One dollar [moving to the graph]. You, like, earn one dollar every time you go up. Like it gets bigger by one every time. So every time you walk one kilometer you get one more dollar, right? Someone want to try that? Um, kilometers times one equals money. So this equation, this table, and this graph are all the same function.

They all mean the same thing. Students frequently know things through experience that they have not been taught explicitly. They can often solve problems in ways we do not teach them or expect if, and this is an important qualification, the problems are described using words, drawings, or notations they understand.

Once the equation for a straight line, , has been introduced, m is defined as the slope of that line and is calculated using the formula. For students to understand slope in these definitional and symbolic ways, they must already have in place a great deal of formal knowledge, including. Knowing algorithms for finding the slope of a function, however, does not ensure that the general meaning of slope will be understood. As illustrated in Figure a , a student can know the algorithm for finding the slope, but not understand that the slope of a line characterizes its relative steepness on a graph and tells something about the rate of change in covarying, dependent quantities.

We have found that younger students have intuitive and experiential understandings of slope that can be used to underpin the formal learning that involves conventional notations, algorithms, and definitions. To illustrate, we gave a class of fifth and sixth graders the following situation:.

## Scaffolding numeracy in the middle years: Multiplicative thinking

Jane is in a walkathon. It is a mystery. We do know that if Jane walks 1 kilometer she will earn 4 dollars and if she walks 3 kilometers she will earn 8 dollars. Students were asked to figure out the slope of the function that tells how much Jane will earn. Half of the students were provided with the formal rise-over-run ratio definition of slope; the other half were given a definition of slope that reflected more familiar, student language, being told that the slope of a function is the amount by which the answer goes up for every change of one in the start value.

They noticed that when Jane walks three kilometers instead of one, she earns four more dollars; thus she earns two more dollars for every extra kilometer she walks. In this way, these prealgebra students identified the slope of the mystery function as 2 without receiving instruction on formal definitions or procedures. In contrast, students who were given the textbook definition of slope were not able to determine the slope in this example. Our point, instead, is that using student language is one way of first assessing what knowledge students are bringing to a particular topic at hand, and then linking to and building on what they already know to guide them toward a deeper understanding of formal mathematical terms, algorithms, and symbols.

In some cases, we may underestimate the knowledge and skills students bring to the learning of functions. Topics and activities we presume to be challenging. The focus of Principle 2 is on simultaneously developing conceptual understanding and procedural fluency, and helping students connect and organize knowledge in its various forms. Students can develop surface facility with the notations, words, and methods of a domain of study e. For students to understand such mathematical formalisms, we must help them connect these formalisms with other forms of knowledge, including everyday experience, concrete examples, and visual representations.

Such connections form a conceptual framework that holds mathematical knowledge together and facilitates its retrieval and application. As described previously, we want students to understand the core concept of a fuctional relationship: that the value of one variable is dependent on the value of another. To this end, we need an instructional plan that deliberately builds and secures that knowledge. Good teaching requires not only a solid understanding of the content domain, but also specific knowledge of student development of these conceptual understandings and procedural competencies.

The developmental model of function learning that provides the foundation for our instructional approach encompasses four levels—0 to 3—as summarized in Table Each level describes what students can typically do at a given developmental stage.

The instructional program is then designed to build those competences. Initially, the numeric and spatial understandings are separate. The initial numeric understanding is one whereby students can iteratively compute within a single string of whole numbers. That is, given a string of positive, whole numbers such as 0,. Notice in a bar graph of yearly population figures that each bar is taller than the previous bar. Spatial and numeric understandings are elaborated and integrated, forming a central conceptual structure.

## Number Sense Series: Developing Early Number Sense

Multiply each number in the sequence 0, 1, 2, … by 2 to get a set of pairs: , , ,…. Notice that a graph of daily plant growth must leave spaces for unmeasured Saturday and Sunday values. Look at the function below. Why or why not? Please show all of your work. The initial spatial understanding is one whereby students can represent the relative sizes of quantities as bars on a graph.

Students can easily see differences in the sizes of bars how tall they are and can use this spatial information to draw inferences about associated quantities.

Students can read bar graphs that, for instance, show daily measurements of the growth of a plant in the classroom. They can see that each bar is taller than the previous one, that the plant is taller on Friday than on Thursday, but cannot easily quantify those changes.

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At level 1, students begin to elaborate and integrate their initial numeric and spatial understandings of functions. They elaborate their numeric understanding in two steps. First, whereas students at level 0 can extend a single sequence of numbers such as 0, 2, 4, 6, …, at level 1 they can operate on one sequence of numbers to produce a second sequence. For example, students can multiply each number in the sequence 0, 1, 2, 3, … by 2 and form the resulting pairs of values: , , , , ….

They come to understand that maintaining equal distances between values on the x-axis is critical to having a meaningful graph of a function. They also progress from understanding graphs with verbal or categorical values along the x-axis, such as cities with their populations on the y-axis , to understanding graphs with quantitative values along the x-axis, such as time quantified as days with the height of a plant on each successive day on the y-axis.

## Scaffolding numeracy in the middle years: Multiplicative thinking

The example of graphing plant growth is an interesting one because it is an activity at the cusp of this transition. Students initially view values on the x-axis as categorical, not sequenced so that Thursday, Friday, Monday is okay.

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Later they come to view these values as quantitative, in a sequence with a fixed distance between the values such that Thursday, Friday, Monday is not okay because Saturday and Sunday must be accounted for. Without being able to view the x-axis as quantitative, students cannot see graphs as representing the relationship between two changing quantities. Drawing a line to join the points provides a visual representation of the relationship between the quantities. The line offers a way of packaging key properties of the function or pattern of change that can be perceived quickly and easily.

For example, students can see how much earnings change per kilometer by looking at the steepness of the line. As their initial numeric and spatial understandings are elaborated, students at level 1 also begin to connect, or integrate, these understandings. They make connections between tables and graphs of x-y pairs, using one representation to generate inferences that can be checked by the other. The overall pattern of a function can be understood both in the size of the increments in the y-column of the table and in the steepness of the line moving from one point to another in the graph.

As these views become integrated, students develop a deeper and more flexible understanding of functions, in this case, a linear pattern with a rise of 1. Grasping why and how the line on a graph maps onto the relationship described in a word problem or an equation is a core conceptual understanding. To ensure that students master the concepts at this level, complex content is avoided. Such limiting of these complicating factors is intended to minimize loads on processing and working memory, thus enabling students to focus on the essence of the integration of numeric and spatial understandings of function.

Students learn more complex content during levels 2 and 3. As students progress to level 2, they begin to elaborate their initial integrated numeric and spatial understandings.

For a full elaboration to occur, it is necessary for students to understand integers and rational numbers and have facility in computing with both of these number systems. Finally, students differentiate families of functions to see differences in the shapes and characteristics of linear, quadratic, and cubic functions. At level 3, students learn how linear and nonlinear terms can be related and understand the properties and behaviors of the resulting entities by analyzing these relations.

Furthermore, students must elaborate their understanding of graphs so they differentiate the four quadrants of the Cartesian plane, understand the relationship of these quadrants to each other, and relate these quadrants to negative numbers.