Is Your Math Class Skimping On Patterns and Functions?
I’ve always been surprised that “functions” are considered an 8th Grade or Pre-Algebra level math skill. I often hear two types of feedback about patterns and functions from math teachers in the elementary and middle grades.
“Patterns are for really young kids”
“Functions aren’t until 8th grade.”
Of course, these responses aren’t universal, but I hear them enough to warrant further discussion. Sure, the standards may not explicitly address patterns and functions in the middle grades. But both play a crucial role in mathematical understanding at all grade levels.
A typical student encounters plenty of patterns in the early grades. They also encounter functions, though they might not recognize the term. In the middle grades, the focus of math class begins shifting to procedural math. This corresponds with the development of math-specific 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” and can start reasoning again. It’s also interesting that when functions are formally introduced (in 8th or 9th grade), there is little attention to the concept of functions. Most of the attention is given to function notation or identifying non-functions.
The loss of function (pun intended) in the middle grades is doubly unfortunate. First because this is the point at which many students begin to conclude either that math is boring or that they are “not good at math.” The second is that functions can be used to introduce dozens of middle grades concepts. Times tables, ordered pairs, and proportions are just a few topics that benefit from a foundation in patterns and functions.
Connecting Patterns and Functions
I highly recommend reading Marilyn Burns’ book, About Teaching Mathematics, cover to cover. This is where I first began to appreciate the connection between patterns, functions, and virtually every K-8 math concept:
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.
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 extend number patterns such as “2, 4, 6″ or “4, 7, 10.”
The first step in connecting patterns to functions is to simply 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 helps students extend their numeracy beyond the counting numbers. For example, “With a ‘times 5’ function machine, how do we get an output of 1?” This approach can also be used to introduce decimals and negatives.
Eventually, students can ‘input’ non-counting numbers. This helps them see that operations work generally the same way with all categories of numbers.
Patterns and Functions in Elementary Math
Patterns are a central component of math in the early elementary grades. By seeing shape patterns, they are beginning to 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.
Working with numbers isn’t about memorizing a list of disconnected facts, it’s about seeing relationships. By seeing operations with numbers as ‘facts’ rather than patterns and functions, students miss out on 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 cordoning off certain number sentences to be memorized, while others are figured on the fly.
Promote Fluency by Emphasizing Connections
Proponents of math facts like to talk about automaticity. 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.’ Give them counters and ten-frames to see 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.’
Times tables are another example of mathematical patterns frequently 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. Having students think of a “7s 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. I start each year with a review of whole number operations, using visual models and number sentences as the vehicles.
Number sentences and visual models highlight the undercurrents of patterns and functions running throughout middle school math content. Patterns provide a foundation for equivalent ratios. A decreasing pattern on a number line is an effective way to “cross the zero” and introduce negative numbers. Patterns can also be tied to proportional relationships, especially rates.
By applying functions to these patterns, 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, they could use a pattern to find how far the car traveled in 1, 2, or 3 hours. To find how far it would travel in 274 hours, ¾ of an hour, or 180 minutes, they will need to recognize the pattern as “times 3.” They can then create in-out machines (tables), graphs, or equations to find the results.
Such generalizing is a crucial element for introducing variables. When 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 is just a brief overview of the role of patterns and functions in K-8 math. The challenge is creating lessons to emphasize each concepts. While it is helpful for teachers to explain these ideas, it is even more valuable for students to discover the connections themselves.
Unfortunately, few textbooks provide the resources to help teach through hands-on exploration. Most break concepts into measurable bits of information, rather than emphasizing connections. This makes it challenging for teachers looking to employ a conceptual approach.
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