mathematics is the language of science. In Galileo’s words:

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is impossible to understand a single word of it. Without those, one is wandering in a dark labyrinth.

Mathematics majors learn the internal workings of this language, its central concepts and their interconnections. These involve structures going far beyond the geometric figures to which Galileo refers. Majors also learn to use mathematical concepts to formulate, analyze, and solve real-world problems. Their training in rigorous thought and creative problem-solving is valuable not just in science, but in all walks of life.

Goal setting involves the development of an action plan designed to motivate and guide a person or group toward a goal. ... As long as the person accepts the goal, has the ability to attain it, and does not have conflicting goals, there is a positive linear relationship between goal difficulty and task performance.

Goal setting influences learning and self-evaluation through its motivational and informational effects. When students set goals they are more likely to attend to instruction, expend effort, and persist. These motivational effects lead to more on-task behavior and more rapid learning. Goal setting also conveys information about performance capabilities when students compare their present accomplishments to their goals. High levels of motivation are sustained and valid capability information is conveyed when goals are specific, difficult but attainable, and near-at-hand. Some educational research examples of these strategies are discussed, along with implications for teaching practice.

Setting goals for Teaching and Learning

Teachers are asked to regularly plan for instruction. Frequently, the lesson plans are focused more on the activities that will be completed for the lesson rather than on the mathematical goals of the lesson (Clark 2003). The plans often focus on a list of events to occur during the lesson; rarely do teachers give thought to a necessary change in the lesson on the basis of student thinking (Kagan and Tippins 1992). By setting clear goals as the basis of the lesson, teachers can plan for and then assess student learning during instruction and make corrections to better meet the needs of students (Huinker and Bill 2017). Stein argues that setting mathematical learning goals provides teachers with guidance on how to design and structure their lesson, making clear to students what they are to grasp and make use of from the lesson (Stein 2017). In the recent IES study of math coach-teacher discussions in Tennessee, when coaches focused on having deep and specific discussions of mathematical goals, teachers had an increased chance of engaging their students in deep and specific math discussions (Russell et al. 2015).

Are all goals created equal?

We recently examined the learning goals we have been setting for our students. We realized that the types of mathematical learning goals that we were associating with our lessons made a difference in how we planned and carried out our lessons. Mathematical learning goals identify the deep, underlying mathematics that students will understand; they also name how students will demonstrate their understanding (Huinker and Bill 2017).

Let us give you two examples of what we describe as mathematical learning goals. Students will discover when making a diagram or drawing a number line model, that when dividing by a number less than one, the quotient will be greater than the dividend, because either they are making groups of an amount less than one or they are making less than one group.

Students will discover when making a diagram or a number line model, that when dividing a fraction less than one by a whole number, the quotient will be less than the dividend because the dividend is being partitioned into additional parts. Therefore, the quotient is less.

These math learning goals state what is true mathematically when (1) a whole number is divided by a fraction less than one and (2) a fraction less than one is divided by a whole number. With a clear sense of the specific mathematical understandings that are being targeted in the lesson, we know what to listen and look for in student responses. Because the mathematical learning goals also name how students will demonstrate their understanding of the math, teachers know what to look for as well as what to listen for.

With the goals set, we can select a task that requires students to grapple with the underlying math ideas targeted in the lesson. When a student is solving 4 divided by 1/3 and  1/3 ÷ 4, we now know, on the basis of the mathematical learning goals, what we are looking for in the responses below that show understanding of each of these ideas.

The diagram of 4 divided by 1/3 shows 4 wholes partitioned into groups that are each 1/3 in size, and there are 12 of those groups of 1/3. The diagram of 1/3 ÷ 4 shows 1/3 partitioned into 4 equal pieces, each of which is 1/12 in size. With these two mathematical learning goals in mind, we are positioned to identify questions that we can ask students to press them to talk about the mathematical learning goal of the lesson.  

Shannese understands the difference between the quotients for each division problem. She notes that the amount being partitioned in each situation differs, and she recognizes that one situation is a partitioning of the four wholes, whereas the other is a partitioning of one-third into four parts.

Keisha’s response indicates that she does not recognize situations in which the dividend can be less than the divisor. Susan has the correct answer; however we do not yet know if Susan knows how to represent the mathematics or knows what the amounts represent. 


The mathematical learning goal can serve as our guide during the lesson, prompting us to continue to press students to talk about the size of the amounts and the underlying reason why the magnitude of the quotient differs in each situation. As you can see, the mathematics learning goal can act as a gauge for us when we monitor student responses.

The instructional process involves five primary tasks:

1. Choosing objectives (content and performances)

2. Understanding student characteristics

3. Understanding and using ideas about the nature of learning and motivation

4. Selecting and using ways of teaching (methods and practices)

5. Evaluating student learning