• by Jack Dieckmann, M.A. • IDRA Newsletter • March 2003

Image of Jack Dieckmann, M.A.This is the second in a three-part series on the role of language in teaching mathematics to English language learners. The first article presented a thumbnail sketch of a specific lesson modeled for a group of middle school teachers who have English language learners combined with other students. (The article was printed in the March issue of the IDRA Newsletter and is available online at: http://www.idra.org/resource-center/learning-angles-with-english-language-learners/).

In the lesson described in the first article, some of the student comments were: “We were right!,” “Let’s hurry up and test the shapes to see if we are right!,” and “All the angles put together make a straight line!” The buzz and the excitement in the lessons illustrated that mathematics can be generated from student actions and reflections on those actions.

English language learners and their teachers can all benefit from the learning inherent in their own actions (i.e., the tearing up and joining of vertices of triangles and quadrilaterals). There are many angles for students to learn and teachers to teach. This second article in the series documents the remarks, perceptions and learning among the teachers observing the lesson that was the focus of the first article. Furthermore, this article illustrates the “discipline of noticing” the reflections of teachers about teaching what they observe (Mason, 2002). These reflections are integrated into teaching practice.

Debriefing the Lesson

The Intercultural Development Research Association (IDRA) hosted an event recently for teachers participating in its ExCELS project. ExCELS (Educators x Communities = English Language learners’ Success) is an innovative professional development program that creates learning communities of schools, families and communities for English language learners’ academic success. Two secondary schools in San Antonio are the partner schools in this U.S. Department of Education Title VII program.

Below is an account of ExCELS teachers that reflects the kind of shared teaching experiences and professional conversations held in the program. These experiences and conversations lead to a deepening of teachers’ ability to notice and serve English language learners.

The lesson, documented in the previous article, happened during a regular 50-minute class period. The trainer was modeling the lesson with another teacher’s sixth-grade class. There were four teachers observing the class. The teachers and the facilitator met immediately after the class for a 45-minute debriefing session. Teacher and facilitator comments for the three lessons – triangles, quadrilaterals and assessment – are summarized below.

Triangle Lesson

The first teacher reflected, “As teachers we often forget to tell students why we are teaching what we’re teaching.”

Establishing relevancy is important though usually excluded from a math lesson. When the teacher does not help students make connections, it perpetuates students’ perception of math as capricious and arbitrary.

Another teacher said, “I didn’t realize that we could integrate metacognitives in the lesson with all types of students.”

Another added, “I see now how a language objective helps me make sure that my students can communicate the mathematics they’re learning.”

Lesson objectives must integrate math content objectives, language objectives and metacognitive objectives. This integration is essential for academic language development.

A teacher stated, “All the students were talking about the math problem.”

Small group work gave students opportunities to practice and listen to English language conversations with peers and with the instructor. Language is the means for learning the content and facility in using the language, but the main focus is on the mathematical ideas. Careful construction and facilitation of mathematical tasks allows for meaningful conversation where language is a vehicle for developing content-based ideas in natural student talk.

Another teacher observed, “It seems that the students’ favorite part was in tearing up the triangles and quadrilaterals.”

Materials that students can use and tear up are useful in teaching geometric concepts.

Quadrilateral Lesson

A teacher commented, “The review was beneficial because it got everybody to retrace their steps and think about what they had done.”

The mini-review helps students connect the activity to the mathematical conjecture and verification process of knowing and doing mathematics.

One teacher said, “The students were hurrying to test out the hypothesis with the quadrilaterals.”

Students’ restatements reinforced predicting-testing-validating as mathematical inquiry and meaning-making.

Another teacher said, “I had never seen these students thinking like mathematicians.”

Routinizing the process of mathematical exploration facilitates grasping increasingly complex mathematical shapes and properties.

A teacher stated, “There was anticipation in the air and the room was buzzing with ideas.”

Another added, “Students were excited because the task was doable and yet not obvious.”

When you awaken students’ mathematical minds with creative challenges, a natural momentum emerges that reflects piqued interest and is student-inspired and student-driven.

Student Assessment

A teacher sated: “As the lesson unfolded we were curious to know if students were really ‘getting it.’ We wrote a series of test questions similar to the required state test to determine if students could apply what they learned to standard measures. Some questions were conceptual and others computational.”

First, the lesson emphasized general pattern identification rather than the computation of the sum of the angles. The lesson worked even if only half of the students answered the conceptual questions correctly. Second, what was taught and what was being measured were not aligned. Third, students will need more time to practice the skills of finding a missing value.

Just as the students were engrossed and participatory in the geometry lesson, so the observing teachers were engaged in the process to the point of co-creating a mini-quiz before the end of the lesson and actually administering it. When staff development allows teachers to co-create and implement a lesson and its assessment, it is of secondary importance whether the teacher additions are totally congruent with the lesson. Although, in this case, the mini-quiz was not well aligned with this lesson, it did represent a sincere collaborative effort and a valid quest for evidence of learning.

The quiz’s emphasis on computation rather than the original plan to have the students write about what they learned was directly connected to the high-stakes testing that teachers are required to consider as a critical indicator of student learning. A more didactic or rote approach would be more efficient for narrow, computational purposes but at the expense of student self-discovery and mathematical discourse.

Along with the fact that some of the students were English language learners and language development had to be integrated into the math learning, the search for meaning and underlying principles has a more lasting and powerful intellectual impact than the short-term memorization of meaningless formulas and disconnected procedures. The third and last article of this series will expand on these ideas.

Discipline of Noticing

The “discipline of noticing” is a groundbreaking approach to professional development that gives special attention to a possibility for the future, highlights a possibility in the present moment, and uses what has been noticed in order to prepare for the future (Mason, 2002). A teacher’s sensitivities and awareness are essential to enhancing his or her professional practice. The discipline of noticing is akin to reflective practice with particular specifications (Schon, 1987).

The debriefing notes above illustrate a systematic way of noticing instructional practices and outcomes. The experience in its proper place is a powerful professional development context that is also practical, accessible and empowering to the teachers as a learning community on their campus.

This focus on noticing is somewhat like experiential learning and can have anywhere from four to six levels of questioning. For our purposes, the observers:

  • Give attention to experienced, common moments in teaching;
  • Connect what is observed with their own experience, distinguishing what is already in their repertoire with what is new or different; and
  • Apply new practices to their own teaching.

Teachers make instructional choices based on their available teaching repertoire. They are, in a sense, limited to this repertoire. The innovation of the discipline of noticing is in having teachers take what they notice as critical incidents, examine them in detail with peers, and plan for new ways that they will confront these issues when they encounter them in the future.

Thus, the noticing is always in relation to planning to behave in a new way in their practice. These opportunities provide a common classroom experience that can become, through a facilitated dialogue, the object of reflection, questioning and experimentation of alternative classroom practices.

Whatever system is applied, it is important that the debriefing proceed from an initial verbalizing of “what happened?,” to “what feelings came up?,” to hypothesizing of “why did these things happen?,” to ultimately “what does this mean for me and my teaching?”

Classroom Teaching as a Medium for Professional Development

Professional development that takes place in a workshop, for example, without students present, can approximate the classroom experience through videos, written cases studies or simulation with some of the teachers role playing students. Those who have a lot of experiences in workshops and seminars know that one big barrier, mostly unspoken, in some teachers’ minds is the assumption that it will not work with “my kids.” Even when a master teacher is an effective communicator, it is not until the participants see that master teacher illustrate “good” teaching with their own students on their own campus that the “ah-ha” of learning happens.

In the ExCELS project, a major component of professional development is to bring the teachers into a colleagues’ classroom and model particular approaches. Though the lesson could possibly not work as planned, – everything observed is material for reconstruction, analysis and application. Classroom teaching itself as a medium for professional development brings the experience closer to home and physically and intellectually reduces the distance from the observation to the application (Bass et al., 2003).

By locating the work of teacher learning within one of their own classrooms, teachers are more open to implementing strategies they have seen in action. This is akin to the attraction of hands-on workshops at conferences, but it also has deeper benefits and consequences. It gives teachers hope when they see students succeeding in ways that they have not seen before. It opens up the infinite possibilities in peer observations, sharing and support. With sufficient guided practice, it can establish a systemic regularity of reflective dialogue that does not require an external expert to enhance the teaching practice.

Community of Learners

Traditional teaching has given teachers extensive independence and autonomy but has also encouraged isolation and has led to the stereotype of a teacher who taught the same way for 20 years. The current dilemmas of classroom teachers include: pressure overload from external accountability measures, isolation, groupthink colored by pessimism, untapped competence and narrowness of teaching roles (Fullan and Hargreaves, 1996).

Teacher isolation has created paranoia about being observed and has also relegated those teachers of poor, minority and English language learning students to despair about the children ever achieving. In this context, teachers cannot be moved easily by brilliant presenters at conferences or by reading books about other teachers’ successes. If change will come about and pessimism will shift to optimism, we have to provide dramatic, immediate and authentic experiences.

A vibrant community of learners then is one that is guided through mutual observation of successes with their own students, mutual aspiration to improved teaching, and ongoing dialogue. The isolation is removed through the collegial experiences. The lowest common denominator mentality and groupthink are replaced by the infusion of new successful practices, mutual challenges for greater success, and an expanded repertoire of teaching approaches that is accessible to the whole learning community.

Specific Relevance to English Language Learners

One challenge for teachers when they are beginning to focus on English language learners is an artificial duality that separates effective language learning techniques from effective content approaches. When considering the needs of English language learners, it is not useful to consider language acquisition strategies without at the same time considering the concepts required in the content-area lesson such as geometry.

When considering a geometry lesson in angles, we must hold several learner needs simultaneously. They have to view language as one of several critical dimensions that interact to form the learning context. In this way, teachers look not to compensate for language differences, but to understand the role of language in the learning of math, and through that understanding consider a wider array of instructional choices that use language to develop meaning in math.

So, in addition to math vocabulary, we need to consider the movement from concrete to abstract and from conjecture to generalization so that a set of paper triangles the student will manipulate, tear and re-glue (which can be considered an English as a second language technique because it is multi-sensory and tactile) is also an important approach for all students regardless of language ability. Having students arrive at an experiential discussion of pattern is based on the premise that all students will understand the definition of a pattern and retain it in long-term memory if it is arrived at from a cumulative sharing of patterns.

Just as students in the lesson on angles were most convinced about the properties of angles when they handled and compared them themselves, so the teachers in this project were able to understand what new elements they could incorporate into their own teaching when they saw various elements of math, language instruction, student experience, manipulatives and small group conversation, seamlessly integrated with a group of their own students. All learners learn deeply through meaningful experiences that are close to home.

Role of the Facilitator or Model Teacher

This process requires a teacher that can walk the talk. He or she must model teaching in other teachers’ classrooms. The teacher must have strong command of the content, a wide spectrum of approaches to teach the content, and the ability to deal sensitively with English language learner needs in the content lesson. To guide the reflection for the community of learners, the facilitator must be reflective in practice, open to feedback, and understand the levels of questions necessary to deepen the level of reflection.

The facilitator has a dual responsibility: to teach the students effectively and to illustrate specific aspects of teaching to the observing teachers. This is more than just teaching a good lesson while being observed by other teachers. The facilitator plans with the consideration of marking critical aspects and incidents in the lesson for didact

ic teacher training goals.

Furthermore, the facilitator or model teacher must notice, mark and bring to consciousness incidents, student responses and anything that took place in the lesson serendipitously that provides a teachable moment for the teachers observing. Both pre-planned and instantaneous in-the-moment instructional decision-making becomes grist for the mill in the debriefing.


During this exercise, teachers observed, shared observations, and reflected and saw new possibilities. As this community of teachers moves toward becoming a community of learners, there is a renewed hope and commitment to improve the achievement of all their students, especially those who are learning English as a second language.
In the next article in this series, we will review the challenges of incorporating this approach in the daily life of a middle school.


Bass, H., and A.P. Usiskin, G. Burrill (Eds). Studying Classroom Teaching as a Medium for Professional Development (Washington, D.C.: National Academy Press, 2003).

Dieckmann, J. “Learning Angles with English Language Learners,” IDRA Newsletter (San Antonio, Texas: Intercultural Development Research Association, March 2003).

Fullan, M., and A. Hargreaves. What’s Worth Fighting for in Your School (New York, N.Y.: Teachers College, Columbia University, 1996).

Mason, J. Researching Your Own Practice: The Discipline of Noticing (New York, N.Y.: Routledge/Flamer, 2002).

Schon, D. Educating the Reflective Practitioner (San Francisco, Calif.: Jossey-Bass, 1987).

Jack Dieckmann, M.A. is a senior education associate in the IDRA Division of Professional Development. Comments and questions may be directed to him via e-mail at feedback@idra.org.

[©2003, IDRA. This article originally appeared in the March 2003 IDRA Newsletter by the Intercultural Development Research Association. Permission to reproduce this article is granted provided the article is reprinted in its entirety and proper credit is given to IDRA and the author.]