2018 Supporting Productive Struggle Conference

Many of you attended our recent Supporting Productive Struggle “Live” Lesson Study Conference in June where we bussed in a class of students and taught two “live” public lessons over two days during the conference. We are thrilled to share that we have been featured by the World Association of Lesson Studies in this recent post about the conference.

 

2018 Supporting Productive Struggle

The team from our region that researched and planned the public lesson study lessons included Shannon Kenyon from Lewiston School District (who also taught both public lessons), Lanna Proctor from Deary High School, Garett Bretz from Orofino Junior High, Pat Zinc from Lewiston High School and Phil Ulhorn from Sacajawea Junior High. The feedback from the conference indicates that teachers in our region strongly desire more of this type of collaboration and learning, and we look forward to providing similar opportunities in the future.

In the meantime, please celebrate the hard work of teachers in our region with us as well as the international recognition received for teachers in our region through the World Association of Lesson Studies!

Algebra Workbooks & Teacher Guides

We’ve been reorganizing the library and getting to know our resources again and wanted to highlight several helpful resources:

This Transition to Algebra set, funded by a grant from the National Science Foundation, was designed to quickly raise the competence and confidence of first-year algebra students by equipping them with a set of broadly applicable tools and strategies. Building on EDC’s hands-on, inquiry-based approach to learning, TTA uses algebraic logic puzzles and explorations to help students shift their ways of thinking from the concrete procedures of arithmetic to the abstract reasoning that success with algebra requires.We can give you a set but you’ll have to make copies of the workbooks for your students.

We also have a series called “Navigating through…” with dozens of books focused on grade bands PreK-2, 3-5, 6-8 and 9-12 and broken into topics like “…Algebra” (pictured below),"Data Analysis,” “Geometry,” “Measurement,” etc, each with a digital resource. The first half of each book provides pedagogy with corresponding worksheets in the back

Our Building PowerfulNumeracy for Middle & High School Students set has a student edition, facilitator guide and a collection of additional activities. Understanding numerical relationships gives students the freedom to choose a strategy. Students become engaged, willing to think, and more confident in their justifications. When we give secondary students this numerical power, we also help them learn higher mathematics with more confidence and more success.

And, finally, the Life ofFred series. They are designed as a home school curriculum, are very engaging with short “chapters.” Each chapter is followed by a review section and then the answers (with conversational banter: “’Wait it don’t get it!’ you say? Well, let’s put it this way…” and gentle admonitions not to peek at the answers)

Geometry Quilts Workbook

Coordinate Graphing: Creating Geometry Quilts

Now available in our resource library. Let us know if you would like to borrow this!

4th Grade Lesson Study

Introductory Lesson on Fraction Concepts and Equivalence

January 11, 2017

Research Question:

What types of questions and patterns of questioning advance the level of classroom discourse?

With NCTM’s Principles to Actions, Ensuring Mathematical Success for All as the primary resource, this lesson study group set out to gain a deeper understanding of how different types of questions and patterns of questioning impact the level of meaningful discourse (NCTM, pp. 29-41). Through the lens of this lesson, the lesson study process illuminated many significant and practical insights in the ongoing quest to advance the level of meaningful discourse amongst students.

This group recognizes that all four types of questions are important (NCTM, pp. 36-37). However, it is important to decipher what classroom conditions and instructional goals are more fitting for different types of questions and patterns of questioning. The group has observed that the “Gathering Information” (Type 1) and “Probing Thinking” (Type 2) questions, are beneficial for providing feedback to the teacher as to what students’ current level of understanding is and can support the formation and wording of “Making the Mathematics Visible” (Type 3) questions and “Encouraging Reflection and Justification” (Type 4) questions to be asked in a focusing questioning pattern, which is more suitable to advancing the level of meaningful discourse involving important mathematical ideas and relationships. In contrast, attempting to use Type 1 and 2 questions to help students understand mathematical ideas would likely be associated with a less-preferred funneling pattern. Using a funneling pattern of questioning could be more appropriate in helping students to understand the context of the problem (what the problem is asking or clarifying what the task is), but not if it were to interfere with student inquiry into deepening mathematical understandings of important concepts that students should discover on their own. While types of questions and patterns of questioning are crucial, they are not the only instructional decision that impacts the level of meaningful discourse. Group dynamics, the structure of the lesson, having multiple entry points and the selection of the task will all impact the level of student discourse. It is also important to attend to which questions should be asked for students to talk about in small groups compared to those questions that are posed in a whole group setting.

There was clear evidence that the visual and concrete representations that students produce strongly support meaningful discourse amongst students, and significantly impacted the level of meaningful discourse in this lesson. In some cases, the visual representations prompted some students to ask questions of each other regarding the mathematical meanings and big ideas exhibited in those visual/concrete representations, indicating an advanced level of meaningful discourse from the Levels of Discourse table (NCTM, p 32).

This group also discussed explicit strategies for enhancing discourse that could include modeling effective conversation between the teacher and a student while the rest of the students observe, with a follow-up whole-class discussion about important elements of sharing mathematical ideas (eye-to-eye contact, listening to others’ ideas, asking clarification questions of each others’ ideas, etc). An additional idea discussed was the use of sentence frames to provide access to begin conversations. 

Perhaps, however, one of the most important conclusions of this lesson study group has is a deepened understanding of, and belief in the power of the “5 Practices” model as an effective structure for promoting meaningful discourse and deeper mathematical understanding. Beyond the implementation of the “5 Practices” in the classroom, the group also believes that using this model as a tool for planning forced them to formally debate and decide upon questions that needed to be asked in different stages of the lesson that may not have been part of the planning discussion in some other model or framework. It was clear that using the 5 Practices for Orchestrating Productive Discussions naturally allows for more meaningful discourse. In this lesson, advanced levels of meaningful discourse were most visible when students were trying to connect different representations and ways of thinking. This model also lends itself to providing the opportunity for the teacher to ask more Type 3 and Type 4 questions. This group went as far as to question whether the “5 Practices” lesson structure had a greater impact on the level of discourse than the exact wording of the questions, question types and patterns of questioning. In particular, displaying students’ representations advanced the level of meaningful discourse and enhanced the quantity and quality of opportunities to use Type 3 and Type 4 questions in a focusing pattern. Additionally, because this structure allows for meaningful Type 3 questions, those same questions can pave the way for Type 4 questions. In other words, when asked to make the mathematics visible, students can then naturally be prompted to justify their reasoning when making connections among mathematical ideas and relationships, thereby advancing the level of discourse amongst students.

Mapping Lesson

Architects in Action

Adapted from http://www.discoveryeducation.com/teachers/free-lesson-plans/architects-in-action.cfm

Show students a map of the United States and point out the scale in the map key. Explain that sometimes we shrink objects or make them larger so they are easier to work with. The map is a scale model of an object that is too large to represent on paper. Other scale models represent objects that are too small, such as a diagram of an atom or a magnified view of a computer chip. Review the scale on the map. For example, the scale may say that 1 inch is equal to 50 miles. Explain that a scale is a ratio used to determine the size of a model of a real object. In this case, the map of the United States is the model.

Illustrate how to draw an object to scale. Use a ruler to draw a square on the board with sides that equal 10 inches in length. Ask students how they might use this square to draw another that is half its size. Now measure and draw a second square with 5-inch sides. Explain that when an object is scaled down, the length of its sides must be reduced by the same amount. Compare the corresponding sides of the two squares. The ratio of the small square to the larger is 5: 10. Show that a ratio can be expressed in three ways: 5: 10, 5 to 10, or 5/10, which is a fraction that reduces to 1/2.

http://francinemassue.weebly.com/map-the-classroom.html

1. Explain that students will use ratio to make a scale drawing on graph paper of the classroom floor plan including the desks, tables, closets, and so on. Divide students into teams of four. The class may choose to use either metric or English measurements.
2. Construct a class data table on the board with three columns labeled "object," "measurement," and "scaled measurement." Students should copy this table in their notebooks and fill in the answers as they measure the objects.  Once teams have recorded all their data, they will decide on the scale of their floor plan. 
3. Students can determine their scaled equivalents by cross-multiplying. Students should recall that when both sides of an equation are multiplied by the same amount, the equation remains balanced. 
4. Have students use their scaled measurement, rulers, and graph paper to draw the floor plan their team measured. Remind them to include a title, labels, and a scale.
5. As students complete their drawings, have them calculate the perimeter and area of their classrooms. What is the relationship between the drawing and the actual classroom? They should notice that the ratio of these areas is the square of the scale they chose. For example, if a scale of 0.5 inch = 1 foot was used, the ratio of areas of the drawing to the actual room will be (0.5 inches)² = (1 foot)² or 0.25 square inches = 1 square foot.

Additional Exercises 

• Compare your classroom floor plan to that of another student. How are they similar and different? Which would be more useful to a construction worker trying to build a classroom in a new school? Why?

• List other instances in which you use ratio to compare objects in your daily life. Why is it important to maintain the same scale for each measurement you record when making your model?

• Debate the merits of using the metric system and the English system to measure lengths. Explain how to convert between the two systems.

• Using what you have learned about ratios, proportions, and scale models, create four word problems for other students in your class to solve. For example: A square carpet measures 8 feet x 4 feet. Suppose the scale of a drawing containing the carpet is 1 foot to 1/4 inch. What are the dimensions of the carpet in the drawing? The answer: 2 inches x 1 inch.

Wordless Geometry Conundrum

From NCTM's September 2016 issue of Mathematics Teacher

What fractional part of the regular pentagram is shaded?

Given Answer: 1/2

A regular pentagram can be decomposed into triangular regions as colored in the figure below. The yellow and turquoise triangles are congruent. The problem's shaded region consists of two turquoise triangles, one yellow triangle and one red triangle, as does the unshaded region.

Glimpse into Lesson Studies in the region

5^th Grade Lesson Study – Introductory Lesson on Division with Decimals

December 7, 2016

Research Question:

What are the essential understandings that students need when dividing decimals, and what can we do to ensure all students learn something new towards 5NBT.7?

By providing students with opportunities to start the problem-solving process on their own without a teacher-prescribed strategy, represent their thinking, share their thinking with their peers, and listen to their peers’ way of thinking, students’ current understandings are illuminated in a way that drives decision-making throughout the lesson.  In this lesson, 19 out of 23 students were able to provide some representation of their thinking and/or problem-solving strategy on a concept that they’ve had no previous experience with. 

Developing expertise in estimating with decimals, paired with students’ reasoning behind their estimates, prior to experiences using computation with decimals strongly supports understanding.  Estimation supports reasoning, elicits evidence of students’ place value understanding and relative size of decimals, which were found to be 2 of 3 key ingredients in developing understanding of decimal computations (per van de Walle).  Given significant evidence that students rarely refer back to their estimate after determining a more precise answer (in this lesson, and based on the experiences of all teachers in the group), there is a need to support students in increasing their buy-in of estimation as a tool for problem-solving.  Also important to initially developing understanding of division with decimals, understanding the relationship between multiplication and division can serve as an entry point for students, and simultaneously supports understanding of both models of division (partitive and quotitive).

A third essential understanding of dividing with decimals lies in the concept of equivalence.  It seemed that in working towards developing generalizable methods for dividing with decimals, equivalence will become increasingly important.  Understanding that 1.1 could also be called 11 tenths will support the generalized method of re-writing the expression that involves decimals (3.6 divided by 0.4) to work with an equivalent expression that does not (36 divided by 4). 

Therefore, there seems to be a potential instructional sequence that involves significant work with estimation of decimal values to determine relative size of decimals (individually and with computation), place value understanding and equivalence, with equivalence having the closest connection to the more common generalizable methods for dividing with decimals.  What is more certain is that introducing procedures or algorithms, such as the long-division algorithm, before these understandings are in place can inhibit students’ reasoning, intuitive thinking and repertoire of problem-solving strategies.

Ideas for further study…

Why are students not transferring use of strategies/models to new situations?

How do we know when students are making meaning of models in a way that they will choose to use it in another situation?