The Effective Mathematics Classroom

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Sharp Tutor
1. Structure teaching of mathematical concepts and skills around problems to be solved
2. Encourage students to work cooperatively with others
3. Use group problem-solving to stimulate students to apply their mathematical thinking skills
4. Students interact in ways that both support and challenge one another’s strategic thinking
5. Activities are structured in ways allowing students to explore, explain, extend, and evaluate their progress
1. The Effective Mathematics Classroom
What does the research say about teaching and learning mathematics?
Structure teaching of mathematical concepts and skills around problems to be solved (Checkly,
1997; Wood & Sellars, 1996; Wood & Sellars, 1997)
Encourage students to work cooperatively with others (Johnson & Johnson, 1975; Davidson, 1990)
Use group problem-solving to stimulate students to apply their mathematical thinking skills (Artzt &
Armour-Thomas, 1992)
Students interaction in ways that both support and challenge one another’s strategic thinking (Artzt,
Armour-Thomas, & Curcio, 2008)
Activities structured in ways allowing students to explore, explain, extend, and evaluate their
progress (National Research Council, 1999).
There are three critical components to effective mathematics instruction (Shellard & Moyer, 2002):
1. Teaching for conceptual understanding
2. Developing children’s procedural literacy
3. Promoting strategic competence through meaningful problem-solving investigations
Students in the middle grades are experiencing important crossroads in their mathematical
education. They are “forming conclusions about their mathematical abilities, interest, and
motivation that will influence how they approach mathematics in later years” (Protheroe, 2007, p.
Instruction at the middle grades should build on students’ emerging capabilities for increasingly
abstract reasoning, including:
Thinking hypothetically
Comprehending cause and effect
Reasoning in both concrete and abstract terms (Protheroe, 2007)
Classroom Observations
Classroom observations are most effective when following a clinical supervision approach (Cogan, 1973;
Holland, 1998). During a classroom observation cycle, the classroom observer and the teacher meet for
a preconference, during which the terms of the classroom observation are established. A focusing
question is selected, and the classroom observer negotiates entry into the teacher’s classroom. Focusing
questions provide a focus for classroom observation and data collection, and could emerge from “big
idea” questions such as:
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2. › What instructional strategy are you looking to expand?
› What are the expected outcomes of the classroom observation?
During the observation, data is collected by the classroom observer while the teacher teaches the
lesson. The observer collects data regarding only the focusing question that was agreed upon during the
preconference. The tool for data collection must match the purpose of the observation.
After the observation, the classroom observer and teacher meet for a postconference. During that time,
the teacher looks at the data that is collected, and the observer asks the teacher what he/she notices
from the data. Based on the teacher’s responses, a conversation focusing on the questions addressed
during the preconference. It is entirely possible (and, indeed, likely) that the focusing question is not
answered, but the postconference conversation results in an additional list of questions that can guide
continuing classroom observations and post-observation discussions.
Classroom observations: What should the teacher be doing?
In an effective mathematics classroom, an observer should find that the teacher is (Protheroe, 2007):
Demonstrating acceptance of students’ divergent ideas. The teacher challenges students to
think deeply about the problems they are solving, reaching beyond the solutions and algorithms
required to solve the problem. This ensures that students are explaining both how they found
their solution and why they chose a particular method of solution.
Influencing learning by posing challenging and interesting questions. The teacher poses
questions that not only stimulate students’ innate curiosity, but also encourages them to
investigate further.
Projecting a positive attitude about mathematics and about students’ ability to “do”
mathematics. The teacher constantly builds students’ sense of efficacy and instills in her
students a belief that not only is the goal of “doing mathematics” attainable, but also they are
personally capable of reaching that goal. Mathematics is not presented as something magical or
Classroom observations: What should the students be doing?
In an effective mathematics classroom, an observer should find that students are (Protheroe, 2007):
Actively engaged in doing mathematics. Students should be metaphorically rolling up their
sleeves and “doing mathematics” themselves, not watching others do the mathematics for them
or in front of them.
Solving challenging problems. Students should be investigating meaningful real-world problems
whenever possible. Mathematics is not a stagnant field of textbook problems; rather, it is a
dynamic way of constructing meaning about the world around us, generating new knowledge
and understanding about the real world every day.
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3. Making interdisciplinary connections. Mathematics is not a field that exists in isolation.
Students learn best when they connect mathematics to other disciplines, including art,
architecture, science, health, and literature. Using literature as a springboard for mathematical
investigation is a useful tool that teachers can use to introduce problem solving situations that
could have “messy” results. Such connections help students develop an understanding of the
academic vocabulary required to “do mathematics” and connect the language of mathematical
ideas with numerical representations.
Sharing mathematical ideas. It is essential that students have the opportunity to discuss
mathematics with one another, refining and critiquing each other’s ideas and understandings.
Communication can occur through paired work, small group work, or class presentations.
Using multiple representations to communicate mathematical ideas. Students should have
multiple opportunities to use a variety of representations to communicate their mathematical
ideas, including drawing a picture, writing in a journal, or engaging in meaningful whole-class
Using manipulatives and other tools. Students, at the middle grades in particular, are just
beginning to develop their sense of abstract reasoning. Concrete models, such as manipulatives,
can provide students with a way to bridge from the concrete understandings of mathematics
that they bring from elementary school to the abstract understandings that will be required of
them as they study algebra in high school. Teachers teach their students how to use
manipulatives, and support the use of manipulatives to solve meaningful problems that are
aligned with the lesson’s objectives.
Classroom observations: What kinds of questions to ask?
Teachers should ask questions that promote higher-level thinking. That does not mean that a teacher
should not be asking questions at the lower end of Bloom’s Taxonomy of cognitive rigor. In fact, it is
important that a teacher begins a lesson with questions at the Recall and Understand levels of Bloom’s
Taxonomy. However, in order to solve meaningful problems, students must be challenged with higher-
level questions that follow the lower-level questions. Students will find difficulty applying their
mathematical ideas or analyzing a mathematical situation if they are not asked higher-level questions in
classroom activities and discussions.
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4. What are some best practices for mathematics instruction?
In general, a best practice is a way of doing something that is shown to generate the desired results. In
terms of mathematics instruction, we typically think of a best practice as a teaching strategy or lesson
structure that promotes a deep student understanding of mathematics.
The Education Alliance (2006) looked at a variety of research studies, and identified a list of instructional
strategies that could be considered to be best practices in mathematics education:
Focus lessons on specific concept/skills that are standards-based
Differentiate instruction through flexible grouping, individualizing lessons, compacting, using
tiered assignments, and varying question levels
Ensure that instructional activities are learner-centered and emphasize inquiry/problem-solving
Use experience and prior knowledge as a basis for building new knowledge
Use cooperative learning strategies and make real-life connections
Use scaffolding to make connections to concepts, procedures, and understanding
Ask probing questions which require students to justify their responses
Emphasize the development of basic computational skills (p. 17)
The National Center for Educational Achievement (NCEA, 2009) examined higher performing schools in
five states (California, Florida, Massachusetts, Michigan, and Texas) and determined that in terms of
instructional strategies, higher performing middle and high schools use mathematical instructional
strategies that include classroom activities which:
Have a high level of student engagement
Demand higher-order thinking
Follow an inquiry-based model of instruction – including a combination of cooperative learning,
direct instruction, labs or hands-on investigations, and manipulatives
Connect to students’ prior knowledge to make meaningful real-world applications
Integrate literacy activities into the courses – including content-based reading strategies and
academic vocabulary development
Additionally, NCEA researchers found that it was important for teachers to create classrooms that foster
an environment where students “feel safe trying to answer questions, make presentations, and do
experiments, even if they make a mistake” (p. 24).
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5. Comparing Effective Mathematics Instruction with Less Effective
Mathematics Instruction
In general, there are two prevalent approaches to mathematics instruction. In skills-based instruction,
which is a more traditional approach to teaching mathematics, teachers focus exclusively on developing
computational skills and quick recall of facts. In concepts-based instruction, teachers encourage
students to solve a problem in a way that is meaningful to them and to explain how they solved the
problem, resulting in an increased awareness that there is more than one way to solve most problems.
Most researchers (e.g., Grouws, 2004) agree that both approaches are important – that teachers should
strive for procedural fluency that is grounded in conceptual understanding. In fact, the notion of
numerical fluency, or the ability to work flexibly with numbers and operations on those numbers (Texas
Education Agency, 2006), lies at the heart of an effective algebra readiness program.
Teachers make an abundance of instructional decisions that can either discourage or promote an
effective learning environment for mathematics. Consider the following examples of instructional
decisions made by some teachers:
Less Effective Instructional Decisions More Effective Instructional Decisions
Mr. Ashley shows his students step by step Ms. Hernandez asks Tim to explain how he
how to solve problems and expects them to do arrived at the answer to his problem.
the problems exactly they way he does.
Mr. Roberts stimulates students’ curiosity and
Ms. Lopez ensures that her students do not encourages them to investigate further by
get lost by requiring them to stop when they asking them questions that begin with, “What
finish an assignment and wait for others to would happen if..?”
Ms. Perkins shows her students how “cool”
To keep them interested in math, Mr. math is and assures them that they all can
Flanagan works problems for his students and learn algebra.
“magically” comes up with answers.
The students in Mr. McCollum’s class are
Two students are working problems on the talking to each other about math problems.
board while the rest of the class watches.
Students are working on creating a graph that
Students have been given 30 ordered pairs of shows the path of an approaching hurricane.
numbers and are graphing them.
Students are conducting an experiment,
Students find the mean, median and mode of a collecting the data and making predictions.
set of numbers.
Students are sharing ideas while working in
The students in Mr. Jones class are sitting in pairs or small groups.
rows and are all quietly working on their
assignment. Students have done their work on chart paper
and are holding the chart paper while
At the end of class Ms. Stark collects explaining to the class how they reached their
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6. everyone’s worksheet and grades them. conclusions.
Students are in groups. One student in the Students are acting out a problem in front of
group works out the problem while the others the class. Others in the class participate in a
closely observe. discussion of the problem.
Mr. Johnson will only allow calculators in his Students are using calculators to determine
classroom during the second half of the year. patterns when multiplying integers.
He believes that students need to learn all
their facts before they use calculators. Mr. Osborne tells his students that their text
book is only one resource that he uses in his
Ms. Brown is showing her students how they classroom. Tonight their homework is out of
can use a formula to easily find the value of that resource.
any term in a sequence.
Students are using color tiles to build the
During the first week of school Ms. Fitzwater terms in a sequence.
holds up the text book and says, “I hope you
are all ready to work very hard this year. This is Some students are working in groups, some in
a very thick book and we will be covering pairs and some individually. Not all students
every single thing in it.” are working on exactly the same thing.
Mr. Swanson believes that all students should Students read about the history of the
get the same instruction at the same time. To Pythagorean Theorem. After reading, they
accomplish this he only uses whole group solve problems using the theorem. Students
instruction. then write about what they did compared to
the original uses of the Pythagorean Theorem.
In Mr. McBride’s class he spends 99% of class
time on skills and computation because his
students have difficulty understanding word
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7. Artzt, A. F., Armour-Thomas, E., & Curcio, F. R. (2008). Becoming a reflective mathematics teacher. New
York: Lawrence Erlbaum Associates.
Cogan, M. L. (1973). Clinical supervision. Boston: Hougton Mifflin.
Davidson, N. (Ed.). (1990). Cooperative learning in mathematics: A handbook for teachers. Menlo Park,
CA: Addison-Wesley.
The Education Alliance. (2006). Closing the Achievement Gap: Best Practices in Teaching Mathematics.
Charleston, WV: The Education Alliance.
Grouws, D. (2004). “Chapter 7: Mathematics.” In Cawelti, G, ed., Handbook of Research on Improving
Student Achievement. Arlington, VA: Educational Research Service.
Holland, P. E. (1988). Keeping faith with Cogan: Current theorizing in a maturing practice of clinical
supervision. Journal of Curriculum and Supervision, 3(2), 97 – 108.
Johnson, D., & Johnson, R. (1975). Learning together and alone: Cooperation, competition, and
individualization. Englewood Cliffs, NJ: Prentice Hall.
National Center for Educational Achievement. (2009). Core Practices in Math and Science: An
Investigation of Consistently Higher Performing Schools in Five States. Austin, TX: National
Center for Educational Achievement.
National Center for Educational Achievement. (2010). Best Practice Framework. Accessed online at , January 24, 2010.
National Research Council (NRC). (1999). How people learn: Brain, mind, experience, and school. J. D.
Bransford, A. L. Brown and R. R. Cocking (Eds). Washington, DC: National Academy Press.
Posamentier, A. S., Hartman, H. J., & Kaiser, C. (1998). Tips for the mathematics teacher. Thousand Oaks,
CA: Corwin Press.
Note: Checkly, 1997; Wood & Sellars, 1996; Wood & Sellars, 1997; Artzt & Armour-Thomas, 1992
are summarized in Posamentier, Hartman, & Kaiser, 1998
Protheroe, N. (2007). “What Does Good Math Instruction Look Like?” Principal 7(1), pp. 51 – 54.
Shellard, E., & Moyer, P. S. (2002). What Principals Need to Know about Teaching Math. Alexandria, VA:
National Association of Elementary School Principals and Education Research Service.
Texas Education Agency. (2006). Mathematics TEKS Refinements, K-5. Austin, TX: Texas Education
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