Crafting a Teaching Sequence for Extended Intervention

The focus of Monday’s session for grades 3-5 math was how to craft a teaching sequence for extended intervention. Participants worked through the entire process of developing a sequence of module lessons that could be utilized for remedial purposes, filling in learning gaps or supporting enrichment. The day started with examining three types of problems encountered in fourth grade. Participants were then asked to focus on just one of the problems and discuss/think of a sequence of related math problems that would lead to a student being successful at the problem at large. Discussions were centered on the idea of how teaching must be collaborative, not an isolated task. Teachers need to play off of the strengths of their fellow teachers in order to help solidify the vertical foundation being built through the Common Core standards. One highlighted belief was that ”A teacher’s pedagogical content knowledge of the grade levels preceding and following his or her own impacts students’ success daily and is the primary engine necessary to meet the needs of all students.” With that in mind, participants started learning how to build a ladder from a point of strength to the objective.

The process for developing the teaching sequence for intervention is based on a cycle that starts with assessing the student, analyzing, developing a plan, teaching and then re-assessing. After assessing the student (using the module assessment), teachers analyze student work using a mathematical practices protocol that helps identify strengths and weaknesses and also aids in developing questions that can be used to help identify the error or where the “lost” has occurred. In other words, teachers need to find where the crack in the foundation is located and where the last point of success is located. Once identified, teachers can read the corresponding module overview and find where in the overview of module topics and lesson objectives the breakdown occurred. At what lesson or lessons did the crack first appear? Once the crack is identified, teachers can now work on constructing a ladder of complexity, but keeping in mind that traveling up the ladder must be able to be done efficiently. Each rung of the ladder is intended for a 20 minute activity, with the top of the ladder being a task aligned to a final objective. Ladders or intervention plans should not exceed 3 weeks in length.

Strategies for finding the vertical links amongst grade levels included looking at the curriculum map, curriculum overview, foundational standards and the Common Core standards checklists found on EngageNY. Much time and energy was spent on researching within topics and lessons across grade levels to find activities or lessons that help aid in teaching the sequence more deeply. Groups made an illustrated poster to share the sequence and then spent time creating “second” chance assessment questions that allow students to experience and see their growth first hand.

Time and pacing came up as an area of concern. Most agreed that the process presented would work well in aiding AIS instruction. The point was really driven home that teachers need to utilize the strengths of their other grade level teachers on where to find foundational lessons in the modules that directly link to the final objective.

Materials for this session are available here.

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Meeting Students’ Needs; Grades 3-8 ELA Module Updates

During today’s 3-8 ELA sessions, Expeditionary Learning focused on meeting students’ needs by providing participants with transcripts of actual moments observed in classrooms where the modules are being used. The transcripts included exchanges between teachers and their students of various levels of language acquisition. Groups discussed the intervention strategies used by the teachers in the transcripts. They also talked about other interventions the teachers could have used to help more students be successful. Discussions resulted in a great exchange of ideas and the sharing of strategies that have been successful for some and not so successful for others. Teachers reflected on the day’s learning about teachers’ questioning, probing and responding habits in the classroom.

Expeditionary Learning also provided an update on some of the ELA modules currently in revision:

  • Grade 4 Module 1A (previously Module 1) is under revision by NYSED and will be posted this summer. This module will still focus on The Iroquois, but it will include Eagle Song as an optional independent read. A new addition will be The Keeping Quilt which will be used as a read aloud and will only require a teacher copy.
  • A new option for teachers will be Grade 4 Module 1B, a module with a focus on poetry. Texts will include A River of Words: The Story by Jen Bryant (teacher copy only) and Love That Dog by Sharon Creech (one per student).
  • Grade 5 Module 4 is also undergoing some revisions. The text Eight Days will remain, but Dark Water Rising has been removed. Unit 2 will be revised during the 2014-15 school year, with no new texts being required.

Module Focus: Grade 4 Math Modules 6 and 7

The focus of Thursday’s grade 4 presentation for mathematics was on the content of Module 6 and Module 7. Module 6, “Decimal Fractions,” allows students to extend prior knowledge of fractions by seeing decimals as an application of fractions. The progression of the module allows students to see that decimal and whole numbers behave the same way, and that working with decimals just increases their sense of number. Participants were reminded that that even though scaffolding is embedded in the lesson content, teachers may need to provide additional scaffolding measures. It is important that throughout any module, teachers amplify language. Teachers need to use academic language and be clear, effective and consistent. Teachers also need to develop conceptual understanding of the content matter by continually going from the concrete to the pictorial to the abstract. Too many visual representations might be ineffective, so teachers need to be strategic when choosing the best modeling techniques to use for the pictorial. Lastly, teachers need to model strong questioning techniques and demonstrate how to speak and write mathematically. Sentence frames and turn/talk opportunities are some examples of how to accomplish this within a lesson.

Participants started off by looking at the end-of-module assessment and working on question 6. They discussed how they could use the assessment as a planning tool and how it would guide the delivery of the lessons.

Module 6 starts off with students exploring tenths concretely through length, weight and capacity. A scale and pre-filled bags of rice was used to demonstrate how students can decompose 1 unit (kg) into 10 bags or tenths. What does the scale say? 0.1. Other decomposition problems are discussed in this module and the number bond is used. The same methodology is used in earlier grades for bundling tens and working with teen numbers. Teachers saw a strong connection here and were excited, saying “We just need to give this process time.” The overall goal of Topic A is for students to build fluency in writing decimal numbers three ways: as a fraction, as a decimal or in words.

In Topic B, students decompose tenths into 10 equal parts to create hundredths. Students model the meter stick with a tape diagram and quickly learn that tenths make us more efficient when counting hundredths. Sometimes we need to adjust the model depending on our learners; therefore, students not only work with tape diagrams but with area models and number disks to see the equivalence of 1 tenth and 10 hundredths.

Topic C gets students to apply their knowledge gained in the first two topics in order to compare decimals. Students continue using tape diagrams and area models to show their conceptual understanding of the decimal comparisons. Participants did an activity from Lesson 11 that involved cutting out decimal flash cards and ordering the decimal numbers from least to greatest. The decimal numbers were represented in various forms. Participants then needed to plot the decimals on a given number line and determine the best endpoints for the number line.

Topic D introduces the addition of decimals with tenths and hundredths, without using any algorithm. Students become more fluent with their conversions between the two in order to add and use decomposition strategies in the process. Lastly, money amounts as decimal numbers are introduced. Money is used to extend the students’ conceptual understanding of decimals while providing an application of the skills learned. Participants ended the morning session by going back to the end-of-module assessment and discussing what they learned as they were going through the lesson that would need to be reflected in the teaching of the modules.

The afternoon focus for grade 4 was on Module 7 that deals with exploring measurement with multiplication. Since fluency for grade 4 is multi-digit addition and subtraction, core fluency differentiated practice sets are used in this module. Lesson 2 contains the master copies for the 4 practice sets. A great feature of these sets is that each one is broken into 2 parts, with part 2 involving problems that do not involve re-grouping. Both parts, however, have the same answer key, which makes for simple grading.

Module 7 allows students to develop an understanding of the two measurement systems (metric and customary) and allows them to become fluent with converting between larger and smaller units. Great application problems that reinforce the RDW process are found throughout the module and students get to connect their problem solving with mixed units. An interesting approach is taken to time, as conversion is taught with the clock being an unwrapped number line. There is a strong connection here with previous work with number lines.

The module ends with 4 lessons that have year-in-review activities that focus on the area of composite figures, more fluency activities and games designed to solidify vocabulary used throughout the year. The presenters gave ample time throughout the day to work on problems and discuss. This was a very informative presentation that once again demonstrated the progressive nature of the modules and how they are written to build off of the prior knowledge of skills.

The materials for this session are available here:
http://www.engageny.org/resource/may-2014-nti-grades-k-5-math-turnkey-kit-for-network-teams

What Can Kids Do? Raising Expectations for Research

The Grades 3-8 ELA Thursday morning session began with a review of the Norms for Collaboration which ensure that participants support each other in a constructive learning environment. The goal of this session is to learn about the principles of teaching research that underlie the Expeditionary Learning (EL) approach to research in the 3-8 ELA modules. The presenters distributed EL’s “Overview of Research in the NYS Grades 3-8 ELA Modules” which highlights how research supports the six shifts and meets the 3-8 Common Core Learning Standards. Participants then looked at a video to examine how the students in the video are engaged in research. The students were working on a case study where they observed an actual snake in the classroom, which was followed by the creation of individual books about the snake for an outside audience. The students were then going to use the research skills they acquired to do a case study of their own snakes.

Conversations around how expectations can impact students took place. The students in the video all rose to the challenge, acquired new research skills and were enthusiastic about the research and the process.

The second part of the morning was devoted to the Research Process. Participants examined the Common Core standards that deal directly with research: W.7-W.9. They looked at the nuances in vocabulary that distinguish one grade level from the next and how the standards evolve in complexity across the grades. This was followed by the reading of various articles in which the authors make connections between research and reading/writing. The participants at each table read each article and shared their findings with their groups. Discussion centered around the role of teachers’ expectations regarding student writing about research and how integral reading and research are to students’ lives now and in the future.

The materials for this session are available here:
http://www.engageny.org/resource/may-2014-nti-grades-3-8-ela-turnkey-kit-for-teachers

Module Focus: Geometry

The focus of the Wednesday morning session for grades 9-10 math was to explore the topics covered in the first two modules of geometry. Congruence is covered in Module 1. The biggest change in geometry with respect to congruence is that two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one on to the other. Rigid motions are first introduced in grade 8 and teachers might want to take a look at Grade 8 Module 2 and Grade 8 Module 3 to see how the properties of rigid motions were explored. Participants discussed how they currently characterize transformations, and most agreed that they associate transformations with a set of rules and they tend to be very coordinate based. Students now need to develop a deeper understanding of transformations and their purpose without the use of the coordinate plane.

Module 1 starts off with 5 lessons on constructions. Students will be performing the same constructions as in the past, but the focus is on not just the figure being constructed, but the steps behind the construction. Students will need precision with their vocabulary, as they will need to be able to communicate clearly the steps behind the construction for all to understand. Focus is on the construction and instruction.

Topic C in Module 1 covers the transformations and rigid motions studied in 8th grade. The progression of intuitive, to the concrete, to formally defining a transformation is developed. Participants took a look at this progression with the concept of reflection in lesson 14 where students explore what they notice about the line of reflection and perpendicular bisectors. They then tie this exploration back to their work in the opening lessons that dealt with constructing the perpendicular bisectors and angle bisectors. Students are then introduced to the formal definition of reflections.

Topic D introduces the concept of congruence through rigid motion. Lesson 22 is the presentation of the proof by rigid motion for the SAS criteria. Students need to know the properties that are preserved with the transformations that are rigid motions (i.e. distance preserving, angle preserving) and need to be able to communicate these properties while writing proofs that involve the use of rigid motions. Once the congruence criteria (i.e. SAS, SSS, HL) have been proven, they then can be used in proofs for congruence as we saw in lesson 26.

Topic G reviews the content of the modules and reinforces the purpose behind the axiomatic system. A math teacher’s story was told: “We have to cover several chapters from the textbook and there are approximately 40 formulas. I may offer you a deal: you will learn just four formulas and I will teach you how to get the rest out of these formulas.” The students gladly agreed.

Module 2 focuses on similarity and right triangle trigonometry. Scale drawings are first introduced in grade 7 and teachers might want to take a look at the content covered in Grade 7 Module 1 for gap purposes. Scale drawings are approached in the geometry module with two methods, the ratio and parallel method. Participants had fun with the parallel method and using the set square to generate parallel lines. After scale drawings are explored, students go on to study the properties of dilations which sets the tone for proving the similarity criteria for triangles (AA, SAS and SSS).

The remainder of Module 2 focuses on right triangle trigonometry. Lessons 16, 21 and 25 set the foundation for the trig functions without officially using the language. These lessons explore the internal relationships within and between similar triangles and how the ratios of corresponding sides can be used to find missing lengths.

The materials for this session are available here:
http://www.engageny.org/resource/may-2014-nti-grades-6-10-math-turnkey-kit-for-network-teams

Supporting Student Growth

The Grades 9-12 ELA Wednesday morning session focused on teachers being able to identify what students can and cannot do with respect to a given standard and identifying next steps and areas for supporting student growth. Teachers began by talking about barriers to using student work as a form of data, and the impact that a teacher can have on students’ ability to read closely, write well, speak and listen effectively and think critically. Teachers talked about not having the time to properly work with individual students, focusing on what students can’t do rather than what they can do, being willing to change the way they do things in their classrooms, and making an effort to teach individual skills rather than focusing on test preparation.

Teachers were presented with the Logic Model: If we analyze our assessments and a particular standard, articulate a learning need in terms of the standard, identify a high impact root cause and create a plan to address the cause, then the student’s skill level will improve. We cannot make an impact on what students do until we thoroughly understand what they can’t do.

To learn how to use the Logic Model, participants looked at student work from the Grades 9-12 ELA modules. This began with examination of the assessment map from Module 10.1 to understand the progression of standards RL.9-10.2 and RI.9-10.2 over the course of the module. Groups then analyzed student work against a rubric focusing on what the student can do in respect to the standard and what the student is struggling with so that teachers could then develop a plan to address the student’s needs. This discussion continued in the afternoon session.

The materials for this session are available here:
http://www.engageny.org/resource/may-2014-nti-grades-9-12-ela-turnkey-kit-for-teachers

 

Modeling with Equations and Functions

In today’s Algebra I session, participants explored the lessons in Module 5 of Algebra I, “A Synthesis of Modeling with Equations and Functions.” The module is packed with experiences that pull together the cohesiveness of the topics covered throughout the year and is loaded with application problems that develop fluency, but not computational fluency alone. This module drives home the fact that students need to be fluent in pulling their prior knowledge to the forefront in a variety of settings. This can be a challenging and interesting task to incorporate into the design of a lesson. Some key problems/exercises that participants looked at were the following:

Lesson 1, Exercise 2:
Students examine a graph of a function and recognize the function type and state the parent function. Students then need to be able to identify what transformation took place to the parent function to produce the graph, a more challenging task and perhaps one that students will struggle with. Lastly, students need to write the equation of the function. Lesson 2, exercise 2 had some concrete examples of the same nature.

Lesson 2, Exercise 4:
This exercise was an excellent example of allowing students the opportunity to communicate their conceptual understanding and critique the reasoning of others.  This problem is highly recommended and generated great discussion amongst the crowd.

Lessons progressed through problems that had students analyzing data sets, verbal descriptions and graphs.

Lesson 4, Exercise 2:
This exercise provided an opportunity for students to determine what type of function best models the data displayed in a graph. The graph appears to be a quadratic, but as participants learned at the last NTI, looks can be deceiving. As it turned out, the graph was quadratic and it provided an opportunity for the sharing of great techniques of solution. These strategies included solving a system of equations, using the second differences (common theme of the day) to find the leading coefficient for the quadratic, and estimating the other root and working backwards to find the quadratic. This problem was well received because of the opportunity it presented for students to be successful.

Another good example of a modeling problem was the opening exercise discussed for Lesson 5 that involved exercise time and rest time for interval training. We quickly learned as a group that part of the modeling process is learning how to handle any assumptions that are made and determining how those assumptions will affect the desired outcome.

Finally, participants looked at problems that involved modeling exercises from sequences and investigated the question: should we believe in patterns? Participants examined an interesting example that involved the appearance of a pattern from points on a circle that crashes after the 6th term. The example reinforced that a pattern can disappear.

One of the biggest takeaways of the session is that students need to be able to recognize whether they have enough information to be sure that the function they have created is an accurate representation of the data being described.

The presenters touched briefly on how to support learning throughout this module and any other module. They shared three key points:

  1. Be attentive to language. Teachers need to be clear with their mathematical vocabulary. They need to accurate and precise with the mathematical language being used in the classroom, so this can transfer to the students.
  2. Teachers need to remember that conceptual knowledge precedes fluency.
  3. Conceptual understanding is achieved through strong questioning techniques, progressing from the concrete-pictorial-abstract, and knowing and showing the progression of the content.

The materials for this session are available here:
http://www.engageny.org/resource/may-2014-nti-grades-6-10-math-turnkey-kit-for-network-teams