In the renaissance years, an educated person dabbled in all fields of knowledge. By the 20th century, an increasing specialization led to compartmentalization of knowledge. Isolated university departments trickled down to isolated school subjects and now teachers label themselves as “math teachers” or “literacy teachers.”
But a fixed set of skills in a particular domain is no longer enough. Our society needs people who learn quickly, adapt old methods to new situations, think outside the box, communicate effectively, and innovate; people who can make sense of the world by looking at it through multiple lenses, not just one.
To help teachers blur the boundaries of subjects, we focus this post on four teaching-practice bridges between the two worlds of learning, mathematics and literacy. Our hope is to plant seeds of reflection. Once educators realize that both literacy and mathematics are ways of knowing and both offer lenses through which we make sense of our world, they will see that their ultimate goal is to teach students to think. The fog then clears and the bridges begin to shine in a new light (see A New Light on Literacy and Mathematics).
Focus on the Big Ideas
So much content, so little time! In order to ensure we cover the breadth of content required, teachers tend to cram as much information as possible into one lesson. This creates a sense of overload in our students, who in turn, shut down. We must ask ourselves, “What is more important: covering all the material or helping students uncover the big ideas?” The Common Core Standards have attempted to facilitate this uncovering by outlining a limited number of goals for each grade. Today, we are looking for more depth. By uncluttering our lessons and focusing on one big idea or strategy, students are much more likely to grasp what we’re teaching, implement it in their own work and retain and transfer it for future use.
To that end, we must be clear about the goal of the lesson, where it fits inside the bigger picture, and how it builds on prior knowledge. As you’re planning your lessons, ask yourself, “What is the one thing I want my students to know or be able to do after this lesson?” Carefully craft your lessons so that every minute focuses on moving your students toward that goal. Maintaining this focus affords your students a more meaningful engagement with the lesson, which will enhance their understanding of the curriculum as a whole. At the end of a lesson, ask them to articulate in their own words what they think they learned, and then compare your teaching goal to their learning perception.
Model Good Questioning
Teacher: What is the sum of the angles in a triangle?
Joey: 280 degrees?
T: You’re warm but you got one digit wrong.
Sarah: Oh, 180!
Such questions, while sometimes useful, do not stimulate any kind of student reflection or meaningful classroom discourse. When our goal in questioning is to seek correct answers, we miss opportunities to delve more deeply and gain insight into student thinking. Even in this single-answer, convergent question, one could probe further to see if “180” was memorized or understood by following up with, “How do you know?” or “Can you convince us that your answer is correct?” rather than saying, “Right!” Arithmetic may be about answering questions, but mathematics is about questioning answers!
But it’s the divergent questions that truly begin opening up students’ minds to deeper thinking. These include what-if type of questions such as, “What if we considered the sum of the angles in a pentagon, how would 180 degrees help you?” [A pentagon can be divided into three triangles, so 3 × 180°=540°.] Questions you can post in your classroom, to show students that you value their thinking include:
How do you know?
How did you figure that out?
How can you be sure?
Does that work in other places?
Is that always true?
Can you convince your peers that your conjecture is true?
Capitalize on the Power of Visualization
You’ve seen it before: Students zoning out during lessons. Students asking you to repeat directions several times. Students who appear to have mastered your objectives for the lesson but forget by the following day. For many of these students, the learning environment we’ve carefully constructed for them may not be best suited to their learning needs. Many learners, particularly our boys and English Language Learners, rely on visuals in order to process and retain what they’re learning.
Incorporate visuals into your teaching through the use of charts containing few words and easily understandable graphics or icons. For example, when teaching embedded numbers to young children (smaller numbers “hidden” inside bigger ones), drawing the staircase in Figure 1 on the board will have a more lasting effect than the symbolic equation 1+2+3+4=10. When asking questions, elicit students’ internal visualizations as a starting point for your teaching, and build on them. For example, ask: “What mental image(s) does the fraction one-half conjure up in your mind? Make a picture or diagram.” When assigning independent or group work, expand what is an acceptable product to include visual, graphical, and artistic expressions of student ideas, thinking, and learning.
Visual literacy is defined as “the ability to evaluate, apply, or create conceptual visual representations” (www.visual-literacy.org). It is becoming an increasingly important skill in our digital world and, because of this, we must guide all of our students, not just those who require visuals for learning, to learn how to process and describe information using a variety of visuals.
Motivate with Memorable Stories
We all love stories. Regardless of our age, gender, race, or walk of life, we are captured by stories because they resonate with us, move us, mark us, and teach us. We are grabbed not only by the story but also by our relationship to it. Stories have a beginning, an end, and a message. Our brains are engaged by their structure, coherence, sequencing, and connections because our brains work the same way. Stories stir emotions. They have the power to uplift us, motivate us, or nudge us into action.
One of the oldest communication methods of all times, storytelling is a pedagogical strategy that captures students’ interest and fosters learning. It seamlessly weaves into the teaching of all facets of literacy: language development, vocabulary acquisition, reading comprehension, and writing. Moreover, we now know the power of children writing their own stories. But in mathematics storytelling is less common and more needed. It helps put a human face on what often seems a set of dry, static, and unrelated facts, rules and procedures. Carefully selecting a story to help students construct a new concept not only engages them but also helps them visualize, find meaning, and remember vividly the process of knowledge construction. Be it math or reading or writing—the joy of learning should be in the journey, not just in the acquisition of knowledge.
Epilogue: A Story
We leave you with a story we hope you will share with your students.
The year was 1786. The place was an elementary school in Brunswick, Germany. The nine-year-old boy was Carl Friedrich. He was a management problem for his fourth-grade teacher who knew not what to do with this mischievous student. So she punished him, made him sit in a corner until he solved the following problem: “Add 1 + 2 + 3 + 4 + … + 98 + 99 + 100.” With no calculators at the time, she hoped to have thirty minutes of peace to teach her lesson to the rest of the class. To her great astonishment, Carl Friedrich was back at her desk with an answer within three minutes, and the correct one to boot. How did he find it? This is what we think he said:
I called S the sum you asked me to compute:
I then wrote the sum again, right under the first, but in reverse order. It’s still equals S:
I then added up the two sums, noticing that each column of two addends equals 101:
And since 2S contains 100 repetitions of 101, I wrote:
But you only asked me to compute S, not 2S, so I divided by 2 to get S:
And so goes the story of the child prodigy, Carl Friedrich Gauss, who turned out to be the greatest mathematician of the 19th century, and often referred to as the “greatest mathematician since antiquity.” The formula he found:
for the sum of the first consecutive 100 counting numbers, 1 through 100, has since been generalized for the sum of the first consecutive n counting numbers, 1 through any number n:
It is known as “Gauss’s formula.”
Test it out for an n value of your choice and enjoy!