A child approximates the number of cubes in collections of increasing quantity. After guessing big number words such as “thousand” and “hundred,” she looks for smaller groups within the larger set. Her estimate of three groups of seven is quite accurate. (There are 27 cubes.)

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

A child estimates the number of cubes in a collection.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

A child directly compares the lengths of two crayons to determine which one is shorter.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

At one learning lab, participants observed a fun project tradition. As has been true for years, the authors of this year’s Idea Book were in the room for its unveiling and discussion. The Idea Book allows program participants to submit write-ups and photos of activities they used to make their math lessons more engaging and to put their own spin on the research lessons discussed in the learning labs. They are organized according to the major content strands and printed in a spiraled book so that even though the program is over, these Big Ideas remain accessible and inspiring for years to come.

Since a recent session addressed important ideas of geometry, the discussion of data analysis didn’t get much attention—in the workshop, that is. However, the Idea Book includes a variety of great ways to make surveys and analyzing data meaningful, engaging activities.

Heather Duncan did a particularly striking job of explaining how she and her kindergartners at South Shore Elementary have made collecting data from surveys and discussing them a regular part of their classroom life.

Heather prefaced her description of their process by pointing out how important it is to take time to really explore the ideas and emphasize the children’s work and thinking. She typically does a process that stretches from three to five days and always ends up with a product everyone can see.

• Day One, they collected results of a survey, usually related to a topic they were already studying. For example, while studying the human body, Ms. Duncan had them compare each student’s height to a cut-out of an average kindergartener hung up in the class.
• Day Two, they synthesized the results. “We used the chart,” she wrote, “to count and determine totals for each designation (taller, shorter, same size) and whether any groups have more/less or the same amount as any others. We also compare totals from the a.m. class with the p.m. class and discuss any similarities or differences in results.”
• Day Three, they further discussed the results and talked about the best way to show them on a graph. In this case, they decided on a bar graph.
• Day Four, they used the resulting graphs to make some conclusions. She wrote, “Which group had more or the most? Which had less, the least, or the fewest? What do we know about the question or our class as the result of our survey?”

They turned the results into a poster, a bulletin board display, or part of a “trivia book” at the end of the school year. Ms. Duncan comments, “[The activity] has since become practice, because we have found it to be a very rich experience. It’s been invaluable using math in context to study things we care about and to answer questions generated from our own interests.”

It is tough learning two new languages at once. In the case of English Language Learners (ELL), the two “languages” are often English and mathematics. Young children in classrooms where the language of instruction is not their dominant language can have challenges when it comes to learning math language, particularly positional words. These children are not only learning concepts about relationships in space, which are challenging on their own account, but they also have to respond to the demands of comprehending in a new language.

The difference between “through” and “between,” for example, could be a difficult distinction for ELLs. Teachers can help students learn these distinctions by amplifying language.

Here is an idea. With small groups of children, create an obstacle course in your classroom where children would need to do the following: walk between two chairs, go under a table, skip around the water table, climb over a cardboard box, and crawl through a tunnel. As you model each movement, use language to describe what you are actually doing: “What am I doing? I am walking between two chairs,” putting an added stress on the positional word “between” and the word “two.” When it is their turn to go through the obstacle course, you can continue to ask all the students what each child is doing.

Then you can invite the students to verbally repeat the sentences, describing their movement as they go through each obstacle in the course. When everyone has completed the entire obstacle course, you can ask, “What did we do in this obstacle course?” They would say: “First we went between two chairs, then under the table, then we went around the water table, then over a cardboard box.” On another day, you can ask the children to create a different obstacle course and once again intentionally amplify language to help them learn the labels for the positional words.

Once children have experienced the action and been exposed repeatedly to the label for each prepositional word, you can continue to amplify the language by matching those words with drawings that illustrate the concept they represent.

Using the above activity enables ELLs to learn not only the labels for the positional words but gives them the linguistic context they need to use them. Doing the language activity with children answering all together provides differentiation in terms of levels of English proficiency. For children who are in the very early stages of English language development, techniques that combine language with action provide repeated exposure to conventional models of English. For children who are more fluent, the verbal component gives them the opportunity to use English in a conventional fashion without fear of being singled out.

Once children have experienced the action and been exposed repeatedly to the label for each prepositional word, you can continue to amplify the language by matching those words with drawings that illustrate the concept they represent. You can make these drawings into cards that can be used for games or to give students directions for lining up or during other transitions throughout the day.

These strategies to make content comprehensible and promote language development are beneficial to all students, but they are essential to ELL students in order to learn foundational concepts and the language we use to describe them.

Watch a lesson like the one described here and check out these other resources for ELL teachers.

Tana Hoban and Ann Morris are both gifted children’s book authors who combine minimal text with wonderful photos that beg to be pored over again and again. Many of them are organized around ideas that call for mathematizing. For example, just the cover of Hoban’s Shapes, Shapes, Shapes or Cubes, Cones, Cylinders, and Spheres will make it easy to engage children in wondering what regular geometric shapes they can spy in the classroom or on a walk around the school. Be sure the children record their findings by drawing or making tally marks so that they have data to analyze when they get back to the classroom. As you guide a discussion into asking what shape seems to be the most common, be sure to include the all-important follow-up: Why? Why do you think that is so? Why do you think you found many more rectangles than triangles?

In the same way, Ann Morris’s Shoes, Shoes, Shoes and Bread, Bread, Bread can jumpstart a survey of the different kinds of shoes the children are wearing or the kind of bread they like, as well as encourage some serious thinking about good ways to categorize them. Make sure you have them show the results. Once again, the best thinking comes as the children and teacher converse together about what questions they have and how they see the data making sense.