One of my biggest “Aha” moments in math education was when I learned about patterns and functions. I knew what patterns were. And I knew what functions were. But until I read Marilyn Burns’ work on Patterns, Functions, and Algebraic Thinking, I didn’t see the connection. Nor did I realize that patterns and functions are central to almost every K-8 math concept.
Now, when I stress the importance of patterns and functions to K-8 math educators, I hear one of two comments:
“Patterns are for young kids”
“Functions aren’t until 8th grade.”
Of course, these responses aren’t universal, but I hear them enough to know they’re worth addressing. It’s true that the standards don’t address patterns and functions in the middle grades. But both play a crucial role in mathematical understanding at all grade levels.
Patterns in Elementary Math
Patterns are central to early elementary math. Beginning with shape patterns, students explore the structure inherent in mathematics. Counting itself is a simple pattern – increase by one.
As students extend their understanding to include skip counting, the patterns are still fairly obvious. The key is to ensure that students continue to notice patterns as they move into operations.
Some educators begin referring to operations as “math facts,” which I find unfortunate and misleading. Operations are actually shortcuts to counting. Adding allows us to ‘count up’ by leaps and bounds, rather than one number at a time. Multiplication is repeated addition, or counting of groups. Subtraction and division are the opposites of addition and multiplication.
Fluency isn’t about memorizing disconnected facts. It’s about seeing relationships. By seeing operations as ‘facts’ rather than patterns and functions, many students miss the connections. Thus begins the steady decline from concepts to procedure and memorization.
I’m not even sure exactly what qualifies as a “math fact.” If ‘5+3=8’ is a math fact, what about 500+300? 579+295? There’s no mathematical basis for deciding to memorize certain number sentences, while others can be figured on the fly.
Connecting Patterns and Functions
I highly recommend reading Marilyn Burns’ book, About Teaching Mathematics, cover to cover. In particular, check out the chapter on Patterns, Functions, and Algebraic Thinking:
Patterns are key factors in understanding mathematical concepts. The ability to create, recognize, and extend patterns is essential for making generalizations, seeing relationships, and understanding the order and logic of mathematics. Functions evolve from the investigation of patterns. -Marilyn Burns
In the early grades, students encounter patterns before they even encounter written numbers. They see a series of shapes and make predictions about which comes next. Later, students discover number patterns such as “2, 4, 6″ or “4, 7, 10.”
The first step in connecting patterns to functions is to assign a value to each term. For example, as students skip count by 3s, begin numbering the terms such that term 1 is 3, term 2 is 6, and so forth. Later (in Algebra), students will connect this to the concept of an “nth term,” and learn to make predictions without finding every term.
The next step is to understand that an input value isn’t always the term number. Start by introducing the idea of a ‘function machine’ — every time you put a number in, another number comes out. The “machine” can perform operations, such as “add 5” or “multiply by 3.”
Building the function machine concept with counting numbers can extend numeracy to other numbers. Pose a question like this one. “With a ‘times 5’ function machine, how do we get an output of 1?” This approach can also introduce decimals and negatives.
Eventually, students can use non-counting numbers as inputs. This helps them see that operations work generally the same way with all categories of numbers.
Patterns and Functions Support Conceptual Math
By upper elementary, the focus of math class shifts to procedural math (arithmetic). This shift corresponds with the development of many math-related ailments. Some students develop math anxiety. Others conclude they just aren’t “math people.”
For many students, this ‘conceptual desert’ in their math education ends with Algebra. By high school, the thinking goes, students will have “mastered the basics.” Then, they can start learning concepts.
It’s interesting that when functions are finally introduced, little attention is given to the concept. In 8th and 9th grades, the focus is on function notation. Or distinguishing functions from non-functions.
The loss of function (pun intended) in the middle grades is doubly unfortunate. First, because this is the point when many students decide that math is boring, or that they are not good at math.
Second, the power of functions could be harnessed to introduce dozens of concepts. Times tables, ordered pairs, and proportions are just a few topics that are easier when seen as patterns and functions.
Promote Fluency by Emphasizing Connections
Proponents of memorizing math facts like to talk about automaticity. But memorization is a short term fix. Students quickly forget what they learn through memorization alone. They are also left underprepared for higher-level math.
Of course, it is important for students to operate fluently, as it frees them up for more complex tasks. One way to do this is through manipulatives and visual representations. Another is by coaching them to see the patterns.
For example, show students how adding ‘8’ to a number increases the tens place by ‘1’ and decreases the ones place by ‘2.’ You can also give them counters and ten-frames to demonstrate it for themselves. Then ask them if this is always true – “What about 1+8?” Also ask if there are similar patterns for adding other numbers, such as ‘7.’ The Sieve of Eratosthenes is another great tool to reinforce this concept.
Times tables are another example of mathematical patterns that are taught as disconnected facts. I can’t tell you how many math teachers complain that their students weren’t taught their times tables. Usually, students memorized each table in sequence, only to forget them as soon as the quiz was over.
Consider using the times tables to connect patterns to functions – counting up by 7s emphasizes the pattern. Using a “Times 7 Function Machine” transfers the pattern to a function. Draw a table with the ‘in’ column on the left and the ‘out’ column on the right. You can then connect multiplication to division by asking what happens when we reverse the columns.
Patterns and Functions in Middle School
Students who come to middle school fluent in patterns and functions have a tremendous advantage over those who don’t. Students who only know facts and algorithms need to develop mathematical reasoning early in the school year.
Creating visual models helps students literally ‘see’ the patterns within the operations.
For example, drawing equivalent ratios can clarify the link between proportions and patterns.
Drawing a decreasing pattern on a number line and “crossing the zero” can help students make sense of negative numbers.
Function Machines and Number Sentences
The next step is to applying function understanding to these patterns. Then, students can skip to any term in a pattern without completing the entire sequence.
For example, to find the distance covered by a car traveling 30 mi/hr, students need to find the pattern. First, they can think about how far the car travels in 1 hour, 2 hours, or 3 hours.
The goal is to recognize the pattern as “times 3.” Then, they can then create in-out machines, graphs, or equations. These tools can find out how far the car would travel in 274 hours, ¾ of an hour, or 180 minutes.
Such generalizing is a crucial element for introducing variables. Once students see that an ‘in-out machine’ multiplies by 3 and adds 7, it’s a small step to think of the input as ‘x’ and the output as ‘y.’
Engage Your Students with Patterns and Functions
This post is just a hint of the role that patterns and functions play in K-8 math. The challenge is creating engaging lessons to emphasize each concept.
It can be helpful for teachers to explain these ideas to students. But it is even more valuable for students to discover the connections through inquiry-based learning.
Unfortunately, few textbooks provide the resources needed to teach through hands-on exploration. Most texts break concepts into disconnected bits of information, rather than emphasizing connections. This can make it challenging for teachers to employ a conceptual approach. If you’re looking for lesson plans or other resources to support conceptual math, visit our online store.
More Engaging Math Lesson Ideas
Teaching students about patterns and functions can increase student achievement and engagement. You can find more stories and ideas in the Room to Discover Educator’s Newsletter. Each week, we share posts like this one with educators around the world.
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