Teaching Math in Authentic Contexts

Our staff recently has been examining the question: "What does math teaching and learning look like in a Primary Years Program (PYP)?"

To answer this question, we explored the text "Mathematics in the Primary Years Programme," one of the subject annexes from Making the PYP happen: A curriculum framework for international primary education. In that document, teachers were able to see the clear vision of how math teaching and learning should look like in our PYP.

To document their thinking, teachers created a Practice Profile (a rubric of teacher behavior) based on what they were reading in the text. As primary and intermediate teachers were working in separate sessions, there were two separate practice profiles created and can be found here: KEC PYP Math Practice Profile - Primary and Intermediate.

An important idea that came up during that professional learning engagement was that in a PYP classroom, teachers should provide students with multiple opportunities to explore relevant problems both inside and out of the units of inquiry. The math annex provides some guidance on concepts that might be best suited for learning in context when they say, "data handling, measurement, and shape and space are best studied in authentic contexts provided by the transdisciplinary units of inquiry," because they represent the "areas of mathematics that other disciplines use to research, describe, represent, and understand aspects of their domain," (p. 85 of Making the PYP Happen).

Reflecting on this new understanding of math instruction in the PYP, one G2 teacher planned for her students to create a timeline, an authentic opportunity to explore the abstract mathematical concepts of measurement, subtraction, space, and time. She knew that the timeline would give her students a better understanding of the heroes they were studying in their unit of inquiry, as they were trying to make sense of the big idea that people influence the world in different ways.

First, the students created the timeline using the scale 1 cm = 1 year. The students worked together to create century strips, each measuring a meter (and alternating in color).

Then, the students needed to figure out how long each of the heroes' strips should be. The teacher modeled how to use a number line to figure out the age of the hero they'd be researching. Using the birth date and death date (or the current date for living heroes), the students figured the difference between the two. It is important to mention that the students did pretty well with this since they have been using number lines and open number lines all year for almost every math topic. This shows how effective it is to give students the opportunity to use the same thinking tools and structures over and over again until they become routine.

Next, students used that information to create strips for each of their heroes, again with the scale 1 cm = 1 year.

Finally, the teacher gave students the opportunity to look at the timeline and document their observations, thoughts, and questions using the thinking routine See-Think-Wonder (from Making Thinking Visible by Ritchhart, Morrison, & Church).

This G2 teacher continues to see other ways that math can be learned in meaningful ways during her unit of inquiry. Recently, she used the data they had collected on heroes' ages to introduce median, mode and range. Using the data the students had already collected made it more authentic and engaging.

The students found that their heroes ranged in age from 37 to 95, a range of 58 years. Kids were surprised and impressed because that seemed like a lot. There were two medians and two modes, so that was confusing for their introduction to the idea of analyzing a data set using those tools, but it was engaging nonetheless.

After reading about how this G2 teacher taught the mathematical concepts of measurement, difference, and data in the authentic context of her unit of inquiry, how could you or have you taught math in meaningful, engaging and authentic ways?

Making Professional Thinking Visible

Effective professional learning is engaging, relevant, connected to long-term goals, and involves participants in interactively constructing understanding. Because thinking routines (especially like those described in Making Thinking Visible by Ritchhart, Church, and Morrison) are tools, structures, and patterns of behavior that are used to initiate, explore, discuss, document, and manage thinking, they can be used by educators during professional development to help actively become engaged in constructing understanding. Furthermore, since they're experienced themselves, those educators are more likely to use those thinking routines with their students, a vital part of creating a culture of thinking in their own classrooms.

Therefore, at a recent pedagogical leadership team meeting, when one of the leaders asked other teams how they guided their students through solving mathematical word problems, I knew that using a thinking routine would help guide us through that exploration.

Here is that story:

Start with a concept.
We started with the timeless, abstract, universal, and transferable idea that solving problems independently is made easier by following a set protocol.

Pick a specific, concrete example of a person, place, situation, or thing that illustrates that concept.
In order to gain an understanding of that abstract idea, we explored different protocols used by teachers throughout our building to help their students solve mathematical problems independently.

Create an opportunity for participants to explore that concrete example.
During this phase of the conversation, we agreed to dialogue and suspend judgement. For more information about dialogue, discussion, and how both can be used to transform conversations we have in schools into meaningful communication, see "Teacher Talk That Makes a Difference" by Robert Garmston and Bruce Wellman.

To guide our dialogue, we followed the thinking routine See-Think-Wonder from Making Thinking Visible by Ritchhart, Church, and Morrison (p. 55).

See: What do we see?

During this phase of the dialogue, team members shared the protocols they use with their students to solve mathematical word problems. Comments and crosstalk were discouraged.

1st protocol shared:

1. What are you trying to find?
2. What numbers will you use?
3. Open number sentence - using a variable for the unknown
4. Decide what to do with a remainder

2nd protocol shared: (students don’t have to follow these in order)

1. Make a prediction
2. Answer
3. Number sentence
4. Strategy
5. Show your thinking

3rd protocol shared: U.P.S. Check

1. Understand: look at the problem and understand what it is asking us to do. What are we trying to find out?
2. Plan: What do we need in the number story to solve the problem? Circle the important points.
3. Solve, using different strategies.
4. Check: Solve it a different way, using another strategy to see if you get the same result.

4th protocol shared: (for behavior problem-solving)

1. What are you doing that I can’t allow?
2. Why can’t I allow you to do that?
3. What will you do instead, next time? 

5th protocol shared: ORID

1. Objective: What do we know about this?
2. Reflective: How do we feel about this?
3. Interpretive: What does this mean for us?
4. Decisional: What are we going to do? 

6th protocol shared: For solving 2-step word problems

1. Equation
2. Representation (including acting it out), to check for understanding
3. 1st step
4. Next step
5. Answer with a label 

7th protocol shared:

1. What do you want to find out?
2. What do you know?
3. What will you do? 

8th protocol shared: Cognitively Guided Instruction (CGI)

1. Read the problem.
2. What do I know?
3. What is the question?
4. What strategy will I use?
5. Organize my thinking so others can understand what I did.
6. Answer the question in a complete thought. 

9th protocol shared: from George Pólya's four key principles to "problem solving" first published the book “How to Solve It” (1945):

1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Look back

Think: What do we think?

Next, leaders were invited to comment on what they had just heard, noting any "big ideas" that surfaced during the sharing period.

• There is more than one way to solve problems.
• There are a lot of similarities, but in different language.
• Protocols range from a lot of steps to a few steps.
• Lots of visual aids, graphic organizers 
• Acting out and using manipulatives helps.

Wonder: What does this make us wonder?

• Do we all have to be doing the same thing?
• Would this be a site-based decision or district-wide?
• Using the various protocols, what’s the level of independent use?
• Can students name the strategy they’re using?
• Do the protocols we’re using lead to innate problem-solving (are students internalizing the steps)?
• Can kids explain why they’re using a particular strategy?
• Does the district math coordinator have any insight?
• What do math experts have to say?
• What language constructs need to be supported?
• How much of this is a vocabulary issue and not a math issue?
• Could the vocab piece be consistent throughout the grades?
• Does our curriculum use the same language as the standardized tests that our students take?
• Should protocols be teacher-given or student-created?
• If protocols are student-created, how can teachers support students? Is this the right way to go?
• Can we use prompts like “show me your thinking” and “did anyone solve this in a different way?” along with these protocols?

Check for understanding by having them write a concept statement.
After our discussion, leaders were asked to respond to this question: "What can you tell me about how we can make solving problems independently easier for our students?"































Reflect on their thinking and decide next steps.
After analyzing and reflecting on the responses above, here are the big ideas I see surfacing. What do you see?

A protocol for solving problems in math should:

• require students to make their thinking visible (or verbal).
• require students to show different strategies to solve the problem.
• help students identify vocabulary that is essential to understand the problem.
• help students visualize the problem.
• have few steps so students can use it independently.
• be universal so it can be used at all ability levels.
• be transferable to all settings where problems need to be solved.
• be student-created (at least partially).
• help students develop PYP attitudes like creativity, commitment (persistence/grit), independence.

Writing conceptual statements with 3rd graders

Among the 22 social studies benchmarks that third graders need to learn, third grade teachers meet the following three benchmarks by guiding their students through an inquiry into why ancient civilizations settled where they did. For this lesson, the teacher follows the simple inquiry cycle Invitation-Investigation-Demonstration.

• Economics Benchmark 3.2.4.5.1: Explain that producing any good or service requires resources; describe the resources needed to produce a specific good or service; explain why it is not possible to produce an unlimited amount of a good or service.

• Geography Benchmark 3.3.1.1.2: Create and interpret simple maps of places around the world, local to global; incorporate the "TODALS" map basics, as well as points, lines and colored areas to display spatial information.

• History Benchmark 3.4.3.7.1: Explain how the environment influenced the settlement of ancient peoples in three different regions of the world. (Early Civilizations and the Emergence of Pastoral Peoples: 8000 BCE—2000 BCE)

Invitation

To invite students to inquire into why ancient civilizations settled where they did, the teacher shows a world map, asking students what they see on the map. The discussion centers on the fact that this is a world map and that the ancient civilizations are concentrated in a particular place on the map (Europe-Africa-Asia).

http://upload.wikimedia.org/wikipedia/commons/4/47/World_1_CE.PNG

Investigation

Then, the teacher shows the students a map that "zooms" in on this concentrated area. This "new" map shows something different. The teacher states that this map will give us information that we need to answer the question, "why did ancient civilizations settle where they did?"

http://www.rcet.org/twd/images/river_civilizations.jpg

Next, the students divide a piece of lined paper into thirds, horizontally. Students label the thirds with the words "See", "Think", and "Wonder". See-Think-Wonder is a Visible Thinking Routine from the book Making Thinking Visible by Ritchhart, Morrison, and Church.

After students write down what they see on the map, they share with someone near them. Next, the students share out with the whole group and the teacher scribes the students' thinking, sorting as students share.

After, students label the categories.

Then, students are given a couple of quiet moments to write down what they are thinking about what the map is telling them, especially regarding where civilizations settled. Like before, they share with a small group before sharing in front of everyone. The teacher again scribes their thinking.

Demonstration

To demonstrate their conceptual understanding, the students are instructed to write down one sentence that sums up all the learning that they did about where ancient civilizations settled. Once everyone has had a chance to synthesize their own thinking, everyone shares their thinking, starting with a simple statement and then revising it until the statement is timeless, abstract (to a degree), universal, and transferrable. In the example below, the quotations between the drafts are the teacher's prompts. The changes on the drafts reflect the suggestion from the 3rd grade students.

"Why did ancient civilizations settle where they did?"

"What else to we want to include? What is missing?"

"What do we mean by 'everything they needed'?"

"I'm a little confused who is doing all of this and what they're doing. Our subject needs a subject and a verb."

"Was it just Egyptians that did this?"

"I'm going to read a page from the text Understanding Civilizations by Stefan Stevens. In the book, the author wrote a page that explains why ancient civilizations settled where they did. Let's check to see if we're missing anything."

http://cdn2.lybrary.com/understanding_civilizations_by_stefan_stevens_1477726225.jpg

"You know a band of the Dakotah-Sioux Indians settled on the banks of the Mississippi River near our school for a lot of these same reasons, but they only settled there several hundreds years ago. Also, the South St. Paul stockyards were started here in 1886 because of the river. How can we change our statement to include these more recent events?"

After reading about how third grade teachers have written a conceptual statement with their students, how have you or could you write generalizations with your students?