Here's the task I'd given several 6th graders during an intervention session on fractions:Work alone or in pairs. Roll two fractions dice, record the fractions shown on individual whiteboards, model the equation with the fractions strips, and then complete the addition problems. Repeat at least five times.
Karla and Shandra filled their boards with more than a dozen equations. At their request, I checked their answers. All were correct. Beside them, Derek quickly completed five problems without touching the fractions strips. I peeked over his shoulder. All his answers were wrong.
It's easy to use labels such as "diligent" and "careless" to describe these students. However, their teachers and I had worked with them to identify their cognitive styles; my knowledge of their styles informed my next moves.
I switched one of the girls' red dice for a blue one so the girls would roll fractions that weren't multiples of each other. I asked, "Do you think you can work without the fractions strips now?" They hesitated, nodded yes, and asked whether they could keep the strips nearby, just in case. They did another dozen problems, including 1/12 plus 1/15. "Now we know we can do the hardest problem. We understand common denominators," they told me.
As the girls worked, I said to Derek, "I think you know how to do these problems, but you just weren't careful. Am I right? Let's see how quickly you can finish five problems correctly. Five in a row. Then I have other tasks I think you'll find more interesting." A few minutes later, he showed me five new problems—all worked out correctly. He immediately tackled word problems and later helped Karla and Shandra understand how to solve them.
Differentiating for Learning Styles
This intervention took place while I was coaching a mathematics team for differentiating instruction using the Jungian learning styles. Normally, I emphasize that by mastering specific strategies, all students can learn in all styles even though one may come more naturally. In fact, school success requires us to learn in all styles. However, the team had noted that many of the intervention materials and much of the curriculum seemed to fit only some of the styles. They wondered whether struggling students might benefit if we differentiated for the styles during interventions.
We based the intervention strategies on research that involved filming 100 students as they completed the same mathematics tasks, coding their strategies and reactions, and then identifying their cognitive styles. The results revealed significant differences in how students with different styles approached mathematics (Kise, 2011).
Jungian Learning Styles
Jungian type theory, popularized through the Myers-Briggs Type Indicator, describes psychological preferences for gaining energy, taking in information, and making decisions, all key processes in education. To illustrate, think about your physical preference for dribbling a basketball with either your right or left hand. We didn't choose that preference; one is just naturally easier. Similarly, we have psychological preferences. And just as basketball players can develop ball-handling skills with both hands—and any good coach ensures that they do—we can learn to use all of the psychological preferences, especially when teachers, parents, or other mentors serve as our guides.
Helping students identify their preferences isn't limiting them. Rather, it's a tool for helping students understand what comes easily and where they might need to develop new skills or strategies.
Gaining Energy, Processing Information
Key to learning are our preferences for gaining energy and for processing information (Kise, 2007; Lawrence, 2009). People gain energy in one of two ways. Those who prefer extraversion are energized through action and interaction with others. These students need to talk and move to think; too much seatwork or listening to lectures drains their energy, and with it their ability to concentrate. In contrast, people who prefer introversion are energized by reflection and solitude. Too much activity or not enough wait time before they're expected to share their answers drains them of energy.
Another set of Jungian preferences describes our starting point for gathering information. People who prefer sensing first pay attention to facts, reality, and past experiences. These students rely on instructions, examples, and hands-on tools or pictures to shape their understanding of mathematical concepts, just as Karla and Shandra did with the fractions strips.
In contrast, people who prefer intuition first pay attention to hunches, connections, and analogies. Derek, for example, did not use the fractions strips. Further, he avoided repetitive practice as evidenced by his first string of wrong answers, preferring to concentrate on more challenging problems. Note that boys and girls are equally likely to hold either preference; there is no gender difference (Myers, McCaulley, Quenk, & Hammer, 1998).
Here's one way to think about the two information-gathering styles. Sensing types build knowledge in an orderly fashion from facts to larger concepts. It's as though they're walking across a planked bridge—and they want every plank in place. If there's a hole, they pause to review what they know for sure, ask questions, or assume that they failed to grasp key information. If they already doubt their math ability, they may need help to begin walking again.
In contrast, intuitive types trust their hunches, making leaps to connect different ideas. They don't notice whether all the boards are on the bridge. In fact, once they've made a leap, they usually resent having to go back and nail any loose boards in place through practice. They may need to learn strategies for testing their hunches or supporting their conclusions.
Chittenden, Salinger, and Bussis (2001) reported similar patterns in reading instruction, with two distinctly different modes of cognition for sounding out words and determining meaning that match Jungian descriptions of sensing and intuitive styles. Similar to the patterns for intuition in mathematics, Chittenden's Cluster A students read fluently, easily guessing words from the context or the pictures without hesitation, sometimes correcting themselves later without losing the pace of the story. Similar to the patterns for sensing in math, Chittenden's Cluster B students did not read ahead, asked for help rather than guess unknown words even when provided with cues or strategies, or remained silent as they attempted to work out the phonetics of a word. They lost momentum rather than make mistakes.
Many intervention programs assume that all students who struggle work best in the sensing style, with gradual learning progressions, practice, and review. Although many students who prefer intuition can adapt to this kind of instruction, others will daydream; do sloppy work (like Derek during the dice-rolling activity); or act out because they're bored.
Four Ways of Learning
- "Let me master it!" (Preferences: introversion and sensing). If these students aren't comfortable with mathematics, they want certainty before proceeding. They like direct instruction and practice work, and they dislike experimenting unless they can receive immediate feedback. Karla and Shandra share this style.
- "Let me do something!" (Preferences: extraversion and sensing). These students learn through movement, interaction, and hands-on manipulatives. For example, instead of using little paper fractions strips, they thrive on working in groups with large magnetized fractions strips at the whiteboard. They often use a purposeful trial-and-error method working with pictures or manipulatives to solve problems until they "see" that their answer matches the mathematics of a problem.
- "Let me think!" (Preferences: introversion and intuition). These students process ideas internally and pride themselves on unique or creative solutions. They are drawn to concepts, not procedures. They prefer working with numbers and may not always benefit from hands-on tools. This is Derek's style.
- "Let me brainstorm!" (Preferences: extraversion and intuition). These students process their ideas best out loud with partners or in groups, transfer new knowledge to new situations easily, and prefer a variety of challenging tasks as opposed to practice work.
How Learning Preferences Apply to Teaching Math
Differences among these styles were visible in four major areas.
Practice
Students who prefer intuition may resent repetitive practice once they understand the concepts. Students who prefer sensing often enjoy such practice, yet they may not retain procedural mastery unless they also grasp the underlying concept. This is often the case when teachers say, "They pass the Friday tests, but by Monday it's as though they've never seen the material before."
One teacher on the mathematics team decided to conduct separate interventions to accommodate students' differing needs for practice. He used several activities to help students understand how to work with mixed numbers and improper fractions. He noted,Even though the two groups started from approximately the same knowledge base, the intuitive group seemed to master the concepts in just a couple of days and wanted to move on. In contrast, it took the sensing group four days to master the concepts.However, when the light bulb went on, it was the highlight of my teaching career. The students eagerly wrote new problems for classmates to try, demonstrated to the principal what they had learned, and asked for more of the problems each day. I have the feeling that since 1st grade, those students have viewed math as magic, something they would never understand. This may have been their first experience in mastery.I've seen an immediate change in their attitude in my regular classroom, as if they now know they can make sense of it. Four days may seem ridiculous, but what if they'd been given that time to catch on in 1st and 2nd grade? Where would they be now?
To enable sensing students to practice to reach mastery while keeping intuitive students engaged, the math team developed families of tasks, such as the dice-rolling game, where changing the dice colors changed the difficulty of the task. We also had index cards with addition problems that involved common denominators for students who still struggled with the basic concept of what it means to add fractions. In the same family were the word problems that engaged Derek. We watched for that light-bulb moment before encouraging students to move to the next level of difficulty.
Instruction and Feedback
There were also major differences in when and how students needed feedback. Although "Let me do something!" students often blurt out questions, "Let me master it!" students may sink lower in their chairs as if to say, "Once again, I can't understand this." Some teachers misinterpret these actions as neediness or a lack of initiative. Students and adults with sensing learning styles point out, "Starting before we're sure we're headed in the right direction is a waste of time."
Both of the sensing groups frequently asked permission before using available materials, drawing figures, or trying a strategy. And if they weren't sure, they frequently said, "I need help" or "I don't understand what to do." During the research project, 50 percent of sensing students asked permission to try a strategy as opposed to only 4 percent of intuitive students. Although one-third of the "Let me master it!" students asked for direct instruction, none of the students with other styles did.
As far as the two intuitive groups were concerned, the "Let me think!" students often worked silently for long stretches of time, working with numbers or staring at a problem. They showed irritation when teachers interrupted their thinking to see whether they had any questions. After pauses that might last as long as 10 minutes, they literally exclaimed, "Oh, I get it now," completed a problem, and then applied that new understanding to later problems. We learned to hold feedback until they asked for it or to ask quietly, "What do you think might work?" and then wait at least 10 seconds for an answer.
Although the "Let me brainstorm!" students also confidently applied new knowledge, they often didn't notice when their solutions actually contradicted their explanations; rather than wait for them to ask for feedback, we needed to be ready with probing questions to help them rethink what they "knew" to be true.
Numbers or Manipulatives?
The sensing ("Let me master it!" and "Let me do something!") students grasped concepts through concrete reality—as did Karla and Shandra using the fractions strips—and not through numbers. In the study, none of the sensing students used numbers to find common denominators or equivalent fractions, whereas 65 percent of the intuitive students did.
For many intuitive ("Let me think!" and "Let me brainstorm!") students, the manipulatives and pictorial representations were unhelpful. In one case, a teacher drew two circles on the whiteboard, dividing one into fifths and the other into sevenths, and asked a student who preferred intuition, "Look at 2/5 and 2/7. See how 2/7 is smaller?" The girl replied, "No, that doesn't help at all." A bit later she raised her hand and said, "I found a common denominator—2/5 is 14/35, and 2/7 is 10/35. I get it now."
However, for the sensing students, understanding came through pictures and manipulatives. Several caught on to the idea of common denominators by physically dividing a set of six wooden tiles into halves, thirds, and sixths and then grasping how they could not divide six tiles into fourths—although a couple of students asked whether they could break the tiles in half!
However, many students hesitated to use the manipulatives. They told us, "Our teachers said we shouldn't use them because we can't on the state tests. And besides, smart kids never need them." To balance the needs of sensing and intuitive students, we emphasized that manipulatives are tools not just for solving problems, but also for explaining reasoning, which increased everyone's willingness to use them.
Student Mistakes
Derek's initial incorrect solutions are a perfect example of how "Let me think!" or "Let me brainstorm!" students' lack of attention to detail can result in mistakes that hide their conceptual understanding. During interventions, tasks need to probe beyond right and wrong answers so that we can tell whether students grasp the concept, are just using a procedure, are making careless mistakes, or still lack conceptual understanding.
A Sensible Approach
Jungian cognitive styles can help educators understand differences in how students approach mathematical tasks. This framework can prove useful for creating equitable instruction during interventions and for supporting students' core needs for feedback, representations, and practice tasks that lead to mastery of mathematical concepts and processes.
