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Table of Contents
Many of us are familiar with the changes in math instruction that have occurred over the last 20 or 30 years. These have largely been influenced by the National Council of Teachers of Mathematics (NCTM), an organization in the United States and Canada that promotes improved mathematics instruction. Its Principles and Standards for School Mathematics (NCTM, 2000), followed later by Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics: A Quest for Coherence (NCTM, 2006), Focus in High School Mathematics: Reasoning and Sense Making (NCTM, 2009), and more recently Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), have served to direct many mathematics initiatives in both countries.
Many aspects of these resources have had a powerful effect, but only a few will be discussed here. Principles and Standards for School Mathematics not only lists standards for math instruction but also articulates the importance of high expectations for all students, a coherent curriculum, teacher understanding of the curriculum, student understanding of the curriculum, formative assessment, and the potential for technology to support mathematics instruction.
Most recently, Principles to Actions (NCTM, 2014) reiterates that a set of standards on paper does not automatically translate to change. It reminds readers that U.S. students are still not scoring at the highest levels compared to students in other countries and that, in general, U.S. students perform well on low-level items on international tests but not as well on items involving higher-level thinking. It also reminds readers that many U.S. teachers work in isolation, without the benefit of collegial or coaching support. It articulates the principle of professionalism in educators as a critical need.
Particularly valuable in this latest NCTM publication is a set of suggested actions to help leaders come closer to achieving NCTM's goals for student math learning. Details will be discussed later in this resource, particularly those involving providing professional development opportunities for teachers, monitoring instructional time, emphasizing "understanding," maintaining a schoolwide culture with high expectations, ensuring resources meet the spirit of the standards, and performing assessment to guide instruction.
In addition, the introduction of the Common Core State Standards in the United States (Council of Chief State School Officers & National Governors Association, 2010), focusing on understanding as well as performing, has changed the focus in many mathematics classrooms. The eight Standards for Mathematical Practice encourage students to dig into the math and not just repeat rote procedures:
Examples of student work based on these processes can be found in many resources (Small, 2017c).
Fleshing out the ideas presented by NCTM helps us imagine what we would see in a school where great math is happening.
Notice that math scores are not the major consideration. High scores do not necessarily mean that math understanding is at a deep enough level. Too many students can successfully repeat what a teacher has shown them but hit a wall when any element of a problem or situation changes. Thus, their work might look good at first glance, but the learning might not be solid enough to transfer to a new or more complex situation.
Deep down, it is the curiosity about math and the ability to solve problems—not just the ability to calculate and solve equations—that we are looking for in our students. Often principals believe that great change is happening in the school without much data to back it up. They usually notice one good thing that is happening somewhere and forget the rest. Figure 1.2 lists specific practices that administrators should look and listen for in their schools to determine whether great math is taking place.
I see great math when …
What I won't see
How will I know?
Students and teachers look forward to math time.
Math is students’ least favorite time in the day.
Directly ask students whether they are excited about math class or are not looking forward to it.
Math is focused on solving problems and thinking and not just recall of procedures.
Children are filling in worksheets.
Worksheets might be assigned occasionally, but their use is not widespread within a class or across classes.
You might ask students to tell you about the most recent math problem they solved. It is only a problem if they didn't already know how to do it, so you need to ask what made it something they had to think about.
Students are communicating mathematically during a large proportion of their learning time and are engaged in the assigned activities.
There is a lot of student silence or lack of engagement.
You might randomly listen to groups of students in different classrooms and evaluate how rich their math conversations are.
Students regularly choose and use visual representations and manipulatives to support and demonstrate their mathematical thinking.
Visual tools are rarely used and are often used in an algorithmic or procedural way, not for solving problems or thinking mathematically.
Visual tools include manipulatives and representations such as number lines, hundred charts, and so on. Students use tools even when not instructed to do so. You might ask students why they chose a particular tool for a math problem. They should not always be choosing because "that's the one the teacher said to use"; they should have their own reasons.
The tools and strategies become more sophisticated as students advance through the grade levels.
New tools and strategies are not introduced as students move from grade to grade.
This can be observed by looking at student work on similar problems at different grade levels. There should be meaningful changes both in the tools and the ways in which they are used.
Students regularly ask substantive questions of their peers and of the teacher.
Basic questions that maintain a focus on organizational or procedural aspects of a task are the only ones students ask.
You might listen for the first five or last five questions in different classrooms on different days and observe the proportion of questions that are substantive.
Students expect daily math challenges and rise to them.
Students expect to be shown everything and do not want to be asked to figure things out.
You might ask students if they are usually shown how to do a problem before they are asked to do it or whether they often figure things out themselves.
Nobody talks about math being "hard."
Teachers openly talk about finding math hard or not liking math to their students or colleagues.
Ask students if they have ever heard other people say that math is hard and ask who those other people were.
Students may obtain good test scores, but, in the end, students are meeting the standards only if the tests reflect both the letter and the spirit of those standards. Consider each of the following pairs of activities; they both relate to the same standard, but only one adequately addresses it.
Grade 3 Standard: Interpret products of whole numbers; e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.
A: How much is 5 × 7?
B: Draw a picture to show what 5 × 7 looks like.
Task B addresses the standard, but Task A does not. Task A asks for a specific number, not for an understanding of the operation; the student could simply recite the fact from memory rather than apply the concept of multiplication.
Grade 6 Standard: Determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one.
A: What is ⅔ × ^{6}⁄_{5}
B: Tell or show what number you might multiply by ^{6}⁄_{5} to obtain a product that is a little more than ⅗.
Task B addresses the spirit of the standard better than Task A. Although the student will find a solution with Task A, it does not ask the student to determine whether a quantity was increased. On the other hand, Task B allows for students to either use calculations or reasoning to consider how the value of a factor changes when multiplied by a particular fraction.
High School Standard: Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions.
A: What are the zeros of y = x^{2} − 5x + 6?
B: How does knowing that x^{2} − 5x + 6 = (x − 3)(x − 2) help you sketch the graph of y = x^{2} − 5x + 6?
Task B addresses the standard better than Task A because it focuses on the required connection between factors and zeros.
Although state or provincial tests are usually constructed with the standards in mind, often teacher-created tests are a mix of questions that adequately test the standards and those that do not.
Questions should attend to the practice, or process, standards as well as the content standards. The standards for practice, or process, need to be assessed regularly and provide important information about whether standards are being met.
For example, one important standard in many states, modeled after the Common Core State Standards, is that students can "construct viable arguments and critique the arguments of others" (Council of Chief State School Officers & National Governors Association, 2010). This ability needs to be assessed. Therefore, a principal might look at teacher tests, quizzes, or assignments for items like these.
Questions should ask for explanations and not just answers.
Questions should ask students to draw a visual to show an idea. For example, a prompt like "Draw a picture to show why √18 equals 3√2" is valuable. A student who realizes that the two right triangles in Figure 1.3 must be similar shows an understanding of why √18 equals 3√2.
Questions should ask students to make connections.
Questions should be open-ended enough to allow students to show their interpretation of ideas.
To meet standards, teachers need to use appropriate resources, whether print or digital. A principal must determine whether the resources that are requested and used by teachers are ones that support the spirit of the standards. What would a principal look for when choosing resources?
The resource should focus as much on understanding and thinking as it does on skill development. There are many examples of resources (computer programs, texts, or supplementary resources) that say they meet standards but actually address only some of the content, and certainly not the spirit, of those standards.
The resource should not regularly model tasks such as how to perform computations, find the rule for a pattern, or count the number of diagonals in a shape. If the resource models for students too much, it is not adequately calling for problem solving. It should leave room for students to figure things out for themselves.
The resource should speak to the teacher as a professional, and the teacher component, at a minimum, should clarify for teachers why certain directions or actions are taken and encourage teachers to reflect on those rationales.
The resource should address differentiation, providing meaningful and thoughtful extensions for strong students and meaningful and thoughtful scaffolds for struggling students (not just easier questions).
The resource should focus students on building connections between ideas they have learned and not treat each math topic as a separate entity.
The value of a resource is not related to the medium (print or digital). There are print textbooks that ask thoughtful questions (e.g., "Use a different strategy to figure out 13 – 8 than to figure out 9 – 2"), and there are digital resources that focus on straightforward, rote procedures.
There is a long history of concern about math instruction being the domain of a small number of interested teachers, whether they consist of the math department in a secondary school or a few keen teachers in an elementary school. Elementary school teachers tell us that they have very little math in their backgrounds and often lack a deep understanding of the curriculum they are delivering; for this reason, they "bow out" of trying to change mathematics instruction. Often they say they feel no pressure to try to improve their math instruction because many of their principals also stay out of math discussions due to their own discomfort in that domain. Most principals readily admit that they defer to their math departments. In addition, many high school teachers today are the products of a more traditional approach to math, so their own experiences as students create an extra hurdle as they try newer approaches in their own classrooms.
With the advent of more current curricula and the desire to improve the math performance of our students on international assessments, these practices will no longer suffice. Underlying decisions that are or are not being made about how mathematics should be taught reflect strong differences in belief about what math really is. It would be interesting for a principal to try this poll in Figure 1.4 with staff (perhaps anonymously) to see just how different their beliefs are; it could be the start of a valuable staff discussion.
Circle your response to each statement.
The big focus in math instruction should be on application.
Strongly agree
Agree
Disagree
Strongly disagree
Covering curriculum matters more than making the math interesting to students.
Developing positive attitudes toward math is challenging.
Math is mostly about developing computational and algebraic skills.
Most of my teaching time in math should focus on problem solving.
Math is more about procedures than concepts.
Most math teaching should be direct instruction, with some guided and some exploratory work.
The weakest students in my class might improve, but they really can't be expected to achieve at top levels.
Open-ended problems should be used rarely because they are too hard to grade.
When teachers have fundamentally different views about the nature of math, it is hard to develop a school culture. However, it remains essential to come to terms with that vision collectively.
There are a variety of ways to obtain data, beyond scores on external tests or teacher marks, that will help you get a pulse on your school. (Some of these will be explored in more depth later in this book.) To use these measures, however, there must be deliberate attention to collecting kinds of data that are often not considered.
Data collected about students reflect teaching practices. Whether the data are taken from individual students or collected from an entire class, student work is a good indicator of what is being taught or emphasized and how the teaching in one class might differ from the teaching in another. Because the principal's focus is on teaching practice, it makes sense to gather data that reflect that practice and show how it has influenced students. The data that you collect can be either quantitative or qualitative.
In this chapter, we showed what a positive math culture looks like in schools that foster students’ mathematical thinking in meaningful ways while meeting rigorous standards, and we outlined a variety of indicators of student success beyond student engagement and test scores. High among these are the following:
We discussed the importance of teacher use of high-quality teaching resources. We also discussed the importance of data as a powerful tool to help administrators determine the strength of the math culture in their schools. The data that can and should be collected to measure math performance include evaluations of both teaching practice and student work, as well as attitudes among staff and students toward mathematics.
This deeper look at positive math culture emphasizes how great math is more than great test scores. It requires much more to give student learning the context, rigor, and depth it needs to make it meaningful. The best first step toward creating this kind of learning in mathematics across your school is with teachers, which is the subject of the next chapter.
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