Our book study of Teaching Student-Centered Mathematics PreK-2 by Van de Walle, Lovin, Karp, and Bay-Williams continues today with a discussion of chapter 13. You can read past posts by visiting our book study archive.
Chapter 13 – Promoting Algebraic Reasoning
There are many aspects of this book that are wonderful. I love how the activities the authors list can be used in my classroom. I love how the mathematical practices are listed on the sides beginning with Chapter 8. I love the expanded lesson at the end of the chapters, and I really love the big ideas at the beginning of the chapters. Chapter 13 focuses on Promoting Algebraic Reasoning with pre-K – 2 grade students. The big ideas of this chapter are:
- Algebra for students this age is more recognizing and extending repeated and growing patterns, and that it should be generalized.
- Algebra is a useful tool for generalizing arithmetic and representing patterns and regularities in our world.
- Symbolism, like the equal sign and variables, must be understood conceptually.
- The structures in our number system can and should be generalized.
When Common Core first came out, I remember hearing, “They want 1st graders to do algebra?” Well, of course they are not going to be doing algebra like many of us remember from high school, but the core idea of the algebraic reasoning, or thinking, these students are doing is looking for and finding relationships and building a structure with those relationships. This type of reasoning prepares them to think mathematically and use math as they grow.
Strands of Algebraic Reasoning
There are 3 strands of algebraic reasoning. Generalizations and symbolism are included in all three strands:
- Study structures in the number system, including those arising in arithmetic
- Study patterns, relations, and functions
- Process of mathematical modeling, including the meaningful use of symbols.
To generalize a concept, students use specific examples to identify things in common that can be used to describe any example of the concept. Students also need the chance to explore and experiment on their own with numbers and problems. Teachers should foster this exploration by asking questions, not telling them exactly how something is done. They will eventually begin to strategically use generalizations to problem solve. The activity One Up and One Down with Addition, on p. 227, focuses on children’s attention to adjacent facts in an effort to help them generalize and reason more strategically.
The hundreds chart can also be a beneficial tool when getting students to generalize algebraic concepts. Students can generalize that one is adding 10 from numbers when asking, “What did we get when we add to get from 72 to 82? From 5 to 15? From 34 to 44?” When students begin understanding that they are moving down exactly one row, they are deepening their understanding of number concepts and how adding 10 changes a number.
Meaningful Use of Symbols
Ah….the equal sign. When asking many 5-8 year olds what the equal sign means, many would say, “It means the answer”. We must change this thinking. In first grade, the Common Core standards say students understand the meaning of the equal sign. It starts early because it is essential that students fully comprehend that what that initial thought was, is not true. In my classroom, I constantly use the phrase “the same as” when reading an equal sign. I also write the majority of equations 8 = 7 + 1. At first some students may say I wrote it wrong, but then I ask them why. I give a little informal assessment when asking this question, it lets me know who still needs understanding of the equal sign. I like how the authors suggest having the students write two expressions on both sides of the equal sign instead of just one equation.
Why is it so important that children in grades pre-K – 2 correctly understand the equal sign?
- Children need to understand and symbolize relationships in our number system and the equal sign is a principal method of representing these relationships
- Helping pre-K – 2 students develop a solid understanding of the equal sign can in turn help them avoid such difficulties in later grades.
Again, because we are working with primary students, the conceptual learning is huge! Teachers need to get those seesaws, balances, and scales out so students can see and explore numbers. The authors give great activities with True or False Equations and Open Sentences.
The other forms of symbolism important at this age, are the variables. Children often think of variables as placeholders for specific numbers instead of representations for multiple or even infinite values. They need experiences that build meaning for both.
A major emphasis in Common Core are the what is missing from word problems (change unknown, start unknown, result unknown). Too often story problems have the result unknown, but children need many experiences with other missing parts. If you need help with creating these types of problems, Greg Tang’s website has a Word Problem Generator with which you just need to select a few specifics and it will generate a word problem for you…I highly recommend you go check it out!
Making Structure in the Number System Explicit
The next step is to have students examine addition and subtraction properties and express them in general terms without referencing specific numbers…first in their own language, and then using symbols. An example from the book…
Teacher: [Pointing at 5 + 3 = 3 + 5 on the board] Is it true or false?
Carmen: True, because 5 + 3 is 8 and 3 + 5 is 8.
Andy: There is a 5 on both sides and a 3 on both sides and nothing else.
Teacher: [Writing 6 + 9 = 9 + 6 on the board] True or false?
Children: True. Same reasons!
Teacher: [Writing 25 + 48 = 48 + 25 on the board] True or false?
Teacher: Who can describe what is going on with these examples?
Rene: If you have the same numbers on each side, you get the same thing.
Teacher: Does it matter what numbers I use?
Teacher: [Writing a + 7 = 7 + a on the board] What is a?
Michael: It can be any number because it’s on both sides.
Teacher: [Writing a + b = b + a] What are a and b?
Children: Any number!
Students need examples to form justifications. Encourage students to try a wide variety of examples.
There is a section that also discusses odd and even relationships. Students, especially in 2nd grade, need to fully comprehend what odd and even actually means, not just know that numbers 0, 2, 4, 6, 8 are even and that numbers 1, 3, 5, 7, 9 are odd. Again, getting that conceptual understanding by using manipulatives and visual representations will aide in students’ exploration of these math concepts.
Patterns and Functions
Patterns are found in all areas of math. Learning to look for, describe, and extend patterns are important processes in thinking algebraically. In fact, two of the eight mathematical practices actually begin with the phrase “look for” which means that children who are mathematically proficient pay attention to patterns as they do mathematics. When possible, activities that involve patterning should use physical materials instead of just drawing or coloring on a page. This allows students the chance to use a trial-and-error approach when exploring patterning, and it also allows students to extend patterns beyond a few spaces on a page. When patterns are built with materials, children are able to test the extension of a pattern and make changes without fear of being wrong.
The authors also go into discussing…
- Repeating Patterns
- Growing Patterns
- Number Patterns
To get the full benefit of each chapter, get yourself a copy of this book. We cannot recommend it enough! Also, please feel free to leave your thoughts in a comment!
There are only four chapters left, so we hope you will stop back and join us this Wednesday for chapters 14 and 15.
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