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Determining Students’ Understandings Through Questions

Monica Hecklinger, Heidi Heimerl, and Bettina Mileur

Masters in Math Education Candidates

School of Education

University of Alaska Southeast

ED 626

Classroom Research

November 2011

Determining Students’ Understandings Through Questions

Math is a complex subject. Throughout the years, concepts build on each other until a new concept is not understood unless the previous ones were learned. However, some teachers do not understand the effects of previous learning when teaching math lessons. Some teachers continue to teach a new concept even when the students are demonstrating that they do not understand the lesson. If the students are completely confused when a lesson is being taught, they will struggle to learn the concept, which will affect their subsequent learning.

As math teachers, we think it is important to learn what the students understand about the concepts we are teaching. In order to determine how well our students understand math lessons, we decided to examine the questions they ask during a lesson. The questions students ask help students share what they understand and what they do not understand. By examining their questions during class, we hope to learn about the concepts students are struggling with most often, whether the concepts are from previous lessons or the new concept.

Research Question and Methods

How do the questions students ask demonstrate their understanding of the math concept or topic?

In order to research this question, we decided to observe math lessons and write down the questions that students ask during the lesson. To remain unbiased, we decided to write down any question that was asked by students during the math lesson, whether it was related to math or not. This would help us identify what students are thinking during the lessons.

After the observations were conducted, we analyzed the types of questions that students asked and divided them into categories. This was our qualitative part of the research question, where we were finding out what students were showing us during the lessons. Our quantitative portion of the research was to count the number of occurrences in each category to assist us in determining where the students are having difficulties most often.

Monica Hecklinger’s Observations

Participants

Monica chose to observe three different leveled general education classrooms in the small Yupik village of Tununak. She observed a third and sixth grade Everyday Math class and a High School Algebra class. The third grade transition classroom contained students in their first year in an all-day English classroom. Previously the students were taught in Yutgen, their native language (kindergarten – third grade). There were six third grade students in that class (4 boys and 2 girls). The sixth grade class of ELL (English Language Learner) students was also taught using SIOP (Sheltered Instruction Observation Protocol) strategies. In this Everyday Math lesson, students learned how to describe a pattern using variables. There were twelve sixth grade students (7 boys and 5 girls). The High School Algebra class was virtually broadcasted from the district office, located in Bethel, via VTC to three rural villages across the Kuskokwim Delta. Students reviewed adding and subtracting polynomials and learned about multiplying and dividing monomials. There were six students at her location (4 boys and 2 girls).

Materials

Monica used the group designed Observation Checklist and a pencil for her observations.

Procedures

Monica observed the elementary classes by taking notes in the back of the room, marking her notes on the group designed Observation Checklist. In the high school virtual class, Monica participated in the lesson and helped by answering questions throughout the lesson and during work time.

Heidi Heimerl’s Observations

Participants

Heidi chose to observe two Algebra/HSGQE classes over the course of a week. The students in her class are all in Special Education, with various cognitive, behavioral, and emotional disabilities. There are 22 students in the first class, which meets during first period. Of those 22 students, 14 are in Algebra while the other 8 are in HSGQE. The class has 12 boys and 10 girls. There are 15 students in the second class, which meets during fifth period. Of those 15 students, 7 are in Algebra while the other 8 are in HSGQE. The class has 5 boys and 10 girls. Recently, there have been several suspensions in both classes, so there were generally three students out of the classroom for suspensions. There is also a low attendance rate for many students; so several students were absent on any given day. Usually, there were about 15 students present for the first class and 9 for the second class.

The students range from sophomores to fifth-year seniors. The students in HSGQE have all the necessary math credits to graduate and are taking the class so that they will gain the necessary skills they need to help them pass the Alaska High School Graduation Qualifying Exam which will allow them graduate with a diploma. Once they pass the HSGQE, they do not have to take the class, so they tend to be more attentive to the lessons when compared to the students who have not taken the HSGQE. The majority of the sophomores are in the first class and the majority of the seniors are in the second class. Therefore, the first class tends to get off task more often and they are more difficult to rein back in once off task behavior begins.

Materials

Heidi only used a lined notebook and a pencil during her observations. After the observations were made, she compared the questions raised to the group designed Observation Checklist.

Procedure

For her observations, Heidi wrote down the questions students asked as they raised them. She sat in the back of the classroom; working as usual and the students were oblivious to her note taking. She only took notes while the teacher was teaching and not during student work time. She was not able to take notes during work time because she has a group of six students she helps, or she wanders the classroom assisting students as needed. As a side note, most questions asked during work time were about the new concept or a previous concept.

Heidi planned to categorize the questions she wrote down into the template that was designed by the group while planning the research, but when she was analyzing her data, she quickly realized that the categories developed beforehand did not correctly address the questions raised by her students. Therefore, she ended up with a new set of question categories.

Bettina Mileur's Observations

Participants

Bettina teaches in a homeschool based learning center. Math is taught as a tutorial program (all school year) and session program (10-week sessions) during the school year. The data she collected was from three different groupings. The first group she collected data from was a computer lab setting of Algebra and Geometry students who meet three times a week for two-hour sessions. Nine to fifteen middle and high school students gather where tutors assist as they progress on individualized computer programs. The second class is kindergarten through third grade and each student has individualized computer work. Part of their time is spent with the teacher in group work. During the last class, numeration, operations, algebra, geometry, measurement, and probability concepts were introduced using literature and math projects. Overall, forty-three students were observed in her school’s Computer Math Lab, which serves as a support to home-schooling families that use online curriculums.

Materials

Bettina used the group designed Observation Checklist and a pencil for her observations.

Procedures

Bettina observed two middle and high school classrooms that use a math curriculum delivered solely by computers. Those programs teach, assess, administer practice problems, monitor students’ skills and advance or repeat their lesson depending on the students’ mastery of the previous lesson. While she observed these classes, she marked the types of questions students raised on the Observation Checklist.

Bettina also observed younger students, who are new to computer work. During this class, she wrote the types of questions raised on the Observation Checklist.

Results

Qualitative

Each group observed varied from the others in this research. However, the questions raised during math lessons fell into similar categories. While planning the research, a group of categories was developed and included the following:

• Asks questions that pertain to the subject (are they on task, or are they trying to get the class off task?)

• Asks questions about new concept

• Asks questions about prior concept (needed for new concept)

• Ask questions for clarification (to make sure they understand correctly)

• Asks for a hint or guidance (to get started, or as a reminder on how to start)

The question categories were developed while thinking about probable questions the students would ask. We thought about the types of questions that are typically asked during a lesson. Except for the first category, each type of question demonstrates a level of understanding needed to master a concept and to demonstrate that particular skill. In this manner, each level of learning builds a broader knowledge base and insures success for more complex math skills.

After the observations were conducted, the questions asked were placed into these categories. For the majority of the classes observed, the questions asked fit neatly into each of these categories. However, these categories were not correctly suited for Heidi's observations when she analyzed her data. The three tables in the Appendix are the tables she developed while analyzing all the questions she observed during her seven class periods (Tables 1 – 3). Upon review after creating the tables, she determined that the second table provides more useful information than the third one. She felt that Table 3 narrowed down the focus too much and it was decided that the categories from Table 2 would be used for subsequent data analysis.

Quantitative

After the questions were categorized into their respective categories, we determined the frequency of each type of question in order to determine which questions occurred most often. Because Heidi's categories ended up differing from the rest of the group, we had to choose which set of categories to use for this step. Upon farther analysis, it was determined that Heidi's questions included the original categories, but were a little more descriptive. Therefore, we decided to use the categories she developed from her analysis for this stage. These categories are:

• New Concept

• Previous Concepts

• Off Task (Worked)

• Off Task (Didn't Work)

• Slow Down/Take Notes/Help Learn

• Future/Real Life

Off task questions were made into two categories, ones that succeeded at sidetracking the class and ones that didn't succeed, because they have a major effect on the data. For the questions that did succeed at distracting the class, only the first question that originally distracted the class counted. All subsequent questions stated during the off task topics were ignored because it was determined that they were still on the topic of the class, even if it was not on the math concept. The data would have been skewed to show that the class was off task more often than they truly were if we included those questions in our results.

Our results are shown in the Appendix. The tables show the percentages of each occurrence for questions in each observation (Figures 1-4).

The results from Monica’s observations show that 75% of her High School students asked questions about the new concept while the remaining 25% of all questions were on a previous concept. She did not observe any questions from the other categories. In her sixth grade observation, 87% of questions were on the new concept, 3% were off task that didn’t work, and 10 % were to help them learn better. The remaining categories had no questions. In her third grade class, 71% of the questions were on the new concept, 7% were on the previous concept, and 21% were off task questions that didn’t work. The remaining categories had no questions.

The results from Heidi’s observations show that the first period asked 24% of the questions on the new concept, 38% on previous concepts, 8% on off task questions that worked, 19% on off task questions that didn’t work, 6% on questions to help them learn better, and 4% on questions about the future. Her fifth period class asked 39% of the questions on the new concept, 26% on previous concepts, 4% on off task questions that worked, 6% on off task questions that didn’t work, 15% on questions to help them learn, and 9% on questions about the future.

The results from Bettina’s observations show that 22% of her High School Computer classes asked questions on the new concept, 16% on previous concepts, 41% to help them learn, and 16% about the future. The remaining categories had no questions. In her K-2 computer class, 38% were on the new concept, 19% were on previous concepts, 6% on off task questions that worked, 13% to help them learn, and 23% about the future. No questions were asked for off task that didn’t work. Her K-2 Math and Literature classes had 38% on the new concept, 16% on previous concepts, 16% on off task that worked, 19% to help learn, and 28% on the future. There were no questions on off task that didn’t work.

Using the data from the tables above, we created pie charts to make it easier to compare our data (Figures 5 – 13). While creating the pie charts, we used the total percentages for each category for each class. Therefore, when a class was observed several times, all the questions observed for that class were calculated into one percentage and one graph was created. When a class was observed only once, the percentages were found for that one class and a pie graph was created. Figure 13 is a pie graph of all the classes and all the periods observed. It represents the frequency for every type of question asked. The pie graphs really helps demonstrate the differences in the frequencies between the classes and make comparing two different classes easier to understand.

Discussion

We found the results of our study interesting. The types of students we studied varied greatly, in age and culture, as well as the type of classroom environments. Due to Alaska’s vast size and diverse regions we ended up collecting data from three different types of classrooms. These schools ranged from small, rural native communities to suburban, medium sized home-schooling resource centers to large, urban Special Education classrooms. We also had everything from kindergarten to high school seniors. The method in which the classrooms were taught also varied, from a physical teacher, to videoconference, to individualized computer programs with tutors. This diversity led to interesting results.

One major result that we noticed was that the older students tended to ask questions during lessons more often than the younger students. This could be due to several issues, such as cultural differences, the types of classes, or the size of the class. However, it was clear that the younger students tended to ask questions that were directly related to the new lesson, while the older students had more to think about. The older students were more active in their education, voicing when the teacher didn't make sense or asking why something was done a certain way. They also had more distractions, such as other students or thoughts about the future. Again, this could be because of many different reasons, including disability needs and the fact that older students will be graduating soon.

The smaller amount of questions asked by Monica’s students might have something to do with language. When comparing rural, English Language Learner (ELL) students with Heidi’s urban, non-ELL students, there were fewer questions, especially when looking at the high school level. This also might have to do with the location of the teacher. In rural Alaska, the lesson was taught virtually compared to the physical presence of the teacher in the urban school. Finally, cultural difference might play a big part in this since Monica’s students were Alaska Natives, who tend to be quieter, and Heidi’s students were a large population of White and African American students, who tend to be more boisterous.

We also noticed that there was a big difference between Heidi's two classes. The first class, which is larger and has more lower grade level students, had greater instances of off task behavior and more questions about previous concepts. The two could be linked, in that the more off task the students are during class, the more questions they will have about previous lessons. This sounds obvious, but more research would have to be conducted to prove this theory. Educational settings and the amount of teaching support affected the number of responses in Bettina's studies. Off task responses were not apparent because of the one on one engagement of the individualized program and high teacher to student ratio. Parent volunteers are utilized in this program, as well.

Something else we noticed was that students tended to ask more questions about the concepts during work time. Most of the observations occurred during the lesson, and few were conducted during work time because the observer was helping students. However, it was noted by all three researchers that students were more apt at asking questions about the new concept or for clarification on old concepts during work time. This might be because students were focused on paying attention to the lesson while it was occurring, or they believed they understood it while the teacher taught. There is also the possibility that students were shy to voice their concerns in front of the class and wanted private help.

When completing further research with this topic, it would be good to look at the types of questions students ask during work time because this is where they are actually doing the work and figuring out what they know and do not know. However, our original question has helped us learn about our students and their learning. The data we have seems to suggest that younger students are learning the needed concepts from previous lessons. Therefore, when teaching, it might be a good idea to check for understanding frequently on the new concept. The older students have more to remember and think about during a lesson. This affects their learning of previous concepts, which in turn affects their learning of the new concepts. Therefore, it would be a good idea to review necessary concepts before introducing new concepts. It would also be wise to complete each step distinctively, even if it is a previous concept, so students have more experience with them. Frequent breaks and checks for understanding would also help the older students.

Appendix

|Table 1 |

|Significant Questions Students Asked During Math Lessons |

|Which axis is which and how are they numbered |

|Which class will they learn a more advanced concept |

|Where the concept is used in real life |

|How much longer until the class ends |

|How is the coordinate plane labeled |

|How to solve a previous homework problem |

|What another students is doing and they are doing that |

|When and where are parent conferences |

|What other students are doing for Halloween |

|How was a problem solved |

|When will the homework be given out |

|How do we a line |

|How were the x-values chosen |

|How did you do that |

|Have we done this before |

|Can I take a picture of the board (for note taking purposes) |

|Can you slow down |

|What do we do when solving an equation |

|Once we figure out a pattern, do we still have to plug in numbers |

|Can you turn the heater on |

|How do you plot negative points |

|How do teachers do this |

|How do classes work in college |

|What are different ways we can plot points |

|Can you turn off the lights, I can’t see the board well |

|When we do that, what are we finding out |

|Would you have the scores from HSGQE yet |

|Can you repeat that |

|Do we learn that later |

|How much math do you need to be in a certain profession |

|Can I use some notes or supplies |

|Do you think someday we will leave Earth |

|What would you do with a million dollars |

|How do you find or plot intercepts on a line |

|What’s the next step |

|Can I borrow someone’s notes |

|Can I change class periods |

|Table 2 |

|Common Themes: Questions Students Asked During Math Lessons |

|Questions about new concept |

|About coordinate planes, plotting points, and graphing lines |

|About how to solve an equation |

|About the next step |

|Questions about previous concept |

|About how to solve previous homework problem |

|About how to simplify an equation |

|Questions to get class off task |

|About remaining length of class |

|About what other students are doing |

|About Parent/Teacher conferences |

|About Halloween |

|About getting homework |

|About money and student jobs |

|Questions to help their own learning |

|About how to solve the problem again |

|About why something was done when solving a problem |

|About taking picture for notes |

|About slowing down the lesson |

|About getting comfortable in class |

|About repeating something |

|About copying notes |

|About changing periods so in a smaller class |

|Questions about future |

|About when will learn a much more advanced concept |

|About where concept is used in life |

|About college |

|About HSGQE |

|About level of math needed for certain jobs |

| Table 3 |

|Major Themes: Questions Students Asked During Math Lessons |

|Questions about math concepts |

|About coordinate planes, plotting points, graphing lines |

|About solving equations |

|About steps for following a problem |

|About is being done through steps |

|About mistakes they made |

|Questions to get off task |

|About length of class |

|About events students are doing after school |

|About other students’ visits |

|About random thoughts |

|Questions to help student learning and future |

|About slowing down |

|About repeating |

|About taking notes |

|About how will use in future |

|About going to college |

|About math needed for potential jobs |

|Figure 1 | | | | | |

|Monica Hecklinger's Observations |  |  |  |  |  |

|  |HS Day 1 |6th grade |3rd grade |HS Day 2 |Total |

|New Concept |5/6 = 83.3% |27/31 = 87.1% |10/14 = 71.42% |4/6 = 66.6% |46/57 = 80.7% |

|Previous Concepts |1/6 = 16.6% |0/31 = 0% |1/14 = 7.1% |2/6 = 33.3% |4/57 = 7% |

|Off Task Worked |0/6 = 0% |0/31 = 0% |0/14 = 0% |0/6 = 0% |0/57 = 0% |

|Off Task Didn’t Work |0/6 = 0% |1/31 = 3.22% |3/14 = 21.4% |0/6 = 0% |4/57 = 7% |

|Slow Down/Take Notes/ Help Learn |0/6 = 0% |3/31 = 9.67% |0/14 = 0% |0/6 = 0% |3/57 = 5.26% |

|Future/Real Life |0/6 = 0% |0/31 = 0% |0/14 = 0% |0/6 = 0% |0/57 = 0% |

| | | | | | |

|Figure 2 |

|Heidi Heimerl's 1st Period |  |  |  |  |

|  | | | | |

|  |Day 1 |Day 2 |Day 3 |Day 4 |Total |

|New Concept |6/19 = 31.57% |5/15 = 33.33% |7/29 = 24.14% |5/32 = 15.63$ |23/95 = 24.21% |

|Previous Concepts |2/19 = 10.53% |5/15 = 33.33% |10/29 = 34.48% |19/32 59.38%= |36/95 =37.89% |

|Off Task Worked |2/19 = 10.53% |2/15 = 13.33% |1/29 = 2.56% |3/32 = 9.38% |8/95 = 8.42% |

|Off Task Didn’t Work |6/19 = 31.58% |3/15 = 20.00% |8/29 =27.59% |1/32 =3.13% |18/95 = 18.95% |

|Slow Down/Take Notes/ Help Learn |0/19 = 0.00% |0/15 = 0.00% |2/29 = 6.90% |4/32 = 12.50% |6/95 = 6.32% |

|Future/Real Life |3/19 = 15.79% |0/15 =0.00% |1/29 =3.45% |0/32 = 0.00% |4/95 = 4.21% |

|Figure 3 | | | | |

|Heidi Heimerl's 5th Period |  |  |  |  |

|  |Day 1 |Day 2 |Day 3 |Total |

|New Concept |12/24 = 50.00% |13/30 = 43.44% |11/38 = 28.95% |36/92 = 39.13% |

|Previous Concepts |6/24 = 25.00% |7/30 = 23.33% |11/38 = 28.95% |24/92 = 26.09% |

|Off Task Worked |0/24 = 0.00% |2/30 = 6.67% |2/38 = 5.26% |4/92 = 4.345% |

|Off Task Didn’t Work |3/24 = 12.50% |1/30 = 3.33% |2/38 = 5.26% |6/92 = 6.52% |

|Slow Down/Take Notes/ Help Learn |2/24 = 8.33% |2/30 = 6.67% |10/38 = 26.32% |14/92 = 15.22% |

|Future/ Real Life |1/24 = 4.17% |5/30 = 16.67% |2/38 = 5.26% |8/92 = 8.70% |

|Figure 4 | | | | | |

|Bettina Mileur’s Observations | | | | | |

|  |HS computer |K-2 computer |K-2 Math and Literature |  |Total |

|New Concept |7/32=21.87% |20/52=38.46% |12/32=37.5% |  |39/116=33.62% |

|Previous Concepts |5/32=15.62% |10/52=19.23% |5/32=15.62% |  |20/116=17.24% |

|Off Task Worked |0/32=0% |3/52=5.7% |5/32=15.62% |  |8/116=6.8% |

|Off Task Didn’t Work |0/32=0% |0/52=0% |0/32=0% |  |0/116=0% |

|Slow Down/Take Notes/ Help Learn |13/32=40.62% |7/52=13.46% |6/32=18.75% |  |26/116=22.4% |

|Future/Real Life |5/32=15.62% |12/52=23.0% |9/32=28.1% |  |26/116=22.4% |

Figure 5: High School (Monica Hecklinger)

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Figure 6: 6th Grade (Monica Hecklinger)

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Figure 7: 3rd Grade (Monica Hecklinger)

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Figure 8: First Period (Heidi Heimerl)

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Figure 9: Fifth Period (Heidi Heimerl)

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Figure 10: Middle/HS Computer (Bettina Mileur)

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Figure 11: K-2 Computer (Bettina Mileur)

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Figure 12: K-2 Math and Literature (Bettina Mileur)

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Figure 13: Total – Everyone, Every Class

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