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?

Learning literacy strategies in authentic contexts

Children should learn literacy metacognitive and comprehension strategies in meaningful contexts. When we give our students the chance to refine their use of these strategies in authentic and relevant contexts, they'll more deeply understand why they're using them and the will more likely use them independently, out of the context they were originally learned.

In the Primary Years Programme, we are obligated to contextualize our students' learning by organizing all our curriculum into six transdisciplinary units of inquiry. How lucky we are that the PYP framework requires us to teach in a way that has been proven to be best practice. What is more, when PYP educators teach literacy strategies in the context of the Unit of Inquiry, they're able to cut through an overcrowded curriculum.

For more information on transdisciplinary learning and learning literacy strategies in relevant contexts see:

• Developing a transdisciplinary programme of inquiry, IBO 2012
• The Primary Years Programme as a model for transdisciplinary learning, IBO 2010
• Lanning, Designing a Concept-Based Curriculum for English Language Arts: Meeting the Common Core with intellectual integrity, K-12
• Erickson & Lanning, Transitioning to Concept-based Curriculum and Instruction: How to bring content and process together

In second grade, students are busily investigating how people influence the world. During this unit, students inquire into the knowledge concepts of hero, making a difference, and personal strengths. During this unit, they are also learning the literacy strategy of summarizing. 

To help students explore both of the big ideas of hero (a knowledge concept) and summarizing (a process concept), the teacher met with guided reading groups to work with texts that were best suited for the students' developmental reading level.

To start, the students in different guided reading groups read a book about a particular hero. To support the students through one of their first times summarizing, the teacher just had the students write one important detail from the text on a post-it note. She then copied those details on the left-inside page of a folded piece of paper. Then, making sure to include all those details, the students told the teacher a summary of the book, while the teacher notated the students' thinking on the opposite-inside page.

Below are two examples.

Next, students were ready to write their own summaries. Below are two examples: 

The back cover of Carlos's summary book offers the reader a teaser to what the book will be about.

This student decided to highlight important vocabulary on the front cover of her summary book.

This student's summary took the form of a listicle of influences, accomplishments, and personality traits (next picture).

After reading about how this teacher taught a literacy strategy in the context of learning concepts from the unit of inquiry, how could you or have you taught students literacy strategies in meaningful, authentic, and relevant contexts?

Making Visualizations Visible

The ultimate goal of literacy instruction is to arm students with the literacy processes, strategies, and skills that they will be able to intuitively and independently use to extract meaning from text. "Text" can include the written word, videos, interviews, speakers, field trips, conversations, discussions, and more.

The teaching and learning of these literacy strategies is best done in authentic contexts. On page 70 of Making the PYP Happen, the authors state that "it is the school's responsibility to provide authentic context for language teaching and learning in all areas of the curriculum," [emphasis added].

In a Primary Years Program (PYP), the program of inquiry provides the authentic, engaging, challenging, and significant context for learners to develop and use language. In other words, teachers should situate literacy instruction right inside the units of inquiry. However, it should be clarified that although comprehension and metacognitive strategies are to be taught within the unit of inquiry, it cannot be assumed that learners will implicitly pick up on these strategies while studying content concepts. PYP teachers still must provide specific time and opportunities for their learners to explore, discover, practice, and construct understanding of these literacy strategies. When we teach literacy in this way, students are learning literacy processes and content concepts simultaneously.

Recently, I took the opportunity to teach a first grade reading lesson from our school's literacy curriculum. The objectives of the lesson were that students would retell story events and visualize & analyze characters during reading. Rather than use the text provided by the series, which would not have been authentic, challenging, or significant for the first graders, I choose a text that would help the learners understand their current Unit of Inquiry's Central Idea: Responsible choices promote balanced living.

A List from Frog and Toad Together by Arnold Lobel.

In the story A List, Toad creates a list of all the things he'll do during the day. I selected this story because the list he makes contains many responsible choices that would represent a balanced lifestyle. (SPOILER ALERT: Toad never completes all the items on the list because it blows away in the wind and he subsequently goes melodramatically ballistic, which should not come as a shock for those of you who know Toad well.)

Before reading the story, I discussed with the students the importance of being able to really understand a story while we read. "After all," I said, "that's the point of reading! To understand what you read." In their book Strategies that Work, Harvey and Goudvis say that "different readers rely on different strategies to help them gain better understanding." One such strategy is visualizing. "Visualizing strengthens our inferential thinking. When we visualize, we are in fact inferring but with mental images rather than words and thoughts." The authors continue, "When we visualize, we create pictures in our minds that belong to us and no one else," (p. 130-2).

However, if we let students keep their visualizations locked up in their minds, students may become mentally overwhelmed. Rather, my goal of the lesson was to get students to make their visualizations visible so that they could more fully engage in thinking. "...it is difficult [for students] to do their best thinking because of cognitive overload, that is, because the thinking demands exceed their capacity. When our thinking is distributed, when we do not have to rely solely on our internal mental resources, we free ourselves up to engage in more challenging thinking," Ron Ritchhart in Intellectual Character: What It Is, Why It Matters, and How to Get It, p. 100-1).

I prompted students to visualize what was going on in the story as I read, but then to draw those pictures from their mind on a blank sheet of paper. After reading, we talked about what their pictures should include, if they really, truly understood the story. Together, we decided that students would need to have the characters of the story, the setting (time and place), and what went on in the story (the events).

To show students how to self-assess their understanding of the text, one student showed his visualization to the group while I modeled the kinds of questions the students should ask themselves.

Did I understand the story?
Did I include the characters?
Did I include the setting?
Did I include the important events of the story?

By reviewing all the students' visible visualizations (I included other examples below), the classroom teacher and I could clearly and quickly see who was able to adequately extract meaning from the text and who was not. Using this powerful information, the classroom teacher could form guided reading groups to support students' unique and specific literacy needs.

After reading about making students' thinking and understanding visible by having them draw out their mental visualizations, how could you make your students' visualizations visible with the "texts" they read, listen to, and view?