## Graphing the Holidays

Originally posted 11/27/2018

Teaching between Thanksgiving and the winter break can be a challenge. How do you keep your students engaged in meaningful math learning while embracing the season? Introducing graphing and data analysis might just be the answer.

Imagine starting your day in first grade with a question about favorite holiday treats. Students can answer the question and instantly you have meaningful data that can be organized into a tally chart, picture graph, or bar graph for students to analyze. Or, students can build a bar graph with post-it notes as they make their choices. Then, spend some time analyzing the results.

Ask 5th graders if they traveled over Thanksgiving break. If so, how far? Now use this data to find mean, median, and mode, or to create a histogram for students to analyze. Or, chart the temperatures over the course of a couple of weeks and use this data to create a line graph.

Third and fourth graders could tally the number of candles in their homes for the holidays and use this data to create a line plot. Fourth graders can use their line plots to explore finding the median.

Planning a holiday party? Survey the students on what should be served and what activities should be included. Students can present the findings in a graph and use the results to determine how much and what needs to be donated or purchased to make the party a success.

The holidays are a great time to share family traditions. Why not use that information to meet some graphing and data analysis standards?

For other ideas to keep students engaged in learning read Mental Math Breaks from December 2017.

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## Who’s doing the talking?

A new school year brings new commitments to improving our practice as teachers of mathematics. One tip I often share with the teachers I coach is, “Ask more and tell less.” Well, that’s easy to say, but what does that look and sound like in the classroom?

Often times, the teacher’s guides are written following a more traditional, lecture-style of teaching. They encourage the teacher to model, or work problems, while the students watch, and then the students are asked to mimic what the teacher did with a similar problem. I challenge you to flip the script and replace the word “show” with “have the students model” and replace “tell” with “ask”. When your teacher’s guide says to show the students the difference or similarities between problems or concepts replace that with, “ask the students what they notice?” It’s these little tweaks that will go a long way toward engaging your students in meaningful discourse and ultimately deepening their understanding.

A fourth-grade teacher from Aurora, Colorado shared her strategies for engaging students in math talk in her classroom.

While this appears to be written for the students to follow, it also suggests some great questions for teachers to ask to generate more discussion.

• How did you solve that?
• How do you know that’s correct?
• Can you solve it another way?
• Can you build a model?
• Can you use numbers and symbols to explain your model?
• Is that the best (most efficient) way to solve that?

So, who’s doing all the talking? Give some of these questions a try and let us know how it goes.

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## Guest Post – Problem Solving: The Heart of Singapore Math

This article originally was featured in the fall 2018 issue of St. Anne’s-Belfield School’s Perspectives magazine and is republished with both the school and author’s permission.

The author, Sara Kronstain, has almost a decade of experience guiding elementary and middle school mathematicians to become critical thinkers and problem solvers.  She teaches fifth-grade math and is the Kindergarten-6th Grade Math Department Chair at St. Anne’s-Belfield School in Charlottesville, Virginia.

## Problem-Solving: The Heart of Singapore Math

When I was in school, my math classes were typical of what one would expect a “traditional” math class to look like. I remember sitting in my elementary and middle school classes, watching as my teachers modeled problem after problem. The class would listen and then practice many of the same types of problems in our notebooks. While this type of teaching may achieve the immediate goal of learning a mathematical procedure, it does not guide students to reach an integral part of learning mathematics: problem-solving (Cai & Lester, 2010).

Singapore Math is comprised of a framework with problem-solving being the center of learning mathematics. This framework is built around five key components – metacognition, process, attitudes, skills, and concepts – all being of equal importance in developing mathematical problem solving in students. Whereas traditional math classes may place primary importance on developing skills and concepts in students, the additional three components of metacognition (self-regulation of learning), process (reasoning, making connections, and applying knowledge), and attitudes (perseverance, confidence, interest) are all key to developing critical thinking and problem solving skills in students (Ministry of Education Singapore, 2006).

A typical Singapore Math lesson is taught with a concrete-pictorial-abstract approach. Where many of my lessons as a math student began in the abstract stage (solving equations), the concrete and pictorial stages allow students to create and solidify their own understanding of a topic. The concrete stage refers to using hands-on materials to model a mathematical situation.  The pictorial stage consists of diagrams and other visuals, thus building students’ learning in a tangible way (Maths No Problem!, 2018). The concrete and pictorial stages allow students to understand why math works the way it does before learning the procedure of how to solve using an algorithm.

Most Singapore Math lessons begin with an anchor task, allowing students to explore these three stages. The anchor task is a question that allows students the chance to deeply explore a topic and develop multiple methods for solving a problem (Ban Har, 2013). Let us say, for example, a group of fifth graders were posed the problem, “The distance of a race is 3km. Lily ran two-fifths of the distance. How many kilometers did Lily run?”  Students would be given the opportunity to freely explore this question by using manipulatives such as fraction bars, fractions circles, or paper (for folding) along with writing materials. Here are a few examples of possible student responses to this question:

Add ⅖ + ⅖ + ⅖Students may use fraction bars, fraction circles, or pictures.  Students become familiar with the phrase “3 groups of ⅖”.

#### Method: Bars

Three boxes are each split into fifths. Two of each of the fifths in all three boxes are shaded in.  The shaded parts are added together.

#### Method: Bar Model

A bar with the length of 3 wholes can be split into five parts.  Each part has a value of ⅗. Then add ⅗ + ⅗ .

Three boxes are split into 5 equal groups, first by placing one half in each group. Then, split the leftover half into five parts (tenths). Each group will have one half of a whole and one tenth of a whole. Combine two groups by adding two halves to two tenths.

In this example, the repeated addition method reinforces addition with fractions, while the last method has students thinking about and manipulating fractions in a much more complex way.  Thinking back to the five key components of Singapore Math, students in this example are refining their process of learning operations with fractions by making connects across operations.  It is powerful that these responses are coming from students, as they are building their understanding of math through collaboration with their peers. In sharing methods, listening to other’s methods, and processing others methods, students are also developing their metacognition. This question could also be modified and challenge students to problem solve in an even deeper way. “What if the total distance was ½ km?  What if the total distance was 3 ½ km?”  Students can then go back to the concrete, pictorial, and abstract stages and continue to build on their problem-solving abilities.

At the end of the day the primary purpose of this math lesson, or any math lesson for that matter, is not simply to learn how to multiply fractions by a whole number. The most important takeaways are the critical thinking, questioning, collaboration, and problem-solving that happens among students. Teachers are not preparing students to go out into a world where they will simply be asked to recite an algorithm. While a goal is for each child to develop a deep love of math, the biggest hope is that students learn to ask questions, logically think through problems, and make sense of the world around them.

#### References

Ban Har, Yeap.  (2013, June 13).  Singapore Math at the Blake School, Hopkins, MN.  Retrieved from http://banhar.blogspot.com/search?q=anchor+task

Cai, Jinfa, & Lester, Frank. (2010, April 8).  Problem Solving. National Council of Teachers of Mathematics.  Retrieved from https://www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Problem-Solving/#brief

Kaur, Berinderjeet.  (2018, March 29).  Building the Maths house: Singapore’s curriculum framework.  Oxford Education Blog.  Retrieved from https://educationblog.oup.com/secondary/maths/building-the-maths-house-singapores-curriculum-framework

## Ask the Experts: What’s the best way to organize my math manipulatives?

The answer to this question is complicated. So much of how to organize materials is dependent upon personal preference with procedures and arrangement within your classroom. One thing that I can say is true in all cases is that they DO NOT belong in the closet!

I highly recommend that you dedicate a shelf or area of your classroom to math materials. It’s equally important for students to choose the most appropriate tool, as it is for them to use them. Having materials out for students at all times will allow for that.

One of the joys of my job is that I get to visit schools and classrooms across the country. So, I will share with you some organizational tips that I have gathered from my journeys.

There are three schools of thought (no pun intended) when it comes to organizing manipulatives; individual kits, group kits or community tubs. You may find it helpful to use a combination of the three, depending on the item.

I’ll mention a couple of manipulatives specifically here.

## Place Value Discs

#### Student Kits

Many teachers prefer to organize discs into student kits. The idea being that students will have easy access to the discs for lessons with minimal time getting discs out and cleaning them up.

This option works great if you have enough discs for each student to have 20 of each place value; 20 ones, 20 tens, 20 hundreds, etc. Students are expected to keep these baggies or boxes of discs in their desks.

Pros: Easy access

Cons: Relies on students to maintain the correct number of discs in their kits. (I was that teacher who couldn’t stand the fact that there was one ten disc on the floor at the end of the lesson that seemed to belong to no one!)

#### Group Kits

Like student kits, you’ll need 20 of each place value in each kit. With group kits, you don’t need as many total discs. The idea here is that students will use discs with a partner or in small groups. These kits can be stored in a community tub and pulled out for use during lessons.

Pros: Easy access

Cons: See above. (Which kit does this disc belong to?!?)

#### Community Tubs

In this case, discs are organized by place value into tubs. So, you would have a tub of ones, a tub of tens, and so on. In each tub, you can keep a set of small cups (Dixie cups work well) for students to take a scoop of the discs when needed. Clean up is a snap. Students simply dump the cups of discs back into the correct place value tub.

Pros: No more mystery missing discs! Very quick set up. (No more evenings spent counting out discs while watching TV.)

Cons: Requires a bit more practice with the procedure of gathering and returning discs to the correct tub.

Linking cubes are a multi-functional manipulative that each classroom should have. For a class of about 20 students, you’ll want to have at least 400 individual cubes. That’s enough for each student to have a set of 20 when needed for instruction. If you’re using them for modeling area or multiplication arrays, you might want double that amount.

#### Student Kits or Group Kits

You’ll want to put at least 20 in each kit.

Pros: Ease of access.

Cons: Whose cube is this?!?

#### Community Tubs

If you are keeping your cubes in tubs, for ease of passing out and cleaning up, organize them in rods of 10, preferably by color. That way you can quickly pass out 2 rods (or more) to each student or partner group.

Pros: Fewer materials in student desks. No more mystery cubes.

Cons: Need to establish procedures for keeping cubes in rods of ten. (Easy, peasy!)

Other manipulatives should be in tubs on a shelf in the classroom available to all students at all times!

If you have any organizational tips from your classroom that you’d like to share, please send us a comment.

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## Ask the Experts: What manipulatives do you suggest for my grade level?

Singapore Mathematics instruction – or, really, just good math instruction – will have students working through three phases of learning, referred to as the Concrete-Pictorial-Abstract approach. In order to teach following this approach, you need to start at the concrete level. Jean Piaget, a Swiss psychologist, believed that in order for students to be able to visualize and abstract mathematics they first must manipulate materials. He called this the concrete operational phase of learning.

So, what do you need to teach concretely? A complete list of recommended materials by grade level can be found here.

Really, though, with a few basic items you can get started…

Kindergarten – used for counting with one-to-one correspondence, measuring with non-standard units, and for modeling basic addition and subtraction situations.

1st – 2nd grade – used for place value understanding, to model story problems and mental math strategies, for measurement with non-standard units, building array models for multiplication, and for beginning bar modeling.

3rd grade – used to model part-whole and comparison word problems involving addition, subtraction, multiplication, and division, for building array models for multiplication and division, and for modeling area.

4th grade and up – used to model word problems for multiplication, division, and ratio, and to model area and volume.

### Base-Ten Blocks and Place Value Discs

Find Base Ten Blocks here and Place Value Discs here.

1st grade – Base-Ten Blocks are used to model place value for numbers to 100

2nd grade and up – Place Value Discs are used as a more abstract (and manageable) model for place value understanding for numbers from thousandths to millions, and for modeling and developing a conceptual understanding of the four standard algorithms. Base-Ten blocks can continue to be used for those students needing a one-to-one representation.

### Paper Strips and Squares of Equal Size

Cut them from paper found in the recycle bin.

1st and 2nd grade – used to model fractions of a whole.

2nd grade and up – used to model the four operations of fractions with the same size whole and for modeling part-whole and comparison word problems.

### Number Cards (Playing Cards) and Dice

Find number cards on our resources page or pick up some playing cards at your local dollar store. Dice can be found here.

All grades – for playing games and making math fun!

Get creative and have fun building your inventory of math manipulatives!

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