Teacher Activities of Before, During, and After phases of a Mathematics Lesson

Before:

Prepare students mentally for the task (i.e. the “pre-activity”):

I feel that this should be done first in order to activate students’ schemata which will allow them to proceed into the activity at hand by placing the concept into a meaningful context. I feel strongly about the mental preparation of any task, as this is an important tool that can be viewed as we would view differentiation. By activating each student’s schema and tapping into their previous experiences and prior knowledge, we bring engage the students mentally on an individual level that allows each student to see their own personal connection to the task at hand (differentiation, in a way!) as well as placing the task into a meaningful context (cultural universals) that will give purpose to the task.

The teacher will be providing some sort of a simple version of the task that will serve multiple purposes: (1) it will activate students’ schemata, (2) it will establish expectations as the teacher is modeling the simple version of the task, some of the strategies and procedures that the task (and lesson task) will demand, and (3) it provides the teacher an opportunity to define vocabulary, giving multiple possible entries into the problem, and/or brainstorming or modeling mental computation or estimation strategies. This pre-activity is the key to engaging students and providing the actual presentation of the task or problem, as well as serving as a pre-assessment to the teacher as to what his/her students already know and where and how s/he can connect to this knowledge in order to provide a basis for students to construct new knowledge upon and modify existing conceptual knowledge.

During:

Provide for students who finish early/quickly:

This is an important aspect of planning that I have often seen neglected in many classrooms that I have worked in, therefore, I am aware of the importance of this step in my planning. Providing for students who finish quickly must also be combined with listening actively, as active listening works as an embedded assessment that reveals the student’s thinking by asking “how” and “why” and encouraging the testing of ideas.

The teacher’s role is crucial in assisting the student who finishes quickly to go beyond the simple solution of the problem/task at hand and extend the task to make it more challenging to those students. This is also a differentiation aspect, as we are to meet the needs of all of our students. The teacher can encourage further exploration by scaffolding questions such as “What if you tried…?” or asking the student validate their solution by asking “Would the same idea/method work for…?”. The teacher can also encourage students who have completed the task early to try to define characteristics or rule construction: “Suppose that you tried to find…” or “Can you find another way to solve this?” These types of scaffolded inquiries work as “hooks” to re-engage the student to go further in their problem-solving task.

After:

Inclusion of all learners!

Summarizing ideas without passing judgment and listing hypotheses that have arisen from students’ discussions is my favorite “after” activity, for I feel that it is through discussion and multiple perspectives that we begin to construct our own ideas and understanding. The subsequent testing of hypotheses will then engage the students in the dynamic nature of mathematical investigation: the doing of mathematics!

All too often I find that elementary mathematics instruction conveys a didactic message of right or wrong ways of solving a problem and the great hunt for the “right answer”, leaving out the exploration of ideas and multiple approaches. This type of instruction inherently excludes a percentage of students in any given group: those who did not share the instructor’s viewpoint or other students’ methodology. I see this learning environment as detrimental to the creative thinking process which is the basis of learning. We not only exclude those students who have not grasped the teacher’s idea, but we simultaneously snuff out any spark of creative individual or group ideas.

It is through the active listening process and the acknowledging of multiple perspectives and strategies that we, as teachers, establish an atmosphere where students are willing to explore ideas and test problem-solving strategies without risking judgment or exclusion. This is the prerequisite learning environment that we must establish through our classroom management and modeling. Unfortunately, many of our students will have expectations of math instruction as being teacher-based and will be hesitant to feel free to fully explore their own problem-solving ideas. Instruction must be anchored in the students’ ideas and understanding. We explore what our students know in each unit’s introductory lessons, using these activities as embedded assessments for planning our problem-solving-based lessons and units. Thusly, students learning of mathematics is actively constructed via problem-solving and de-construction and re-construction of their existing conceptual knowledge and understanding. Teaching is then full integrated with the students’ learning process.

Discussing math concepts

I have always been fascinated by listening to young children explain concepts of regrouping and trying to grasp their understanding of our base ten system. Our readings, the video clips, and in-class discussion have all underscored the importance of the discussion of mathematical processes for me. I feel that way too much emphasis is put upon the omnipotent one and only correct answer, as opposed to the process itself. I feel that it is through our understanding of a student’s processes and conceptual knowledge that we can gain insight into misunderstanding and/or mis-connected ‘discrete’ concepts that underlie a student’s poor understanding of a big idea or concept. Hundreds of worksheet problems cannot tap into a conceptual misunderstanding or lack of understanding if the ‘right answers’ are all that are asked for by the assessment tool. Through conversation and dialogue, we negotiate our mathematical understanding and question the reasons why we need to find efficient problem-solving methods and tools. The process of sharing and comparing methods and ideas is the journey to a deeper, more integral understanding of a technique or concept. This personal realization leads me to question why we do not spend more time on this aspect of teaching elementary mathematics. By understanding exactly what our students’ conceptual understanding is based upon, we can then truly connect to their prior knowledge and build cognitive networks of mathematical meaning that make sense to them in a real-world context. Knowing how to calculate a correct answer using a traditional algorithm is not an accurate predictor of a student’s ability to apply a mathematical concept in their daily life. I wonder why we do not explore other number base systems and concepts at the elementary level in order to develop the ‘template’ of numerical base systems. It seems as though base systems are not explored until middle school and algebra-level coursework. Why do we wait so long to play with the concept of bases and pattern repetition?