|Creating 2D, 3D, and animated math models is an integral|
part of teaching and learning math today.
Jo Boaler, author of Mathematical Mindsets, is not a big fan of homework, however she recommends that reflective homework assignments can be very helpful and positive. To begin the year, I may include lessons related to social development, work habits, and math effort during each lesson. I also may ask students to reflect on these elements as part of their nightly homework to build capacity as well as practice and knowledge of reflective practice--a skill that's a valuable element of any learner's toolbox.
As a new or existing math teacher, take some time this summer to review and analyze your school's report card or progress report. Think about the standards students will be scored on. In systems like the one I work in scores are simply progress notes such as "progressing towards expectations, meets expectations, or exceeds expectations" and in other systems students are still scored with old-time traditional grades. I am a fan of standards-based reports as they foster a progressive, learner's mindset. The kind of mindset that demonstrates to learners that we're all on the continuum and our good efforts help us to move closer to our goals (standards).
Below I analyze each element of the report card that I have to grade in the year ahead. The analysis demonstrates to me a good order and process for teaching each concept. Not all required standards are included on our reports, but the standards with greatest priority are included.
Perseveres when faced with challenges:
What does this look like? How does one persevere? When have you persevered in the past? To introduce this topic, I may present an open task and have students work on it with a focus on their needed perseverance. We'll reflect later on what that felt like, and how they motivated themselves and each other to keep going.
Uses positive strategies to resolve conflict:
We'll review the "Take the first step" process of conflict resolution which is to talk to the person you're having the conflict with. The second step is to get help. We'll brainstorm the types of conflicts that happen at school and the many ways one can resolve those conflicts.
Accepts responsibility for own actions:
What do you do when you make a mistake or do something wrong? How do you deal with that?
Follows school and classroom rules:
First we'll create a class constitution with our rules and protocols. Then we'll talk about ways one can work to follow those rules.
Plays cooperatively with peers:
What does it mean to play cooperatively? What does this look like?
What is initiative? How does one demonstrate that in school?
Seeks help as needed:
Don't stay stuck! When you need help, seek it out. How can one do this?
Manages time well:
What does it mean to manage time? What strategies do you use to do this.
Works independently and productively:
What id independent work? When might a student be asked to work independently? What does "working productively" mean? What does that look like?
Transitions in a timely and appropriate manner:
What does transition mean? When do we transition? What is appropriate behavior for transition?
Follows directions, routines, and procedures:
First we need to establish routines and procedures and make those routines and procedures explicit. Similarly all directions need to be clear and explicit. So to follow directions, routines, and procedures, first students need to understand what they are. Next students need to brainstorm together the strategies they use to follow directions, routines, and procedures well.
What does optimal participation look like in the math classroom? How can we encourage one another to participate? What hinders student participation?
Asks for Clarification:
What does the word "clarification" mean? What are the ways that you might ask for clarification during class or after class? What happens if you ask, and you're still not clear?
Attends to Work:
What's expected with regard to "attending to work" and what does that look like in a classroom? Will this look different during different kinds of learning experiences?
Perseveres when Challenged:
What does perseverance look like in the math classroom? What behaviors are exhibited when you are persevering, and what activities are not exhibited in this regard?
Attends to precision while applying mathematical concepts:
Why is precision important in mathematics? What strategies can one use to be precise?
Math Concept, Knowledge, and Skill
Communicate mathematical thinking orally and in writing using math vocabulary:
Math vocabulary study is integral to math understanding. Every unit should begin with a review of the vocabulary. Vocabulary should be clearly visible in the classroom and accessible for at-home study too. Students should regularly express their mathematically thinking with written or spoken words. Creation of math scripts and videos support this work which ties in nicely with the Standards of Mathematical Practice.
Represent and interpret data using graphing skills:
Initially this is a great standard to meet when making early year infographics to help a class get to know one another. The line plot is the featured graph for fifth grade at this time. Later in the year this standard can be reviewed as you solve fraction problems and chart fraction/decimal data related to real-world problem solving.
Automatically recalls multiplication and division facts through 12:
What do we know about the numbers one through twelve? What factors and multiples can we identify related to these numbers? Which of these numbers are square numbers, composite numbers, prime numbers, or perfect numbers? Where are these numbers present in our culture, and what other names do we have for these numbers? How can we gain automaticity with facts? Why is this important? How do we do with this now, and how can we develop this skill?
Coordinate Grid Knowledge and Use: Able to graph points on the coordinate plane to solve problems.
Understand how to graph points, and then how to use graphing to determine numerical relationships. Student enjoy learning this and practicing it. This study can be embedded throughout the year as students analyze numerical relationships and solve problems.
Classifies two-dimensional figures based on their properties with particular attention to parallel, intersecting, and perpendicular lines:
Students enjoy this study which can be taught well using drawing, tangrams, geoblocks, and online games/venues.
Understand the concepts of volume and applies formulas to solve problems:
Since volume problems essentially include simple numbers, this is a great unit to introduce as you review basic facts. It's also good to review area and perimeter at this time before you get to area models of multiplication and division with larger numbers. Further this unit comes nicely after an initial review of two-dimension geometry since understanding those shapes and how to talk about them informs the volume, area, and perimeter unit.
Use positive and negative numbers to describe quantities:
This concept fits nicely into initial number talk at the start of the year when we look carefully at the numbers 1-12 and where those numbers fit on the number line. Again this can be reviewed during money-related word problems.
Writes and interprets numerical expressions using parentheses:
This is a great skill to teach as you teach students to use calculators effectively. This is also a great teaching goal to master when you are working with "easy numbers" and warming up math year skills. Though it is scheduled on our reports for Term Two, most teach this early in the year as it can be useful and practiced throughout the year as number concepts, knowledge, and skill become more sophisticated.
Generates, extends, and compares patterns and relationships:
Again this is scheduled for term two, but should be a focus of every math lesson with the following questions: What patterns do you see? How do the patterns compare? How might you extend the pattern? What relationships do you see? How would you define the relationship? Can you represent the pattern and/or relationship using a numerical or algebraic expression? Can you graph the relationship or pattern?
Understands the base ten place value system from millions to thousandths?
What are the parts of the place value system? How do these parts work together? Why was the base ten place value system invented? What value does it have in our culture? How do the values of numbers change as you move up and down the place value system? How do the values of numbers change when you add, subtract, multiply, or divide by base ten numbers?
Perform addition, subtraction, multiplication operations with multi-digit whole numbers?
Why do we use the word "operation" when we talk about addition, subtraction, and multiplication? Why do we add, subtract, and multiply? How can we perform those operations effectively and efficiently? Is there one way to do this? Is there a best way or does this depend on when and why we are performing the operation? What 2D, 3D, and animated models can we make to demonstrate these operations?
Perform division operation with multi-digit whole numbers?
What does it mean to divide? When would we use this operation? How can you divide one number by another? What words do we use to describe the numbers associated with division? What models can we make to depict division?
Performs operations with decimals to the hundredths:
When do we need or want to add, subtract, multiply, or divide decimal numbers? Why is this important? How can we perform these operations? Working with money and problem solving is an important art of this study.
Convert standard measurement units within a given measurement system:
This study fits nicely with new science standards that included working with matter. Many experiments can be done that find students converting measurements while exploring matter. This also alines well with base ten system study as students look at models of the base ten system and match those models to metric models.
1/2 = 1 divided by 2 which equals .5, both 1/2 and .5 are equivalent to half of the whole. Proper fractions like decimals demonstrate part of a whole. We can easily turn a fraction into a decimal by dividing the numerator by the denominator. Sometimes it's easier to work with fractions and sometimes it's easier to work with decimals. A good way to practice this skill is to begin by making a fraction, decimal, percent, and ratio equivalency chart. Later it's good to look at how this skill relates to meaningful fraction numbers with real-time problem solving and project work.
Add and subtract fractions with unlike denominators and solve word problems with addition and subtraction of fractions:
Use a signature story to describe and discuss this skill: "Amy and Paul were making cookies. They each had a table of ingredients. The cookies called for 3/4 cup of sugar. In the end, Paul had 1/8 cup of sugar let and Amy had 2/6 cups of sugar left. Together do they have enough sugar left to make one more batch of cookies?" How would we figure this out? What can we do to figure out if they have enough sugar left? Later students can make up their own stories and demonstrate this skill conceptually. Lots of fraction model wok supports this effort as well.
Applies and extends previous understandings of multiplication to multiply a fraction or whole number by a fraction and solves related real-world problems.
This work also benefits from signature stories. Begin with multiplying a whole number by a fraction: "Emily wanted to give every child 1/2 a piece of paper for the art project. There were twelve children. How many whole pieces of paper did Emily need? What if Emily wanted to give each child 1/3 piece of paper, then how many sheets of paper would she need?" Draw a model of this story. Solve the problem with a model and mathematically. Make up your own problem, solve with model and numbers. Present your problem to the class.
Next, work with multiplying a fraction by a fraction using the area model. Again use a signature story: "I bought a big rectangular cake for my dad's birthday. After the party I had 1/2 of the cake left. I wanted to give my brother 1/4 of the half cake. What portion of the original cake did I give to my brother?" Show how this problem looks with a model, then solve the problem mathematically. Have students make up their own problems to solve with the area model (similar to the cake), picture model, and using the fraction multiplication algorithm.
Applies and extends previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions:
Introduce with a story: "I was going to the fireworks with my family. I had three pieces of licorice, but there were eight people in the car. I decided to split the licorice into 1/4 pieces. If I did that, would I have enough licorice pieces for all people in the car?" How can we write this problem as an expression? What would this problem look like as a model? Do I have enough pieces if I divide 3 by 1/4? Or "I had 1/4 of a pizza left. I want to split it into 3 equal pieces. What size pieces will I end up with?" What does this problem look like as an expression and model? Students can make up their own fraction division problems to solve with friends and present to the class.
Understand what a mixed number is and solve related world problems:
Discuss mixed numbers. Brainstorm when we use mixed numbers in real-time. Look at the many ways we can write and/or model a mixed number. Add, subtract, multiply, and divide mixed numbers. Solve and create real-world mixed number problems.