Equal Sign Means Relationship

Blog post #3 in the series
Lies We Tell Our Students

The precision of our explanations matters! The words we choose and the way we put those words together can make all the difference between teaching children “tricks” and teaching for understanding. Imprecise language is another way we inadvertently lie to our students.

Today, we are turning our attention to the commutative property. Without thinking about it, we may very well say something similar to the quote in the illustration above, “With the commutative property, order doesn’t matter.” Hmmmmm…is that the truth, or is there an inadvertent lie hiding somewhere?

The Story

Here’s the story I like to use to get things started. Let’s say my doctor tells me to take 3 pills per day for 9 days. So he gives me 27 pills. Now…is that the same thing as taking 9 pills per day for 3 days? The total is still 27 pills. In this context, however, 3 x 9 is certainly not the same thing as 9 x 3. In this context, order does matter! (Disclaimer: please do not use this context with children!)

The (Inadvertent) Lie

Without intending to do so, the teacher who says that “order doesn’t matter” with the commutative property is teaching an arithmetic trick that could very well lead students astray later on.

The Truth

Instead, we want students to understand that the commutative property allows us to mentally add or multiply in either order to get to the same total (sum or product). When decontextualized, the numbers can be reversed and still provide the same result: the order can be changed to find the total of two addends, and the order can be changed to find the total of two factors. The language we use needs to be precise enough to indicate that the total is what stays the same, not necessarily the behavior.

Using the story from above, I like to represent 3 x 9 and 9 x 3 on a ten frame. In the pictures below, you can see 9 groups of 3 on the left and 3 groups of 9 on the right.

9 x 3 = 27

You can clearly figure out that the total is 27 for both 9×3 and 3×9. However, are they really the same thing? Is taking 3 pills per day for 9 days the same thing as taking 9 pills a day for 3 days? Of course not! The total is the same, but the context leads to entirely different behaviors!

In Summary

To sum up this week’s Lie We Tell Our Students:

• The (inadvertent) lie: According to the commutative property, the order of the addends (or factors) doesn’t matter.
• The truth: According to the commutative property, you can add (or multiply) two addends (or factors) in any order and get the same total.

As always, we love hearing from you! Do you think your students fully understand the meaning of the equal sign? Have you caught your students (or yourself) using imprecise language about the commutative property? Please share your stories in the comments box below.

Next Steps for Teachers: Take time to think through your mathematical explanations beforehand. During your planning time, consider the many concepts you will teach during the upcoming lesson. Mentally rehearse your explanations and listen for imprecise language that might get in the way of understanding.

Next Steps for Leaders: Initiate discussions during collaborative planning time. Ask groups of teachers to discuss concepts for which they might be using imprecise language. Here are a few ideas to get you started: rectangles have two long sides and two short sides, squares have four equal sides, the perimeter is the outside, “top number” and “bottom number” for numerator and denominator, reducing fractions, borrowing & carrying, using the word “makes” for “equals.” 

Blog post #4 in the series, Lies We Tell Our Students

Let’s be clear…this “lie” is not really about counting. The misconception we address today centers on the notion of unit. When young children are learning to count, whether by rote or by counting objects by ones, fives, tens, etc., the pattern of the count results in a quantity that is larger (when moving from left to right on the number line) than the previous number.

What we’re really talking about in this post is this: When students are associating numerals and quantities, it’s important that they realize very early on that the unit matters. Whether they are counting marbles or elephants or apple halves or boxes of pencils, it’s important that the total amount in the set be identified with the common unit size of each item in the set.

The Story

Christina’s kindergarten class regularly engaged in “cooking” as part of their reading and math lessons. One day, she brought in the ingredients to make s’mores and posted the recipe for all to read. The class read the recipe together: each child needed 2 graham crackers, 4 chocolate pieces, and 1 large marshmallow.

Christina said to the class, “I brought 2 boxes of chocolate pieces for us to use for today’s recipe. Is that enough chocolate for everyone?” As was their routine, the students paired up and discussed this question with a partner for 30 seconds. When they came back together, one of her students said, “We don’t think you have enough! The recipe says we each need 4 chocolates, and you only brought 2.”

The (Inadvertent) Lie

We teach children that when they’re following the counting sequence, a number that comes “first” is always less than the number(s) that come next. Of course, this is true with the counting sequence when we’re rote counting or counting objects that represent a common unit. However, young students (or even older students working with fractions) have often overgeneralized the notion of counting, so if they ignore the unit, they will misunderstand how to make the comparison.

The Truth

What we really mean is that when children are counting “up” (with or without context), subsequent numbers are greater than previous numbers, assuming we’re counting from left to right on the number line and only when the units being counted are either the same or are directly comparable.

However, when the units change, the relationship between the numbers change, as well. For example, Which is greater, 2 dozen pencils or 5 pencils? As adults, we make this shift quickly and notice that we have to convert the “2 dozen” to “24” in order to make the comparison. However, young children may simply pick out the numbers and try to compare them directly, not realizing that they must make a unit conversion before they can determine which is number is greater. Therefore, they may think that “5” is greater than “2,” not realizing that the 2 dozen actually represents 24. Once they realize they are comparing 24 pencils and 5 pencils, the task is once again rudimentary.

Next Steps for Teachers: Provide students with many opportunities to deal with comparing quantities when the objects are described by different units. Examples include pencils & boxes of pencils, eggs & cartons of eggs, apples & bags of apples, cookies & bags of cookies, crayons & boxes of crayons, paper & packages of paper…the sky is the limit. Present the problem in writing first, to see if students recognize the need for “unpacking” to get to a common unit. Then, be sure to provide the real objects so they can see how the objects in one set are grouped differently than in the other set. Then, students can mentally or physically manipulate the objects so they are comparing items described by a common unit.

Next Steps for Leaders: Include a 5-minute math minute at the beginning of a grade-level meeting or staff meeting. Use the idea of “unit” to stimulate a conversation about how to help students at different levels grapple with the notion of unit. For K-2, it’s a matter of packing and unpacking  objects of different units so they can add or subtract. In grades 3-5, this is an important way to help students understand multiplication and the order of operations. In grades 3-8, teachers will want to venture into the world of fractions, where unit plays a major role in fraction comparisons and operations. And in all grades, K-8, context problems rely heavily on this notion of unit.

In Summary

Counting is not just for kindergarteners!!!! The notion of unit resurfaces multiple times during the elementary years. Solving word problems, comparing and ordering whole numbers, performing multiplication in context, working with measurement units, value-counting money, telling time, working with fractions…all of these require a deep understanding of units and how to “fairly” compare them. We love hearing from you. Please share your thoughts on how you might introduce this idea to the students you work with. You can write your ideas in the comments box, below.

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.

Blog post #5 in the series, Lies We Tell Our Students

Many of the mathematics “rules” we teach students in the early grades come back to haunt them later on. Far too often, we teachers neglect to account for number sets beyond which our students are working in at the moment. Unfortunately, this disconnection leads us to inadvertently teach procedures that are misleading and fall apart in later grades.

Take, for example, ways in which we teach students to compare numbers to determine their relative values. We teach them to “line the number up on the right” and then look to see which number is “longer” (aka, has more digits).

Yikes! I’m sure you already see the error in this logic. However, our young children follow the rules we teach them to a tee, generally without question. And then, when they get to the point in later grades where this “rule” has expired (e.g., when decimal fractions are introduced), they don’t know what to do.

The Story

Ms. Jackson, fifth grade teacher, just started her unit on decimal fractions. She did a great job of setting her students up for success by teaching important concepts about fractions, units, and the difference between decimal and fraction notation. However, every time she asks her students to compare two decimal numbers, she notices that the majority of her students keep lining up the number using the digits on the furthest right-hand side rather than lining up the numbers at the decimal point, the point that separates the whole number from the fraction. She is reaching an unexpected level of frustration as her students just don’t seem to “get it.” What’s going on here?

The (Inadvertent) Lie

In the early grades, students are only exposed to whole numbers and simple fractions. Furthermore, they only see fractions written in fraction form (aka, with a fraction bar), rather than written as base-ten fractions (aka, with a decimal point). Therefore, when comparing base-ten whole numbers, their teachers frequently teach them to compare the values of numbers by “lining them up at the right.” Although this process works for whole numbers, it is not generalizable to all numbers. This strategy becomes inappropriate once the base-ten numbers are extended to fractional amounts.

The Truth

Part of the issue here is that students are introduced to number sets incrementally. We tend to focus on whole numbers in K-3: 0-20 in kindergarten, 0-100 in first grade, and 0-1000 in second and third grade. Fractions are introduced in grades 1-3, but only in fraction form, and usually only halves, thirds, fourths, sixth, and eighths. Therefore, when students are introduced to decimal fractions for the first time in fourth grade, they have not grappled with the nuances of working with decimal numbers with digits on both sides of the decimal point.

Nor have students grappled with the decimal point itself. Even though they are introduced to money concepts and notation in K-3, students have not had the opportunity to develop an understanding of decimal notation. They’ve only worked with whole numbers with little to no exposure to decimal numbers and decimal notation. The truth is, we are leading them to an inaccurate understanding of the number system and inadvertently teaching misconceptions about using decimal points.

So…what can we do about this?

Next Steps for Teachers

• Show young children that the decimal point goes to the right of the whole number portion of a decimal number. Be sure they understand that it’s not necessary to include the decimal point when writing whole numbers — it is implied. This is about exposure, not mastery.
• Use the example of money with young children to emphasize what a decimal point is and how it separates dollars and cents. Help students understand that dollars are whole numbers and cents are parts (fractions) of a dollar. You don’t need to get too specific here – just introduce them to the idea that a decimal point separates wholes from parts.
• Assist fourth- and fifth-grade students with the transition. Acknowledge that since they’ve only worked with whole numbers, they may have generalized a rule that is no longer appropriate.

Next Steps for Leaders:

• Facilitate vertical articulation conversations about how number sets are introduced across grade levels and how to talk about decimal numbers in the early grades. The conversation should include developing common language for talking about whole numbers, decimal fractions, decimal notation, and number comparisons.
• Lead a conversation about how teachers in grades K-3 will talk about decimal numbers and notation to avoid teaching in ways that lead to misconceptions. This conversation should also include ways in which 4-8 teachers might connect and expand upon what was taught in the primary grades.
• When observing math lessons, stay attuned to how teachers talk about whole numbers, decimal numbers, and decimal notation. Remind teachers about the agreed-upon language to help students work on these concepts across the grades.

In Summary

When we teach students to compare numbers, the way we talk about those numbers matters. In grades K-3, students work only with whole numbers. In this case, primary teachers often use an over-simplified rule such as “line up the digits on the right-hand side” as a means for comparing numbers. This approach can lead to a false generalization and misconceptions about how decimal numbers are compared once there are digits on both sides of the decimal point. Finding ways to talk about decimal numbers and decimal notation in grades K-3, prior to working with decimal fractions, plays an important role in facilitating learning across the grade levels.

What are your thoughts about teaching decimal number comparisons in the early grades? How can we connect whole number work with decimal fraction work in a way that builds student understanding over time? We would love to hear from you – please leave your comments in the box below.  

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.