Math Hierarchy of Needs

To have a system in which ambitious, high-quality mathematics instruction occurs, students need to experience each of the layers (Newman, Smith, Allensworth, and Bryk, 2001). Instructional improvement at scale is only possible in practice when district leaders deliberately coordinate each layer of the math hierarchy so that they constitute a system in the true sense of the term (Cobb & Jackson, 2011).

http://bit.ly/MCOEPlanningMatrix

Backwards Teaching

(aka You Do, We Do, I Do aka Bansho)

Adapted from 5 Practices for Orchestrating Productive Math Discussions

By Margaret S. Smith & Mary Kay Stein

NCTM & Corwin Press, 2011 www.nctm.org

Step 0: Choose a good word problem.

• Look at the next lesson in your textbook for a good resource of problems to use
• Choose one of the word problems that are likely at the end of that lesson

Step 1: Anticipate possible student solution methods

• Do the problem yourself
• What are students likely to do to solve this problem?
• Which solution methods will most likely lead to the desired mathematics?
• What level is your textbook aiming for: Conceptual or procedural? Use models or an algorithm?

Step 2: Introduce the problem to students (Three Read Protocol) - 5 to 10 minutes

• First Read - What is this situation about?

• Teacher reads the problem stem orally.
• Students turn-and-Talk to discuss what the problem is about
• Teacher calls on students to share what the problem is about.

• Second Read - What are the quantities in the situation?

• Class does choral read of the problem stem.
• Students turn-and-talk to discuss quantities (and their units)  in the problem. Explicit and implied quantities (and their units).
• Teacher calls on students to name quantities. Teacher records these quantities and units on the left side of a poster. Example: “25 cats”, in which 25 is the quantity and cats is the units.

• Third Read - What are possible mathematical questions we can ask?

• Students read the problem stem with partners or as whole class.
• Students turn-and-talk to discuss mathematical questions that we might ask about this problem stem.
• Teacher calls on students to share possible questions that can be asked with this problem stem. Teacher records these questions on the right side of the poster that was used during the Second Read.
• Teacher choose the question(s) that the class will solve. If the desired question was not shared by students, then the teacher writes the question that students will work on.

Step 3: Monitor students as they work on the problem (Kikan-Shido…“teaching between the seats”) - 15 to 20 minutes

• Listen, observe, identify key strategies
• Keep track of the various approaches students are using
• Ask questions of students to get them back on track or to think more deeply
• When students are stuck, try to nudge them rather than offering too much help. Suggest a change in representation: concrete, pictorial, or abstract.
• Student solutions need to be readable by others. Provide at least 11” x 17” paper in landscape orientation, using markers.

Step 4: Select two or three solutions that will be shared

• Think about the lesson in your textbook. What math concept are you trying to highlight?
• Purposefully select student solution methods that will advance mathematical ideas
• Look for three student methods that build upon each other.

Step 5: Sequence the two or three solutions and share them - 5 to 10 minutes

• In what order do you want to present the student work samples?
• Common ways to order the work samples

• Concrete, Representational, Abstract

• Concrete: a solution using manipulatives
• Representational: a solution in which the student drew pictures to make meaning of the problem
• Abstract: a solution in which the student used numbers and algorithms

• Frequency: Most common method, then 2nd most common method, 3rd most common
• Common Mistake and Solution: A solution attempt with a common mathematical mistake, followed by two successful solution methods

• Students will share their work by referring to their 11x17 paper. Other options include: put under doc cam or drawing the work on the board in a clearly defined area.

Step 6: Connect (compare and contrast) the two or three solutions - 10 to 15 minutes

• The teacher asks the class questions to make the mathematics visible.
• The goal of this step is to have students identify and describe two or three key ideas and/or strategies.
• Compare and contrast 2 or 3 students’ work – what are the mathematical relationships?
• What do parts of student’s work represent in the original problem? The solution? Work done in the past?

Step 7: Practice problem - 5 to 10 minutes

• Give students a nearly identical problem to solve. Leave the context the same, but change the numbers. Or change the context in a small way: football becomes soccer, chickens and goats becomes geese and pig, etc.
• Teacher should roam around the class to formatively assess students; who is demonstrating understanding and who seems to need help?

Step 8: Write/Reflect

• Students produce some sort of written communication. What and how much depends on the grade level of the students.

A Sample Bansho Board

This is how you might organize your whiteboard/chalkboard rather than using a document camera.

NOTES about the board sample

These three were the original three student solution methods that were shared. I sorted them from least sophisticated to most sophisticated.

Two additional methods were shared later, so we arranged them underneath the method that most closely resembled them.

When I need to annotate on student work, I always use orange to indicate my annotations.

A 5th grade example

There are many resources that recommend this approach to teaching mathematics. Here are just a few…

http://ww2.kqed.org/mindshift/2014/02/25/bigger-gains-for-students-who-dont-have-help-solving-problems-struggle-to-learn

https://news.stanford.edu/news/2013/july/flipped-learning-model-071613.html

http://ww2.kqed.org/mindshift/2012/11/15/struggle-means-learning-difference-in-eastern-and-western-cultures/

Backwards Teaching - A better way of teaching math

http://bit.ly/InventtoLearn

https://m.phys.org/news/2015-06-maths-schools.html

• Students prefer it.
• The project found students preferred to work out solutions for themselves, and determine their own strategies for solving problems, rather than following instructions they have been given.
• Lead researcher Professor Peter Sullivan, Faculty of Education, said the study suggested that students learn mathematics best if they engage in building connections between mathematical ideas for themselves rather than being told how to by the teacher.

http://web.math.ucsb.edu/department/cmi/IBL_History.html

• Warren Colburn (1793-1833) who wrote several arithmetic texts emphasizing student invention of computational procedures and mental arithmetic
• Later, in 1819 educator Samuel Goodrich, author of The Child’s Arithmetic, argued that teaching arithmetic by rote actually prevented children from understanding arithmetic and that they should discover rules by manipulating tangible objects.
• About the process of teaching and learning Everybody Counts says “Evidence from many sources shows that the least effective mode for mathematics learning is the one that prevails in most of America’s classrooms: lecturing and listening” and adds a bit later “students learn mathematics well only when they construct their own mathematical understanding”
• Student achievement is significantly enhanced by systematic opportunities to engage in inquiry.

http://ww2.kqed.org/mindshift/2014/02/25/bigger-gains-for-students-who-dont-have-help-solving-problems-struggle-to-learn/

• Allowing learners to struggle will actually help them learn better, according to research on “productive failure” conducted by Manu Kapur, a researcher at the Learning Sciences Lab at the National Institute of Education of Singapore.
• Two groups of students: the first were scaffolded carefully by the teacher and successfully solved the problems. The second group collaborated with one another without any prompts from the teacher. The second group was unsuccessful in solving the problem. “And when the two groups were tested on what they’d learned, the second group “significantly outperformed” the first.”

https://drive.google.com/drive/folders/0B1rMdLby1poGQVF2OE9RNzVDcmM

• Research evidence suggests that students need opportunities for both practice and invention.
• The findings from a number of research studies show that when students discover mathematical ideas and invent mathematical procedures, they have a stronger conceptual understanding of connections between mathematical ideas.
• Teachers should regularly allow students to build new knowledge based on their intuitive knowledge and informal procedures.

CHECKLIST: You Do, We Do, I Do (aka Backwards Teaching)