The Mathematics Curriculum Framework is a tool districts can use to develop their own curriculum. A curriculum must be coherent, focused on important mathematics, and well-articulated across the grades. The mathematics it focuses on must be mathematics that is worth the time and attention of the students and that will prepare them for continued study and for solving problem in a variety of settings. Students should also have opportunities to learn increasingly more sophisticated mathematical ideas as they progress through the grades. They should not spend a significant part of their time reviewing topics already learned in previous grades.
The framework contains the big ideas that are aligned to the PA and NCTM standards, so it reduces the workload of the districts. Curriculum is not just the what but the how. The framework is also a tool to assist in making sure that there is not unnecessary repetition in district curriculum. For example, if two-digit addition shows up in grades 2, 3, 4, 5, 6, 7, and maybe even 8, the district needs to take a closer look at where students should have secure knowledge of this idea.
Standards-based curriculum and instruction includes content linked to the standards, instructional practices that foster problem solving, communication, reasoning and connections through multiple representations, and assessments that enable students to show what they know and can do in mathematics. Importance of high-quality teaching – teachers need to know mathematics in two ways. They must know mathematics to do mathematics by themselves. They must also know mathematics and how to use it to help their students learn. How do teachers learn to know mathematics in ways that enable them to organize the content, create/use activities, and adjust the activities to address the goals of the lesson as well as students’ needs, problems, and difficulties?
As you can see, the U.S. intends to “cover” many more topics than top achieving countries. How can students learn 25 or more topics in depth?
This slide shows more detail about the intended topics. How does the United States compare to the high-performing countries? The TIMSS study characterized U.S. curriculum as “an inch deep and a mile wide,” and attributed that to a splintered vision. So every effort was made to ensure that the framework was focused and coherent and that it addresses the PA and NCTM Mathematics Standards.
Even though some of the high-performing countries use tasks that focus on procedures, what happens during implementation? [They raise the level of the cognitive demands of the task for students.] What happens in the United States? Why do you think this happens?
The Superintendents of Allegheny County gave the Allegheny Intermediate Unit the charge to develop a regional mathematics curriculum framework. Rather than expend the resources, time and money, in each district doing the task, they saw value in developing the curriculum regionally. Then each district could use its resources for professional development and for instructional materials.
Based on the work of Liping Ma, the framework’s knowledge networks show a big idea, key components of that idea, and building blocks that student need to begin to understand that big idea. It is not hierarchical. We are calling them networks because we want to emphasize the connections between the ideas.
The first task for the team was to identify the big ideas at each grade level. These are the big ideas for grade three.
It looks like there isn’t much to do; let’s take a closer look. Remember, the oval is the big idea, the essential learning. As with the individual webs, there may be an order one wishes to follow among the big ideas. However, no particular order is suggested.
Big Idea on Top linked to Key Components and building blocks
This slide shows an example of High School Probability web. Even though Understand and Apply concepts of Probability appears as a big idea at earlier grade levels, this web describes how the concepts will be treated at a higher level, thus spiraling student knowledge in a way that is in line with PA and NCTM standards for this content strand.
Curriculum is not just the WHAT. It is also the HOW. By identifying the big ideas, teachers won’t be expected to teach everything every year. There will be time for students to use manipulative materials. Time to allow students to struggle with problem solving. Time to hold classroom discussion and give students a chance to talk in class about the mathematics. This may require different instructional materials that expect greater depth of understanding.
Use to complete your district’s strategic plan. Use for professional development.
K-12 Regional Mathematics Curriculum Framework Math & Science Collaborative Allegheny Intermediate Unit
Curriculum Frameworks Are Resources To: <ul><li>Assist districts in developing curriculum consistent with the PA Standards for Mathematics and PA Assessment Anchors </li></ul><ul><li>Indicate what students should know and be able to do at each grade level </li></ul><ul><li>Provide high quality learning experiences for students so that they may develop deep conceptual understanding of the essential learnings </li></ul>
Curriculum Frameworks Are Not: <ul><li>Required by law </li></ul><ul><li>Detailed lesson plans </li></ul><ul><li>Items for which students must be tested </li></ul><ul><li>Directives for uniform programs and textbook adoption </li></ul><ul><li>Curriculum formats that all districts must follow </li></ul>
The Process <ul><li>Team members identified ‘big ideas’ at each grade level </li></ul><ul><li>Discussed essential learnings across grade levels </li></ul><ul><li>Eliminated unnecessary repetition while assuring there were no gaps </li></ul><ul><li>Confirmed that PA and NCTM Standards are addressed </li></ul><ul><li>Provided a chart with alignment of Big Ideas to PA Standards and Assessment Anchors </li></ul>
Background and Rationale <ul><li>Implement PA and NCTM Standards </li></ul><ul><ul><li>What is standards-based curriculum and instruction? </li></ul></ul><ul><li>Glenn Commission Report: “Before It’s Too Late” </li></ul><ul><ul><li>Emphasizes the importance of “high-quality” teaching </li></ul></ul><ul><li>TIMSS Data </li></ul><ul><ul><li>U.S. curriculum lacks focus and coherence </li></ul></ul>
FIGURE 1 Types of Math Problems Presented FIGURE 2 How Teachers Implemented Making Connections Math Problems TIMSS Video Study
So…. <ul><li>How can we identify and organize a few big ideas at each grade level? </li></ul><ul><li>What might the curriculum look like as it’s organized around big ideas in mathematics? </li></ul>
These ideas represent the essential learning at this grade level. All learning is correlated with the Pennsylvania Mathematics standards. Problem solving, reasoning, communications, connections, and representations are integrated throughout the curriculum. Teachers should encourage students to demonstrate and deepen their understanding of numbers and operations by solving interesting, contextualized problems and by discussing the representations and strategies they use . Understand the Base 10 Number System Understand the Relationship between Addition and Subtraction Analyze Data Understand Multiplication and Division Classify Figures Select and Use Appropriate Units and Systems to Measure Understand Patterns, Relations, and Functions Third Grade
Understand Multiplication and Division within 100 Understand multiplication as repeated addition, combining equal groups, using arrays, area models, linear models, Cartesian models. Represent multiplication and division problems using many different models. Use different strategies to solve multiplication and division facts and problems, including multi-step problems and determine reasonableness of answers. Know multiplication and division facts to 10. Apply multiplication and division to non-routine problems Find missing factors. Problem solving, reasoning, connections, communication and representations should be integrated throughout the web. Understand division as repeated subtraction, finding equal shares. Understand the relation between multiplication and division. Grade 3
Read interpret, and construct charts, tables, line plots, pictographs and bar graphs. Analyze Data Form and justify an opinion on whether a given statement is reasonable based on analyzed data. Gather, organize, analyze and display data using tables, charts, pictures, graphs, etc. Predict the likely number of times a condition will occur based on analyzed data. Problem solving, reasoning, connections, communication and representations should be integrated throughout the web. List or graph the possible results of an experiment, (e.g. design a fair or unfair game). Grade 3
Grade 6 Investigate and Understand the Processes of Measurement Understand the Meaning and Effects of Arithmetic Operations with Decimals, Fractions and Percents Recognize and Create Examples of Geometric and Numeric Patterns Construct and Interpret Data Displays in order to Make Inferences, Predictions and Arguments Understand Number Theory with respect to Factors and Multiples
Grade 6 Recognize and Create Examples of Geometric and Numeric Patterns Demonstrate Transformations on the Coordinate Plane Identify Examples of Geometric and Numeric Patterns Identify and draw lines of symmetry Represent, analyze, and generalize patterns with tables, graphs, words, and/or rules Problem Solving, Reasoning, Communication, Connections, and Representations should be integrated throughout the web. Determine and describe sizes, positions, and orientations of shapes, after rotations, dilations, reflections, and translations Relate and compare these different forms of representations for a relationship Use some transformations to create tesselations Identify and construct similar figures that model proportional relationships
E.3 Understand and/or apply basic concepts of probability or outcomes <ul><li>Solve problems involving independent simple and compound events. </li></ul><ul><li>Use experimental and theoretical probability distributions to make judgments about the likelihood of various outcomes in uncertain situations. </li></ul>9-12 Understand and Apply Concepts of Probability E.4 Develop and/or evaluate inferences and predictions or draw conclusions based on data or data displays <ul><li>Draw and justify a conclusion regarding the validity of a probability or statistical argument. </li></ul>E.2 Select and/or use appropriate statistical methods to analyze data E.3 Understand and/or apply basic concepts of probability or outcomes <ul><li>Apply probability and statistics to perform an experiment involving a sample and generalize its results to the entire population. </li></ul>9-12 Select and Use Appropriate Statistical Methods to Analyze Data 9-12 Develop and Evaluate Inferences and Predictions Based Upon Data Analysis 9-12 Understand and Apply Concepts of Probability E.3 Understand and/or apply basic concepts of probability or outcomes <ul><li>Compare odds and probability. </li></ul>9-12 Understand and Apply Concepts of Probability PA Assessment Anchor 2.7.11 Grade 11 PA Standards Knowledge Webs from Framework
Reasons to Use Framework <ul><li>To avoid feeling overwhelmed with so much content </li></ul><ul><li>To move from concrete to graphic to abstract at all levels </li></ul><ul><li>To shift from a focus on teaching procedures to engaging students in problem solving </li></ul><ul><li>To lead students to a deep conceptual understanding of the essential learnings </li></ul>
Standards-Based Classrooms <ul><li>Curriculum that includes: </li></ul><ul><ul><li>Content linked to standards </li></ul></ul><ul><ul><li>Instructional practices that foster problem solving, communication, reasoning and connections through multiple representations </li></ul></ul><ul><ul><li>Authentic assessments that enable students to show what they know and can do in mathematics </li></ul></ul>
Using the Framework <ul><li>As an aid in writing District-level curriculum </li></ul><ul><ul><li>Adapt/adopt </li></ul></ul><ul><li>As a Professional Development tool </li></ul><ul><ul><li>Use to start discussions within and across grade levels </li></ul></ul><ul><ul><li>Use activities and assessments in professional development </li></ul></ul>