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Tuesday / February 20

The Bridge to the Learning: What It Means in the Math Class

This post is an excerpt from Pearse, Margie and K. M. Walton. Teaching Numeracy: 9 Critical Habits to Ignite Mathematical Thinking. Thousand Oaks, CA: Corwin, 2011.

What is a Bridge to the learning? Let’s begin with the definition of the word bridge:

A connecting, transitional, or intermediate route or phase between two
adjacent elements, activities, conditions or the like.

The key words here are connecting and transitional. The Bridge to the learning, in classroom-speak, is when students shift from where they are to where they need to be in order to access the upcoming learning. Each time students have the occasion to call up a memory and broaden a connection, we, as teachers, make learning more accessible. Modifying what is known to what is new is central to how we deepen understanding.

It is important to first differentiate an Anticipatory Set from a Bridge. They have a similar function; a Bridge, however, focuses more on actively involving the learner in the process, whereas an Anticipatory Set is more of a teacher-driven approach to “setting the stage.” A Bridge transfers the emphasis from teaching to learning. It effectively guides students’ thoughts and ideas into the fresh arena, welcoming those thoughts and making students feel comfortable, ready to stretch and deepen.

The Bridge is the opportunity to activate and build prior knowledge, present an inquiry moment, promote alternative strategies, allow for discovery, frontload vocabulary, make personal connections, fill some potential gaps, and provide necessary cues.

To summarize using the definition above, a bridge simply allows us to get from one place to another safely, directly, and as easily as possible.

What is the purpose of a Bridge to the learning?

  • To provide guided inquiry and discovery: When students learn by dis­covery, they are much more likely to understand, remember, and apply their learning to other situations (Allen, 2000, p. 123). Guided inquiry requires students to work together to solve problems, derive patterns, or generate hypotheses as opposed to the teacher deliver­ing direct instructions on what to do. Guided inquiry fosters a numeracy-rich classroom. The teacher’s job in guided inquiry is not to impart knowledge but, instead, to facilitate students’ movement along the path of discovering knowledge themselves.
  • To activate and build background knowledge: During the Bridge, you either activate background knowledge or build necessary knowl­edge so that students can better understand the upcoming lesson. Students need some background knowledge before they know enough to want to learn more (Allen, 1999, p. 132).
  • To preview: One highlight of a comprehensive approach to numeracy across the curriculum is that students perform some form of pre­viewing prior to the actual presentation of content. Previewing refers to any activity that starts student thinking about the content they will encounter in the lesson. These activities appear to be par­ticularly useful for students who do not possess a great deal of back­ground knowledge about the topic (Mayer, 1979).
  • To supply cues: A cue is a trigger to initiate thinking. Cues help bring the learning to mind. With cues, teachers provide students with direct links between new content and content previously taught (Marzano, 2007, p. 32).
  • To facilitate predictions: Making predictions keeps students engaged in the learning. Students care more about their learning when they make predictions. In mathematics, making predictions empowers students to navigate successfully through the textbook.
  • To build interest and make connections: Students are motivated to think when the context of a problem interests them. Working in a mean­ingful context can help students build an initial understanding of a concept. Students’ knowledge is generally organized around their experiences, not around the abstract concepts within the discipline of mathematics (Hyde, 2006, p. 49).
  • To scaffold comprehension: Students build up an understanding of con­cepts gradually. We can use the innate, pattern-creating, meaning-making, inductive reasoning that students bring to school to good advantage. We can provide many examples of the target concept in a particular situation so that students develop a solid initial, local ver­sion of the concept. Then we can deliberately provide experiences with the same concept in different contexts (Hyde, 2006, p. 49).
  • To assess: You can use the Bridge to judge the readiness of your stu­dents. Will you need to adjust the pace of your lesson? Who might need additional scaffolding? Who would benefit from small-group instruction?

Key Ideas

Students need lots of help getting started. The Bridge is what scaffolds many students toward understanding the purpose of your lesson. Cris Tovani supports the idea that students depend on opportunities to wrestle with the upcoming learning when she states that “meaning does not arrive, it must be constructed” (Tovani, 2004, p. 104). The Bridge is often the pathway to success for many students; without it, some students do not have access to the learning. Studies have shown that true learning takes place only when students engage with information and structures deeply enough to merge that content into their personal views and under­standings of how the world works (Harlen, 2000).

There may have been a time when transferring facts from a math text­book to students was the sole objective of a math lesson. In those days, a cursory understanding of the mathematical concepts was probably consid­ered sufficient. But to succeed in the 21st century, we are well aware that simply possessing a mental bag of facts does not guarantee that a person is truly numerate.

In mathematics especially, simple “telling” does not work. Comprehension comes through integrating knowledge into a meaningful whole, for mathematics is no more simple computation than literature is typing (Paulos, 1992). The Bridge to the Learning offers students the opportunity to merge content with strategic thinking.

We all realize that the role of a teacher is to help students learn. How do students learn? Psychological research suggests that learning is a con­structive process. It entails making connections, relating new knowledge to what is already known, and applying knowledge to new contexts. Only by working with content do we internalize it and make it our own. Instructors who understand how learning occurs use active learning tech­niques in the classroom. The Bridge forces students to manipulate the material to solve problems, answer questions, formulate questions of their own, discuss, explain, debate, or brainstorm (http://gradschool.about .com/cs/teaching/a/teachtip_4.htm).

The Bridge is a key component to active, numerate learning in the mathematics classroom. It is the transition into the learning.

Bridge to the Learning

The Bridge to the Learning is where students:

  • Preview new material
  • Participate in guided inquiry and guided discovery
  • Activate background knowledge
  • Build background knowledge
  • “Buy into” the learning
  • Develop a curiosity
  • Generate questions
  • Make predictions about the learning to come
  • Preview text
  • Make a personal investment
  • Validate their thinking
  • Engage in an interesting activity
  • Discover meaning
  • Brainstorm ideas
  • Experiment with manipulatives
  • Connect yesterday’s lesson to today’s lesson
  • Collaborate
  • Organize and categorize mathematical concepts
  • Formulate a hypothesis
  • Generate a formula by discovering how and why it works
  • Generate questions they would like answered by the learning to come
  • Solve a problem that introduces the lesson
  • Make real-life applications to the learning

Rick Wormeli (2009) challenges us to “get out of our students’ way and let them soar” (p. 75). Teaching that emphasizes inquiry helps students pro­cess and retain information. It leads to self-questioning, deeper thinking, and problem solving. Shelley Harwayne (1992) attests to this when she says, “Students will learn most when the journey is really theirs.”

Allow your students to notice their thinking before and after the les­son. Lay down a foundation of thinking, and then help them merge the content with their thinking (Tishman, Jay, & Perkins, 1993). Begin with an activity that invites student questions, and then Bridge those questions to inferences as a way of scaffolding the concepts (Gear, 2008, p. 78).

Students often have to touch, physically manipulate, and if possible, smell and taste new concepts to integrate them into long-term memory (Wormeli, 2009, p. 61), and the Bridge is the perfect time to allow for sensory exploration. Wormeli asserts that there are physics involved in teaching students to become critical thinkers. As he paraphrases Newton’s First Law on Inertia: “A student at rest stays at rest unless acted upon by an outside force” (p. 75). We need to be that force; and the Bridge provides the platform upon which we can stand.

How can I do this in my math class… tomorrow?

Scrolling and Skimming

Scrolling: To move your eyes quickly over an article to determine what the information is about or if the information is relevant to what you need to find. What supports are there to help you find out what it’s all about?

  • Boldface words
  • Headings
  • Photographs
  • Illustrations
  • Captions
  • Sidebars

Skimming: To read quickly for main ideas or supporting details in a text.

  • What is the topic?
    ____________________________________________________________
  • Why might you be learning this (make connections)? ____________________________________________________________
  • What question(s) come to mind from the skimming? ____________________________________________________________

Interactive Cloze

  1. Select a paragraph from the text that is about to be taught or assigned.
  2. Delete important keywords and duplicate copies for students. You can also create your own worksheet. Important! Make sure key­words are deleted—not words such as is, the, and, and so on.
  3. Have students predict what they think the missing words might be. Have them justify their choices.
  4. Ask students to read the selection quietly.
  5. Have students complete the cloze activity on their own or with a partner.
  6. Allow students to discuss their answers with a partner.
  7. Students can then reread the actual text to find out if their answers are correct.

Word Toss

  • Identify major concepts that will be taught in the lesson. Write 7 to 10 words and “toss” them on the board or overhead.
  • Ask students to write a few sentences using some or all of the terms. The sentences should show how the students predict the terms will be related to the lesson they are about to learn.
  • Have students share their sentences with partners.
  • After the lesson, challenge the students to check on the accuracy of their predictions and make revisions, if necessary. (Hollas, 2005, p. 88)

Focus Question

Determine the Purpose of the lesson. Create a Focus question or two that will help craft the flow of the lesson to suit that Purpose. Present the question(s) to the students at the Bridge. Ask them to think, write, and discuss their responses. Then put up the same Focus question at the Debrief and have the students use their new knowledge to add and revise their original responses. Allow time for quiet conversation. Make the ques­tions about real life rather than simply about the behavioral objective of the lesson. You will get more buy-in this way because the students will find relevance in the learning.

Central Question

Ask students to preview the text and generate and record a Focus question they think will be answered by the end of the lesson. While learn­ing, have them revise their question, answer it, or continue with the ques­tion. During the Debrief, discuss strategies and answers. Be sure to share those questions that were not answered. If the question cannot be answered at that point, record it visibly for the class to answer at some point.

Greet and Go

This method is an interactive way for students to use their background knowledge to make predictions about the vocabulary to be covered in the day’s lesson. First, give each student an index card with a term or phrase from the lesson they are about to learn. Ask the students to circulate around the room and read their cards to each other. After a couple of min­utes, call “freeze.” While returning to their seats, ask the students to be thinking about what they predict today’s lesson will be about. Have the students return to their small groups and jointly write a prediction on what they will be learning about that day. This activity is a great one when the lesson is “loaded” with vocabulary (Forsten et al., 2002b, p. 46).

Three Facts and a Fib

This activity forms a perfect setup for learning math. The students are instantly engaged. You present three facts about what they are about to learn and one fib (Forsten et al., 2002a, p. 63). They discuss which one is the fib and why. It is fun and worthwhile. Sometimes at the Debrief, have the students make up their own facts and a fib as an exit ticket. They iden­tify the fib on the back and write a reflection on what they learned from the lesson. The exit tickets become the next day’s Ignition, and the students did all the work.

Three Facts and a Fib

1. All squares are rectangles
2. All rectangles are squares
3. All parallelograms are quadrilaterals
4. All rhombuses are parallelograms

Preconcept and Postconcept Checks

Your purpose with this activity is to have students quickly self-rate their understanding of certain vocabulary that will be introduced in the lesson. The first check happens before the learning as a Bridge, then after the learning as a Debrief.

At the Debrief, ask the students to prove their understanding by creat­ing representations or explanations. In this way, the teacher guides student thinking but still allows for a self-discovery of connections, always bring­ing missed connection opportunities back to the whole class. This method also helps differentiate the lesson. Quickly circulate around the room and see which students might need to touch base with you during the lesson or which students need to be pushed beyond the original plan (Forget, 2004, p. 230).

08.19.15_Pearse_Figure 1

Example and Nonexamples

The idea here is to ask the students to formulate a definition by observ­ing examples and nonexamples. Before the students enter the room, write several examples and nonexamples of a particular math concept you are going to teach. Have them discuss and formulate a definition. Then chal­lenge them to revise and add to that definition during the lesson.

Discovery Through Manipulatives

Put a little cup of M&M’s on the tables before the students enter. On the board write, “How could you use the M&M’s to teach ____________?” (e.g., ratios, percentages, equivalent fractions). Let them discuss and share. Now bring them to the learning. They already have a solid understanding.

Concept Circles

Concept circles are circles with words placed in the sectors (Vaca et al., 1987). Ask students to discuss and write about the connections they see between the words (see one version here). You could also write three of the words and challenge the students to identify the connection, then add a fourth word that would also fit in the category. You can write three words that “belong” and one word that doesn’t and ask the students to find the word that doesn’t fit. Make sure they can justify their decisions. One variation that is valuable as a Debrief is to have students write the words and challenge other students to find the connections (Allen, 1999).

08.19.15_Pearse_Figure 2

 

References:

Allen, J. (1999). Words, words, words: Teaching vocabulary in Grades 4–12. Portland, ME: Stenhouse Publishing.

Allen, J. (2000). Yellow brick roads: Shared and guided paths to independent reading 4–12. Portland, ME:

Stenhouse Publishing.

Forget, M. A. (2004). Max teaching with reading and writing: Classroom activities for helping students learn new subject

matter while acquiring literacy skills. Victoria, BC, Canada: Trafford Publishing.

Forsten, C., Grant, J., & Hollas, B. (2002a). Differentiated instruction: Different strategies for different learners.

Peterborough, NH: Crystal Springs Books.

Forsten, C., Grant, J., & Hollas, B. (2002b). Differentiating textbooks: Strategies to improve student comprehension and

motivation. Peterborough, NH: Crystal Springs Books.

Gear, A. (2008). Non-fiction reading power: Teaching students how to think while they read all kinds of information.

Portland, ME: Stenhouse Publishing.

Harlen, W. (2000). Teaching, learning and assessing science 5–12. London: Paul Chapman Publishing/Sage.

Harwayne, S. (1992). Lasting impressions. Portsmouth, NH: Heinemann.

Hollas, B. (2005). Differentiating instruction in a whole-group setting (Grades 3–8). Peterborough, NH: Crystal Springs

Books.

Hyde, A. (2006). Comprehending math: Adapting reading strategies to teach mathematics, K–6. Portsmouth, NH:

Heinemann.

Marzano, R. J. (2007). The art and science of teaching. Alexandria, VA: Association for Supervision and Curriculum

Development.

Mayer, R. E. (1979). Twenty years of research on advance organizers: Assimilation theory is still the best predictor of

results. Instructional Science, 8(2).

Paulos, J. A. (1992). Beyond numeracy. New York: Vintage.

Tishman, S., Jay, E., & Perkins, D. (1993, Summer). Teaching thinking disposition: from transmission to enculturation.

Theory Into Practice, 32.

Tovani, C. (2004). Do I really have to teach reading? Content comprehension, Grades 6–12. Portland, ME: Stenhouse

Publishers.

Vaca, J., Vaca, R., & Gove, M. (1987). Reading and learning to read. Boston: Little Brown.

Wormeli, R. (2009). Metaphors & analogies: Power tools for teaching any subject. Portland, ME: Stenhouse Publishers.

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Written by

Margie Pearse has over 30 years of teaching experience with certifications in mathematics, elementary education, English as a Second Language (ESL), and Pennsylvania Quality Assurance Systems (Certified Instructor – PQAS 2014). She is presently at First Philadelphia Preparatory Charter School as their K-12 Math Coach and in higher education, training pre-service teachers how to create deeper, more numeracy based lessons.

Margie’s educational philosophy can be summed up as such, “Why NOT reinvent the wheel! Yesterday’s lessons will not suffice for students to succeed in tomorrow’s world. We need to meet students, not just where they are, but where they need to be. There is great potential in every child. It is our job to empower students to discover that potential and possess the tenacity and self-efficacy to reach it.”

Published Books: Teaching Numeracy: 9 Critical Habits to Ignite Mathematical Thinking, released by Corwin in 2011; Learning That Never Ends, released by Rowman & Littlefield in 2013; and Passing the Mathematics Test for Elementary Teachers, by Rowman & Littlefield, February 2015.

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Latest comments

  • Wow! Amazing strategies!

  • Hi S. Dean. Thank you! I really believe that incorporating a Bridge to the Learning everyday provides equal access and really levels the playing field. I think it’s okay for students to borrow background knowledge in order to find a pathway to understanding. We just have to be intentional in our planning so that can happen.

  • Thanks – this is great CPD for my team.

  • Thanks – this is great CPD foe my team.

  • Awesome! Thank you for such great information!!

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