Webinar Files
Key Understandings
The National Council of Teachers of Mathematics states, “In understanding equality, one of the first things students must realize is that equality is a relationship, not an operation” (2000–2007).
https://www.learnalberta.ca/content/mepg7/html/pg7_preservationofequality/step1.html
What Is Equality?
In Mathematics, “equal” refers to a relationship not an action or an operation. Equal is used to describe a comparison. We compare amounts, values, and measurement like length, volume, capacity, etc. The equals sign should be read as “is the same amount as” or “represents the same amount or value as”.
If a unit is not identified, the comparison between “how much” or “how many” in each set or how much each set is worth. For example, 1 = 10 would be not equal whereas 1 cm = 10 mm is equal because it has units.
The understanding of equality as a relationship forms the basis for all number properties. The understanding and application of number properties is a significant factor in students building efficient strategies for computation and forms a foundation for success in algebra.
Caution about the “Balance” Metaphor:
Teachers need to be cautious as students’ understanding of weight may interfere with their ability to consider whether or not 2 sets or expressions balance. When 2 lengths or 2 volumes are being compared, the metaphor “balance” has no meaning. We really use balance when we’re talking about equations that are balanced. In order to address all aspects of equality, teachers need to use a variety of models and/or metaphors. When using the balance metaphor, we want to ensure students understand that we use the term as an adjective to describe the state of equality rather than a verb.
Why Is Equality Important?
Determining if relationships are equal or not equal forms the foundation for almost everything we do in math. Applications include work with number lines, situations that require comparing, ordering and determining magnitude and the development of an understanding of the commutative, associative and distributive properties. Distractors that interfere with understanding include color, size, shape, etc.
According to the Alberta Mathematics Kindergarten to Grade 9 Program of Studies (2016), Change and Relationships are two of the components that define the Nature of Mathematics. Equality is about change and relationships.
CHANGE
It is important for students to understand that mathematics is dynamic and not static. As a result, recognizing change is a key component in understanding and developing mathematics.
RELATIONSHIPS
Mathematics is one way to describe interconnectedness in a holistic worldview. Mathematics is used to describe and explain relationships. As part of the study of mathematics, students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form.
https://education.alberta.ca/media/3115252/2016_k_to_9_math_pos.pdf
Mathematics Kindergarten to Grade 9 Program of Studies, Alberta Education, 2014
Why Begin with Equality?
It is important that students first understand relationships between numbers before they are asked to operate with them. Operations emerge as a thinking and communication tool to help us in our quest to determine and prove equality and inequality. The equal sign naturally arises through a need for the formal presentation of determining equality.
The Big Ideas of Equality
- Mathematically, equality refers to a relationship between objects that can be quantified.
- Mathematically, equality can refer to a relationship between units of measure.
- Operations emerge as a way to balance an inequality.
Teacher Background Knowledge
What do I need to know as a teacher in order to be able to teach the concept(s)?
Equality Key Ideas
Program of Study Outcomes
Mathematics Draft K-4 Curriculum (PDF)
Below you will find outcomes from k – 6 related to Equality. There is room for conversation about how/if different outcomes are related to equality…. did all teachers choose the same outcomes?
Evidence of Learner Understanding
What level of understanding do your students have regarding Equality? Use the “Big Ideas of Equality” to guide your teaching. Sample evidence is provided for each Big Idea in order to guide you when assessing your students’ level of understanding.
Research Links
What researchers say about equality:
1. Students in elementary school often misinterpret the equal sign (=) as an operational (i.e., do something or write an answer) symbol even though the equal sign should be viewed as a relational symbol (Sherman & Bisanz, 2009). Students should understand the equal sign as relational, indicating that a relationship exists between the numbers or expressions on each side of the equal sign (Jacobs, Franke, Carpenter, Levi, & Battey, 2007). The number or expression on one side of the equal sign should have the same value as the number or expression on the other side of the equal sign. If the equal sign is interpreted in an operational manner, this typically leads to mistakes in solving equations with missing numbers (e.g., 5 − ___ = 1) and difficulties with algebraic thinking (e.g., x − 2 = 2y + 4; Lindvall & Ibarra, 1980; McNeil & Alibali, 2005b). Research has shown, however, that ongoing classroom dialogue (e.g., Blanton & Kaput, 2005; Saenz-Ludlow & Walgamuth, 1998) or explicit instruction (McNeil & Alibali, 2005b; Powell & Fuchs, 2010; Rittle-Johnson & Alibali, 1999) can change students’ incorrect interpretations of the equal sign.
This misconception, according to researchers at Texas A&M University, can inhibit a student’s mathematical achievement. According to their research, students who exhibit the correct understanding of the equal sign show the greatest achievement in mathematics and persist in fields that require mathematic proficiency like engineering (Capraro & Capraro 2010).
2. TEACHING COMPUTATION WITHOUT ATTENTION TO RELATIONAL AND ALGEBRAIC THINKING ERECTS A ROADBLOCK TO STUDENTS’ LATER PROGRESS. Students must see all math as a search for patterns, structure, relationships, as a process of making and testing ideas and in general making sense of quantitative and spatial situations (Schoenfild, 2008)
3. Liping Ma “Changing one or both sides of an equal sign for certain purposes while preserving the “equals” relationship is the secret of mathematical operations”.
4. Equivalence obeys three properties:
- reflexivity, the relationship relates a thing to itself, A is related to A;
- symmetry, if A is related to B, then the opposite direction is also true, i.e; B is related to A; and
- transitivity, if A is related to Band B is related to e, then A is related to C.
- The most used equivalence relation in mathematics is ‘equals’.
- For the equivalence properties, transitivity appeared to be well understood, symmetry reasonably understood and reflexivity not recognized.
- Equality relationships were commonly seen in terms of balance and students were skilful in returning .relationships to balance when they were unbalanced;
- Equality as transformation was less familiar and older students appeared unable to interpret examples such as x+3=5 in terms of change and reversing change, even’ when an example was worked through with them..
- Researchers found NO EVIDENCE of recognition of reflexivity. Cooper, Rixon, Burnett
More Readings:
Powell, S. R. (2012). EQUATIONS AND THE EQUAL SIGN IN ELEMENTARY MATHEMATICS TEXTBOOKS. The Elementary School Journal, 112(4), 627–648.
Knuth, E., Alibali M. W., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2008). The Importance of Equal Sign Understanding in the Middle Grades. (PDF)
Try This
These activities were provided in the Equality webinar. Choose at least one activity to give to your students. Have them record their responses. Pay special attention to how students get to their answers. Do they have an operational or relational understanding of Equality?
Resources
Programming decisions and the selection of learning resources are made by school authorities, schools, teachers and students. The use of authorized resources is not mandatory. A broad range of learning resources may be used to meet the needs of all students.
The links below will provide you with a variety of resources that may come in handy when teaching the concept of equality. These resources are suggestions and are meant to complement what you are already using. They are not necessarily aligned to Alberta Curriculum. These resources come from a variety of sources and are not affiliated with Alberta Education.
“Understanding the Equal Sign Matters at Every Grade” in the Delta K Journal
Lorway, G. (2017). Understanding the Equal Sign Matters at Every Grade. Delta-K,54(1), 39-43.
Parent Communication
Ideas for Using the Following Information:
The information included below can be used in multiple ways.
- Portions can be included monthly parent bulletins/newsletters
- Items can be discussed at Parent-Teacher Interviews
- Ideas can be used in a Parent Information Night
Message 27 – A Math Message to Families (Seeley, 2009)
NCTM – Support for Families – Helping Your Math Students
NCTM – Algebra – Connect it to Student Priorities!
Newsletter
Alberta Education’s Parent resource area:
https://education.alberta.ca/mathematics-k-6/program-of-studies/
What is equality?
- Equality is when you have an equal number of items in two groups
Why is it important? Children need a strong understanding of equality. It allows them to
- compare quantities and decide if one group has more, less or is equal.
- move from arithmetic to algebra
Simple Activities you can do with your child
How equality changes from year to year?
In Kindergarten
- an understanding of equality starts with students recognizing and naming familiar arrangements of 1 – 5 objects or dots (this is called subitizing)
- students compare two groups of items (groups from 1 – 10) by matching up the items in each group. This allows students to talk about which group has more items or less items or do the two groups the same number of items
In Grade 1
- Students will recognize, name and write numbers 1 – 20, as well as being able to draw objects showing the number (for example the number 7 has 7 objects to draw out), and if they have items (like beads) then students will show that the number 16 means there are 16 beads.
- Students continue to compare two groups of items, and now the groups may have 1 – 20 items in each group.
- Students will talk about the groups, if each group has the same number of items, the groups are “balanced”, if one group has more or less, then the groups are “imbalanced”
- Students will be able to look at groups of items and write the math equation that shows what is happening with the groups. For example, if you have one group with 7 items, and then another group with a pile of 3 and a pile of 4 items, the student will be able to write: 7 = 3 + 4
- If you were to switch the piles around in a different order, for example if you had the first group with 7 items, and then in the second group you changed the piles into a pile of 2 and a pile of 5, the student would know that the two groups are still equal (still have the same number of items)