5.4: Comparison and Patterning

Course Competency 5. Examine strategies that teach early math skills.

5.4. use attributes and objects for comparison and patterning.

Algebraic Thinking

Pause to Reflect

Before reading this section, did you think that algebra was something that should be considered in the preschool classroom? Why or why not?

 

As with Geometry, one might think Algebra is too difficult of a subject for young children. However, early skills in comparison, patterning, and explaining thinking when combining sets (addition) or decomposing sets (subtraction) lay a strong foundation for algebraic thinking." The Wisconsin Model Early Learning Standard that aligns most closely with Algebraic Thinking is B.EL. 4 Uses the attributes of objects for comparison and patterning.

What is algebraic thinking?

As you read the Kansas Mathematics Standards, you will notice the Domain “Operations and Algebraic Thinking” in all grade levels preK grade 12. Furthermore, algebraic thinking and concepts permeate all areas of mathematics. Algebra is more than manipulating symbols or a set of rules, it is a way of thinking. When looking at the Wisconsin Standards for Mathematics, you will also see that standards for Algebraic Thinking are present in all grade levels K-12.

According to the K-5 Progression on Counting & Cardinality and Operations & Algebraic Thinking (2011), algebraic thinking begins with early counting and telling how many in a group of objects, and builds to addition, subtraction, multiplication, and division. Operations and Algebraic Thinking is about generalizing arithmetic and representing patterns.

Algebraic thinking includes the ability to

recognize patterns,

represent relationships,

make generalizations,

and analyze how things change.

In the early grades, students notice, describe, and extend patterns; and they generalize about those patterns.

Completing puzzles in preschool can indirectly relate to algebraic thinking by fostering foundational skills that are important for later mathematical concepts. Here's how:

1. **Pattern Recognition:** Many puzzles require children to identify and match patterns. This skill is fundamental in algebra, where recognizing patterns in numbers and operations is crucial.

2. **Problem-Solving:** Solving puzzles encourages critical thinking and problem-solving skills. These skills are transferrable to algebra, where students must analyze problems and develop strategies to find solutions.

3. **Spatial Reasoning:** Some puzzles, especially geometric puzzles, enhance spatial reasoning abilities. Spatial reasoning is important in algebraic thinking, especially when dealing with graphs and spatial representations of mathematical concepts.

4. **Logical Thinking:** Puzzles often require logical thinking and deduction. This logical approach is valuable in algebra, where students must follow logical steps and rules to solve equations and problems.

While completing puzzles directly addresses skills like pattern recognition and problem-solving rather than specific algebraic concepts, these foundational skills play a role in preparing children for more advanced mathematical thinking, including algebra, as they progress through their education. (OpenAI. (2024, May 10). Completing puzzles in preschool can indirectly relate to algebraic thinking [Response to "Does completing puzzles in preschool relate to algebra?"]. OpenAI. https://openai.com)

Connecting Number and Operations and Algebraic Thinking

Kindergarten

Students in kindergarten solve addition and subtraction problems in various ways as they make sense of and understand the concepts of addition and subtraction. Kindergarten students should see addition and subtraction equations, and should be encouraged to write equations, such as 4 + 3 = 7 and 7 – 3 = 4, but do not require this of students at this level. It is critical that students see the relationship between numbers, and teachers need to provide students those experiences to manipulate numbers using objects, drawings, mental images, etc. so that students can progress from the concrete, to the pictorial, to the abstract levels.

Preschool

Preschool teachers can connect number and operations to algebraic thinking in several ways:

1. **Pattern Recognition:** Introduce children to patterns in numbers and shapes. For example, use simple sequences like 1, 2, 3, 4, and encourage children to predict the next number. This lays the groundwork for understanding number sequences and algebraic patterns later on.

2. **Problem-Solving with Numbers:** Present age-appropriate word problems that involve basic addition and subtraction. Encourage children to use strategies like drawing pictures or using objects to solve these problems. This helps develop problem-solving skills, which are essential in algebra.

3. **Exploring Equality:** Teach children about the concept of equality using balance scales or visual representations. For instance, show that 2 + 3 is equal to 5 and explore different ways to represent this equation (e.g., 3 = 5 - 2). This introduces the idea of balancing equations, a fundamental concept in algebra.

4. **Introduce Variables:** Introduce the concept of variables using simple symbols (e.g., using a star "*" to represent an unknown quantity). Encourage children to solve problems where they need to find the value of the variable, such as in simple equations like 3 + * = 5.

5. **Use Concrete Manipulatives:** Utilize physical objects like blocks or counters to represent numbers and operations. This hands-on approach helps children understand abstract concepts by making them more tangible and relatable.

6. **Encourage Mathematical Discussions:** Foster discussions about mathematical concepts during group activities or circle time. Ask open-ended questions that promote critical thinking and reasoning, such as "How do you know that 2 + 2 is equal to 4?"

7. **Incorporate Math into Everyday Activities:** Find opportunities to integrate math into daily routines, such as counting during snack time or incorporating math concepts into storytelling or games.

By incorporating these strategies into their teaching practices, preschool teachers can effectively bridge the gap between number and operations and the foundational concepts of algebraic thinking, setting a strong foundation for future mathematical learning. (OpenAI. (2024, May 10). How preschool teachers can connect number and operations to algebraic thinking [Response to "How can preschool teachers connect number and operations to algebraic thinking?"]. OpenAI. https://openai.com)

Supporting Algebra and Functions (Classification and Patterning)

Obviously, preschool is not the time to teach traditional algebra, but this is the period when foundational algebraic concepts evolve and gradually develop. Children observe the environment and learn to recognize similarities and differences. They learn to sort, group, and classify objects. They learn to recognize ordering relations, such as large to small, and to identify patterns. They develop the ability to make predictions, form generalizations, and derive rules.

Figure 9.7: As she built this tower, this young girl made a pattern with the colors red and blue.[6]

The information below is from the California Preschool Standards and is again provided as a reference to the age you may see these algebraic thinking skills appear. However, remember the Wisconsin Model Early Learning Standards include the skills for comparison and patterning in order of development but do not assign a specific age to any skills as all children develop at their own rate and time. The standards for Wisconsin emphasize that children should develop skills in categorizing, matching, sorting, using positional words, and patterning. See WMELS B.EL. 4 Uses the attributes of objects for comparison and patterning

Examples in 1.0 and 2.0 would correlate to WMELS B.EL. 4 Uses the attributes of objects for comparison and patterning.

At around 48 months of age	At around 60 months of age
1.0 Children begin to sort and classify objects in their everyday environment.	1.0 Children expand their understanding of sorting and classifying objects in their everyday environment.
1.1 Sort and classify objects by one attribute into two or more groups, with increasing accuracy.	1.1 Sort and classify objects by one or more attributes, into two or more groups, with increasing accuracy (e.g., may sort first by one attribute and then by another attribute).
2.0 Children begin to recognize simple, repeating patterns.	2.0 Children expand their understanding of simple, repeating patterns.
2.1 Begin to identify or recognize a simple repeating pattern.	2.1 Recognize and duplicate simple repeating patterns.
2.2 Attempt to create a simple repeating pattern or participate in making one.	2.2 Begin to extend and create simple repeating patterns.

Teachers can support children’s development of the algebra and functions foundations with the following:

• Organize the classroom into different categorized storage areas to facilitate classification
• Include materials and objects for sorting in the environment
• Identify opportunities for sorting and classifying in everyday routines
• Recognize and extend sorting in play
• Plan for children at different levels
• Integrate sorting into children’s current topic of interest and study
• Point out patterns in the environment
• Engage preschool children in conversations about patterns
• Play with patterns in various formats such as
  • Objects
  • Movement
  • Sounds
  • Rhymes and stories[7]

Figure 9.8: Providing sets of materials in different colors sets up natural opportunities for classification and grouping.[8]

Vignettes

As part of a curriculum unit on the seasons, the children went for a nature walk and collected various types of leaves. During the walk and later in the classroom, the children explored the leaves and were encouraged to describe different attributes of the leaves such as shape (pointy, round, long, needle), size (small, tiny, wide, big), color (red, green, yellow, orange, brown) and texture (smooth, soft, hard, wet, dry, rough). Children were then asked by the teacher to sort the leaves: “Put leaves that belong together in groups.”

The teacher asks Enrique, “Why did you put these leaves together and those leaves together?” Enrique responds, “They are same.” The teacher asks, “How are these the same?” Enrique points and says in Spanish, “Café aquí, amarillo aquí, y hojas rojas.” (“Brown here, yellow, here, and red leaves here.”). The teacher points to each group of leaves and says in English, “Great! Brown, yellow, and red leaves. What other ways can we sort the leaves? How about putting all the big leaves here and all the small ones there?” The teacher models for the child, sorting leaves by size. “Where do you think this leaf would go?”