as teachers read the book, I Spy Shapes in Art (or other shape composition book, like The Shape of Things), to the class. This book asks students to find shapes in various paintings and the class will go through this book and talk about the shapes they find on the pages. Teachers will name the different shapes and discuss the shapes characteristics throughout the book. Students will be thinking about shapes, their characteristics, and how to look for them in their environment, especially looking for shapes related to growing things, like trees and flowers. This helps students link their past knowledge and creates interest for the focus of the unit.

Explore. Teachers will provide students with a paper showing different shapes, especially focused on quadrilaterals, which can be problematic for students (Oberdorf & Taylor-Cox, 1999). Students will be instructed to go on a shape hunt around the classroom, working to find all the shapes on the paper. Once they find the shapes, students will draw the object they found. They will also write the number of sides for each shape, and any other attributes they notice, demonstrating their understanding of attributes that distinguish shapes. Students will share their findings in a whole group discussion. Later, they will create their own shape art, inspired by the book and the shape hunt. They may first be creative with their art, then they’ll be asked to focus their creations on a plant of some sort, like a flower or tree.

Explain. To extend learners knowledge of shapes, teachers will show the video “What are Composite Shapes” to begin talking about how some shapes are composed of other shapes. Teachers will draw composite shapes and discuss the shapes used to create them. Teachers will answer students’ questions throughout the discussion and continue to point out different attributes of the basic shapes to reiterate characteristics that define the shapes. This activity will extend students’ understanding of shapes and provide them with additional formal vocabulary.

Elaborate. On this day, teachers will give students access to a variety of pattern blocks and pattern block puzzles that are shaped like plants. Students will choose a puzzle and then use the patterns blocks to complete it. They will then be asked to label the puzzle with the plant parts, and explain to a peer how they know it is that plant part. After students finish their puzzles, they will be asked if they can fill in their plant puzzle in another way – such as two trapezoids instead of one hexagon. Students will be able to apply their new learning about composite shapes by observing how shapes make up other shapes and experience changes in surface temperatures. If students need a challenge, they can create pattern block puzzles for their peers to complete.

Evaluate. Learning will be assessed individually with a teacher and in pairs. First, they will work in pairs to create composite shapes with a partner and they will discuss characteristics of the shapes and describe how they are building them. Teachers will have students create composite shapes as well as other shapes from the composite shapes and ask them various questions about what the shapes were created from. Students should be able to identify different attributes of the shapes, such as the number and length of sides, perhaps starting to notice differences in angles. They should also be able to create a variety of composite shapes and identify what shapes were used to create them. Teachers will take anecdotal notes and use a checklist to assess what students learned.

4th grade- Mirror Maze Experiment

DCI • Math • Science Standards

Math

4.MD.C.5a: Understand that an angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle.

4.MD.C.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

Engineering and Technology

4.ETS2: Links Among Engineering, Technology, Science, and Society

1) Use appropriate tools and measurements to build a model.

2) Determine the effectiveness of multiple solutions to a design problem given the criteria and the constraints.

Author- Adapted from the Unit Plan created by Taylor Spence, 2020

Summary

This unit explores the relationship between the angles of mirrors and the number of reflections shown. Students will be introduced to the idea of a mirror maze through a video which shows how difficult it is to complete the maze. Then, learners will conduct a hands-on experiment using an object, mirrors, and a protractor to see how changing the mirror’s angle affects the number of reflections of the object. Once they have completed the experiment, students will discuss their findings with peers and graph their data using bar graphs and Venn diagrams. Students will also revisit the video they watched at the beginning of the unit in order to see if they are able to give a more accurate explanation for why the maze is so difficult. Teachers will assess students’ learning through a test which involves a Claim, Evidence, Reasoning explanation  (CER), math concepts about angles, and hypothetical changes to the experiment and what their affect could be on the number of reflections. An overview video can be found here.         CER Checklist     CER Student Graphic Organizer

5Es

Engage. Students will begin this unit, watching the Ellen Game of Games Video, called Mazed and Confused. In this video, contestants are put in a maze filled with mirrors. Their objective is to find a ball and get out of the maze. Throughout the video, contestants bump into the mirrors and find it difficult to get out. Following the video, the students will engage in a discussion about what they saw. These questions will get the students thinking about what was happening in the maze, especially why it was so difficult for them to get out. Students will also share their own experiences if they have ever been in a mirror maze. This activity generates interest in the unit while also allowing students to share prior knowledge they may have related to the topics associated with the unit.

Explore. Students will then perform an experiment about angles and reflection with mirrors. The teacher will have a variety of small objects for students to choose from that can be reflected in a small mirror. These objects might include a small toy car or a figurine. Students will also select 2 mirrors and a protractor to use during the experiment. Students will be instructed to set the mirrors up like they were in the video (see project overview video) they had watched. They will place the protractor underneath the mirrors and adjust the mirrors along the protractor to produce different angles. Students will create 5 angles, with the requirement that there are 2 acute and 2 obtuse angles. Then, students will place the object they chose between the mirrors and measure the number of real and reflected objects they see. They will repeat this process for each angle. Students can experience the key concept of this unit and make connections between the size of the angle and the number of reflections observed.

Explain. On this day, students will demonstrate their learning and discuss their findings from their experiment with mirrors conducted during the Engage phase of the unit. Students will be given questions to assess their basic knowledge of angles as well as questions asking them to express their findings from the experiment and graph them using bar graphs and Venn diagrams. Students will work with a partner for discussion and comparison of their data. They will also explain why the dimensions of their objects affected the amount of reflections they saw in the mirrors. This activity asks students to communicate their new knowledge and use formal language associated with angles.

Elaborate. Students will revisit the mirror maze video they watched at the beginning of this unit and engage in a discussion to see if they are able to more accurately explain why the mirror maze was so difficult for the people in the video. They will also estimate what the measures of the angles of the mirrors that were in the video and identify the angles that would produce different numbers of real and reflected objects. This discussion asks students to revisit their original thinking and apply their new learning.

Evaluate. Students’ learning will be assessed through a test that asks students to discuss math and science concepts they have explored, as well as a CER for the experiment they completed. Students will be asked to discuss what they learned through the experiment and their data, math concepts related to angles, and what might happen if there were changes to the experiment (object size, flat mirrors, tools that could work instead of mirrors, etc.). This will assess students’ understanding of angles while also showing their evidence of accomplishment in the activity.

Reference

Lange, Alissa A.; Robertson, Laura; Price, Jamie; and Craven, Amie. 2021. Teaching Early and Elementary STEM. Johnson City: East Tennessee State University.
https://dc.etsu.edu/etsu-oer/8

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License