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.

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:

Grade 5 Math Module 3: Addition and Subtraction of Fractions

During this morning’s grade 5 math session, participants worked through the Module 3: Addition and Subtraction of Fractions.  The first two lessons (topic A) review grade 4 standards and the concept of equivalent fractions.  Students use paper folding activities to demonstrate equivalent fractions, which helps make the concept very concrete. They use visual modeling via arrays and number lines to help make meaningful connections and the relationships that fractions have to one another.

Topic B is where students encounter fractions with un-like denominators.  They know the language of 1 apple +1 apple, 1 third + 1 third, so when they encounter 1 third +1 half, they know the units are not the same and that they need to make the units the same (common units).  Lots of work with arrays is involved here and there are some struggles, but overall the visual model helps the transition from concrete to abstract.

The group spent a lot of time on a two-step word problem from Lesson 7 that involved subtracting from a whole with uncommon units.  The solution was presented two different ways using visual models (arrays).

For topic C, participants took a look at three student solutions to the problem 3 3/5 – 2 ½.  It was interesting and exciting to see students use unbundling and decomposition that they saw in previous grades with whole numbers, and applying those concepts to their work automatically with fractions.

Topic D focuses on the problem-solving practice and application of the concepts learned. Multi-step problems are tackled with various strategies. The module does a nice job of showing the progression of equivalent fractions, making units pictorially, making units numerically, and then being able to apply knowledge to solve problems that involve addition and subtraction of unlike fractional units.

Module Focus: Grade 4 Math Module 3

The morning session for grades 4 and 5 math focused on Grade 4 Module 3: Multi-Digit Multiplication and Division, where students see multiplication and division in action.  The standard algorithm is introduced in grade 4, but it is not a fluency for this grade level.  Multiplication is a fluency in grade 5 and division is a fluency in grade 6, so the intent of the module is to allow for the deep conceptual understanding of the process through the use of modeling techniques.

Participants took a look at division through the use of number bonds, array/area models, and place value/number disk charts.  This exercise provided a great visualization of the process.  Language plays a key role, with the correct use of the terms, “whole,” “quotient,” and “remainder” (how many are left). Phrases like “distributing evenly” and “decomposing” are important in explaining the process.

Teachers need to pick fluency activities that tie into the lesson well.  Participants looked at an example:

How many groups of ____ are in _____? Prove it by counting by ______.

The fluency activity will lead into the concept of “remainders,” an important link to the lesson.  Teachers need to make sure there is a connection with the fluency to the lesson objective.