why are these students two years behind in math?

Why Are These Students Two Years Behind in Math?

Jane Kise, Ed. D.

Differentiated Coaching Associates, LLC www.edcoaching.com [email protected]

Presented at ASCD, March 24, 2012

Philadelphia, PA Note: This handout contains research background, examples, and information so that attendees can share what they learn with colleagues. During the actual session, participants will see film clips that illustrate how students with different mental processes approach mathematics tasks, discuss the needs of these students during instruction and interventions, and learn about the different activities that were effective in helping students master concepts.

Why do students fall behind in mathematics? To begin to answer this question, I

filmed 100 6th grade students at a high-poverty school as they tackle tasks involving fractions, looking for evidence of learning styles and whether they affected comprehension. I was also coaching the school’s mathematics professional learning community (PLC) on effective teaching strategies. This PLC chose to focus on fractions instruction because of its foundational importance to algebra.

We used the information from the filming project to inform instruction in the

school’s heterogeneous math classrooms and to plan interventions. Most of the students I selected for filming were about two years below grade level in proficiency, according to standardized tests. The filming project and the intervention work convinced the teachers of the importance of addressing learning style needs when students fail to grasp mathematics concepts during regular instruction. The Learning Styles

The school where I conducted the research uses the theory of Jungian type to frame learning styles and differentiation. Jung observed that people have natural preferences for how we are energized, gather information, make decisions, and approach work and life, all of which affect how one teaches and learns. Handout 1 contains a further explanation of these preferences:

Extraversion and Introversion: This natural preference defines how people are energized. Extraverted types are energized through action and interaction. Introverted types are energized by time for reflection. Sensing and Intuition: This natural preference defines how people gather information. Sensing types attend first to reality, what their five senses and prior

Dr. Jane Kise “Why Are These Students Two Years Behind in Math? 2 ASCD Session 1224 March 24, 2012, Philadelphia, PA

Copyright 2012 by Jane A. G. Kise. All rights reserved. Please contacted [email protected] for permission to reproduce this article.

experiences can tell them. Intuitive types attend first to the unseen world of hunches, connections and analogies.

All normal people can use all four of these preferences, just like we can hold a pencil and scribble something with both hands. However, we prefer one preference over the other in each pair, just as most of us write with a preferred hand.

These four preferences combine to form four different learning styles, as described in Handout 2, which address diverse student needs for energy and information, two key factors in any classroom experience. To succeed at school, students need to develop skills with all the preferences; it isn’t helpful to always learn in one’s own style. However, confidence in one’s abilities often results from learning new concepts in one’s own style. Here is a quick summary of what we saw in this project:

• The differences in the four learning styles are visible, even to those with little

formal type training • Students who are proficient in mathematics can learn in any style • Students have biases about which styles are “smart” that sometimes block

them from effective learning strategies • Teachers may misinterpret the work of students whose learning style is

different from their own • Students who are multiple years behind benefit greatly from interventions

conducted in their own learning styles. The Filmed Tasks

The fractions tasks chosen were designed to show student thinking as they worked. The tasks were:

Task 1. Using fractions, name the colored areas of a rectangle that has 1 red

square, 3 green squares and 4 blue squares. Is there another fraction that describes the blue part of the rectangle? [Students were shown a drawing of the rectangle]

Task 2. Make a shape that is ¼ red and ¾ yellow Task 3. Make another shape, with more parts, that is still ¼ red and ¾ yellow Task 4. Make a shape that is 1/6 red, 1/6 green, 1/3 blue and 1/3 yellow Task 5. Make a bigger shape, with more parts, that is 1/6 red, 1/6 green, 1/3 blue

and 1/3 yellow Task 6. A rectangle is ½ red, 1/3 blue and has one green square and one yellow

square. What might the rectangle look like? What fractional part is green?

Students were free to choose from a variety of supplies: markers, colored pencils, color tiles (one-inch squares of wood painted red, yellow, blue or green), blank paper, one-inch gridline paper, and isometric graphing paper ideal for drawing triangles, hexagons, and other shapes which have angles other than 90 degrees.



While the two students with high standardized test scores (filmed for control

purposes) completed the three tasks in less than ten minutes, some students were not able to complete the first two in 40 minutes, even with individualized instruction. There were striking differences in how students approached the tasks, shown in Handout 3; misunderstandings were similar but how they tackled the problem and the resulting solutions clearly tied back to their learning styles. Students from different cultural backgrounds (African American, Hmong, white European, and East African) were present in each of the learning styles.

The immediate application of the filming, though, was to help the teachers

understand what might block student understanding and to plan for interventions. The entire team watched a film of one student, Rashan1, who preferred Extraversion and Sensing and revealed on an attitude survey that he was afraid he just couldn’t do math.

When the filming facilitator informed Rashan that he’d be completing fractions

tasks, he murmured, “Oh, dear. I’m not good at any of it. Especially dividing,” and buried his face in his hands. Given that division of fractions had not yet been covered—in fact the class was still working on addition of fractions with like denominators—his lack of self-confidence was especially striking. Below are two excerpts from the film transcript:

Excerpt from Student 21 (Extraversion and Sensing), January 2008 F: Record a geometric shape that is 1/4 red and 3/4 yellow. S: Okay. [Draws a 2”x4” rectangle on blank paper. Whispers to self] Oh boy, let’s

see, about… [Grabs red pencil and hesitates.] Does it have to be 2 different shapes?

F: Nope, all in the same shape. S: Okay… [Draws red square in 1 corner of rectangle equal to approximately 1/8 of the

shape, colors it, looks back at problem card—Figure 1] Oooooh, okay, now I’m starting to get it. [Turns paper over] Now I get it. [Draws new rectangle quickly with blue pencil, uses red pencil and draws 2 vertical lines that create thirds, rereads card and hesitates before coloring—Figure 2] I need a different sheet of paper [reaches for clean one and completes task correctly—Figure 3]

F: Okay, so tell me how you thought about that. S: ‘Cause I thought about that, there, there’s not another part about a

different color being in there anywhere. So then you have 3 plus 1 is 4 and in each one of the numbers you have a 4 and it’s telling you that you have just 1 square and it’s gonna have 4 squares in it and they’re supposed to be those two colors in both of them.

1 All names have been changed.



Figure 1 Figure 2 Figure 3

The way the student compared his work to the stated problem and then self-corrected illustrates purposeful experimentation, not random attempts, that increased the student’s understanding of the problem. Note below how his understanding of the meaning of “4” affects his ability to complete Task 3.

F: Record a geometric shape that is 1/6 red, 1/6 green, 1/3 blue, 1/3 yellow. R: Okay…[gets all the right pencils] okay, let me see…[quickly draws 3x5

rectangle on blank paper, divides it in 6ths] 1/6 red [colors 1 square red] 1/6 green [colors 1 square green, 1 square blue, hesitates, colors 1 square yellow. Looks back at problem and taps pencil.] Oh my goodness.

F: Tell me what’s going on in your head. R: [grabs and shakes head] I actually thought it would be 6 squares and this

one red and these two, and then there should be 3 blocks left over. F: Why did you think there should be 3 blocks left over? R: Because there’s 3’s at the end of them and I think that when it says 1/3 you

should color one of these two colors out of the three that was left over. But obviously not.

F: So you thought that if you colored this one out of six and this one out of six, then there would be three left because of the 3 in these 1/3’s.

R: Yeah, and then, but obviously not ‘cause I thought you were supposed to have 6 squares for the sixths ‘cause you’ll have to color in one out of the six squares. And, that didn’t happen.

Rashan’s methods and struggles were similar to other students we filmed who preferred Extraversion and Sensing2, summarized in Handout 2.

The teachers also viewed part of a film of a student with the opposite learning style, Introversion and Intuition, who ignored the tiles and paper and built her first shape of one red and three yellow markers. One of her teachers commented how this reflected her constant drive for individuality, remarking, “I thought her inability to finish worksheets was due to laziness or apathy, but they bore her to tears, don’t they?” The contrasting films of these two students with identical test scores focused the ensuing discussion on the need to differentiate instruction for learning styles as well as ability.

2 Student learning styles were assessed through the MMTIC®, published by CAPT (www.capt.org), and student confirmation through self-discovery activities (Kise, 2007).



The Interventions While mathematics instruction took place in heterogeneous classrooms, the

teachers decided to form small groups by learning styles for the interventions. Handout 4 provides information on how students with different type preference combinations approached two of the intervention activities and supports the following findings in the overall study:

• Students who preferred Sensing asked for more feedback and directions • Students who preferred Intuition were less willing to engage in repetitive tasks

once they grasped a concept • Students who preferred Intuition were more likely to make mistakes in

counting, use the wrong colors, or miscalculate when they in fact understood a concept.

One teacher worked with a group that preferred Introversion and Sensing and

another group that preferred Extraversion and Intuition. His experiences illustrate the effectiveness of the groupings.

One of the first concepts I tackled was the relationship between mixed numbers and improper fractions. I used a variety of activities—fraction strips, drawing models, working on individual whiteboards, and so on. Even though the two groups started from approximately the same knowledge base, the Extraverted and Intuitive group seemed to master the concepts in just a couple of days and were ready to move on.

In contrast, it took the Introverted and 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 each other to try, demonstrated to the principal what they had learned, and asked if they could do more of the problems the next day. I have the feeling that since first 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 classrooms, 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 first and second grade? Where would they be now?

This is not to say that all Introverted and Sensing students need this much time to grasp concepts, but that these students flourish when they are allowed to cement concepts before moving on. In contrast, the Extraverted and Intuitive students flourish with a more rapid pace and a wider variety of activities.



Rashan was part of one of our small intervention groups. By the second hour of the intervention, he was up at the document camera, defending his placement of 5/4 on a number line. Later, when he completed a fractions addition problem, he asked, “Can I take this paper home to show my dad? He isn’t going to believe I did this.” Deep, rigorous knowledge and use of learning styles can result in students believing that they can do the math.

Conclusion

This study documented differences in how students approached mathematical tasks that could be accounted for through the framework of Jungian type preferences. The data indicates that using Jungian type preferences as a framework for determining the kinds of support available could be important to responding to students’ strengths and motivational needs. A limitation is that while this project demonstrated visible differences in how students approach mathematical tasks that are related to type preferences, further study is needed to determine whether classroom structures and strategies that honor these differences would result in improved student mastery of mathematical concepts. The intervention experiences that teachers engaged in following the data collection suggest that further research is warranted. Handout 3 summarizes learning style characteristics observed during the project.

The following is a summary of possible implications for instruction from the

scope of the current study. Allow Students to Control When They Receive Feedback. With differences in

the amount of facilitator interaction sought by students who preferred Extraversion or Introversion, and Sensing or Intuition, classroom strategies that allow students to control when they receive feedback are important. While teachers may want students to monitor their own thinking, the students in this study who preferred Sensing seemed to need more confidence in their ability to master mathematics concepts, as shown in the interventions, before they were ready to work independently on challenging problems. In contrast, some of the students who preferred Intuition seemed to resent the facilitator’s attempts to check on their progress as they worked independently. It appeared that his questions interrupted these students’ thought processes.

An interesting aside: When teachers are shown a film clip from Student 21, the

student who used trial and error, they frequently react to the amount of time the student spent on the tasks. One teacher commented in a workshop, “I would have intervened after his first mistake, not wanting him to fail. That would not have helped, would it? The student needed time to make meaning.” When asked whether they wanted to intervene, approximately 80 percent of the teachers who have seen the clip say that they probably would have, but now understand that the student was not asking for help and that feedback would have hindered his progress in mastering concepts.



“Norm” The Use of Manipulatives and Representations. The differences in how students used and made meaning from the available manipulatives and representations also have classroom implications. Frequently, the use of manipulatives was key to making meaning for students with a preference for Sensing. If students were unable to complete Task 4 during the filming, the researcher worked with them afterward, showing how 6 tiles could be separated into equal groups of 2, 3, and 6. “I can see the fractions in the tiles,” was their frequent response. However, when the researcher asked some of them why they did not use tiles during the filming, several said, “Only dummies use the hands-on stuff. Our teachers don’t want us to.”

The teachers involved in the project discussed how they often limited use of

manipulatives since students would not be allowed to use them during the state accountability tests. However, after seeing the films, they realized that many students benefited greatly from manipulatives. The teachers agreed that to normalize the use of them in class so students did not view manipulatives as something that only struggling students used. They also noted that it was important to identify manipulatives that could easily be represented pictorially so students could transfer the way they used the concrete objects to representations they could use during testing.

However, the teachers also discussed how some students actively resisted using

manipulatives with comments such as, “Why do I have to move around fake pieces of pizza when I can see it all in my head?” To also honor the needs of the students who seemed to benefit less from manipulatives or representations, the teachers discussed how they could emphasize their usefulness when explaining concepts and solutions to others. Preliminary use of this strategy during small-group activities indicated that framing manipulatives as a tool for explaining one’s thinking did increase students’ willingness to use them.

Develop Families of Tasks versus Repetitive Tasks. Both the results of the

study and the feedback from the intervention classes indicate that students who prefer Intuition are more comfortable transferring knowledge from one situation to another that is similar but not identical, whereas the students who preferred Sensing sought more reinforcement. Again, during the intervention classes the students who preferred Sensing sought additional practice even after they had mastered concepts. This suggests that to meet the needs of students with either type preference, teachers might prepare families of tasks, where students can take control of how many practice problems they need to do before moving on.

This may be especially important due to the tendency of the students who

preferred Intuition to make what appeared to be mistakes unrelated to lack of conceptual



understanding when they were asked to do repetitive tasks. Further, literature on characteristics of gifted children (Webb et. al., 2005) 3often include items such as

• Interest in experimenting and doing things differently • Tendency to put ideas or things together in ways that are unusual, not obvious,

and creative (divergent thinking) • Learn basic skills more quickly, with less practice (p. 4). This study suggests that these are characteristics of students who prefer Intuition

rather than giftedness, regardless of their mastery of mathematical concepts. If instruction and intervention strategies assume that students who need extra support do not wish to show uniqueness and need more practice with basic skills, struggling students who prefer Intuition may be at a disadvantage in those settings.

Question to Probe Mathematical Understanding. Repeated viewing of several

films revealed that the facilitator had not noted that in order to complete Task 3, several students had literally drawn bigger squares to “grow” their shape whereas the task was designed to reveal whether or not they understood equivalent fractions in just this manner. The representations the students produced technically showed a correct solution, but the students had arrived at their answer without understanding the concept. This points out the need for carefully designed tasks and follow-up questions to assess student mastery of concepts. In this case, the facilitator could have asked a question such as, “What other denominators could be used to describe your solution?” or “What fraction names could you give each part now?” For Task 6, the question, “What fraction name would you give the green part,” which was in the script but which the facilitator in some cases forgot to ask (see the Student 45 excerpt) was designed to test conceptual understanding.

Another implication can be derived from the frequent use of trial and error, or

“guess and check” by students who preferred both Extraversion and Sensing. While they arrived at correct solutions, they often did not have a clear understanding of the concept or process they used. Conceivably, assessment methods that do not ensure that students have built efficient processes as well as a sound understanding of the concept could put students with a preference for Sensing at a disadvantage; without this understanding it would be more difficult for them to apply the concepts in new situations.

Given the tendency for students who preferred Intuition to make mistakes with

colors or counting squares, assessment and questioning strategies also need to differentiate between incorrect answers resulting from misunderstandings versus inadvertent mistakes. The tendency for assessments to concentrate on right answers,

3 Webb, J. T., Amend, E. R., Webb, N. E., Goerss, J., Beljan, P., Olenchak, F. R. (2005): Misdiagnoses and dual diagnoses of gifted children and adults. Scottsdale, AZ: Great Potential Press, Inc.



which allows for efficient grading, can result in the conceptual knowledge of some students being underestimated.

In summary, while more research is needed, using the type dichotomies of

Extraversion and Introversion, and Sensing and Intuition, as a framework for understanding differences in how students approach mathematical tasks may prove useful in creating equitable instruction. The framework could take differentiation beyond changing content, process and product to supporting students’ core needs for feedback, representations, pathways to conceptual understanding, and practice tasks that lead to confidence and mastery of mathematical concepts and processes.

Copyright 2007 by Jane Kise. All rights reserved. Reprinted from Differentiation Through Personality Types: A Framework for Instruction, Assessment and Classroom Management by Jane A. G. Kise, pages 174-175. Thousand Oaks, CA: Corwin Press, www.corwinpress.com. Reproduction authorized only for the school site or nonprofit organization that has purchased this book. 10

Handout 1 Extraversion or Introversion:

Where do you get your Energy? Extraversion (E) Gaining energy through action and interaction, the outside world Introversion (I) Gaining energy through reflection and solitude, the inner world

EXTRAVERSION INTROVERSION Thinks out loud (talks!) Thinks inside (quiet!) Likes to work in groups Likes to work alone or with close friend Likes noise Dislikes noise Prefers to speak Prefers to read or write Lots going on One activity at a time Says what they’re thinking Keeps thoughts inside

Circle which describes you best: E (Extraversion) I (Introversion) U (Not Sure)

Sensing or Intuition: What Information gets your attention?

Sensing (S) First paying attention to what is, to information you can gather through your five

senses—the facts. INtuition (N) First paying attention to what could be, to hunches, connections or imagination—a

sixth sense.

SENSING INTUITION Likes facts and concrete things Likes ideas & imagination Experience first Explanation first Sees the trees—details Sees the forest—big ideas Wants clear expectations Wants room to roam Step-by-step learning Random learning Practical, common sense New insights

Circle which describes you best: S (Sensing) N (INtuition) U (Not Sure)

Note: There are two other preference pairs: Thinking and Feeling (two approaches to decision-making) and Judging and Perceiving (two approaches to our desire for openness or closure in work and life in general). These also play a role in mathematics instruction but were not as significant in the film project research and could not be covered in the short presentation framework of this conference.

Copyright 2007 by Jane Kise. All rights reserved. Reprinted from Differentiation Through Personality Types: A Framework for Instruction, Assessment and Classroom Management by Jane A. G. Kise, pages 174-175. Thousand Oaks, CA: Corwin Press, www.corwinpress.com. Reproduction authorized only for the school site or nonprofit organization that has purchased this book. 11

Handout 2 Learning Styles

While all eight psychological preferences are important for teaching and learning, concentrating on Extraversion, Introversion, Sensing and Intuition ensures that students have the energy and the information they need to learn.

If teachers begin by planning in their normal style, or through the curriculum, and then adjust for the quadrant whose needs are least met, they will in fact meet the needs of all four quadrants. This creates a manageable process for differentiation. As teacher type knowledge increases, they can make further adjustments.

IS: Let me know what to do

• Set clear expectations and goals • Show me examples • Provide the steps in the process • Answer my questions as I have them • Give me time to think • Let me work with and memorize facts • Avoid too many surprises • Build on what I know

• Let me know along the way if I’m doing things right

• Connect content with past efforts and experiences

IN: Let me follow my own lead

• Let me delve deep into things that interest me • Avoid repetition and routine • Let me figure out for myself how to do things • Give me choices • Listen to my ideas • Let me learn independently • Let me start with my imagination • Help me bring what I envision into reality • Give free rein to my creativity and curiosity

• Provide references for me to build my own knowledge base

ES: Let me do something

• Start with hands-on activities • Give me steps I can follow • Let me think out loud and work with others • Tell me why I’m learning something • Give me chances to talk and move • Set a realistic deadline • Give me examples • Provide clear expectations • Go light on theory • Let me use my experience and skills

EN: Let me lead as I learn

• Start with the big picture, not the details • Let me dream big without penalties • Let me find a new way to do it • Let me interact with others • Give me choices • Keep changing what we do

• Let me teach or tell someone what I’ve learned

• Le me be in charge of something • Let me talk or work in groups • Let me come up with my own ideas

Copyright 2012 by Jane Kise. For workshop use only. Please contact [email protected] for permission to reproduce. 12

Handout 3 Learning Style Characteristics Seen In Math Task Films

For this project, 100 6th grade students were filmed while completing fractions tasks. All had access to markers, plain paper, 1” square paper, isometric graph paper, and color tiles. The following information summarizes the differences in ways students with different learning styles approached the same tasks.

Introversion and Sensing (IS) At their best:

• Asked questions to clarify understanding • Checked work before explaining answer • Applied a literal interpretation of some of the

problems, leading to quick solutions Struggles

• Were unwilling to experiment if they were uncertain

• Were easily confused by examples if they were not chosen carefully

• Hesitated before asking questions Task behaviors

• Used squares paper and markers; none used tiles unless the facilitator suggested it

• Only one used numbers to find common denominators

Introversion and Intuition (IN) At their best:

• Showed perseverance when exploring their thinking

• Looked for innovative ways to solve a problem • Applied learning from one problem to next

Struggles • Made careless mistakes that interfered with

their thinking • Trusted own hunches, seldom sought

clarification or new information Task behaviors

• Often drew shapes other than rectangles or used isometric graph paper

• One student built shapes with markers rather than the tiles

• Worked quietly for up to nine minutes on a task

• Many used numbers to find common denominators or equivalent fractions

Extraversion and Sensing (ES) At their best:

• Experimented with tiles and drawings to find solutions

• Quickly asked for clarifications • Clearly articulated ideas and questions

Struggles: • Asked for feedback continually • Inaccuracies in drawings affected their thinking • Struggled to transfer concepts to new problems.

For example, if they understood how to make a bigger shape including fourths, that knowledge did not transfer to the problem concerning sixths

Task behaviors: • Altered the materials to make sense of

problems (only ones who shaded tiles, divided graph squares in half, etc., to fit in thirds and sixths)

• Used trial and error without asking for help in between experiments

• None used numbers to find common denominators

Extraversion and Intuition (EN) At their best:

• Applied learning from one problem to the next • Confidently proceeded on their own

understandings Struggles

• Accuracy in counting and in explanations • Struggled to unlearn something they inferred or

a conclusion they drew • Communicated in general terms that increase

difficulty of clarifying their misunderstandings Task behaviors

• Careless mistakes; used colors that did not match problem or counted tiles and squares incorrectly

• Unaware of the denominator they were illustrating, i.e., talking about 12ths while illustrating 10ths.

• So confident in their answer that they didn’t see mistakes even while explaining their solution

• Long verbal explanations

Copyright 2012 by Jane Kise. For workshop use only. Please contact [email protected] for permission to reproduce. 13

Handout 4 Influence of Type Preferences on Students

During Instructional Intervention The following chart illustrates the different ways students with different type preferences engaged in the same activities during intervention sessions. Activity 1: Goal: Understanding the meaning of the numerator and denominator. Students built shapes with specific fractional parts, using color tiles. Activity 2: Goal: Finding equivalent fractions and adding fractions. Students, working in pairs, rolled fractions dice, recorded the fractions on individual whiteboards, modeled the equation with fraction strips, and then added the fractions. Introversion and Sensing • Activity 1: These students often asked for extra

practice. For example, after they built a 3-tile shape that was 1/3 blue, then a 4-tile shape that was 1/4 blue, many asked for additional simple tasks (1/5 blue and so on) before moving on to more complex tasks.

• Activity 2: Although they were only asked to complete and record at least 5 different dice rolls, many of these students completed 10 or more. They also kept using the fractions strips until the instructor asked them whether they could do the problems without them.

Introversion and Intuition • Activity 1: These students quickly mastered the

concept of building shapes with different fractions. Several asked to draw rather than build shapes, and also quickly moved to working independently on tasks that involved fractions with different denominators.

• Activity 2: Several students rushed through the task, getting most of the 5 required problems wrong. We then required 5 correct equations before they could move on. This seemed to motivate them to be more careful.

Extraversion and Sensing • Activity 1: These students frequently said how much

they liked working with the tiles, saying, “I can see the fractions.”

• Activity 2: These students loved the chance to sit on the floor to roll the dice, coming up with their own rules for the toss. They also enjoyed working together with the large magnetized strips on the classroom whiteboards rather than individual desk-sized sets.

Extraversion and Intuition • Activity 1: These students quickly mastered the

concept of building shapes with different fractions and enjoyed working together to solve problems that involved fractions with different denominators. They frequently miscounted their tiles or used the wrong colors.

• Activity 2: These students enjoyed rolling the dice, but made frequent mistakes until required to solve a few problems with the fractions strips.

why are these students two years behind in math?

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