Teach Math Well: Are We Asking the Right Questions?

Emails circulated about the value of using an online unit assessment versus the paper/pencil unit assessments of the past. It's a good question, and one that should be posed and discussed. Personally, I'm a fan of using multiple formal and informal formative and summative assessments, and I support the use of online assessments as part of that overall assessment approach.

I also believe that we have to be mindful to have a good assessment balance and not make teaching and learning all about the test--we have to make space for children to enjoy their learning. While it's important to discuss assessment types and the role of assessment when teaching math, there are also many more questions that beg our attention and discussion.

For example, one of the greatest math question that we face in our standards-based school communities, and how do we continually uplift and teach the students who are one or more years behind the expectations. Of course, I believe that math education should be progressive where every child is moving ahead from where he/she is and that the focus should be on that movement rather than if everyone is at the same level. For example if I have students at fifth grade who are testing at a second grade level and others who are testing at a seventh grade level, they all should be working to move up. I don't want those "seventh grade level" fifth graders to think they've achieved all there is to know in math and sail along without a good challenge, and I also don't want those "second grade level" fifth graders to get discouraged and always feel behind. So the question, How do we challenge all in positive, proactive ways resulting in good math growth, engagement, and knowledge?, is a great question to ask and discuss amongst peers.

Another good question to discuss in math is what are the best ways for specialists, assistants, coaches, and classroom educators to work together to maximize math teaching and learning for all. We have instituted Responsive to Intervention in creative ways that help in this regard. We also differentiate during core instructional times. The way we collaborate matters when it comes to uplifting children's skills, knowledge, and concept.

Further we can ask questions about content specific teaching and learning. For example, at a recent meeting, it was acknowledged that teaching the "behavior" of the base ten place value system is challenging. Students at fifth grade find it difficult to grasp that numbers increase by X10 as they move up each place in the place value system and decrease by 1/10 as they move down in value place by place. Fluency with that concept takes time.

Time is an important consideration. Many on the outside would like teachers to work like soldiers and be at the same place in teaching at the same time. This, in many ways, negates the where students are at and what they need. It also disregards the need for educators to make the curriculum their own and teach it with the artistry, experience, and analysis needed. Good teaching is not a robotic function, and good teaching makes good use of time. Unlike a robotic response, good teaching analyzes deeply and maximizes the use of time to teach all well.

Too often we spend time talking about questions that don't matter that much in education. Instead we should dig a little deeper and spend time discussing what matters--the big, deep questions that truly impact what we can do to better teach all children.

Teamwork Matters: Making Math T-Shirts

Start the year by focusing on the fact that students learn better when they work together. Emphasize that your math class is a math team, and as a team the way you work together matters. 

Ask students if they belong to any teams. Then ask them to tell you what's important when it comes to successful teamwork. After that, you may want to show one of my all time favorite teamwork videos, and ask students to look for the attributes of successful teamwork as they watch the video:

After the film, acknowledge that like the Panyee soccer team, most teams have T-shirts. Say that the class will have some fun with numbers as they make paper T-shirts and a class bulletin board that celebrates your dynamic math team.

To make the T-shirts, do the following:

1. Set up a computer and printer in the classroom. Have students take a good picture of their face and print it. I use PhotoBooth for this. Students go to the picture stand, take their photo, and print it one after another.
2. Hand out the number activity guide which includes a T-shirt template.
3. Allow students to work with calculators to complete and check their work. 
4. As students work, observe their efforts and listen to their conversation. This observation will serve as a good initial assessment of student effort and need--an assessment that will inform future lessons.
5. Have students complete the task in pencil, check the work with a friend, and then check with you. Coach students with regard to correct completion of the work. When complete, have students write over the numbers and words with a fine line pen, write their number on their t-shirt and add the fact data to the front of the T-shirt or just below, tape their picture on top and hand it in for a class bulletin board that will display all the students pictures with a title such as "Teamwork Matters: Marvelous Mathematicians."
6. When students are done, they can work together on the enrichment activities included on the number activity guide. 

This project is a great start-of-year activity, one that provides assessment, fun with numbers, vocabulary/fact review, and an emphasis on teamwork. I recommend. 

Maker Math: Volume Exploration - The Candy Box

There's been many iterations of the volume related candy box exploration, and this is one I'll use with students in the days ahead to teach and practice finding the volume of three-dimensional figures. Prior to teaching this lesson, students will study and learn to use rulers.

Story

Bon Bon Recipe and Picture Site

Angela liked to make candy. She especially enjoyed making bon bons the old fashioned way. Her candy bon bons became so popular that people started asking if they could buy them from her to sell at their shops. Angela decided to put each bon bon in an individual box, and then put 24 individual bon bons boxes in each big box. That way people could display the big box while selling each individual bon bon.

First, she designed an individual box for each bon bon. She created a net made up of squares that measured one inch on each side. That was the perfect shape for her bon bons as they fit tightly, but did not squish the candy. Angela worked with a local packaging company to find the just right paper--a kind of paper that wouldn't melt the candy and keep its shape too.

Next she wanted to make a good strong and beautiful box to fit all 24 bon bon boxes. That's where she needs your help. Follow the directions below to help Angela.

Net Maker

Math Investigation

1. Take a look at the bon bon (real or pretend (a clay version)).
2. Create the one-inch box. One-inch box net available via this website. 
3. Go to the Maker Station and choose the kinds of paper(s) and other supplies you'll use to make a box that can easily fit 24 bon bon boxes. The Maker Station for this project includes a variety of paper weights, cardboard, tape, and decorative materials. 
4. Design your box net and create the box.
5. Complete the box data sheet and try out the enrichment options listed on that sheet.

Follow-Up

• Have students bring in a number of recycled boxes. Measure and figure out the volume of those boxes.
• Make a collection of nets available for students to make, measure, and describe. 
• Practice figuring out the volume of a variety of three dimensional prisms: simple sheets,  more complex Common Core practice. and word problems.
• Show this Khan Academy volume video:

Real World Connections
Tell students that there are companies that's goal is to make just right packaging for products. There's even a packaging magazines and websites. Happi is one example. Ask students what skills they think a packaging expert would need, and how a business like that would be important. Show the videos below that tell how cardboard boxes are made:

More about Packaging and its effect on the environment. 

Packaging Design

TEAM Matters: Global Cardboard Challenge

We've named our grade-level TeamFive and we'll use this graphic to lead their TEAM studies and collaboration.

Jo Boaler's book, Mathematical Mindsets, Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching, is a terrific well-researched book to read and use as you teach math in 2016-2017.

One study in the book that most caught my attention was a study that demonstrates the wonderful impact collaboration had on the success of all students. That's why it's important to start the school year with an emphasis on TEAM: Together Everyone Achieves More.

A good way to do that is to promoted the Global Cardboard Challenge as an initial team building activity. I've designed a number of activities leading up to the project to promote understanding of TEAM and one's place and responsibility on a team.

In years past students have remarked that this project was one of their favorite events of the year, and I believe that the project created a strong sense of TEAM across the grade-level too. Try it out. Let me know what you think. Year after year, I update and grow this project to best meet student's needs and interests.

Teaching and Learning: Numerical Expression

It's great to start the school year with a challenging, engaging math lesson. Children love to come to school at the start of the year and learn something new. This sets the stage for a vigorous, engaging year ahead.

Teaching numerical expression lends itself to engagement and provides students with a terrific skill to use as they learn, practice, and apply mathematical operations with numbers simple and complex.

Introduction: What Do These Expressions Equal?
Evaluate one or more of the expressions below in any order by hand or with a calculator. Giving students the chance to use a calculator takes away the pressure to remember facts and to calculate correctly. This frees up all students to reach for higher level learning in this lesson. 

8 + 6 X 5 - 1/2 + (8-3)

99 - 1 + 2 X 3

Let students play with it. Some will finish quickly, others will take a bit of time, and some may not know where to begin. Give the students about 10 minutes, then ask for children to come up to the board to explain their thinking about each expression.

Teach Order of Operations

Show Flocabulary's PEMDAS movie. While some mathematicians are not in favor of PEMDAS, I still think it's a good tool for early learning with regard to Order of Operations. As students work becomes more advanced, they quickly understand where PEMDAS doesn't exactly fit, hence I still believe it's a good starting point. 

Also introduce this graphic shared by a former student:

Study and Practice
Net talk with students about what they can do to remember, study, and apply the Order of Operations. List their ideas. Then let them work with classmates to do all or some of the following activities plus any good ideas they've come up with that you want to add to the list:

• Create a mnemonic, poem, picture, or mini poster that will help you remember the Order of Operations.
• Watch the PEMDAS rap over again, then make up one of your own.
• Make up your own expressions and evaluate them. Check with the teacher.
• Write 5 different expressions and 5 different equations that equal 50 (remember expressions do not have equal signs while equations do). Use at least two different operations in each expression. Make sure you use every operation at least once. Use parentheses, brackets, and/or braces in at least two expressions.
• Use exponents in at least two expressions. 

More Practice

Study the chart below. Then determine where you might add parentheses, brackets, and/or braces to simplify the challenging expression and puzzle below. Then solve the puzzle and expression.

ASSESS

After a day or two of practice and some Khan Academy review and study related to the topic too, it's a good time to assess students' skills. This is one assessment that provides the teacher with a good understanding of what students learned and still need to know.

Related Grade Five Standards

Teaching and Learning: Coordinate Grids at Fifth Grade

Students truly enjoy learning and applying all information related to coordinate grids.

I like to introduce some of the fifth grade coordinate grid concepts and skills early in the year so that we can apply those skills to most units throughout the year. Plus it's the kind of skill that's highly accessible to most math learners making it a positive start-of-the-year unit, one where all students are able to work together with confidence and engagement.

Lesson Introduction: Whet Their Appetite, Make it Meaningful
We'll take a look at the aerial photo of the playground. It's easy to find an aerial view photo of your school playground and this immediately makes the lesson interesting to students.

Say, can anyone locate the "back swing set." Hands will raise quickly. Students describe where it is. They say things such as "in the back," "next to the trees," and "far from the building."  I write their words down, and underline the location words such as next to, far from, and in the back.

After that I'll explain that mathematicians aren't satisfied with vague descriptors like next to, far from, and in the the back since they typically want to be as specific, precise, and exact as possible. Math brings order and organization to our world, and that helps us to see accurate patterns, determine relationships, present facts and figures in meaningful, explicit ways, and solve problems.

I'll then explain that one way to order or organize a map is to overlay the map with a coordinate grid. Then I'll show the coordinate grid map overlay and explain how using the grid provides us a more accurate, specific, and exact way to discuss a map.

Learning Experience

Pass out the grid map to all and we'll identify the vocabulary.

Link for Handout

Complete the worksheet. Then graph a few significant coordinates or ordered pairs on the school map. After that practice making a picture(s) with ordered pairs. This is one free resource for practice pages. A quick Internet search will identify many more.

Finally, at the end of the lesson, review the vocabulary with this catchy Flocabulary rap.

Provide Opportunities for Review and Study for Homework
Encourage students to sing the Flocabulary rap for homework. Also ask them to practice drawing a number of pictures using ordered pairs and practice the concepts using Khan Academy.

Enrichment

• Have students create their own treasure hunt maps with coordinate pair clues to find the treasures in real time or on a digital or hard copy map.
• Have students create the coordinates for a wonderful ordered pair picture. Make sure they test the picture, then make copies of the ordered pair path for the class to use as they create the picture.
• Have students write real world problems like the ones they've completed on Khan Academy and then solve those problems using coordinate grids. 

Coordinate Grid Learning Standards:

CCSS Standards Language Used in This Lesson and Others

• 5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate)
• 5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

CCSS Standard Included in Khan Academy Review and Later Lessons

• 5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Teaching and Learning Divisibility Rules

Important Knowledge, Concept, and Skill: Factors, Multiples, and Divisibility Rules

Introduction: First Impressions
The introduction to a learning experience can last anywhere from 5-15 minutes. It's meant to wake students up to the topic, get them thinking, and make the lesson "sticky" or brain-friendly by connecting the concept to something they know, are interested in, and/or excited and even alarmed by.

"What does a first impression mean?' A first impression is what you notice first.

Students will likely list a number of descriptors of a first impression. Then say what can we know about a person from a first impression? And after that, can we know all about a person by a first impression, essentially, "can we judge a book by its cover?"

Then hold up the number above and ask, what's your first impression of this number--what do you notice? List all that students notice.

After that ask, "What can we know about a number just by looking at it?" And after that short discussion, can we know everything about a number by just looking at it?

Rationale:
After the introduction, share the rationale for the lesson. 

"Numbers are ingredients for mathematical thinking and process. The more you know and understand numbers, the better you'll be able to work with those numbers to describe situations, solve problems, and reason mathematically. Learning about the divisibility rules will help you to understand numbers better. What do you think the term "divisibility rules" means?" Take some ideas, then show the definition card.

One definition you may use is this one:

A Divisibility Rule is a way to figure out the factors of a whole number without performing division, usually by examining the digits.

Teaching and review of multiples and factors is an engaging and valuable focus of the fifth grade math curriculum. Though this focus is more closely matched to fourth grade standards, it's integral to fifth grade standards related to numbers and operations, fractions, measurement, and more.

Teaching/Learning

Day One
Once you've introduced the learning experience and main definition, it's time to explore. A good way to explore is to teach and play Factor Captor online or off. Encourage students to think about their "first impressions" of numbers as they play. Start by playing a simple version of the game with the whole class, then give students time to play at the level they desire with friends.

At the end of the time available gather students, and ask what "first impressions" of numbers they had as they played. Also ask them to share any strategies, patterns, or other observations they made as they played the game.

Days Two and Three (an more possibly)
Review the definition of a divisibility rule, then review the rules below. 

image reference

Watch the video below and singing the song. After that students can practice using this divisibility rules exercise and chart above with classmates.

Once that's complete, students will be ready to play the Factor Game game again. You may even want to encourage students to play the game at home with friends and family members to practice more. At the end of the lesson, add to the chart students made yesterday about game the strategies, patterns, and number observations they made during the game. Ask students if they used the divisibility rules while they played and if those rules helped them to play better.

At this point, you may want to sponsor a Factor Captor Tournament to give students plenty of practice. This tournament provides students with the opportunity to teach one another, practice, and deepen their knowledge and skill. The tournament also provides the teacher with lots of time to coach students with this knowledge, concept, and skill as well as others.

Day Five: Learning-to-Learn Mindset, Behavior, and Science Connection
Before students assess their skills, engage in a conversation about how students can move this knowledge from "short term memory" to "long term memory." This is a good opportunity to teach students about the limited space and accessibility our short term memory has while our long term memory ensures, for the most part, that facts, figures, and concepts are well rooted and easily accessed. You may ask students to think of information they never forget and facts/figures that are easily forgettable, and after that, the students and you can list ways that help us to learn information well--ways to embed that information into our long term memory. Your list will likely include learning tasks such as writing notes, making flash cards, testing one another, making an mini poster or infographic, turning the concept into an image, singing a song, creating a rhyme or poem, acting it out, and teaching the concept to others. Give students time to choose a strategy, work with friends, and learn/study divisibility rules.

Assessment
Following that, it's time for a quick online or offline assessment like this example. I suggest using Google forms now that they have the new quiz feature embedded right on the forms. That's a quick way to create an assessment to see who knows their divisibility rules and who still needs to study. Allow students to take the assessments as many times as needed until they grasp the learning. Provide extra support where and when necessary.

Blended Learning Resources

Related Videos:
These were a couple of great videos that added to the teaching.