About the Revised Edition A s we developed this revised edition of Bridges Intervention, we made several types of changes to the original materials The first three revisions listed in the following table are found in all nine volumes The final two revisions also occur throughout the materials, but are particularly relevant and noticeable in Volumes 2, 4, 5, and 7, as noted Bridges Intervention Revised Edition Volumes Enhancing teacher questioning to support student sensemaking √ √ √ √ √ √ √ √ √ Promoting positive mathematics identities for all students √ √ √ √ √ √ √ √ √ Improving usability and support for teachers √ √ √ √ √ √ √ √ √ Supporting fact fluency through updated strategy language and sequences of activities √ Incorporating more accessible and inclusive problem contexts © The Math Learning Center 0622 √ √ √ bridges.mathlearningcenter.org ds, each struction lled Pair It ce-down in you win dents win ds face-up nts have and set Bridges Intervention: About the Revised Edition Enhancing Teacher Questioning to Support Student Sensemaking Questioning Strategies The action steps and sidebars now include explicit support for purposeful teacher questioning This is a move away from direct instruction to more effectively support student sensemaking A greater focus on teacher questioning is intended to provide opportunities for all students to access and understand the mathematics of each session Teacher questions are highlighted through italicized text in the action steps and sample dialogue Further, sidebars featuring questioning strategies, such as this one from Volume 1, provide banks of questions that teachers can use repeatedly during activities and games to promote students’ development of conceptual understanding and procedural fluency Questions to ask students during the Pair It Up, Fives game include: What did you turn over? How many more to make 5? Is it a match? How you know? Is there a combination of numbers that make 5? Promoting Positive Mathematics Identities for All Students Throughout the volumes, timed activities have been removed from placement assessments, progress monitoring, and session activities By focusing on fluency instead of memorization, the materials support positive mathematics identities for students by providing time and opportunity for sensemaking and strategy development Bridges Intervention | Print Originals 02 rom the top ur collecn line with le Count The team udents aren’t P11 ath Learning Center | mathlearningcenter.org © The Math Learning Center | mathlearningcenter.org udent © The Math Learning Center 0622 Further, placement assessment and progress monitoring scoring guides have been updated to be more strengths-based; that is, they now focus on Progress Monitoring what students understand and are able3-9 to do.Scoring This changeGuide is reflected most concretely in the removal of zero-point indicators, which described what students could not yet instead of indicating what students were doing successfully An example of a strengths-based progress monitoring guide from Volume is provided here Student Name more than s into the Skill Assessed Scoring Part Written Progress Monitoring 1a–b Counts two different sets of base ten pieces and records a 3-digit number to match each set 229, 124 pt (½ a point for each correct response) 2a–d Writes 3-digit numbers in expanded form 600 + 30 + 4; 400 + 20; 200 + 10 + 3; 300 + pts (½ a point for each correct response) Part Individual Interview Counts forward by 1s from 786 to 801 pt Counts fluently across this range Correctly writes a number given orally Identifies the digit in the tens place, and tells how much it’s worth 385; points to the 8; tens or 80 (either response is acceptable) pt When given the written number, is able to point to the digit in the 10s place and tell how much it’s worth pts Completes all parts of the task correctly Correctly writes a number given orally Identifies the digit in the hundreds place, and tells how much it’s worth 517; points to the 5; hundreds or 500 (either response is acceptable) pt When given the written number, is able to point to the digit in the 100s place and tell how much it’s worth pts Completes all parts of the task correctly Correctly writes a number given orally Identifies the digit in the ones place, and tells how much it’s worth 708; points to the 8; ones or (either response is acceptable) pt When given the written number, is able to point to the digit in the 1s place and tell how much it’s worth pts Completes all parts of the task correctly TOTAL SCORE 10 pts bridges.mathlearningcenter.org Bridges Intervention: About the Revised Edition Volume Volume | | Module Session 16 Module Session 16 Finally, we have made numerous language changes within the Teachers 11 Guides—removing Repeat with the following problem situations: references to “struggling students” and moving to 11 Repeat with the following problem situations: gender-neutral language—when describing students student actions.9 Maria brought 16 pieces of fruit for the class to eat forand snack She brought apples, and the rest wereoforanges many did Maria for Taken together, these changes promote interactions with Maria brought 16 pieces fruit forHow thestrengths-based class tooranges eat for snack She bring brought snack? [16 = 9 + _ ] apples,supporting and the restthe were oranges How many oranges did Maria bring for students, development of students’ positive mathematics snack? [16 = 9 + _ ] identities Josiah had 20 fish crackers in his snack cup Twelve fish crackers were gold, and thehad rest20were Howinmany red ficup sh crackers have in his cup? Josiah fish red crackers his snack Twelvedid fishJosiah crackers were gold, [20 = 12 + _ ] and the rest were red.and HowSupport many red for fish crackers did Josiah have in his cup? Improving Usability Teachers Volume | Module [20 = 12 + _ ] The revised materials are designed so that any educator—from experienced Practice Solving More Snack Problems teachers Page to new paraprofessionals—can quickly identify key instructional Session 17 Practice Page Solving More Snack Problems goals for instruction as well as questions ask students that focusand on those Assign a Solving More Snack Problems page.toReview the instructions example goals Further, additional just-in-time support has been added for educators Assign a Solving More Snack Problems page Review the instructions and example • Note that the number circled as the answer in the example is not at the end of the equation, in the form of additional Teacher Note sidebars, such as the examples from the the middle • but Noteinthat number circled as the answer in the example is not at the end of the equation, Materials but in the middle Volume shown at right Theyand provide information so thatinteachers can • Complete the first problem together, support students as needed completing the others & Mats Materials Print Originals anticipate thinking and use models, identify areas of potential Complete thestudents’ firstto problem and support students as needed in completing the others • Cards Remind students draw atogether, lineOther to divide theof bar representing each problem into parts that are roughly proportional the• student addends challenge, and provide additional support for(1 per students Organizer to number racks P3–P4 Solving into Moreparts Pet that • • Graphic Remind students draw to a line to divide the bar representing each problem Teacher Note Teacher For the firstNote problem, students may start with For the first problem, on their number rack and students may start with bring over the amount on their number rack and on theover bottom that will bring the amount make 16 on the left side on the bottom that will of the number rack Or theyof make 16 on the left side maynumber begin with the the rack.16Oronthey left of their number mayside begin with 16 on the rack andof move back left side their9number to the right to see how rack and move back many left to the are right to Ask see them how to share strategies and many their are left Ask them to comment on how they are share their strategies and the same and different comment on how they are Put Together/Take Apart Problems: Pets student and for display)and solveProblems are roughly proportional to the addends SUPPORT Have students use their number racks to model each problem SUPPORT Have students use their number racks to model and solve each problem Copy instructions are located at Fluency the top of each print original Supporting Fact Language, strategy Warm-Up Two names, Less and activity sequences have been updated to reflect current research, in support of students’ development of fact Explain that you’ll play a game similar to the previous session but that this time fluency In making these changes, we drew specifically on the work of Baywhen you say a number, students will respond with the number that is less than Williams and Kling (2019) Volumes and have been organized around yours For example, if you say “5,” they would say “3” because minus is the development of foundational facts, followed by a focus on derived fact Saystrategies a numberThese and randomly callintended on individuals or the whole group to answer changes are to develop students’ accuracy, flexibility, and efficiency with facts, which within are the20 defining characteristics Begin within 10, then expand to numbers if appropriate for your group of fluency (NCTM, 2014) In Volume 2, this was accomplished through revisions to2the language and the focus of some activities In Volume 5, a Warm-Up Part-Part-Whole on the Number Rack of the oftogether modulestowas required; in fact, it isnumber the only reordering Explain that yousequence will work build numbers on the rack volume in which the sequencing of content has changed from the original again Today, though, you’re going to give the instructions in writing by drawmaterials The lists from (below) illustrate the ing a number treeofonmodules the board andVolumes writing 2anand equation to match progression of the volumes from foundational facts to derived fact strategies Discuss both representations and help students connect the two Teacher Instead of telling you what the target number is and using Volumemy Modules Volume number rack to show you how many to putModules in the top row, I’m going to draw a number tree andEqual write an equation show you &the Structuring Five Groups of to Two, Five Ten target number and how many are in the fi rst part Ready? Structuring Ten Doubles (×2 Facts) Tens & Half Ten Facts Part-Part-Whole Doubles & Halves Doubling with Fours & Eights Near Doubles Add a Group with Threes & Sixes Ten & More, Pretend-a-Ten Subtract a Group with Nines Pretend-a-Ten Multiplication & Strategies Early Subtraction Strategies Division Experiments Teacher What’s the target number, and how many beads will you start Subtraction to Ten Array Model for Division withStrategies: on the top rowUp of your number rack? Fact Families, Fact & Division Fact Families Student OK,Strategies we’re supposed to Multiplication make 7 5+ the same and different Teacher Note Instructional Goals Teacherthe Note Dividing whole into Solve addition and parts maythe bewhole challenging Dividing subtraction probleminto for some students who parts may be situations withchallenging sums and are uncomfortable with for some students who minuends to 20 involving estimating Ask them are uncomfortable situations of puttingwith to decide which part is estimating Ask them together and taking apart larger andwhich to draw theis to decide part line toaddition demonstrate Solve problems larger and to draw the an understanding by counting on andof line to demonstrate which part isproblems largerof subtraction an understanding and which smaller by counting which part isisback larger and which is smaller Add and subtract within 20 Teacher Note Some students will benefit from having a number line or hundreds chart to help visualize the beginning number and the “two less” number This might be available in the room, or the student may need one to touch =7 Teacher Note After sliding to the left on the top, students might slide one over at a Teacher What does the equation I wrote have to with this problem? time on the bottom while counting on from They Student It’s kind of the same—like, “What you have to put with © The Math Learning Center 0622 bridges.mathlearningcenter.org stop when they reach | | mathlearningcenter.org Bridges Intervention Guide © The Math Learning Center will to makeTeachers 7?” the target number, If Student We have to put in the top row and then figure out how many more to put in the bottom row Bridges Intervention: About the Revised Edition Incorporating More Accessible and Inclusive Problem Contexts The contexts of some problem situations have been modified to be more equitable and inclusive, particularly in terms of gender and socioeconomic class In Volume 7, Module 8, for example, the original materials include a series of sessions previously focused on children raising hundreds of dollars to fund their travel to summer camp These have been revised to a more inclusive context: students raising funds for a school trip In Volume 4, Module 7, problem situations that involved categorizing and sorting boys and girls have been modified to focus on categorizing and sorting according to other characteristics: There were 32 children playing basketball at the park There were more children playing soccer than playing basketball How many children were playing soccer? 28 children played on the slide 12 more children played on the swings than on the slide How many children played on the swings? The focus is now on the activities in which the children are engaged, rather than the gender of the children Conclusion The changes made in this revised edition of Bridges Intervention enhance teacher questioning, support student sensemaking, promote positive math identities, improve usability, support fact fluency, and provide more accessible and inclusive problem contexts Although the number of changes to the scope and sequence were minimized, significant improvements were made to the content throughout the volumes to support implementation of more equitable and effective teaching and learning References Bay-Williams, J., & Kling, G (2019) Math fact fluency: 60+ games and assessment tools to support learning and retention Alexandria, VA: Association for Supervision and Curriculum Development NCTM (2014) Procedural fluency in mathematics: A position of the National Council of Teachers of Mathematics Reston, VA: National Council of Teachers of Mathematics © The Math Learning Center 0622 bridges.mathlearningcenter.org ... example • Note that the number circled as the answer in the example is not at the end of the equation, in the form of additional Teacher Note sidebars, such as the examples from the the middle • but... on the slide 12 more children played on the swings than on the slide How many children played on the swings? The focus is now on the activities in which the children are engaged, rather than the. .. begin with the the rack.16Oronthey left of their number mayside begin with 16 on the rack andof move back left side their9number to the right to see how rack and move back many left to the are right