Warm-up Problems and Their Effect on Comprehension and Motivation in Math

Introduction

The beginning of my teaching was an exciting and educational time for me. I learned and discovered many things, but one of the biggest struggles I saw with myself and with other teachers was the start of every class. I struggled to completely engage the students while doing all of the administrative tasks that were assigned to me as well as making sure the students had completed their homework. Even if I completed the tasks quickly and could start class at a reasonable time I still wasted some time. With only a few wasted minutes it was still difficult for me to turn the students from what they had been doing previously into what I wanted them to be doing. To combat this problem, I started giving students problems or thought provoking questions on the board to start the period. I soon realized that these questions helped keep students engaged with mathematics while I completed these tasks, and it turned them from what they were doing into the world of mathematics. As I started to implement these questions on a daily basis, I began to wonder what type of questions were the most beneficial. Also, how other teachers were effectively using the beginning of their class periods. This led to my desire to research and study warm-ups. Included in that would be researching different types of warm-ups and there overall effects on students motivation and understanding in the mathematics classroom.

Problem of Practice

As more and more items vie for our students' attention it is vital now more than ever teachers have the ability to engage their students and maximize their time with them. Mathematics continuously builds upon itself so it is important that students retain and be able to use their prior knowledge along with arriving motivated to the classroom everyday. But, because students' attention is demanded by so many things this is becoming harder and harder to do. Also, grading standards are about to change with the implementation of common core so teachers will have a new group of standards to try and achieve in the classroom. Teachers do not always know a student's background, therefore all we can do is maximize their time and learning while they are in the classroom. We do not have any control over what they should have learned in previous years, but we do have control over the now. Presently aiding them in fixing their mathematics problems and helping them learn the material in our classroom. Using warm-ups everyday in the classroom has the ability to help with all of these things. According to Williams (1984) "a teacher's success - a good lesson - depends on a warm-up activity."

The questions for this study are:

1. How do different types of bell ringers affect student understanding and motivation in math?
2. Do students view bell ringers as helpful to increasing their understanding?
3. What type of bell-ringer increases comprehension the most?

Literature Review

Many teachers and organizations stress the importance of the beginning of each lesson across all content areas. One of the ways in which teachers find success is in using warm-ups. The use of warm-ups are not prevalent in mathematical research so I looked to warm-ups used in other subject areas especially the sciences since those subjects have similar ideas to mathematics. I found articles on four ideas; the benefits of using warm-ups in the classroom, qualities of great warm-ups, effective implementation, and incorporating mathematical reasoning games.

Benefits of Warm-ups

One thing that stood out in the literature was no matter what the purpose of a warm-up is in the classroom many of them still have the same benefits. They do not just have comprehension benefits but also motivational and administrative as well. That does not mean warm-ups can teach themselves and a teacher can leave the students alone during this time, but it does allow time for the teacher to work without directly instructing the students. Jones, (2007) in his article that focuses on the benefits of different types of warm-ups and warm-ups as a whole, provides a list of benefits he found in his classroom for warm-ups if used correctly and on a consistent basis in the classroom. The first point that is made is warm-ups allow students to get into a daily routine and understand the teacher's expectations of them. Research has proven that when students become part of a routine they tend to be more comfortable and therefore better students. Also, throughout research if reasonable expectations are given to the students then they also tend to become better students. Warm-ups help facilitate high expectations from students by setting the tone for a class. Next, warm-ups "capture the student's attention and start them thinking." Students come to classrooms from a variety of different subjects and disciplines. Therefore, it is important to focus their mind on learning mathematics and transition from what they were thinking about when they walked in my classroom. Through this capturing of the students attention discipline problems are decreased. Due to the fact they are engaged to what is going on in the classroom instead of things happening outside of it. Finally warm-ups allow teachers time to complete all of the tasks which are mandated, examples include: taking attendance and checking homework while the students are working and learning mathematics.

Through new education programs such as Common Core there is a push towards the higher levels of Bloom's taxonomy. This is a movement towards a deeper understanding and higher comprehension for students. Bell-work when created and implemented correctly can, "require students to synthesize ideas, find personal relevance, and create products that can be shared and defended with peers [to] encourage students to work at the deepest levels of Bloom's taxonomy," according to Slater (2006). The deepest levels of Bloom's taxonomy are synthesis and evaluation. Slater in his article focuses on why the first few minutes of class are so important and how warm-ups can be used to maximize this time. Therefore, using a warm-up not only benefits the student's learning, but follows the desired outcomes for teachers in the classroom. Another benefit stated by Slater is that many teachers travel to different rooms throughout the day. Using a warm-up allows the teacher to set up the classroom in which they will be teaching to their liking while also engaging the students.

Qualities of Great Warm-ups

Warm-ups have all of these benefits, but how are they achieved? Like many other things when not created and facilitated correctly warm-ups cannot have the desired effect to a highly stimulated generation. But when created and facilitated correctly warm-ups have the benefits previously mentioned. Williams, (1984) in his article focuses on warm-ups supplementing the lesson and connecting to the formal part of the lessons. The first of three keys to an effective mathematics warm-up is the warm-up should be connected to the actual lesson in some way. Therefore warm-ups should not be used for ACT prep or other similar ideas but should be used to complement your lesson. Next, warm-ups should last between five to eight minutes. This is important because a warm-up shorter than five minutes does not seem important or relevant to the students. A warm-up lasting over eight minutes takes time away from the actual lesson. Again, the idea of the warm-up complementing the actual lesson arises. The final key to a successful warm-up is the warm-up needs to increase a student's motivation in learning mathematics. This follows the idea that students are coming to a classroom from all different places and they need to understand why mathematics is essential and/or relevant to them.

Hegna, (1997) article focuses on the importance of the first couple minutes of class and how when used correctly warm-ups effect permeates throughout the whole lesson. This particular article had some additional thoughts to what makes a good warm-up. Hegna found that the beginnings of his classes were wasted for a majority of reasons. He started to target that time and implemented changes to try and correct these problems. He believes that starting as soon as a student walks in is very important because the first minutes of class are the most important. Therefore not a single initial minute of class time should be wasted. This also allows for the students to value the first few minutes of a class making them less likely to be late which is another benefit of a warm-up used correctly. Another point is waiting to perform the administrative tasks until all of the students are working on their warm-up. This allows a teacher to be engaged with the students and answer any quick questions they have. If a teacher were to give the warm-up then go straight to their administrative tasks then it would seem as if the warm-up were busy work. If the teacher waits a couple of minutes to start their teacher related duties this shows the students that the warm-up is a learning tool for them. Hegna provided projects for students who did not complete the warm-up during the beginning of class to encourage accountability in working on them. Students seem to work harder on items that we put value on. This can be done by grading or looking at so it is important to build accountability in working the warm-ups each time allowing for the students to again value warm-ups and therefore the beginning of class.

Warm-ups to Review Material

Warm-ups can be used for a variety of different reasons. One of the ways I am going to use them in my classroom is for reviewing previous material. This will be review material in the subject I am teaching and also material covered in previous mathematics classes. Wilburne and Peterson, (2007) whose article discusses the need for a more organized lesson to keep the students' engaged and help support their learning, makes the point that reviewing previous material, either the current mathematics subject or another one, allows a teacher to do three things. They found these ideas because they saw a need for a more organized lesson. Therefore, they implemented a before, during, and after format in their classroom lesson plan. The first thing a warm-up allows is activating the prior knowledge needed for the day's lesson. The next is supplementing the knowledge that should have been previously learned, so they can understand today's lesson. Finally, it allows the teacher to see if the students have the prior knowledge to learn new material. An example of this would be if students do not understand how to graph lines, then, it would be impossible to teach them linear programming. It would be important to either supplement their knowledge on graphing lines or to go back and teach this subject to avoid wasting a whole day trying to learn something that they do not have the tools to learn.

Other ideas involving reviewing previous material can be found in an article by McDonald (2010) in which she states that warm-ups can allow time for student's to ask questions about the previous material. McDonald, whose article purpose was to encourage teachers to implement warm-ups in the classroom but also includes ideas on how warm-ups could be used ineffectively, studied four eighth grade science teachers all of which did warm-ups. Three of the four used textbook suggested warm-ups and one of them did warm-ups involving relevancy to the real-world, common misconceptions, and questions from the students. Ways in which the teacher studied reviewed material while creating relevancy was to ask the students to write opinion questions relating to the last topic. For example, in a mathematics class a possible way to incorporate this idea would be asking the students their opinion on what type of savings account would be the best investment if they were studying compound interest and continuous compound interest. McDonald also states that teachers should not rush to help students when reviewing previous material, but rather should allow their students to work through the problem or to ask for assistance from other students and allow them to work collaboratively. This allows the students to challenge themselves and talk through the solutions with other students. The conclusion McDonald came to was warm-ups used to review still need to contain all of the same principles that make good warm-ups which was established in the previous section.

McDonald touched on using collaboration as part of reviewing previous material where Gobbo (2004) discussed thoroughly why collaboration is important when reviewing material. Gobbo, in his article which focuses on the positive benefits of a review over the previous lessons material before the lesson begins, used collaboration every day in his class to review material. He discussed how reviewing for the first few minutes of a class allows students to complete three items: First, it allows them to compare their notes with that of other students. This would be helpful because it allows students who were not there or could not comprehend their notes to work on them and therefore be ready for the day's lesson. Next, it "re-establishes key ideas and allows students to elaborate on them". The more times information goes through a person's brain and the more connections they can make to it, so the more likely they are to remember it. Finally, a warm-up used as a review allows the students to share ways they remember the material with other students. Examples could be mnemonic devices, original examples, and visual versions of the material. The final point stated by Gobbo was students learn in different ways, so when they share this information with each other they present it differently. This allows for differentiation from the teacher for another style of learner.

This differentiation comes back to the article by Wilburne and Peterson in which they state the reviewing previous knowledge allows the teacher to find misconceptions and supplement prior knowledge. Once the misconceptions are found then the teacher has the ability to differentiate instruction for each individual student. Being that the teacher understands where and what the student is struggling with.

Mathematical Games and Reasoning

Another category of warm-ups is mathematical games and puzzles. There is a great deal of research on games and puzzles in the classroom. This is a part of my action research, but it is not the main concept that I want to focus on. Therefore, the articles I selected about games and puzzles were very broad. The first point I found is the use of fun activities captures the student's attention at the beginning of a class. According to Romano, (2011) whose article focuses on helping new teachers use the first few minutes of a class period, games and puzzles help the students turn from passive observers to participators in their learning of the material. I would assume this will help with their motivation in the classroom which will have its own set of benefits. These benefits include "build[ing] positive attitudes, self-esteem, enhanc[ing] motivation, promoting mathematical learning, social interaction, mathematical discussion, and risk taking," according to Bragg (2006). Bragg, whose article was written to discuss the importance of games in the teaching of mathematics, used calculator games to explore misconceptions in her classroom. These benefits go along with the benefits of being a correctly created and implemented warm-up which were established in the previous section. Bragg also discusses not using games as a reward but as a part of the lesson. This is how I will be implementing them because warm-ups are the beginning part of a lesson. Also, in this unit I will be using puzzles as the warm-up. Logic puzzles according to Wanko (2009) "develop additional problem-solving strategies while practicing deductive reasoning. Wanko's article discusses how logic puzzles can be used to help students' development of proof readiness. My hope is through the logic puzzles and games the students will utilize their critical thinking and problem solving skills. Furthermore, the students will hopefully enjoy working them therefore receiving all of the benefits mentioned above.

Literature Review Summary

Warm-ups throughout research in other subject areas have been shown to be a great tool in motivating and aiding in student comprehension. There are many types and many ideas of what makes a great warm-up throughout research but many of the principles and ideas are the same. Warm-ups in mathematics need to incorporate different items than other subjects, but the core principles need to remain the same to receive the benefits listed above. When creating a warm-up they need to have a purpose and need to supplement the actual lesson. When creating my warm-ups I have three goals allowing me to have a purpose for the warm-up for each unit. When doing bell ringers the students can become more comfortable with the type of questions a teacher can ask and also learn to read and interpret the teacher's directions. McDonald stated that warm-ups the majority of the time should be done individually or collaboratively without the help of the teacher. This allows the students to start to become comfortable with reading something and then acting upon those directions. Warm-ups have the ability to make or break a lesson, but I know not getting the students engaged at the beginning of the period destroys the potential of the learning environment.

Methodology

My goal of this action research project was to determine the effects of warm-ups on students in the classroom and to gauge their overall feelings on how completing warm-ups effected their motivation in the mathematics classroom. To determine if warm-ups aided in comprehension for students their assessment grades and survey questions were assessed. To determine warm-ups effects on student motivation survey questions were assessed.

Setting

My action research project was conducted in a College Prep Finite Mathematics course for seniors at a high school located in east Tennessee during the spring semester of 2013. The population of the school is approximately 2,000 students, and it is located in a suburban area. The high school is located in a county in which the three biggest ethnic groups are Caucasian, African American, and Latino. The population is predominately Caucasian at 83.9%. Following that the county has an 8.7% African American population and a 3.5% Latino population.

Participants

The students who participated were in a College Prep Finite Mathematics which consisted of 25 students in which 11 were males and 14 were females. All of the students were seniors who are almost certainly going to graduate high school. Finite Mathematics was a new course in the 2012-2013 school year so all of the students were taking it for the first time. The demographics of the students in the class were 1 Hispanic male, 1 African American female, all the rest were Caucasian. The students in the study were not the most prolific students at mathematics but they were not the biggest strugglers either. Many students were told they were going to be taking this class but some others chose to take it or moved into it after the beginning of the semester so therefore there was a wide range of students in the classroom.

Data Collection 

I collected data in three different ways: student surveys, teacher reflections, and assessment grades.

Surveys. After each unit in the chapter the students completed surveys on what they thought about the warm-ups in that unit. Each survey included the following questions:

• What are your observations about the beginning of each class? (What happens? What do you do, while homework is getting checked?)

The first survey I created that the students took was after the first unit which involved no warm-ups. Questions on the survey included:

• What warm-ups they believed would have helped their understanding.
• Did the warm-up in this unit motivate you to learn or aid in their comprehension?
• The survey's I created included both rating questions and free response questions.

Teacher Reflections. These were completed four different times. My reflections included information about what the students did during the warm-up, how they transitioned to the lesson, and their participation level. This was done in a note format.

Student Assessment. After each test I recorded the students' cumulative assessment and recorded it. The first test was not used as part of the research because the first test was used so the students could become comfortable with my testing format. The next unit involved no warm-up and therefore was used as a control for the students. 19 students were present for the test in the unit with no warm-up so therefore the assessment data collected on the remaining tests for the study was done only on these 19 students. I collected their assessment grades on the units involving no warm-up, warm-ups reviewing the previous day's lesson, warm-ups involving previous mathematics concepts involved in the day's lesson, and warm-ups involving games and puzzles. This assessment data was used to see if students' comprehension of the unit was aided by the warm-ups done in this unit.

In assessing this data I used a paired t test. I used this test because I wanted to compare 2 paired groups whose population was approximately normal. I looked for normality by looking at each assessment's scores on a normal probability plot. If the normal probability plot is approximately linear then normality is a valid assumption.

In summary the data collected was qualitative and quantitative. Students' surveys included both qualitative and quantitative data while teacher observations were strictly qualitative and students' assessment results was strictly quantitative. The following table summarizes the data collection and analysis process.

Student Survey 

Do students view bell ringers as helpful to increasing their understanding?

How do different types of bell ringers affect student understanding and motivation in math?

Once at the end of each unit or four times total

Descriptive Analysis. Responses are categorized and frequency within each category was calculated.

Teacher Reflections

Do students view bell ringers as helpful to increasing their understanding?

How do different types of bell ringers affect student understanding and motivation in math?

What type of bell-ringer increases comprehension the most?

Once at the end of each unit or four times total

Typical Comments - These were "coded" by significant events or ideas for each of the different warm-ups.

Assessment Grades

What type of bell-ringer increases comprehension the most?

Once at the end of each unit or four times total.

Statistical Comparison tests

To read the results section click here

Discussion:

The goals of my action research were to determine the most effective warm-up to implement in a mathematics classroom in terms of comprehension and motivation. The study identified three different types of warm-ups and how each one affected students' comprehension and motivation in a mathematics classroom. I did not find any of the warm-ups used in the study to have implications for students' learning experiences in the classroom. In my observations I found the students believed that it did as I also did. Also it has tangible benefits for me as the educator. It allowed me to set up the classroom and do all of the management tasks that I had to do while the students were actively engaged into learning or dealing with mathematics. From a motivational aspect I believed the students were more excited to come into my classroom because of warm-ups. They really enjoyed that they got the chance to review material in the classroom as well as spend ten minutes of class playing games. On those days the students were more likely to participate in the following lesson.

There were some limitations to my study which prevented by study from being as clear as possibly. One of the limitations was the number of absences of some of the students. Some students were absent during test day and therefore had to make up the test at a later date. The next limitation was the differences in each of the units in the subject. Finite mathematics was built as a sample course and therefore the topics throughout the course are not very similar. This factor may have caused students' assessment scores to vary a bit from unit to unit, which skewed the data somewhat. The final limitation in my study was the student's participation on the warm-ups. Warm-ups where diligently worked on by many of the students but in some cases student's did not participate very well even with my walking around and encouraging them to do so.

To correct some of these limitations in future studies I will select a subject in which units are very similar in difficulty and subject matter. Also I would have collected and graded each warm-up which was done in class to increase student participation. I could not have corrected the number of absences but could have only taken data on students who were their almost everyday of the study. This would have caused my sample size to be extremely small but would have corrected some of the skewness of my data. 

This study found that incorporating warm-ups into the classroom has tangible benefits for the teacher and has motivational benefits for the student. From my perspective warm-ups used on a regular basis has the potential to increase motivation in students. The study does not show that student comprehension is increased when using warm-ups on a regular basis but it did indicate that students believe their comprehension is increased. This helps with the overall enjoyment and fulfillment of a mathematics class.

In my future as a teacher of mathematics I will always use warm-ups in my classroom. I found warm-ups allowed me time to set up my classroom while engaging the students. In addition, I observed increased enjoyment and belief of learning from the students. Therefore warm-ups have value for both the student in the classroom as well as the teacher making them a worthwhile investment.

Works Cited

Bragg, Leicha. "Hey, I'm Learning This." APMC 11.4 (2006): 4-7. Print.

Gobbo, Ken. "The Usefulness of In-Class Review." College Teaching 52.4 (2004): 142. Print.

Hegna, Gerald. "Reducing Tardies: A Possible Alternative." NASSP Bulletin 81.114 (1997): 114-116. Print.

Jones, Fred. "Warm Ups - They're Not Just for Gym." Classroom Connection NA (2007): NA. Print.

McDonald, Judith. "Warm Ups: A Brain Stretcher or a Brain Sleeper?." na 1 (2010): 1-4. Print.

McDuffie, Amy, Kay Wohlhuter, and M Breyfogle. "Tailoring Tasks to Meet Students' Needs."Mathematics Teaching in the Middle School 16.1 (2011): 550-555. Print.

Romano, Micael. "The Beauty of Bell Ringers." The Science Teacher 1234235 (2011): 14. Print.

Slater, Timothy. "The First Three Minutes ... Of Class." The Physics Teacher 44 (2006): 477-478. Print.

Wanko, Jeffrey. "Japanese Logic Puzzles & Proof." Mathematics Teacher 103.4 (2009): 266-271. Print.

Wanko, Jeffrey. "Japanese Logic Puzzles." Mathematics Teacher 103.4 (2009): 266-271. Print.

Wilburne, Jane, and Winnie Peterson. "Before-During-After: Model to Plan Effective Secondary Mathematics Lessons." The Mathematics Teacher 101.3 (207): 209-213. Print.

Williams, David. "Warm-Ups - Keys to Effective Mathematics Lessons." The Arithmetic Teacher 32.1 (1984): 40-43. Print.

Zammit, Tiffany. Warm-ups, Why? Just Tell Me the Answer!. Ann Arbor: ProQuest LLC, 2010. Print.