Do you work in a school where the vast majority of students come to you without the necessary foundation in mathematics to work at the level the standards are demanding for your grade level? If so, you are not alone.
Out of desperation and a pervasive sense of urgency, many schools that serve large numbers of struggling students make choices that are unlikely to improve student learning; choices that often inhibit learning, such as more test preparation or more testing without systematic ways to use test results to address student needs. In these schools there is often more focus on skills and procedures in math; more worksheets; more drill as practice, rather than practice that includes reflection and analysis; less focus on sense-making; and more mechanical or technical solutions (e.g. rigid pacing calendars, “teacher proof” textbooks) rather than more investment in teacher learning and growth.
If the pressure of high stakes testing has led your school or district to test more using standardized tests, unit tests from text books, or previous exams (for the record, a policy that I disagree with, but one that we often confront at Metamorphosis), then we need to find ways to use these tests to deepen student learning rather than reinforcing the despair we feel when the tests tell us what we already know: “These kids are behind.” How can you use tests and testing to yield results in conceptual understanding and develop learning habits of mind? By focusing the test prep on the capacity to analyze test questions, examine errors, articulate ideas, and question the thinking of others and/or the text. If you do this, you will be building the capacity to learn, and this skill is the most important one of all.
What would this look like? All test prep would be conducted to assist students in developing their capacity to make sense of and analyze test questions/problems, rather than just focusing on correct answers.
Here are a few techniques that might work for tests:
- Remove the Question
Try removing the question from a given word problem and asking the students to make meaning of the situation and information given without the question.
Use a “see/wonder” approach to develop student capacity to describe and make meaning of the situation in the text and then to wonder about it. The wonder aspect of the approach gets students to ask questions related to the situation. That sounds like this: “Please look at the chart on the board (or the problem or the graph) and tell us what you notice.” Encourage students to describe what they see, accept all answers as valid, and jot them down on the board. This is a way to give all students access to the task, develop confidence, and bring more student voice into whole class discussions. Next, ask students to wonder, “What do you wonder about this information? What questions could you ask?” Then write down student questions. (Oftentimes, the students ask the very question the test makers pose.) Together you could sort the questions into ones that can be answered using mathematics and ones that cannot. The teacher and/or the students can select a few of the student questions for the students to solve, including the actual test question without revealing which one is the test question. After students work on the problems, the teacher can share with students the question that is actually asked. By that point, all students would at least have some understanding of the context and perhaps have ways of solving the question. This approach brings together effective literacy practices (e.g. close reading, questioning the text) and mathematic practices (e.g. meaning-making, posing questions).
- Remove the Choices
If there are going to be multiple choice questions on the test, remove the choices and have students work through the question, including a full written response. Then share the choices on the actual test and have the students analyze the probable error in each of the three wrong choices. This practice helps students begin to analyze their own errors since some of the students in the class are likely to have made the very errors the test makers included in the choices. Time spent on analyzing the thinking behind the errors is very valuable in developing learning habits of mind and improving understanding of the mathematics under discussion.
- Present a Problem with Mixed Characteristics
Give students a set of problems that have mixed characteristics (e.g. require different operations; single verses multistep problems). Have them sort the problems into categories based on some criteria such as “ones I know I can solve” and “problems I don’t understand” or problems that require several steps and problems that can be solved using one step, or whatever categories that would help students construct an understanding of the types of problems they will face on tests. Absolutely do NOT focus on key words since we cannot understand which words are key unless we understand the whole—the context itself. The words “all together do not always mean to add, for example. The phrase, “How many were left?” does not always indicate subtraction. Many a teacher has lead her students astray by focusing on key words and asking, “What operation is needed?” rather than on making sense of the scenario described in the problem. Visualizing the scenario or acting out the problem are more useful techniques in helping students make meaning than key words.
In addition to tests, we have a tendency to use teacher-made quizzes throughout a unit. These quizzes have limited value, especially if we grade them and hand them back to students. Once you put a grade on a quiz, the learning stops. Quizzes can help us, however, get a more in-depth picture of what individual students understand if we add a metacognitive element. For example, you can design the quiz using a format that has two columns—in the left column put the problem for the student to answer and in the second column provide space for students to share their thinking about their answer choice. Asking “Why did you select this answer?” would give us insight into student thinking (e.g. make thinking visible/audible). This will help you learn to analyze the different reasons why students got an answer wrong, and will help you move away from lumping students into an amorphous “they.”
Curiosity about how students think increases teacher capacity to listen better to students. By attending to student thinking, we are better able to determine each student’s individual issues and confusion. Getting specific opens up the possibility of designing a mini-lesson to address the needs of small groups of students who are making the same errors for the same reasons.
We know that learners struggle and grapple with different challenges—one might be confidence, another might be a misunderstanding, a third might be over generalizing, and so forth. Each challenge requires a different intervention. Effective interventions allow teachers to engage with students in a more relaxed and curious manner. It also implies that the teacher is able to effectively assess each student through observation, conversation, exit tickets, analysis of work samples, and other informal forms of assessment. Teaching this way requires teachers to understand the content well enough to anticipate and recognize common misconceptions or confusions, and to have a large enough pedagogical repertoire to offer an appropriate learning strategy to the student. Yes, teaching is complex! No amount of test prep will compensate for ineffective teaching.
This is why it is imperative that teachers who work in schools with struggling learners receive useful, timely support in two forms—on-site content coaching in mathematics and participation in content-focused seminars. The seminars or workshops give teachers the opportunity to immerse themselves for a day or more in understanding better the content they teach, the probable misconceptions students will have in that content area, and the interventions that would best assist students. Coaching provides the support needed for teachers to put the content they learn in the seminars to use in their classrooms with the support of a knowledgeable partner. By combining these two types of support, teachers can increase their pedagogical repertoire and deepen their own mathematical understanding as they increase their capacity to improve student learning.
Investing in educators is the surest way for schools to increase the capacity of their teachers and principals. The investment needs to be focused on teachers learning more math, as well as knowing more about how students learn math, how to better assess what students know, and how to work from what students do know.
The complexity of teaching cannot be overstated. Though I have provided lots of ways to utilize tests and quizzes more effectively, I believe it is time we stopped trying to fix teaching through extensive student testing, teacher proof text books, pacing calendars, and other technical approaches that do not improve teacher content or pedagogical content knowledge. It is time we start engaging with teachers in identifying how best to support each teacher individually to address issues like fragile or limited content knowledge, or an insufficient pedagogical repertoire to meet the varying needs of students in order to ensure all students have the support they need to succeed.