A rigorous instructional program is possible in even the most challenging context.  ISA's math program provides a great example of how coaching brings rigor and engagement.

Developing Mathematical Thinkers: Moving Students from Rote to Rigor

This is the second in a three-part series on the ISA Whole School Reform Approach. This month the focus is on ISA’s college-preparatory instructional program that provides all students with rigorous inquiry-based curriculum and instruction aligned with the Common Core State Standards (CCSS) and which also embeds content and numeric literacy in the content areas. Mathematics will serve as the lens through which we view ISA’s inquiry-based approach to high level teaching and learning.

ISA math coach Julie Arcement describes a typical scenario that she often observes in high school math classrooms during the early stages of her coaching:

The teacher is at the front of the class demonstrating how to solve a quadratic equation. He has placed the following equation on the board: y=ax2+bx+c and is modeling for the students how to solve the problem. The teacher begins the carefully planned lesson by walking his students through the steps used to solve the equation. On several occasions, the teacher asks students if they have questions and if they understand the steps that he is walking them through. One student asks a question to which the teacher responds. Some students are trying to make sense of the skill that the teacher is modeling but seem hesitant about responding to the teacher’s question about what step comes next. One or two have their heads on the desk, and some are talking to classmates about unrelated topics. The teacher then calls on three students to come to the board and work similar exercises. One student is able to replicate the problem-solving steps with some prompting from the teacher. Students are then instructed to solve the quadratic equations on the worksheet that the teacher has passed out to them. A few students are able to work the exercises completely and correctly. Several students struggle but try to work through the exercises by following the steps that the teacher modeled as best they can. Some students try to begin to solve the problems, but then quickly give up and sit silently, talk to classmates, or bring out books for other classes.

Arcement points out that the first step in helping students to develop into mathematical thinkers is to support teachers as they reflect on their own beliefs and practices about mathematics and how they teach it. She partners with teachers as they learn to shift their mathematics instruction to an inquiry-based approach that focuses on developing students who can:

  • Construct new knowledge through investigations using interesting tasks and activities.
  • Talk deeply about the meaning of mathematical concepts.
  • See the connections and interrelationships between mathematical concepts.
  • See the relationship between mathematical concepts and procedures.
  • Speak about what a procedure means and why it makes sense.
  • Grapple with all types of mathematical problems.
  • See multiple strategies to solve all types of problems.
  • Defend their process and answer with explanation, justification, and proof.
  • Critique the reasoning of others and refine their own reasoning based on others’ feedback.
  • Reflect on their thinking and learning as a means of self- evaluation and growth as a learner.

ISA’s math coaches are the key to helping teachers shift from an instructional approach that is often limited to student memorization of facts and procedures or knowledge of isolated skills. “The journey of growing as an educator has a different starting point for every teacher,” says Arcement. “I always bear in mind my own journey of growth as a teacher as well as the great benefit I received from working with an ISA coach myself.” The ISA process of coaching teachers to use an inquiry-based approach to math instruction ultimately results in the development of students who are proficient mathematical thinkers who can comprehend concepts, operations, patterns and relationships and can have the procedural fluency to carry out math procedures flexibly, accurately, efficiently, and appropriately.

Click here to see a student-centered lesson using the inquiry-based approach to teaching mathematics. Reflect on the following questions as you watch the video: What do you notice about the questions that the teacher asks? How do they engender inquiry on the part of students? What do you notice about the teacher’s silences? What learning do you notice occurs with students as a result of this process?

I am hopeful that you will find the sample inquiry-based math lesson, the video on the inquiry approach to teaching math, and the impact of ISA math coaching on two high school math teachers informative and enjoyable. Watch out for the next newsletter for the third and final articles in the series. The focus will be on distributed counseling. Click here to see the previous newsletter.



The Inquiry-based Approach to Teaching Math: Teaching Students to be Mathematical Thinkers

Mary Ellen Tyrell and Sapphira Hendrix do not teach at the same schools.  They do not know each other. They do not even teach the same grade levels.  However, they are both extraordinary math teachers who are totally committed to using ISA’s inquiry-based approach to teach math.  Tyrell is the STEM Department Team Leader at the Institute for Health Professions at Cambria Heights, and Hendrix teaches ninth grade Algebra at Brooklyn Preparatory High School.  What these two teachers do have in common is a strong relationship with their ISA coaches that has facilitated their development and implementation of inquiry in their math classrooms.
“When I started teaching,” Hendrix said, “I had an idea of the kind of teacher I would be.  I thought that I would teach the way that I was taught.  I would lecture and model how to work a problem.  Students would take notes, solve practice problems quietly at their desks, do homework, and love math like I did.  I was totally unprepared for 30 ninth graders who were not naturally in love with math.  I don’t know how I would have survived my first year without the consistent help and support of my ISA coach.”  Tyrell agrees. She primarily used the I do; we do; you do pedagogical strategy during her pre-ISA coaching years.  “Now,” she says,” my classrooms are 100% student-centered, not teacher-centered.  The students take ownership of their learning.  My students predict, make inferences, pose ideas and defend them, and have discussions with their peers about why they used a certain process to solve a problem.   Their voices, thoughts, and ideas are valued and respected.  Using the inquiry approach to teaching math is the best preparation that they could possibly get to succeed at the college level.”
Tyrell says, for example, that prior to support from her ISA coach she taught quadratic equations in a more teacher-centered way.  Now she presents her students with tasks such as: How long after you take a time-released medication will you have to wait to take another dose?  Such tasks require students to represent their solution graphically and as an algebraic equation as well as orally share their thinking with the class.  The inquiry approach requires students to integrate knowledge, skills, and concepts and apply a body of knowledge to solve a real-life problem.  “They have to demonstrate deep understanding of math, and they have to think critically to perform well on these kinds of tasks,” Tyrell states.  Hendrix points out that math skills, procedures, and processes are embedded in the tasks.  “Using inquiry really promotes procedural fluency because the math procedures and skills make sense to the students,” she states.  

See an inquiry-based math lesson by clicking here.

Both teachers attribute their development of inquiry pedagogy to their partnership with their ISA coaches.  “I can’t imagine being able to meet the demands of Common Core without having had the excellent coaching that I received through ISA,” stresses Hendrix.  She explains that it was the consistent weekly feedback based on observations in her own classroom and subsequent forward planning that helped her shift her instructional practice.  “My coach and I went over every lesson before I presented it to students.  One of the biggest benefits of this process was discovering my own knowledge gaps and working with the coach to close them before I tried to teach the students. ”Having access to the ISA curriculum was also a huge benefit Hendrix recalls.  “It took me eight hours to design a 50-minute lesson.  There is no way that I could have designed a lesson from scratch every day.”
Tyrell says that the benefits for students of using the inquiry method are not just confined to math.  She is proud of how confident and courageous her students have become.  “They are not afraid to try even if they do not know exactly how to do something.”  Tyrell states that she gives each student an ISA performance task so they can demonstrate what they have learned.  “I always include one element of something that is unknown, and they are never stymied by this.  Not one student pushes the paper away and says, ‘I don’t know how to do that.’ The question that I constantly ask myself is: What do I want my students to do with their lives?  I want them to be prepared to be CEOs, not cashiers.  I expect them to function at the highest levels possible so that they have real options in life.  Using the inquiry-based approach to teaching and learning fosters attitudes and beliefs that will serve them well over their lifetimes.  There is nothing in the world I would rather do than this.”

Gerry House is the President of the Institute for Student Achievement and a Sr. Vice President at ETS. She is also a board member for the Woodrow WIlson Foundation, Adelphi University, and the Alliance for Excellent Education.



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Coaching for Rigorous Math Instruction

The First Coaching Session

ISA math coach Julie Arcement often begins the initial coaching session by sharing background information about herself and asking questions about the teacher’s background.  “Many teachers have never had a coach before and are curious about the coaching relationship.   Their first questions are usually:” ‘Are you here to evaluate me?  Why am I getting coaching?  Are other teachers being coached also?’”  Arcement assures them that her coaching role is not that of an evaluator and explains that as a coach she is there to serve as a resource, support, and thought partner.  She knows that she must first build a trusting relationship with teachers before any true coaching can take place.  After establishing the intent of the coaching relationship, Arcement gets to know more about the teacher and how he views his classroom.   As part of the conversation, she also highlights the teacher’s strengths on which they can build.  She and the teacher talk about the students in the class—their backgrounds, strengths, and their growth areas.  The teacher also agrees to introduce Arcement as his coach to his students the following week when he will also let them know that his coach will be a regular visitor to the classroom and that things will go on as usual.   Arcement ends this first session by sharing ISA’s curriculum resources and asking the teacher to think about the following questions, and they will discuss them at their next session: What is math and why do we teach it?  What do you want your students to understand about math?  What do you want your students to be able to do with their mathematical learning?
Read the full article.



An Inquiry-based Math Lesson

After several months of working together, ISA math coach Julie Arcement is ready to observe a teacher she has coached deliver an inquiry-based math lesson on linear regression centered around the following three activities.  The students are engaged with each other as they work collaboratively in groups.
Activity 1
Shaquille O’Neal is 7’1” tall and has a 15” foot.  What size shoe do you think he wears?  How did you come up with your conjecture?
Activity 2
The shoe size a person needs varies linearly with the length of his or her foot.  Use this relationship to determine a way to predict the shoe size of any man. 

  • First, gather data (shoe size and the foot length of at least four young men in the class.}
  • Next, graph the data.
  • What do you observe?

Based on the points you graphed, do you agree that there is a linear relationship between shoe size and foot length?  How do you think you can use this graph to determine Shaquille O’Neal’s shoe size?  Explain.
Activity 3
We already know that the graph should be linear.  On your own, plot the points and create the line of best fit (the straight line that best models the data in your graph}.

  • Describe how you did it.
  • Compare the line with the lines of other students in your group.  Your task here is to have a group discussion as to why your line better fits the data than the others in your group.  You will come to a consensus for your group’s line of best fit.  Be prepared to share your group’s graph with the class and how you determined it was the best from your group.
  • Write the equation of the line of best fit that you created.
  • Explain what the different parts of your equation represent.



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