Teaching Sequences for Short-Term and Instant Interventions

At the beginning of Tuesday morning’s session for grades 3-5 mathematics, participants described conditions that they would want to have in place in their “ideal” teaching community. Answers included the no “blame” game, common language approaches, vertical and horizontal teaming, time for data driven instruction and an endless supply of manipulatives.The goal of the day was to continue discussing intervention methods, specifically short-term and instant intervention within a lesson, and how these methods could help the participants’ ideal community become a reality.

Short-term intervention is based on the same cycle as the extended intervention: assess, analyze, plan and teach. After assessing, the analysis focused on different types of questioning strategies that could be utilized to determine where the error occurred, where the last place that the student seemed successful was, and what gaps might exist that could make the next objective difficult. Questioning or “break it down” techniques included providing an example, providing a context, providing a rule, providing a missing or first step, the roll back, or narrowing/eliminating false choices. Teachers need to exercise restraint during the questioning so as not to take too much time out of the lesson and lose the focus.

Strong questioning techniques come from a solid knowledge of the content, not just at grade level, but across the board. Content knowledge is obtained through text study, collaborative planning, peer coaching and professional development. Participants discussed how to plan short-term interventions, which used the same process described in Monday’s post, but on a much smaller time scale. Teachers need to decide when a short-term intervention is more appropriate for supporting a student than extended intervention. Teachers also need to determine what they need to develop in themselves so that they can quickly craft effective short-term interventions.

The session concluded with reflection on a professional reading by T.R. Wang. This one quote seemed to summarize the objective of the presentation: “…one is to study whom you are teaching, the other thing is to study the knowledge you are teaching. If you can interweave the two things together nicely, you will succeed.”

Materials for this session are available here.

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:
http://www.engageny.org/resource/may-2014-nti-grades-k-5-math-turnkey-kit-for-network-teams

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

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

Module Focus: Grade 5 Math Module 5

The Grade 5 presentation focused on the material covered in Module 5: Addition and Multiplication with Volume and Area. Participants practiced lots of hands-on activities in order to experience the progression students experience with developing their concept of volume and area. Students first experience calculating volume by building figures and counting unit cubes. Students construct open boxes and calculate volumes by “filling” in the box. They then experience volume pictorially through the use of dot paper and constructing cubes.

There is much discussion in the module about composing and decomposing right rectangular prisms using layers, which helps with students’ conceptual knowledge of what volume actually means. There is no mention of a volume formula in Topic A. Topic B is where the multiplication formula is introduced with the concept of layers. Students also explore the connection between volume in cm and liquid volume in mL. We were able to see the liquid volume increase by 1 mL after the dropping of a cubic centimeter – very cool!

Application problems were presented at this point, such as:

  • A small fish tank is filled to the top with water. If the tank measures 15 cm x 10 cm x 10 cm, what is the volume of the water in the tank? Express answer in Liters. What if after a week, water evaporates so that the water level in the tank is 9 cm high? What effect does that have on the volume of the water? How many Liters?
    • This is an interesting problem in that students can just take off the “layer” from the original water level, or they can re-calculate with a new height of 9 cm. 
    • A shed in the shape of a right rectangular prism measures 6 ft. long by 5 ft. wide by 8 ft. high. The owner realizes that he needs 480 cubic feet of storage. Will he achieve this goal if he doubles each dimension? If he wants to keep the height the same, what could the other dimensions be for him to get the volume that he wants?
      • This problem lends itself to discussing what happens to volume when you double one dimension, two dimensions, or all three. The “create a sculpture” activity in lessons 8-9 is an opportunity for students to express their creativity, while at the same time apply the concepts and formula of volume to design a sculpture within a given set of parameters. The activity is graded with a rubric used by the students. Participants discussed the value of having students use a rubric. Peer review always holds students more accountable, but the peer review also ties into the Mathematical Practice of critiquing the reasoning of others. 

Topic C shifts the focus from volume to calculating the area of rectangles with fractional side lengths.  Once again, this demonstrated an excellent transition from concrete, pictorial to abstract. Students tile a rectangular region using patty paper, then draw the image on white paper (area model), and then use prior knowledge of area (partial products) and the multiplication of fractions to calculate the area. Participants practiced this transition using mystery rectangles. The topic ends with application problems that ask students to decide which process is more efficient and whether they should deal with improper fractions or convert to mixed numbers.  We want them to say “It depends.”

Topic D uses the cutting apart of trapezoids and parallelograms in order to take a look at the properties that exist for each, leading the student towards success in being able to create a hierarchy of quadrilaterals that go from most general to specific. Excellent visual activities were done here with parallelograms constructed by the group so that participants had a wide range of parallelograms. Activities showed the angle relationships that exist within these shapes (consecutive angles supplementary, all four angles add up to 360). Participants looked at diagonals for parallelograms and great questioning techniques were modeled in regards to answering the question, “will the diagonals always be bisected, or are they ever the same?”  Angle measurement was recommended as a fluency activity. An excellent end to an excellent week here at NTI.

Module Focus: Grade 4 Math Module 5

The Grade 4 Math session on Thursday focused on the Module 5: Fraction Equivalence, Ordering and Operations. Number bonds, tape diagrams, and area models were used consistently throughout the module to strengthen conceptual understanding and develop confidence when working with fractions. This “fractional” confidence allows students to transition to concepts/problems of higher complexity, which was demonstrated as we went through the lessons.

Students start their fraction work off with experiencing problems that involve decomposing fractions using number bonds, similar to the work they did with number bonds and whole numbers in the earlier grades.  How many ways can you represent 5/6 as an addition problem? When answering this problem, students encounter how to express a non-unit fraction as a whole number times a unit fraction. The work here is extended to fractions that are greater than 1, such as decomposing 7/4 = 4/4 + 3/4 .

Fraction equivalence is explored using tape diagrams (paper folding) and area models.  Participants look at an application problem from lesson 5:

A loaf of bread was cut into six equal slices.  Each of the slices was cut in half to make thinner slices for sandwiches.  Mr. Beach used four slices.  His daughter said, “Wow, you used 2/6 of the loaf.” His son said, “No, you used 4/12.” Work with a partner to explain who was correct using a tape diagram.

This problem pulls in all content discussed so far, and solving the problem does not require a fractional algorithm. Fraction equivalence is extended to fraction comparison. Students combine knowledge of benchmark fractions with fraction equivalence to handle comparisons that involve fractions with common numerators, fractions with denominators of related units. The final goal is comparing fractions with denominators of unrelated units.  Topic D shows once again the link of the work done previously with decomposition and composition to the addition and subtraction of fractions with common denominators.  A new visual is added here, the number line with arrows.

Topics E and F add a layer of complexity to what has been learned by extending fractional equivalence and operations to fractions greater than 1. Based on their knowledge, students devise their own strategy for handling problems like 3 3/5 – 4/5. Some might decompose the 4/5 to be 3/5 and 1/5, and then solve the simpler problem of 3 1/5, which is 2 4/5. Others might decompose the 3 as 2 5/5 and now look at the problem 2 8/5 – 4/5, which is 2 4/5. Students practice converting between improper fractions and mixed numbers based on the context of the problem. Never is the traditional algorithm of how to convert a mixed number into an improper fraction discussed.  The module shows that the algorithm is not necessary.  The module ends with a re-visit to repeated addition of fractions as multiplication and shows the connection to the distributive property when solving problems like 2 x 3 1/5.  Visuals are used again here to help make the connection.

The remainder of the session presented educators with a plan on how to make choices with implementing the lessons within the given time frame that remains between now and the assessment. Pacing is a huge area of concern and many teachers are behind the timeline. So how do we adapt the lessons to support successful pacing while bridging gaps in prior knowledge, but not sacrifice the rigor?

A planning protocol was introduced that encourages teachers to look at the lessons further out, not day to day. Reading the module overview and studying the module assessments is the place to start in order to keep the purpose, sequence and delivery fresh in the mind. Next, teachers should read through the lesson and ask what major concept is necessary to successfully complete the exit ticket. Pay attention to the subsequent lesson and examine the exit ticket there. What is the relationship between the two exit tickets and what will be the impact of what gets cut out of the lesson to those two tickets? Teachers also always need to consider the needs of specific students in their classroom.