Math Practices Posters

If you've been looking for posters to help illustrate the mathematical practices, you might want to take a look at the posters posted on the Utah Education Network website.  They have developed sets of posters for grades:

• K-1
• 2-3 
• 4-5 
• 6 and a second 6th grade version here

Here's one of the math practice posters for K-1.

And here's the same math practice standard for 6th grade.

I'll let you know if I find additional posters -- particularly for middle and high school.

Smarter Balanced Item and Performance Task Development

Smarter Balanced (the Common Core assessment system California is part of) has released a series of training tools for those who will be writing assessment items.

Six types of tasks are described in their Introduction to Evidence-Centered Design powerpoint:

1) Selected Response:  Our students have been exposed to Selected Response items on the CSTs for years.  With the new technology SBAC will use, selected response items will allow for more than one correct response.  As can be seen in the following item, students are responsible for thinking about number of sides, angle measures, parallel side relationships, and side lengths all at one time.  

As can be seen in the example below, students will need to carefully examine all of the possible response options, because more than one can be correct.  

2) Constructed Response:  Now that students in California will be responsible for calculating and providing their own answers, publishers and teachers will need to move away from multiple-choice only classroom assessments.  In addition, students will need to learn to organize their work and their thinking to provide legible, complete, and convincing evidence to support their solutions.

3) Extended Response:  Students will need to developing their reading comprehension skills for this type of task, as well as their ability to work with multiple parameters.

4) Performance Tasks:  When a complex claim is being assessed, a performance task may be required to support the full range of the claim.  "A Performance Task is used to assess a set of assessment targets as opposed to a narrow focus on just one or two targets, as is typically the case with selected and constructed response items.  As an example, this performance task contains multiple parts, each designed to collect specific types of evidence that are combined to make a claim about student ability to read, synthesize, and communicate in writing."

5) Technology-Enhanced Tasks:  These tasks capitalize on technology to collect evidence through a non-traditional response type.  For instance, in the item below, the students will be using technology to draw a line on the grid -- the green line is an example of what the student might produce.

Additional examples of technology-enhanced tasks from the Mathematics Selected Response, Constructed Response, and Technology-Enhanced Items powerpoint

6) Technology-Enabled Tasks:  These tasks incorporate multi-media or other interactive elements in the assessment target -- what the student sees/hears before they respond.  The students will then respond to this type of task by either selecting one or more responses or by producing text or numerals.  In this task, the students are able to use a dynamic geometry environment to construct the desired floor.  Once a student is satisfied, he can then type in the response to the question.

With more SBAC assessment examples available, teachers will definitely want to think about their own assessment practices.  Some issues that quickly come to mind include:

• Preparing students for multiple correct answer selected response tasks
• Preparing students to check for accuracy before answering constructed response tasks
• Preparing students to show their work and explain their reasoning in constructed, extended, and performance tasks
• Preparing students for the reading and writing academic language demands
• Providing students with dynamic geometry and graphing technology experiences, as well as spreadsheet experiences (based on sample items that I've seen so far).  
• Ensuring that students have ongoing experiences with visual representations of mathematics concepts and operations

As you think of additional issues that relate to teacher assessment practices, please share them in the comments below.

Thanks for all that you do for students!

Great Tasks Sampler from NCSM

As we're thinking about how to engage students with tasks that incorporate both the CCSSM content and practices standards, it's helpful to have tools that allow us to analyze potential tasks and rubrics to help both teachers and students identify the important elements of the mathematical work being done.

NCSM has put together some sample tasks along with tools and rubrics that are helpful starting points:
2012 Great Tasks Sampler at http://www.mediafire.com/view/?2qlwurmgdc0bbve.

For more resources from NCSM around CCSSM tasks, take a look at http://www.mathedleadership.org/ccss/materials.html.

Thanks for all that you do for students!

Core Math Tools for Secondary Students

NCTM has recently released a fascinating set of Java-driven math tools, including CAS, dynamic geometry and trigonometry, spreadsheet, data analysis, and simulation tools.  You can find introductory information and the download link at http://www.nctm.org/resources/content.aspx?id=32702.

I haven't had a lot of time to get acquainted with these tools yet, but it is clear that this is an important addition to the freeware we can make available to students, particularly as we incorporate mathematical modeling as well as career readiness into our courses.

I suspect that this will be an excellent companion tool to Geogebra (www.geogebra.org) and WinPlot (math.exeter.edu/rparris/winplot.html).  


I'll share more about all of these tools in future posts.


Thanks for all that you do for students!


Note:  I just ran across a list of technology tools the Park City Math Institute used in 2011 http://mathforum.org/pcmi/hstp/tech.resources.html.

Sample Questions Highlight Common Core Expectations

New York has recently posted sets of sample questions for grades 3 through 8 that are designed to help teachers think about the instructional shifts that will be needed as we move to the Common Core State Standards for for both ELA and Math http://www.p12.nysed.gov/apda/common-core-sample-questions/
   * Multiple Choice
   * Short Constructed Response
   * Extended Constructed Response


As you look at these sample questions, the New York site recommends you do the following:

* Interpret the way the standards are conceptualized in each question.
* Note the multiple ways the standard is assessed throughout the sample questions.
* Take note of numbers (e.g., fractions instead of whole numbers) used in the samples.
* Pay attention to the strong distractors in each multiple-choice question.
* Don’t consider these questions to be the only way the standard will be assessed.
* Don’t assume that the sample questions represent a mini-version of future state assessments.

I've selected a few samples from each grade level that seemed particularly noteworthy.

3rd Grade

4th Grade Fractions -- a challenging multi-step problem..  Many students would benefit from knowing how to use bar diagrams to help conceptualize this problem.  To learn more about bar diagrams, take a look at www.thinkingblocks.com.

4th Grade Operations and Algebraic Thinking -- this problem requires students to move beyond step-by-step arithmetic thinking.  They'll need to have lots of experiences "telling the story from beginning to end" using numbers and symbols.

5th Grade Number and Operations in Base Ten -- this is a very different kind of place value question than most textbooks currently ask.  Instead of asking the value of the 5, they're asking for a relative comparison that involves a clear understanding that every place value is 10 times larger than the place value to its immediate right.

5th Grade Number and Operations: Fractions -- Students will need experiencing creating their own fraction models and also understanding what is meant by "expression".  From examples in the Progressions fraction document, it seems clear that rectangular area models will be an important tool for students, along with some experience with circle area models.  Also, we will all need to make sure we understand the difference between a "visual model" and "mathematical modeling".  

5th Grade Measurement -- another multi-step word problem, with the additional challenges of metric conversion with decimal quantities.  A nice approach to metric conversions can be found in the Primary Math materials from Singapore, where students learn to think of 5.6 meters as 5 meters and 60 centimeters, which can then be more easily converted to 500 cm + 60 cm.  After working with undergraduates who have only learned to convert using the King Henry ... mnemonic, I would highly recommend moving away from this procedure without understanding to something more like what Singapore has students understand.

6th Grade Expressions and Equations -- This evaluation of a 4-term expression with exponents above 2 might more typically have been seen in 7th and 8th grades.  This would indicate that 6th grade teachers will want to add algebra tiles to their teaching repertoire.  Notice also that there is much more room for student error in this constructed response format, so having students develop the habit of mind of double checking their work (Accuracy and Precision Math Standard) will be important.

6th Grade Measurement and Expressions/Equations -- Another question that anticipates that students will be able to add like terms, and use the distributive property with algebraic expressions.  It also challenges students to understand that when linear measures are scaled (such as perimeter) the scale factor remains the same, but when area measures are scaled, the scale factor is squared.  In this problem, though, students can use calculations in Part C if they are unsure of this high-level understanding.

6th Grade Geometry -- This is a multi-step geometry problem on a coordinate plane.  Notice that students will need experience with coordinate axes where the interval is greater than 1.  Many textbooks provide primarily single-step problems in geometry, where students are either asked to plot points, or identify a triangle on the coordinate plane, or find the area of a triangle or other quadrilateral -- but not all of these at one time.

7th Grade Sample Questions -- note the incorporation of fractions and mixed numbers that increase the complexity of the items.

8th Grade Sample Questions -- note the expectation that students will be able to "explain" and "justify" in their constructed response questions.  Also, the questions about slope will require students to have a strong conceptual understanding of slope, and to be able to move comfortably between the concept of slope as seen in tables, graphs, and equations.

These are some of my observations after a first look at the New York Sample Questions.  Once you've had a chance to look at the items, it would be great to get your additional observations -- feel free to add your comments to this post.

Thanks for all that you do for students!

Mathematical Modeling: Exciting and Challenging!

Mathematical Modeling is one of the most dramatic changes across the Common Core Math Standards.  Most of us haven't experienced this kind of mathematics inside the classroom -- either as students or as teachers.  Fortunately, most of us have experience in our personal and, possibly, professional lives.  Because mathematical modeling is all about using mathematics to solve complex, real-world problems.  If you've ever planned a party or vacation, bought a car, or renovated a house, you've most likely been engaged in mathematical modeling.

(many more cars and features were included in this task)

Even trying to make healthy eating choices generally involves mathematical modeling, where we have so much nutritional information that we need to prioritize and simplify the information that we decide to focus on as we put together our food selections.  

Today I talked with UCI Department of Education alumni about Mathematical Modeling, and what teachers can start working on with students to help them begin to develop some of the concepts and skills involved in mathematical modeling.

Four areas that seem like good starting points:

1. Identifying important quantities, and discussing possible assumptions and approximations to simplify complex real-world situations
2. Helping students to learn how to strategically use spreadsheets, dynamic inquiry software, and calculators to map and analyze relationships mathematically.  Geogebra is a particularly useful and FREE technology tool that can be used to model geometric, algebraic, and statistical problems.  
3. Having students learn about and get more practice with writing about mathematics.  As you look at sample Smarter Balanced assessment items (see Assessment page at the top of this blog), you'll see the kinds of writing students are likely to be asked to do.  
4. Make sure that students are getting experience with all four Depth of Knowledge levels.  When students are formulating mathematical models, they will probably be working at DOK Level 3.  When they are analyzing, interpreting, and reflecting on their mathematical models, they will be working at DOK Level 4.

If you're looking for more details about how to strategically introduce mathematical modeling to your students, I'd recommend two short papers by Lyn English.  These papers include several math modeling problems, and explain how following their problem sequence can help scaffold student understanding and skill development around mathematical modeling.  

• Mathematical Modelling with Young Learners
• Mathematical Modelling in the Primary School  

If you'd like to see concrete examples of math modeling tasks, I've started a collection of resources on the Domain: Modeling page at the top of this blog.  If you find more good tasks that you think I should add, feel free to include your link in the comments section.

Thanks for all that you do for students!  

Val

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Welcome to the Common Core Math Library

Welcome to the Common Core Math Library blog.  I've started it to provide an organized location for some of the many Common Core math resources.

If you find a resource that you'd like me to add, feel free to post it in the comments section, and I'll do my best to transfer it to one of the resource pages.

And, if you are working with K-12 students, pre-service student teachers, or classroom teachers, and would be willing to share what you are trying and learning about Common Core math standards, I hope you will add your comments to this blog.

Thanks for all that you do for students!

Valerie Henry, Ed.D.
Lecturer, Math Education
UC Irvine