BIG IDEA 1
Understanding the Physical Quantity of Number

The most important goal for any teacher of mathematics, in my view, should be the development of curiosity and wonder in students. When students are curious, they become motivated and inspired to learn anything they set out to learn. There is probably no more important time to spark this journey of curiosity than when students are beginning kindergarten. This becomes particularly important if students are in homes where parents think they should get their children ahead in school by teaching mathematical methods to memorize. If students think their role is to remember inflexible sets of rules, they will be hampered in their mathematical journeys. There is a very strong danger that this approach leads them to believe that this is what math is-a set of rules-that they do not need to make sense of, only memorize. In an international survey given to over 13 million students as part of the international PISA testing, from the OECD, it was found that students who took a memorization approach to mathematics were the lowest-achieving students in every country (Boaler & Zoido, 2016). Unfortunately, the elementary school years are often when students develop the idea, from damaging practices such as timed tests (Boaler, 2014), that mathematics is all about memorization. The opposite of a dry, unappealing, memorization approach to mathematics is one that encourages curiosity and wonder.

We invite students to wonder in all of the activities in this book, starting with Big Idea 1. In the Visualize activity, we ask students to discuss times when they have wondered "How many?" and then give students different groups of objects and invite them to work out how many there are.

In our Play activity, we have chosen photographs for students that include several different things to count, and different ways to count them. We have chosen photographs that offer different ways to count in order to help students know that in mathematics there are usually many different ways to see things and varied ways to approach them, and that different approaches can all be correct if students justify their thinking and give reasons for their approach.

In our Investigate activity, we invite students to make their own books. We recommend that students work in pairs. Each pair is given a number for which to make a page for the class counting book and, if possible, a digital camera. Students can then be taken to spaces where they look for numbers in the world. Together the class makes a whole book that they can refer back to during the rest of the year.

Jo Boaler

References

1. Boaler, J., & Zoido, P. (2016, November 1). Why math education in the US doesn't add up. Scientific American. Retrieved from https://www.scientificamerican.com/article/why-math-education-in-the-u-s-doesn-t-add-up/
2. Boaler, J. (2014). Fluency without fear: Research evidence on the best ways to learn math facts. Retrieved from http://youcubed.org/teachers/2014/fluency-without-fear/

Count a Collection

Snapshot

Students stretch their counting capacity by counting collections of classroom objects and coming to agreement with a partner about how many objects there are.

Connection to CCSS

K.CC.4, K.CC.5, K.CC.1

Agenda

• Collections of various classroom or everyday objects in containers
• Optional: sticky notes or index cards

• Collection to count
• Tools for recording, such as sticky notes or index cards

To the Teacher

Counting concepts are at the heart of kindergarten mathematics, and educators who both research mathematics learning and write for teachers have made counting central to considering early mathematics. The notion of repeated, ongoing counting of collections of objects is the subject of two books that we recommend: Franke, Kazemi, and Turrou's (2018) Choral Counting & Counting Collections and Liu, Dolk, and Fosnot's (2007) unit of study Organizing and Collecting, which is part of the Contexts for Learning curriculum series. Both of these resources provide clear ideas for how to incorporate counting collections of classroom and everyday objects into the long-term trajectory of the early childhood classroom. In this activity, we outline a skeleton of what this work can look like, but the research that Franke et al. have conducted in this area makes clear that counting is less an activity than a routine.

One of the key ideas in counting collections is that the size of the collections can and should be tailored to students' development of counting concepts, such as the oral counting sequence and one-to-one correspondence. When students are getting started, collections will probably need to contain fewer than 10 objects, and for some students perhaps no more than 6. But this number can grow toward and beyond 20 objects as students' counting capacity grows. Indeed, we return to this idea in the final big idea in this book with the premise that students can count toward and beyond 100 objects later in the year. But this growth is only possible with ongoing opportunities to count, recount, and learn to organize objects for counting.

The collections you provide for students to count should be ordinary classroom objects, such as books, cubes, bears, crayons, markers, counters, tiles, or clothespins. Collections should be offered in some kind of container, such as a bag, bowl, bucket, or tray. You'll want to have plenty of collections ready to offer students when they are ready to try a new group of objects. Be sure these collections have a varied number of objects, depending on what your students might be ready for, so that you can direct students toward collections that will stretch their counting capacities.

Activity

Launch

Launch the activity by asking students, Have you ever wondered, "How many?"? Ask students, What kinds of "How many?" questions have you wondered? Give students a chance to turn and talk to a partner about the kinds of situations where they have wondered, "How many?" Listen in as students talk, and revoice some of the questions you hear. You might hear such questions as "How many cookies can I have for dessert?" "How many people are in our class?" "How many stairs are in the staircase?" "How many pets do you have?"

Examples of small collections for students to count

Tell students that asking "How many?" is something that we all do every day. Tell students that today they are going to start figuring out ways to answer those questions by counting groups or collections of objects.

Show students a small group of objects from your classroom. The group should contain more than 5 and fewer than 10 objects. Ask, If my partner and I want to figure out how many there are, what could we do? Model how you and a partner count the group and check with each other by asking, Do we agree? Tell students that they will be working with a partner to count a collection of objects together and that they should come to agreement about how many there are. This might mean counting the group several times until they can agree.

Explore

Provide each partnership with a group of objects to count. Partners work together to count the group of objects and figure out how many there are. Students work to come to agreement about the count, resolving any differences through recounting together.

As you observe students counting, pay attention to how they are counting and look for strategies you might want to have them share with the class. You'll also want to attend to opportunities to sit down with students and help them with the number sequence or to resolve disagreements.

Offer students new collections to count as they finish with and agree on the number objects in the group they've been given. Over time,...