Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Teaching Strategies

This curriculum unit integrates Social Studies Pennsylvania State Standards about  the United States (history and geography), and 5^th grade Math Common Core Standards about place value, decimals and fractions. Teachers could choose to teach all or the part that is relevant to their students. The unit is designed for grades 3^rd to 5^th, but can be modified for grades 1^st to 8^th.

Progression of the Unit

The 1^st group of lessons will focus on observing, classifying and researching U.S. coins, locating National Parks from the 50 states, and historical individuals, especially women trailblazers. Real and fake coins as well as paper illustrations will encourage students to be more active and hands-on when learning about decimal fractions. For instance, students can model $0.30 with 3 dimes and the equation 3 x $0.10 = $0.30, or with 6 nickels and the equation 6 x $0.05 = $0.30. As part of the unit, I will encourage and challenge students to collect as many different types of  U.S. coins as they can, and coins from other countries as well. These class challenges are relevant, especially when schools have a diverse student body like my school [See School Demographics].

The 2^nd group of lessons will focus on using coins to learn more about decimal fractions, the base-10 number system and place values. Different number systems will be discussed to illustrate why understanding the concept of place value is so critical. For instance, the two most commonly used number system are: the decimal (Base-10), and binary (Base-2) mainly because both are easy to manipulated for human use and computer applications. Starting in the school year 2023-2024, my school district is implementing Illustrative Math (IM) as a district-wide math curriculum. The IM framework uses the following teaching strategies: opening routine, formal assessment, whole group guided practice, small group instruction, independent work, and whole group reflective closure. I believe that it is important to use direct instruction, guided instruction, peer tutoring, and cooperative learning, to develop a community of math learners, and independent learning centers for students to practice and master what they are learning. In addition, number lines, tape diagrams, and graphic organizers are excellent tools to enable students to visualize their math thinking as well as build their academic math vocabulary.

Independent Learning Centers

Embedded throughout the unit will be opportunities for students to understand how numbers are written more logically in certain languages, and more misleadingly in some other languages, and discuss the major advantages and disadvantages of the different number systems. For example, I’ve witnessed Spanish-speaking students who struggle to remember the English words for numbers from “11 to 19,” and often say numbers such as 13 as “thirty” instead of “thirteen.” One of my former students used to say “one three” for 13, “one four” for 14, and so on. The words “eleven” and “twelve” give little clues about the based-10 number system and place value. In other languages like Chinese, 11 is written 十 一(tens one), and 12 is 十 二 (tens two) while twenty is 二 十 (two tens). In Chinese, most fractions are written with the denominator before the nominator, example: 1/3 (one-third) is written as 三 分 之 一 (three portions of one) where three is not a whole number, but three parts that make up a whole. Once students learn how to say whole numbers in Chinese, they are able to talk about rational numbers in decimals, fractions, and percentages with the addition of a few more vocabulary words.¹⁹ I would argue that most Asian  languages have a built-in system of relative linguist clarity to make numbers easily to learn and manipulate mathematically. Other examples are: in Khmer, the number six is written as five-one (ប្រាំមួយ); and in Swahili, two-digit number like eleven (kumi na moja) are written with the word “na’ meaning “and” between the two digits.

I plan to set up a learning station for students to work collaboratively or independently with coin collection, and another learning station for students to learn how to write and say the numbers 0 to 20 from different languages. Below are some resources for the two learning centers.

• https://polyglotclub.com/wiki/Language/Multiple-languages/Vocabulary/Count-from-1-to-10-in-many-languages
• https://mathlair.allfunandgames.ca/languages.php 1 to 10 in different languages https://mathlair.allfunandgames.ca/languages2.php 11 to 20 in different languages
• https://www.mentalfloss.com/article/31879/12-mind-blowing-number-systems-other-languages

Opening Routines and Reflective Closures

In recent years, there has been a big push for teachers to use “teacher noticing” and “leading discussion” as an opening routine to allow students multiple entryways to the lesson.¹⁸ Encouraging students to use the strategy of mental math (without the use of pencil and pen) to make reasonable approximation is a powerful tool. Opening routine questions can also be modified to serve as reflective closure activities at the end of a lesson to assess the precision of what students have learned. Below are some approximating questions to help students to build fluency with decimal fractions using coins:

• What is the least number of coins you need to make a total of $1.20?
• What is the greatest number of coins you need to make a total of $1.20?
• Estimate what fraction of 20¢ is 15¢? What fraction of $0.75 is $0.25?
• What is half of $0.50 versus $0.50 divided by one-half?
• Using the fewest number of coins possible, how close can you come to 1/3 of dollar?
• Is 1/3 (one-third) of $1.00 the same as 33% of $1.00? Which value is greater?

Formative Assessment Versus Reflective Closure

I think both formative assessment (FA) and reflective closure serve similar purposes. Their core role is to enable teachers to assess students’ understanding of the learning objective of the lesson. While FA is continuous, typically informal and constant throughout a lesson with teachers giving as many immediate feedbacks as possible to as many students as realistically doable, reflective closure (usually at the end of a lesson) provides time for all students to absorb what should have been learned, and how the concepts are learned through individual writing (sometimes sharing by verbalizing with peers or in group discussion). Reflective closure is an effective way to debrief the lesson in order for teachers to plan the next instructional steps. Sometimes it can be productive to assign homework for reflective closure in order to give students ample time to process what they have learned.

Guided Instruction: Whole Group and Small Group

Guided instruction can be done with a whole group or in small groups. It is a structure whereby a teacher supports each child’s development of math proficiency at increasing levels of difficulty in order to reach all students where they are at, and to take them to the next level. During guided instruction, teachers can use guided questioning, manipulatives like coins, base-10 blocks, as well as visual representations like number lines, tape diagrams and hundredth grids (not the same as hundred grids for whole numbers) to deepen student understanding.