Support for Families: An Orientation to Bridges

If your school or district is considering Bridges in Mathematics we offer answers to some common questions below. For parents of students in classrooms currently using Bridges the grade level links in the left-hand channel provide additional information and resources you may find useful.  If you have questions that are not addressed here, your child’s teacher or principal may be able to help, or you can contact us directly at MLCsupport@mathlearningcenter.org.

Curriculum Origins & Philosophy

 

Curriculum Details

 

About the Math


Curriculum Origins & Philosophy

 
What is Bridges in Mathematics?   (Back to Top)
Bridges in Mathematics is a complete elementary mathematics curriculum for use in kindergarten through fifth grade. Bridges is published and distributed by The Math Learning Center, an Oregon-based nonprofit that has been working for more than 30 years to improve mathematics education for students, teachers, and families. Originally published in 1999 and updated in 2007, Bridges is now used in thousands of classrooms nationwide. See more about The Math Learning Center and its approach.
 
 
Who developed the program?   (Back to Top)
 With initial funding from the National Science Foundation, Bridges was developed by experienced teachers who know what works with real students in real classrooms. The program was refined and completed by a team of curriculum developers, math education leaders, classroom teachers, and professors of mathematics and mathematics education. Annotated bibliography
 
What makes Bridges different from other math curricula?   (Back to Top)
Like other strong elementary math curricula used in classrooms today, Bridges in Mathematics is based on the following goals for students and teachers:
  • Provide opportunities for all students to be successful in math through the use of research-based teaching methods and visual models
  • Help students master both essential skills and mathematical concepts so that they can solve a wide range of mathematical problems, from basic calculations to complex problems in real-world situations
  • Foster all students’ interest in and enjoyment of mathematics
  • Help students develop the skills and confidence they need to be successful in middle-school math and beyond
  • Help teachers improve their knowledge of mathematics and their ability to teach it
What makes the Bridges curriculum different is 1) the extensive, careful use of visual models and 2) consistent attention to both basic skills and conceptual understanding.
 
1.  Visual Models Make New Ideas Easier to Understand and Remember.
Many people are accustomed to seeing pictures when students are studying geometry, but Bridges helps students use pictures to understand concepts in all areas of mathematics, including algebra and computation. For example, fourth graders use rectangles to represent multi-digit multiplication problems. Although students ultimately calculate using numbers alone (either mentally, on paper, or with a calculator), the pictures help them understand why certain procedures work, and many students find it easier to remember a single picture than a set of steps. View an example using multiplication (below) or visit this blog article for links to several examples of a Bridges teacher demonstrating multi-digit division on video.
 
Using a Picture to See Multiplication
13 x 12
First, students use the rectangle to multiply the ones and tens in each number and then add them to find the product. Then, they see  how these numbers can lead to a computational algorithim, which they practice first with pictures and numbers and then with numbers alone.
 
2.  Basic Skills and Conceptual Understanding Are Both Essential.
Students must use their understanding of mathematical concepts and their mastery of computational skills when they solve almost any problem. The examples below are drawn from Grades 2–4 of Bridges in Mathematics. You’ll see that in all cases, students must apply both their conceptual understandings and their computational skills to solve the problems correctly. Because conceptual understanding and skills go hand-in-hand, Bridges teaches them together, while also offering skills practice that helps students keep their mastery of facts and procedures current: this practice takes the form of games (used more frequently in the lower grades) and paper-and-pencil assignments (used more frequently in the higher grades).
 
Problem
Concepts
Skills
Mrs. Brown is the gym teacher. She has 15 soccer balls and 8 footballs.

a. How many more soccer balls than footballs does Mrs. Brown have?

b. How many soccer balls and footballs does Mrs. Brown have in all?   (Grade 2, middle of year)
• Understand that finding the difference involves subtraction (not just taking away).

• Understand that finding the total involves addition.
• Recall a basic subtraction fact
(15 – 8 = 7).

• Recall a basic addition fact
(15 + 8 = 23).
Frank was measuring out some peanuts. He wanted exactly 1 kilogram of peanuts. So far, he has 300 grams. How many more grams does he need to get exactly 1 kilogram of peanuts?   (Grade 3, middle to end of year) • Understand that you need to convert 1 kilogram to 1,000 grams to determine the difference between what Frank has and what he needs.

• Understand that you can add up or subtract to determine how many more grams Frank needs.
• Recall metric conversions (1,000 grams in 1 kilogram).

• Recall or find the difference between 1,000 and 300 (1,000 – 300 = 700).
Mina went up to the fence to get a closer look at the animals and spotted 47 adorable piglets running around. The farmer was building pens to hold them and told Mina it would take 1 pen to hold every 6 piglets. How many pens will he need to build?   (Grade 4, end of year) • Understand that this situation (equal groups) calls for division.

• Understand how to handle the remainder in this problem situation: 7 pens will hold just 42 piglets, so another pen is needed for the remaining 5 piglets. This means that the answer to the problem is 8, even though the answer to the calculation (47 ÷ 6) is 7 r5.
• Complete division with a remainder
(47 ÷ 6 = 7 r5) by recalling basic multiplication/division facts (6 x 7 = 42) and subtracting (47–42 = 5).

 

Curriculum Details

 
Does Bridges use a spiraling approach to teach mathematics?  (Back to Top)
A spiraled approach to mathematics instruction is not the defining feature of Bridges in Mathematics. However, Bridges does incorporate elements of a spiraled approach by revisiting topics and skills that have been addressed in previous grade levels or in earlier units of the same grade level. Some teachers and parents are concerned that spiraling can become repetitive, use classroom time inefficiently, and prevent students from mastering mathematical skills and ideas. To avoid these pitfalls, Bridges returns to topics in a deliberate way with the following purposes in mind:
  • To provide the ongoing practice that is required for students to keep their skills sharp and their conceptual understandings fresh.
  • To deepen students’ understanding of a familiar topic by exploring it in a new context or in relation to another idea. For example, students revisit fraction concepts in the context of data analysis when they interpret pie graphs.
  • To provide multiple opportunities for students to master difficult topics. For example, a student who struggles to understand fraction concepts at the beginning of fourth grade benefits from a return to those concepts toward the end of the year, while those who are already proficient have the chance to do more challenging work and deepen their understandings.
Bridges does not repeat instruction while sacrificing mastery. Students are expected to demonstrate proficiency with essential skills and mastery of key concepts, but parents and teachers know that students learn and develop at different rates. By giving students multiple chances to master mathematical content, Bridges provides more opportunities for all students to succeed and excel in mathematics.
 
Teachers at all grade levels are provided with a Math Skills & Concepts Student Report that they can use to show where your child is on the continuum toward mastery of specific skills. They may need to adapt this report to match your state standards exactly, but it provides a clear way of showing which skills and concepts your child has mastered at different times during the school year. Click here to see the Fourth Grade Math Skills & Concepts Student Report as a PDF.
 
How does Bridges use homework?  (Back to Top)
Homework is a chance for students to practice what they have learned and for families to see what students are doing in math class. Homework is assigned with increasing frequency as students progress from kindergarten through fifth grade. In the lower grades, assignments are sent home about once a week; by fifth grade, you can expect to see assignments two or three times a week. In addition, teachers may send home supplemental practice pages if students need more practice with a particular skill or if there is a desire for more frequent homework.
 
To view all regular homework assignments for a particular grade level, select the grade-level link under Support for Parents in the left-hand channel at the top of this page. Scroll to the bottom of the grade-level page, where youll find links to PDFs of all homework assignments.
 
How can I help my child with Bridges homework?  (Back to Top)
Most homework assignments include a note for families that explains what students have been doing in class, what the assignment is, and what kind of help students may need to complete the assignment. In grades K–3, students will need more help from adults to read and complete the assignments, many of which are mathematical games. These games provide more practice with skills students have already learned in school.
 
In grades 4 and 5, students will do more homework independently, although we know that fourth and fifth graders often need help from an adult to complete their homework. Many assignments include notes and examples that facilitate family members' work with students. The grade-level links under the Support for Parents link in the left-hand channel at the top of this page provide additional resources that families may find useful when helping students with Bridges homework. To view all regular homework assignments, scroll to the bottom of the grade-level page, where youll find links to PDFs of Home Connections.
 
How can Bridges help students who struggle in math?   (Back to Top)
Bridges helps teachers address the needs of struggling students in the following ways:
  • Using visual models that make new ideas and skills accessible to all students, particularly those who are visual learners. See also "What makes Bridges different from other math curricula?"
  • Incorporating manipulatives that give all students a hands-on way to understand new concepts and skills.
  • Allowing time to work with small groups or individual students who need more attention.
  • Encouraging students to solve problems in the ways that make the best sense to them.
  • Giving students more than one opportunity to master new concepts and skills. See also "Does Bridges use a spiraling approach to teach mathematics?"
  • Providing additional support materials (e.g., practice games and assignments) to help struggling students become proficient with new concepts or skills.
 
 
How can Bridges help students who are very talented in math?  (Back to Top)
In a Bridges classroom, students practice basic facts and learn to do calculations with larger numbers, but they also spend a lot of time working through complex problems that require more mathematical sophistication and creativity. All Bridges students are invited to extend their thinking, make generalizations, and support their answers on a regular basis. For example, third graders are challenged to consider whether the sum of two odd numbers will always be even and explain why. Students who answer this question quickly and easily might be invited to predict whether a wide variety of number combinations will have an odd or even sum. The teachers' guides include many specific suggestions that help the teacher prompt students to take a problem to the next level.
 
In the higher grades, students can solve challenge problems that are identified with a special icon; the same icon is used in the teachers' guides to help teachers find challenges for their students. Challenge problems appear on assignments students do in class and on homework. Some students enjoy number puzzles and calculations with large numbers, so the challenge icons sometimes identify problems of this nature. However, students who are talented mathematically benefit from working on more complex problems like the examples from Grade 5 Bridges shown below. Problems like these don’t just keep gifted students busy: they call upon students’ computational skills, ability to find patterns and make generalizations, and mathematical creativity. The goal is to keep gifted students engaged, deepen their understandings, and foster their love of mathematics by presenting problems that they have the tools to solve, but which require them to work at a higher, more interesting level.
 
Three Fifth Grade Challenge Problems
Circle the two numbers whose product is 1,274.
 
26
34
49
61
If you made a circle that was 16 inches around (had a circumference of 16 inches), do you think it would have an area that was greater or less than a square with a perimeter of 16 inches? Explain your answer.
 
4, 7, 10, 13, 16, 19, 22, 25 ...
 
a. What would be the 30th number in this sequence? Show all your work.
 
b. What would be the 100th number in this sequence? Show all your work.
 
c. Would the 876th number in this sequence be odd or even? Explain how you can tell.

About the Math

How does Bridges teach basic facts?  (Back to Top)
As students develop their ability to recall basic facts, it makes good sense to address both mastery of the skill (quick recall of facts) and understanding of the concept (the properties of the operation and the relationships between facts). Bridges in Mathematics teaches basic facts by first having students explore the operation (addition, subtraction, multiplication, or division) in the context of story problems or situations, which ensures that students understand what it means to add, subtract, multiply, or divide. Students then learn strategies for solving basic problems; these strategies illustrate properties of the operation and can be used to calculate mentally with larger numbers (as well as to help recall facts when needed). Finally, students practice the facts until they can recall them from memory.
 
The table below shows when students are expected to master their basic facts.
 
Facts
When Mastered
Addition and Subtraction Facts to 10
Grade 1
Addition and Subtraction Facts to 20
Grade 2
Multiplication Facts (by 0, 1, 2, 5, 10)
Grade 3
Multiplication and Division Facts (through 12 x 12 and 144 ÷ 12)
Grade 4
 
How does Bridges teach computational algorithms (e.g., multiplying larger numbers)?  (Back to Top)
Bridges teaches students to compute with larger numbers by first establishing conceptual understanding of the operation, then using visual models to learn different ways of calculating, and finally helping them become proficient with efficient algorithms. When computing with larger numbers, students are frequently encouraged to make an estimate first. Estimation promotes good number sense, helps students evaluate whether their final answers are reasonable, and encourages them to develop mental math skills that are useful in so many real-world situations.
 
The examples below illustrate the sequence of instruction that helps fourth graders master multi-digit multiplication.
 
Conceptual Understanding
Models for Calculating
Practice
Step 1
Students use base ten pieces to solve multiplication story problems.
Step 2
Students quickly sketch rectangles and use them to see how and why algorithms work.
Step 3
Students calculate with numbers alone, using estimation to gauge the reasonableness of their answers.
Maggie’s 13 chickens each laid a dozen eggs in two weeks. How many eggs did they lay altogether?
 
 
 
 
1. Multiply the numbers.
 
 
2. Think about rounding to estimate the answers to the problems below. Then rewrite each problem vertically and solve it using the standard algorithm. (Hint: Use the answers above to help with your estimates.)
 
Why does Bridges ask students to explain their answers and show their work?  (Back to Top)
Asking students to show their work provides more information for teachers and improves student learning: when students explain how they solved a problem, they come to understand the mathematical concepts more deeply. Showing their work also provides detailed evidence that teachers can use to see what students know and where their misconceptions lie. This evidence is essential: it allows teachers to adjust the way they teach to meet students’ needs, and it allows them to document student learning over time, which helps them communicate with families about students’ progress. For similar reasons, state tests often require students to explain how they solved a problem. Students are better prepared for such test items when they explain their solutions on a regular basis.
 
I heard that students play a lot of games in Bridges. Why would students play games in math class?  (Back to Top)
Decades of research have shown games to be an effective way for students to practice their skills such as counting, recalling facts, telling time, counting money, and working with fractions and decimals. While worksheets are one effective way to practice such skills, games reward children for speed and strategy, and are especially engaging to young learners. We also find that games motivate students to do their best and that adjusting the game rules or parameters is a quick way for teachers to make them easier or more difficult for different groups of students. Games are also an effective way to practice more than one skill at a time: for example, counting money and adding multi-digit numbers.
 
How does Bridges prepare students for middle-school math?  (Back to Top)
Below are the expectations middle school teachers have of their incoming sixth graders accompanied by an explanation of how Bridges ensures students are prepared.
 
1.  Fluency with Basic Facts
Middle school teachers want students to be able to recall basic facts from memory so that they can concentrate on solving more complex problems. Bridges includes the clear expectations, explicit instruction, and opportunities for practice that students need to master their facts by the end of fourth grade.  See also "How does Bridges teach basic facts? "
 
2.  Ability to Calculate with Larger Numbers
By the end of fifth grade, students will have learned efficient ways to add, subtract, multiply, and divide larger numbers. They practice using standard algorithms and other efficient methods of computing.
 
3.  Estimation Skills and Good Number Sense
Bridges requires students to make estimates that help them evaluate the reasonableness of their answers. This frequent use of estimation promotes very good number sense and mental math skills. The visual models students use to represent different kinds of numbers ensure that Bridges students have a clear sense of larger numbers into the millions as well as fractions and decimals.
 
4.  A Good Attitude about Math
Bridges students tend to enjoy math class and find doing mathematics to be an interesting and worthwhile activity. Through playing mathematical games, engaging in mathematical discussions, and solving all kinds of problems, Bridges students develop the skills, confidence, and persistence required for success in middle school and beyond.  See also "What They're Saying About Bridges."