Creating Rubrics

I can't take the credit for designing this sample rubric, thanks go to Cait Camarata from Edutopia. Here's a quick view. (Double click image to enlarge it.) If this works for you, go to Edutopia Sample Rubric to download an editable version.

What are the big ideas students should understand about measurement?

From: Van de Walle, John A. & Lovin LouAnn H. ; Teaching Student Centered Mathematics
Grades 3-5; Pearson Education Inc. 2006.

• Measurement is about comparisons of an object's attribute with a unit that has the same attribute. Ex. An object's length is compared to units of length, an object's area to units of area, time to units of time, etc.

• Meaningful measurement and estimation of measurements depend on personal familiarity with the unit of measurement being used. (It has to make sense).

•Estimation of measures and the development of personal benchmarks for frequently used units of measure help students become more familiar with units. Developing personal benchmarks help prevent errors in measurement and help students make sense of the action of measuring.

• Measurement instruments are devices that replace the need for actual measurement units. It is important to understand how measurement instruments (rules, scales, thermometers, etc) work so that they can be used correctly and meaningfully.

•Area and Volume formulas provide a method of measuring these attributes by using only measures of length.

•Area, perimeter and volume are related to each other, tho not precisely or by formula. For example, as the shapes of regions or three-dimensional objects change, but maintain the same areas or volumes, there is a predictable effect on the perimeter and surface areas.


Students should learn that when measuring something, there are three steps:

1. Decide on the attribute to be measured
2. Select a unit that has that attribute
3. Compare the units by filling, covering, matching or some other method, with the attribute of the object being measured.

How Should We be Teaching Mathematics?

What are the big ideas we should be teaching in Mathematics?

• Mathematics is a language we use to communicate ideas

• Pattern is the underlying theme of mathematics

• The goal of teaching mathematics is to help students develop mathematical power (autonomy, confidence in problem solving) and 

mathematical thinking abilities (ability to recognize patterns, identify errors, generate alternative strategies for problem solving; generalize common problem situations)


The Case for Teaching What to Do and Why: An article by Marilyn Burns


What are mathematically worthwhile tasks? (Designing Mathematically Worthwhile Tasks)
The NCTM Brief (April 8, 2010) outlines the following criteria to guide the creation of a worthwhile mathematical task:

1. The problem has important, useful mathematics embedded in it.
2. The problem requires higher-level thinking and problem solving.
3. The problem contributes to the conceptual development of students.
4. The problem creates an opportunity for the teacher to assess what his/her students are learning and where they are experiencing difficulty.
5. The problem can be approached by students in multiple ways using different solution strategies.
6. The problem has various solutions or allows different decisions or positions to be taken and defended.
7. The problem encourages student engagement and discourse.
8. The problem connects to other important mathematical ideas.
9. The problem promotes the skillful use of mathematics.
10. The problem provides an opportunity to practice important skills.

What real world mathematicians do:
reason, investigate, conjecture, refute, justify, question, calculate, organize, look for patterns, prove, sort, change, generalize, compare, explain, discuss, verify, delete, collect, explore patterns and relationships, formulate questions, make sense of ideas, think in flexible ways, formulate and solve problems

Mathematical Habits of Mind or Dispositions:
• willingness to accept challenges, to explore ideas
• perseverance
• curiosity

What is Mathematical Literacy?
The National Research Council has produced an influential document, 
Adding It Up: Helping Children Learn Mathematics, that provides one way 
of describing mathematics literacy. 

A mathematically literate person is described as one who demonstrates 

• Conceptual Understanding: understanding mathematical concepts, 
operations, and relations 

• Procedural Fluency: skill in carrying out procedures flexibly, 
accurately, efficiently, and appropriately 

• Strategic Competence: the ability to formulate, represent, and solve 
mathematical problems 

• Adaptive Reasoning: the capacity for logical thought, reflection, 
explanation, and justification 

•Productive Disposition: habitual inclination to see mathematics as 
sensible, useful, and worthwhile, combined with a belief in diligence (perseverance) and one’s own efficiency

Breen and O' Shea: Mathematical Thinking and Task Design
http://www.maths.tcd.ie/pub/ims/bull66/ME6601.pdf


What is the role of teachers in developing mathematical thinking?
From: NCTM Principles and Standards for School Mathematics
http://www.fayar.net/east/teacher.web/Math/Standards/Previous/ProfStds/TeachMath1.htm

The teacher of mathematics should pose tasks that are based on-

 sound and significant mathematics;
 knowledge of students' understandings, interests, and experiences;
 knowledge of the range of ways that diverse students learn mathematics;

and that:
 engage students' intellect;
 develop students' mathematical understandings and skills;
 stimulate students to make connections and develop a coherent framework for mathematical ideas;
 call for problem formulation, problem solving, and mathematical reasoning;

 promote communication about mathematics;
 represent mathematics as an ongoing human activity;
 display sensitivity to, and draw on, students' diverse background experiences and dispositions;
promote the development of all students' dispositions to do mathematics.




What is the responsibility of students in learning to think like mathematicians?
• to make sense of the task,of mathematical ideas
• to wonder, be curious, ask questions
• to estimate,make predictions (I think...; I predict...;
• to justify thinking (Here's how I know I am right.. I know this is true because...)
• to make connections among ideas and prior knowledge (How does this connect with what I already know? Is it a new idea? Does it expand on something I already know? Why is this idea important to me?)
• to assume ownership of their learning: (How will I use this idea in my life? How can this idea help me? Why is this idea important to me?)
• to work collaboratively as well as independently (My thinking.../ My friend's idea...My questions.../ My friend's question...)



Criteria for Mathematical Thinking: A Message to Students
1. Everything you do in mathematics should make sense to you
2. Whenever you get stuck, you should be able to use what you know to get yourself unstuck.
3.You should be able to identify errors in answers, in use of materials, and in thinking
4.Whenever you do a computation, you should use a minimum of counting
5. You should be able to perform calculations with a minimum of rote pencil-paper computations
6.When the strategy you are using isn't working, you should be willing to try another strategy instead of giving up.
7. You should be able to extend, or change, a problem situation by posing additional conditions or questions.

Assessing Mathematical Thinking:

Process Standards: Besides content, students need to develop these process skills:
• Problem Solving 
• Reasoning and Proof 
• Communication 
• Connections 
• Representations 

Problem-Solving Standard 
Instructional programs from pre-kindergarten through Grade 12 should 
enable students to 
• build new mathematical knowledge through problem solving 
• solve problems that arise in mathematics and other contexts 
• apply and adapt a variety of appropriate strategies to solve problems 
• monitor and reflect on the process of mathematical problem solving  9 


Reasoning and Proof Standard 
Instructional programs from pre-kindergarten through Grade 12 should 
enable students to 
• recognize reasoning and proof as fundamental aspects of 
mathematics 
• make and investigate mathematical conjectures 
• develop and evaluate mathematical arguments and proofs 
• select and use various types of reasoning and methods of proof 


Communication Standard 
Instructional programs from pre-kindergarten through Grade 12 should 
enable students to 
• organize and consolidate their mathematical thinking through 
communication 
• communicate their mathematical thinking coherently and clearly to 
peers, teachers, and others 
• analyze and evaluate mathematical thinking and strategies of 
others 
• use the language of mathematics to express mathematical ideas 
precisely 

Connection Standard 
Instructional programs from pre-kindergarten through Grade 12 should 
enable students to 
• recognize and use connections among mathematical ideas 
• understand how mathematical ideas connect and build on one 
another to produce a coherent whole 
• recognize and apply mathematics in contexts outside of 
mathematics 

Representation Standard 
Instructional programs from pre-kindergarten to Grade 12 should 
enable all students to 
• create and use representations to organize, record, and 
communicate mathematical ideas 
• select, apply, and translate among mathematical representations to 
solve problems 
• use representations to model and interpret physical, social, and 
mathematical phenomena 

Reference: 
National Council of Teachers of Mathematics. Principles and Standards for School 

Mathematics. Reston, VA: National Council of Teachers of Mathematics, 2000.




Having Mathematical Conversations: The Teacher's Role in Discourse
From: NCTM Principles and Standards for School Mathematics
http://www.fayar.net/east/teacher.web/Math/Standards/Previous/ProfStds/TeachMath2.htm

The teacher of mathematics should orchestrate discourse by-

 posing questions and tasks that elicit, engage, and challenge each student's thinking;
 listening carefully to students' ideas;
 asking students to clarify and justify their ideas orally and in writing;
 deciding what to pursue in depth from among the ideas that students bring up during a discussion;
 deciding when and how to attach mathematical notation and language to students' ideas;
 deciding when to provide information, when to clarify an issue, when to model, when to lead, and when to let a student struggle with a difficulty;
 monitoring students' participation in discussions and deciding when and how to encourage each student to participate.

Revoicing: a teaching tool to promote and support "math talk".

"In a revoicing move, the teacher essentially tries to repeat some or all of what the student has said and then asks the student to respond and verify whether or not the teacher’s revoicing is correct, as in the dialogue below."

1. Philipe: Well, if we could use three, then it could go into that, but three is odd. So
then if it was . . . but . . . three is even. I mean odd. So if it’s odd, then it’s not even.
2. Ms. D: OK, let me see if I understand. So you’re saying that twenty-four is an odd number?
3. Philipe: Yeah. Because three goes into it, because twenty-four divided by three is eight. 

This excerpt from: http://mathsolutions.com/wp-content/uploads/0-941355-53-5_L.pdf

Resources:
The following online resources are starting points for gathering mathematically worthwhile tasks.

1. NCTM Illuminations: http://illuminations.nctm.org/


2. Howard County Public Schools Common Core Curriculum -
https://secondarymathcommoncore.wikispaces.hcpss.org/

3. Annenberg Learner: http://www.learner.org/


(Important ideas to know about Measurement)
http://www.learner.org/courses/learningmath/measurement/session2/part_a/index.html


4. Inside Mathematics: http://www.insidemathematics.org/


5. B.C Mathematics K-7 IRP
https://www.bced.gov.bc.ca/irp/course.php?lang=en&subject=Mathematics&course=Mathematics_K_to_7&year=2007



Math Resources from the Richmond District Resource Center (DRC)

Lesson Starting Points:

• Start with a rich problem that has multiple entry points and allows a variety of strategies
• Engage students in dealing with the problem
• Discuss, compare, interact ensuring a supportive environment that encourages respect
• Help students connect and notice what they’ve learned

Sample problems:

50 Problem Solving Lessons




More Problems:

Add them up!
Directions: Place the # 1-6 on the triangle game board so that three numbers on each side add up to the same total. How many different ways can you do this?

Is 7 really a lucky number?

How many different rectangles (arrays) can you make with 24 tiles? 23 tiles? What do you notice?
Which number between 1 and 99 has the most arrays? What do you wonder? How can knowing this be useful in your everyday life?

Literature Starting Points:

• Anno, Mitsumasa and Tsuyoshi Mori. Anno's Three Little Pigs. London, UK: The Bodley Head, 1986.
• Burns, Marilyn. The Greedy Triangle. New York, NY: Scholastic, 1994.
• Clement, Rod. Counting on Frank. Milwaukee, WI: G. Stevens Children's Books, 1991.
• Enzensberger, Hans Magnus. The Number Devil: A Mathematical Adventure.New York, NY: Henry Holt, 1998.
• Schwartz, David M. How Much Is a Million? New York, NY: Lothrop, Lee & Shepard Books, 1985.
• Scieszka, Jon and Lane Smith. Math Curse. New York, NY: Viking, 1995.
• Tahan, Malba. The Man Who Counted: A Collection of Mathematical Adventures. New York, NY: Norton, 1993.
• If the World Were a Village: A Book about the World's People by David J. Smith

An Algorithm for Solving Problems:

1. Ask yourself, "What am I trying to find out?"
2. Ask yourself: "What do I already know?" (What information is given in the problem)
3. Choose an appropriate strategy or strategies: (make a diagram, chart or table; use manipulatives; talk it over with a buddy; look for patterns; use calculations: addition, subtraction, multiplication, division; work backwards; use the guess and check method; think of a similar problem; simplify the problem or break the problem into parts, etc.)
4. Ask yourself "Does this solution make sense?"/ Ask yourself, "How do I know I am right?"
5. Share your solution with a partner, compare solutions; share with the class


Math Journals: Sentence Starters for daily writes (5-7 minutes at the end of the math lesson)
Today I.. (describe a problem you solved, an investigation you conducted or a game you played)
I noticed that....
I think....
Something interesting was...
Something challenging was...
I'm not sure about..
Here is a diagram, graph, picture of how I solved the problem...
If I had to tell a friend how to solve this problem, I would say...
An new idea I had was...
This reminded me of...
I wonder.....
I know I am right because...
I was a good mathematician because I ( asked questions, made some predictions or estimated, I made some connections, I looked for and found some patterns, I persevered, I compared, I....)

A Collection of Math Gems

Math Gems

Ah me oh my, but these do keep us smiling!



 





and last but not least...