Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Learner Log

A place to explore teaching and learning

Eadweard Muybridge: Photography Pioneer

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Eadweard Muybridge portrait, by photographer Frances Benjamin Johnston, LC-USZ62-33083 (b&w film copy neg.)
Eadweard Muybridge portrait, by photographer Frances Benjamin Johnston, LC-USZ62-33083 (b&w film copy neg.)

English expatriate Eadweard Muybridge, born on April 9, 1830, took daring steps, cutting down trees and venturing into dangerous places, to get landscape photographs that would distinguish him from his contemporaries. See the story of his shot, Falls of the Yosemite, taken in 1872 while on a six-month trip West in Art Through Time, program 10, “The Natural World.”

Read how Muybridge developed photography techniques that captured human and animal movements in new ways in American Passages, unit 8, “Regional Realism.” Muybridge also invented the zoopraxiscope (image #8245 in the archives), a device that projected a moving image from still sequences.

In the video for workshop 6, “Possibilities of Real Life Problems,” of Private Universe Project in Mathematics, ninth graders are asked to solve how fast a cat, captured in a series of photos by Muybridge more than 100 years ago, was moving in frames 10 and 20.

Find a slideshow of 17 of Muybridge’s images of Guatemala in Teaching Geography, workshop 2, “Latin America.” Below each slide is information about the content of each photo and questions to compare the past with the present.

Caught Reading: Measuring Penny

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Engaging students in measuring objects and space around us is active and fun! Remember, measuring is always doing – and we all measure, all the time. Consider how often you hear questions that start like this:

Lauradaughterreading
My daughter and I recently got caught reading Measuring Penny. While reading, we talked about how we measure qualities of the world every day, including length, volume, and time.

How much…

When will it be time to…

How big…

Will this fit…

Do I have enough…

How old…

Is this too much…

How do these compare…

We ask these kinds of questions every day at home, at school, at work, at the store, on the bus, and at the park. Be sure to tell your students how answering these kinds of questions involves mathematical thinking and increases our reasoning and analytical skills. Use examples and point out the connections between investigation, measurement, and number.

While measurement is often viewed as a matter of procedure, there’s really more to it. Think about what features of certain objects and spaces are measurable, and why we even care about size and scale. Think about comparisons. Think about magnitude. Think about precision and estimation; whole units and parts. All of these ideas are fundamentally related to measurement, and contribute in important ways to our awareness of and actions in the world.

Determining the amount, size, or degree of something is necessary and useful. This is clearly illustrated in Loreen Leedy’s Measuring Penny. In this story, Lisa learns a lot about her dog, Penny, by measuring. Lisa measures Penny’s height, compares Penny’s weight to other dogs’ weight, considers how much food and water Penny needs every day, and calculates how fast Penny runs. Lisa is identifying many different measurable attributes of Penny and being her care-taker. She uses different tools and both standard and non-standard units to measure and explore. She is deepening her understanding of quantities and number and enriching her perspectives on her special pal, Penny, and what it means to take care of a pet. Illustrations in the book show us how to measure and why measuring can be so powerful!

Consider using books like this in your classroom. Be sure to highlight these big ideas related to measurement that help us bring conceptual understanding and procedural understanding together.

Big Idea #1: Identifying, describing, and comparing measurable attributes of objects around us helps us make sense of everyday life.

Big Idea #2: Specific techniques, tools, and formulas are used to determine measurements.

Measure and explore with your students, and share your thoughts and reactions in the comment section below. Tell us how you develop understanding of measurement in your work with children.

For additional ideas and materials on this topic, check out

Annenberg Learner’s Learning Math: Measurement for K-8 teachers

Eames Powers of Ten

Teach Students to Like Math

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Teaching Math Library, K-4, program 6, "Animals in Yellowstone" Fourth- and fifth-graders develop number sense and meaning for large numbers by estimating how many bison, elk, and pronghorn they saw on a field trip to Yellowstone National Park.
Teaching Math Library, K-4, program 6, “Animals in Yellowstone”
Fourth- and fifth-graders develop number sense and meaning for large numbers by estimating how many bison, elk, and pronghorn they saw on a field trip to Yellowstone National Park.

No Math = No Fun

Know Math = Know Fun!

Math has a special kind of beauty and appeal to the person who is willing to look. Yet we know there are students who don’t know how to look, and there are students who look and are unable to see. There are also students who don’t want to look because they think they don’t like math or won’t “be good at it.” It can certainly be hard to help all students see math as interesting and approachable, but if we can shift our students’ perspectives, we can make a big difference.

Before you design ways to motivate and inspire your students to like and do math, you have to consider your own perspective on mathematics. What does mathematics mean to you? What appeals to you about mathematics? When is doing math fun for you?

To me, math is patterns, puzzles, and polygons. It’s the path of a ping-pong ball and the structure of a palace. Math is a model of the moon that we can study. Math is really big and really small. It’s interconnected. It’s practical. It’s fun. The power of math is seen in objects around us, processes, and predictions. Without math we wouldn’t have efficiency, production, and invention.

Ask your students, What does mathematics mean to you? What appeals to you about math? When is doing math fun for you? What they say and how they say it brings awareness, and from there you can take action.

Here are several steps you can take to promote new and different ways for your students to see math and to have fun doing math. Let’s get all kids to look at math with interest and enthusiasm!

1: Promote curiosity about how the world works.

Start a list of students’ curiosities. Record questions, confusions, and criticisms. Ask and explore real world situations: What math is relevant here? How might some aspect of mathematics help us understand and describe this situation? Can math help us determine ways to change or improve this situation?

2: Ask students to stop and really look at ordinary objects and events around them.

Take time to notice details. Look for structure. What are the parts of that ordinary object? Look for patterns, relationships, and magnitude. What grows? What shrinks? What repeats? How is this connected to that? What features of objects are measurable? Mathematical ideas are revealed when we look more and see more.

3: Tell your students the story of mathematics. Think characters, setting, plot.

Help students make sense of the origins and evolution of mathematical ideas. Who wanted to know what? Why? Who did what? What happened as a result? The interaction between mathematical thinking and human activity is incredible! Push students to make connections between the past and the present. Use the story of mathematics to help them think about the future.

I have used the ideas above in my own classrooms with students in kindergarten and students in college. I think they’re powerful and exciting. They increased engagement and productivity in my classroom, and led to some of the most interesting learning experiences I ever had as a teacher.

If these ideas are interesting to you, have fun and try them out! Ask those questions. Explore and learn with your students, and share your thoughts and reactions in the comment section below.

The main idea in this blog is inspired by the work and writing of Harold R. Jacobs. For more information, check out his book, Mathematics: A Human Endeavor.
Some other ideas in this blog are inspired by the work and writing of G. Polya. For more information, check out his book, How to Solve It: A New Aspect of  Mathematical Method.

For additional ideas and materials on this topic, check out

Mathematics: What’s the Big Idea? This video workshop offers motivation and tools for K-8 teachers who want to explore ways of changing how they teach math.
Teaching Math Libraries, K-4, 5-8, and 9-12 These video programs demonstrate how teachers guide and assess student understanding, and offer strategies for keeping students motivated and engaged.

9 Ways to Encourage Play for Kindergarten Day and Every Day!

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ArtsEvery_11Time to pull out the blocks and finger paints. Kindergarten Day recognizes the importance of play, games, and creative activity in children’s education. In 1837, Friedrich Froebel, born April 21, 1782, established the first kindergarten in Germany. German immigrants brought the idea to the U.S. in the 1840s. In 1873, the first public kindergarten was started in St. Louis, MO.

Kindergarten classrooms of the past provided oodles of time for students to use their imaginations, develop social skills, and learn to love learning. As the arm of standardized testing reaches into the earliest years of childhood development, concerns are raised about the disappearance of play experiences. Read about why playtime is important for young students in this report from the Alliance for Childhood.

Meanwhile, in the spirit of Friedrich Froebel, we present the following ideas for using play to teach literacy and math skills, as well as concepts for social studies and science:

1. Students learn to appreciate different cultural backgrounds as they explore holidays such as the Chinese New Year and Valentine’s Day in Teaching Reading K-2 Library, program 3, “Building Oral Language.” Sensory activities and crafts are combined with reading and writing activities to help students make connections.

2. Chuck Walker pairs kindergartners with 6th graders for counting activities located inside and outside of the classroom in Teaching Math, A Video Library, K-4, program 3, “Math Buddies.”

3. Students learn about story structure and engage their imaginations when theatre artist Birgitta De Pree visits the classroom in The Arts in Every Classroom: A Video Library, K-5, program 10, “Bringing Artists to Your Community.”

4. Thalia’s teachers tap into her interests and add whimsy with song and drawing to literacy lessons for this energetic kindergartner in Teaching Reading K-2 Library, program 4, “Thalia Learns the Details.”

5. Young students learn mathematical concepts while playing with different types of manipulatives in Teaching Math, A Video Library, K-4, program 7, “Cubes and Containers,” program 12, “Dino Math,” and program 43, “Beans, Beans, Beans.”

6. Students understand economic concepts of supply and demand while working together to make bread in Social Studies in Action, A Teaching Practices Library, K-12, program 6, “Making Bread Together.”

7. In Ms. Mesmer’s classroom, students participate in a variety of fun activities to compare holidays, while learning about seasons and the earth’s rotation around the sun. See Social Studies in Action, A Teaching Practices Library, K-12, program 8, “Celebrations of Light.”

8. Watch students practice their French vocabulary using song, movement, and cut-and-paste activities in Teaching Foreign Languages, K-12: A Library of Classroom Practices, program 4, “Chicken Pox.”

9. A kindergarten class mixes with a 4th-grade class to create an original performance based on Quidam by Cirque du Soleil in The Arts in Every Classroom: A Video Library, K-5, program 11, “Students Create a Multi-Arts Performance.”

What are ways you are using play in your kindergarten classrooms?

Using Representations in the Math Classroom

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TM K-4 marshmallows copy
Second-graders create and discuss a bar graph based on the number of marshmallows they estimate each person in their class would eat on a camping trip. From Teaching Math Library, program 10.

“Show your work.” Teachers ask students to show their work to get a glimpse of how they are thinking through a problem. However, showing work is not just useful to the teacher. When students create representations (by drawing charts and diagrams, using manipulatives, etc), they work through their understanding of math concepts and develop problem-solving skills.

According to the NCTM Representation Standard, instructional programs . . . should enable all students to:

•    Create and use representations to organize, record, and communicate mathematical ideas
•    Select, apply, and translate among mathematical representations to solve problems
•    Use representations to model and interpret physical, social, and mathematical phenomena

The Teaching Math online courses look at various ways students from grades K-12 may represent their mathematical thinking. Observe student examples of representations and practice using representations to solve problems. See the session 5, “Representation,” section of each grade band: K-2, 3-5, 6-8, 9-12.

See examples of different manipulatives in use for the algebra classroom in Insights into Algebra, workshop 1, “Variables and Patterns of Change.” Examples include teacher-made manipulatives –– cups used to represent the coefficient of the variable and chips used to represent the constant terms –– or virtual manipulatives in the form of a computer program allowing students to examine graphs of change.

Students estimate the number of elk, bison, and pronghorn in Yellowstone Park in program 6, “Animals in Yellowstone,” of Teaching Math: A Video Library, K-4. At 11:50 in the video, the teacher asks students to explain and adjust their placement of estimates on a number line.

Students work in groups to identify a pattern and find a rule that determines that pattern in Teaching Math: A Video Library, 9-12, program 6, “Staircase Problem.” Students must explain their thinking using charts and paper blocks.

More resources for using representations in math classrooms:

Private Universe Project in Mathematics, workshop 6, “Possibilities of Real Life Problems”

Learning Math: Number and Operations, session 4, “Meanings and Models for Operations”