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Mathematics: What's the Big Idea?

Teaching for Meaningful Understanding

The following piece was created by Professor Kathleen Fisher for the Interactive Science Workshop Series which aired on the Annenberg Channel last spring. Although originally directed at science teachers, we feel that much of the information presented within is applicable not only to science education, but to mathematics and other disciplines as well.


This workshop series will look at various facets of teaching for meaningful understanding. It may be useful for you to have this overview for reference. The description below may also clarify for you what we mean by terms such as 'meaningful understanding,' 'personal knowledge construction,' and constructivism,' which are likely to be mentioned in the workshops.

To make sense of ideas, students need to put those ideas together in their own minds and in their own ways. This is the process of personal knowledge construction. Teaching that promotes knowledge construction by students is often described as constructivist. Among the features of a lesson that contribute to creating a constructivist orientation are the following:

a) Framework. Lessons are organized into a coherent framework that aims to build bigger ideas over time. Sequence and connection are important.

b) Lessons. Each lesson is selected to illustrate an important idea. It is also useful if the lesson gives an outcome that is surprising or unexpected to students, and therefore gets their attention and challenges their assumptions.

c) Framing. The teacher begins each lesson by asking students to summarize what went on before and what was learned. The teacher draws links that lead students from the previous lesson(s) to the current one.

d) Prior Knowledge. Before doing a lesson, it is important to find out what the students already know about the topic. This is often done via whole class discussion, with the teacher eliciting ideas from students and organizing them on the board. Solid starting points, gaps in knowledge, disagreements, and important alternative conceptions can be identified in this way.

e) Prediction. Students are asked to predict outcomes before performing activities, to discuss their predictions with their group, and to write down their predictions. This prompts students to build a mental model of the event, run a simulation, and commit themselves to the anticipated outcome. In this way, they become invested in observing the actual outcome.

f) Group Work. Working in groups (typically of three or four students each) involves students in collaboration and dialog. They struggle to formulate and express their ideas to one another, listen to one another, and negotiate meanings. They teach to and learn from each other.

g) Context. Each lesson involves students in observing a phenomenon or event. This provides a concrete illustration of the key scientific principle being studied. It also draws students into the problem, generating interest and motivation, and gives students an opportunity to think of their own questions. The phenomenon provides a context in which the scientific ideas make sense.

h) Embedded Questions. Questions are inserted throughout a lesson to prompt students to focus on key issues and think about deeper levels of interpretation.

i) Data Collection. Support is provided to guide students in making observations and collecting data. Although they work in groups, students are expected to keep their own notes and record the relevant data about each lesson.

j) Data Summary. At the end of a lesson, data from the whole class is summarized on the board or on transparencies. This allows teacher and students to assess the accuracy of the data. Generally there is a high level of agreement among all groups. This is valuable because when students' observations disagree with their expectations, they have a strong tendency to conclude that "I did the experiment wrong." That is, they hang onto their ideas and dismiss their observations. However, when students see that the whole class got the same results, they tend to be more willing to reconsider their ideas. Sometimes students do not obtain the same results, and then an effort is made to identify the reasons.

k) Data Manipulation. It is often useful to manipulate the data after it is collected, as in calculating survival rates or summarizing changes over time in a line graph. This manipulation of the data is intrinsic in science.

l) Post Knowledge (Data Interpretation). One hears a great deal these days about eliciting prior knowledge. But it is equally important to elicit post knowledge. What have students learned from the lesson? The conclusion may seem obvious to the teacher, but do the students see it? How many still believe their naive ideas? A whole class discussion of the data and its interpretation is essential to find out what was achieved in the lesson and what still needs to be addressed.

m) Links to Everyday Knowledge. Throughout the lesson and especially during the post knowledge discussion, students are prompted to bring in examples from everyday life, as well as to apply their knowledge in interpreting everyday situations brought up by the teacher.

n) Reflection. One good rule of thumb is that students spend at least as much time making sense of a lesson as they do in performing the activity. Students are typically engaged in organizing their ideas into larger frameworks and making the connections between their ideas explicit, through activities such as concept mapping, semantic networking, writing, or making presentations.

o) Safe Environment. When students are treated with respect, as thoughtful young adults, that is how they behave. Creating an environment conducive to thoughtful discussions is often challenging for teachers who have learned the art of 'controlling' their classes. Among the strategies of successful teachers are: talking softly, talking in an authentic and thoughtful way, listening carefully to students' ideas and responding with interest, encouraging students to respect one another's ideas, getting to the same physical level as the students to talk to them, and giving interesting and challenging assignments.

p) Embedded (on-going) Assessment. Teachers use each phase of the lesson to assess students' progress, identify problems in understanding, and alter instruction as needed. They monitor groups, listen in on conversations, review journal comments, and ask probing questions.

q) Authentic (summative) Assessment. At the end of a lesson, teachers try to devise assessments that reflect the kinds of activities and skills used in the lesson, and that focus on deep understanding of larger issues rather than small details. Multiple choice items are used much less frequently, and when used, are often two-tiered. The first item is a content question (e.g, Where does most of the weight of dry wood come from, a) soil, water, & sun, b) CO2, water & sun)? The second item is a reason question (e.g., The reason this is true is because a) this is what the plant needs to grow, b) these are converted into sugar in photosynthesis, c) a gas couldn't possibly make up the weight of a tree).





Mathematics: What's the Big Idea?

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