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Simplify each expression. If the expression does not represent a real number, say so.
− 16 - \sqrt { 16 } − 16
Find the mean, variance, and standard deviation for each data set. 232 254 264 274 287 298 312 342 398
Solve the equation by factoring.
x 2 − 6 x + 5 = 0 x^2-6 x+5=0 x 2 − 6 x + 5 = 0
The accepted symbol used to denote the slope of a line is the letter m. Investigate the origin of this symbolism. Begin by consulting a French dictionary and looking up the French word monter. Write a brief essay on your findings.
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5 Teaching Mathematics Through Problem Solving
Janet Stramel
In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)
What is a problem in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.
According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.
There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.
Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.
Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.
Teaching through problem solving focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.
Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):
- The problem has important, useful mathematics embedded in it.
- The problem requires high-level thinking and problem solving.
- The problem contributes to the conceptual development of students.
- The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
- The problem can be approached by students in multiple ways using different solution strategies.
- The problem has various solutions or allows different decisions or positions to be taken and defended.
- The problem encourages student engagement and discourse.
- The problem connects to other important mathematical ideas.
- The problem promotes the skillful use of mathematics.
- The problem provides an opportunity to practice important skills.
Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.
Key features of a good mathematics problem includes:
- It must begin where the students are mathematically.
- The feature of the problem must be the mathematics that students are to learn.
- It must require justifications and explanations for both answers and methods of solving.
Problem solving is not a neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.
But look at the b ack.
It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.
When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!
Mathematics Tasks and Activities that Promote Teaching through Problem Solving
Choosing the Right Task
Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:
- Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
- What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
- Can the activity accomplish your learning objective/goals?
Low Floor High Ceiling Tasks
By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].
The strengths of using Low Floor High Ceiling Tasks:
- Allows students to show what they can do, not what they can’t.
- Provides differentiation to all students.
- Promotes a positive classroom environment.
- Advances a growth mindset in students
- Aligns with the Standards for Mathematical Practice
Examples of some Low Floor High Ceiling Tasks can be found at the following sites:
- YouCubed – under grades choose Low Floor High Ceiling
- NRICH Creating a Low Threshold High Ceiling Classroom
- Inside Mathematics Problems of the Month
Math in 3-Acts
Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:
Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.
In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.
Act Three is the “reveal.” Students share their thinking as well as their solutions.
“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:
- Dan Meyer’s Three-Act Math Tasks
- Graham Fletcher3-Act Tasks ]
- Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete
Number Talks
Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:
- The teacher presents a problem for students to solve mentally.
- Provide adequate “ wait time .”
- The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
- For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
- Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.
“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:
- Inside Mathematics Number Talks
- Number Talks Build Numerical Reasoning
Saying “This is Easy”
“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.
When the teacher says, “this is easy,” students may think,
- “Everyone else understands and I don’t. I can’t do this!”
- Students may just give up and surrender the mathematics to their classmates.
- Students may shut down.
Instead, you and your students could say the following:
- “I think I can do this.”
- “I have an idea I want to try.”
- “I’ve seen this kind of problem before.”
Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.
Using “Worksheets”
Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?
What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.
Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.
One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”
You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can
- Provide your students a bridge between the concrete and abstract
- Serve as models that support students’ thinking
- Provide another representation
- Support student engagement
- Give students ownership of their own learning.
Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.
any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method
should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning
involves teaching a skill so that a student can later solve a story problem
when we teach students how to problem solve
teaching mathematics content through real contexts, problems, situations, and models
a mathematical activity where everyone in the group can begin and then work on at their own level of engagement
20 seconds to 2 minutes for students to make sense of questions
Mathematics Methods for Early Childhood by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.
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Initial Thoughts
Perspectives & resources, what is high-quality mathematics instruction and why is it important.
- Page 1: The Importance of High-Quality Mathematics Instruction
- Page 2: A Standards-Based Mathematics Curriculum
- Page 3: Evidence-Based Mathematics Practices
What evidence-based mathematics practices can teachers employ?
- Page 4: Explicit, Systematic Instruction
Page 5: Visual Representations
- Page 6: Schema Instruction
- Page 7: Metacognitive Strategies
- Page 8: Effective Classroom Practices
- Page 9: References & Additional Resources
- Page 10: Credits
Research Shows
- Students who use accurate visual representations are six times more likely to correctly solve mathematics problems than are students who do not use them. However, students who use inaccurate visual representations are less likely to correctly solve mathematics problems than those who do not use visual representations at all. (Boonen, van Wesel, Jolles, & van der Schoot, 2014)
- Students with a learning disability (LD) often do not create accurate visual representations or use them strategically to solve problems. Teaching students to systematically use a visual representation to solve word problems has led to substantial improvements in math achievement for students with learning disabilities. (van Garderen, Scheuermann, & Jackson, 2012; van Garderen, Scheuermann, & Poch, 2014)
- Students who use visual representations to solve word problems are more likely to solve the problems accurately. This was equally true for students who had LD, were low-achieving, or were average-achieving. (Krawec, 2014)
Visual representations are flexible; they can be used across grade levels and types of math problems. They can be used by teachers to teach mathematics facts and by students to learn mathematics content. Visual representations can take a number of forms. Click on the links below to view some of the visual representations most commonly used by teachers and students.
How does this practice align?
High-leverage practice (hlp).
- HLP15 : Provide scaffolded supports
CCSSM: Standards for Mathematical Practice
- MP1 : Make sense of problems and persevere in solving them.
Number Lines
Definition : A straight line that shows the order of and the relation between numbers.
Common Uses : addition, subtraction, counting
Strip Diagrams
Definition : A bar divided into rectangles that accurately represent quantities noted in the problem.
Common Uses : addition, fractions, proportions, ratios
Definition : Simple drawings of concrete or real items (e.g., marbles, trucks).
Common Uses : counting, addition, subtraction, multiplication, division
Graphs/Charts
Definition : Drawings that depict information using lines, shapes, and colors.
Common Uses : comparing numbers, statistics, ratios, algebra
Graphic Organizers
Definition : Visual that assists students in remembering and organizing information, as well as depicting the relationships between ideas (e.g., word webs, tables, Venn diagrams).
Common Uses : algebra, geometry
Before they can solve problems, however, students must first know what type of visual representation to create and use for a given mathematics problem. Some students—specifically, high-achieving students, gifted students—do this automatically, whereas others need to be explicitly taught how. This is especially the case for students who struggle with mathematics and those with mathematics learning disabilities. Without explicit, systematic instruction on how to create and use visual representations, these students often create visual representations that are disorganized or contain incorrect or partial information. Consider the examples below.
Elementary Example
Mrs. Aldridge ask her first-grade students to add 2 + 4 by drawing dots.
Notice that Talia gets the correct answer. However, because Colby draws his dots in haphazard fashion, he fails to count all of them and consequently arrives at the wrong solution.
High School Example
Mr. Huang asks his students to solve the following word problem:
The flagpole needs to be replaced. The school would like to replace it with the same size pole. When Juan stands 11 feet from the base of the pole, the angle of elevation from Juan’s feet to the top of the pole is 70 degrees. How tall is the pole?
Compare the drawings below created by Brody and Zoe to represent this problem. Notice that Brody drew an accurate representation and applied the correct strategy. In contrast, Zoe drew a picture with partially correct information. The 11 is in the correct place, but the 70° is not. As a result of her inaccurate representation, Zoe is unable to move forward and solve the problem. However, given an accurate representation developed by someone else, Zoe is more likely to solve the problem correctly.
Manipulatives
Some students will not be able to grasp mathematics skills and concepts using only the types of visual representations noted in the table above. Very young children and students who struggle with mathematics often require different types of visual representations known as manipulatives. These concrete, hands-on materials and objects—for example, an abacus or coins—help students to represent the mathematical idea they are trying to learn or the problem they are attempting to solve. Manipulatives can help students develop a conceptual understanding of mathematical topics. (For the purpose of this module, the term concrete objects refers to manipulatives and the term visual representations refers to schematic diagrams.)
It is important that the teacher make explicit the connection between the concrete object and the abstract concept being taught. The goal is for the student to eventually understand the concepts and procedures without the use of manipulatives. For secondary students who struggle with mathematics, teachers should show the abstract along with the concrete or visual representation and explicitly make the connection between them.
A move from concrete objects or visual representations to using abstract equations can be difficult for some students. One strategy teachers can use to help students systematically transition among concrete objects, visual representations, and abstract equations is the Concrete-Representational-Abstract (CRA) framework.
If you would like to learn more about this framework, click here.
Concrete-Representational-Abstract Framework
- Concrete —Students interact and manipulate three-dimensional objects, for example algebra tiles or other algebra manipulatives with representations of variables and units.
- Representational — Students use two-dimensional drawings to represent problems. These pictures may be presented to them by the teacher, or through the curriculum used in the class, or students may draw their own representation of the problem.
- Abstract — Students solve problems with numbers, symbols, and words without any concrete or representational assistance.
CRA is effective across all age levels and can assist students in learning concepts, procedures, and applications. When implementing each component, teachers should use explicit, systematic instruction and continually monitor student work to assess their understanding, asking them questions about their thinking and providing clarification as needed. Concrete and representational activities must reflect the actual process of solving the problem so that students are able to generalize the process to solve an abstract equation. The illustration below highlights each of these components.
For Your Information
One promising practice for moving secondary students with mathematics difficulties or disabilities from the use of manipulatives and visual representations to the abstract equation quickly is the CRA-I strategy . In this modified version of CRA, the teacher simultaneously presents the content using concrete objects, visual representations of the concrete objects, and the abstract equation. Studies have shown that this framework is effective for teaching algebra to this population of students (Strickland & Maccini, 2012; Strickland & Maccini, 2013; Strickland, 2017).
Kim Paulsen discusses the benefits of manipulatives and a number of things to keep in mind when using them (time: 2:35).
Kim Paulsen, EdD Associate Professor, Special Education Vanderbilt University
View Transcript
Transcript: Kim Paulsen, EdD
Manipulatives are a great way of helping kids understand conceptually. The use of manipulatives really helps students see that conceptually, and it clicks a little more with them. Some of the things, though, that we need to remember when we’re using manipulatives is that it is important to give students a little bit of free time when you’re using a new manipulative so that they can just explore with them. We need to have specific rules for how to use manipulatives, that they aren’t toys, that they really are learning materials, and how students pick them up, how they put them away, the right time to use them, and making sure that they’re not distracters while we’re actually doing the presentation part of the lesson. One of the important things is that we don’t want students to memorize the algorithm or the procedures while they’re using the manipulatives. It really is just to help them understand conceptually. That doesn’t mean that kids are automatically going to understand conceptually or be able to make that bridge between using the concrete manipulatives into them being able to solve the problems. For some kids, it is difficult to use the manipulatives. That’s not how they learn, and so we don’t want to force kids to have to use manipulatives if it’s not something that is helpful for them. So we have to remember that manipulatives are one way to think about teaching math.
I think part of the reason that some teachers don’t use them is because it takes a lot of time, it takes a lot of organization, and they also feel that students get too reliant on using manipulatives. One way to think about using manipulatives is that you do it a couple of lessons when you’re teaching a new concept, and then take those away so that students are able to do just the computation part of it. It is true we can’t walk around life with manipulatives in our hands. And I think one of the other reasons that a lot of schools or teachers don’t use manipulatives is because they’re very expensive. And so it’s very helpful if all of the teachers in the school can pool resources and have a manipulative room where teachers can go check out manipulatives so that it’s not so expensive. Teachers have to know how to use them, and that takes a lot of practice.
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Original research article, preservice mathematics teachers’ perceptions of mathematical problem solving and its teaching: a case from china.
- 1 School of Mathematics and Statistics, Hunan Normal University, Changsha, China
- 2 School of Mathematics, Yunnan Normal University, Kunming, China
- 3 The High School Attached to Hunan Normal University, Changsha, China
- 4 School of Mathematical Sciences, East China Normal University, Shanghai, China
- 5 Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai, China
Preservice mathematics teachers’ accurate understanding of mathematical problem solving and its teaching is key to the performance of their professional quality. This study aims to investigate preservice mathematics teachers’ understanding of problem solving and its teaching and compares it with the understanding of in-service mathematics teachers. After surveying 326 in-service mathematics teachers, this study constructs a reliable and valid tool for the cognition of mathematical problem solving and its teaching and conducts a questionnaire survey on 26 preservice mathematics teachers. Survey results reveal that preservice mathematics teachers have a good understanding of mathematical problem solving and its teaching and are more confident in the transfer value of problem solving ability. By contrast, in-service teachers are more optimistic that problem solving requires exploration, continuous thinking, and the participation of metacognition. This article concludes that preservice mathematics teachers should focus more on the initiative and creativity of students and put students at the center of education. In addition, teacher educators should provide more teaching practice opportunities for preservice teachers. The findings also show that in-service teachers’ understanding of problem solving and its teaching is inferior to that of preservice teachers on some indicators, implying the importance of post-service training for in-service teachers.
Introduction
Mathematics learning is essential in cultivating the philosophical thinking and rational spirit of students ( Jiang and Xiong, 2021a ). Many countries, including China, attach great importance to mathematics education ( Jiang and Xiong, 2021b ). Problems are the heart of mathematics. Mathematics is developed and perfected by constantly solving various problems, and the mathematician’s main reason for existence is to solve problems ( Mason, 2016 ). Therefore, mathematics consists of problems and solutions. Learning how to solve mathematical problems is at the heart of mathematics education ( Polya, 2002a ). Solving mathematical problems helps students better understand mathematics ( Munzar et al., 2021 ). To a large extent, mathematics education cultivates the problem solving abilities of students ( Polya, 1990 ). Learning to solve problems and think mathematically requires continuous reflection on the nature of this activity ( Arcavi et al., 1998 ). Therefore, the problem solving teaching ability of mathematics teachers is vital. Nowadays, the relevance of problem solving in teaching and learning mathematics has become commonplace ( Libedinsky and Soto-Andrade, 2016 ).
When teachers invite students to solve a problem, they do not know as much about how to solve the problem as many people think ( Andrews and Xenofontos, 2015 ). Improving problem solving requires focusing on some recommendations, mainly for teachers and their education ( Zimmermann, 2016 ). Teachers’ knowledge of teaching content affects their classroom practice, which involves student learning and achievement ( Peterson et al., 1989 ). Problem solving is getting from where you are to where you want to be by continuously reformulating the problem until it becomes something you can manage ( Kilpatrick, 2016 ). The cognition of mathematical problem solving affects mathematical problem solving and teaching behavior. Teachers’ beliefs about mathematics impact their teaching, and teachers with different views about mathematics teach differently ( Philipp, 2007 ; Lester and Cai, 2016 ). Teachers are central to advancing the affective atmosphere and social interaction of the class ( Pehkonen et al., 2016 ), and their beliefs have a considerable impact on the nature of classroom practice ( Kayan Fadlelmula and Cakiroglu, 2008 ). Therefore, teachers need to have a good understanding of mathematical problem solving and its teaching.
Preservice mathematics teachers are prospective teachers who will teach mathematics after graduation ( Jiang and Jiang, 2022 ). Many preservice teachers complete advanced mathematics courses with a limited interpretation of critical terms, incorrect beliefs about the nature of mathematics, and a failure to recognize that mathematics stimulates analytical thinking and creativity ( Paolucci, 2015 ). They have difficulty raising and solving problems ( Işık and Kar, 2012 ; Mallart et al., 2018 ). Teacher education can alleviate negative attitudes or beliefs about mathematics and teaching mathematics to college students preparing to become teachers ( Looney et al., 2017 ). Preservice mathematics teachers with proper training will have better problem solving and problem solving teaching performance ( Crespo and Sinclair, 2008 ; Karp, 2010 ). They need a teacher preparation program that focuses their attention on the learning of the students they are teaching ( Kilpatrick, 2016 ). They must understand mathematics, teaching, and pedagogy ( Register et al., 2022 ). Proper understanding of mathematical problem solving and its teaching is an essential part of the professional quality of preservice mathematics teachers and can effectively guide their future teaching of mathematical problem solving.
Significant advances have been made in understanding the affective, cognitive, and metacognitive aspects of problem solving in mathematics and other disciplines ( Lester and Cai, 2016 ). Research on the correlation between the use of various problem solving strategies and problem solving success has been plentiful over the last century ( Schoenfeld, 2007 ), and there have been many suggestions for teaching problem solving effectively ( Mason, 2016 ). However, empirical studies of preservice teachers’ understanding of mathematical problem solving and its teaching are still rare. The big question facing current mathematical problem solving research and teaching practice is this: How do we make meaningful problem solving a regular feature of mathematics classrooms ( Leong et al., 2016 )? It is necessary to train many outstanding mathematics teachers, and a feasible method to achieve this is to pay attention to the education of preservice teachers. We should understand the mathematical problem solving and teaching knowledge of preservice teachers. On the basis of this knowledge, we can develop educational strategies to improve their problem solving teaching skills.
Improving the mathematical problem solving teaching ability of teachers requires understanding the mathematical problem solving and teaching perception of in-service and preservice teachers. For high-quality mathematics (problem solving) teaching, preservice teachers think that “developing students’ thinking ability” and “mathematical communication ability” is more critical. By contrast, in-service teachers think “learning arrangement” and “building connections” are more important ( Clooney and Cunningham, 2017 ). Age and work experience may shape beliefs related to mathematical problem solving ( Metallidou, 2009 ). It is helpful for the training of preservice teachers to understand how in-service teachers view mathematical problem solving and its teaching. Therefore, comparing the cognition of preservice and in-service teachers toward mathematical problem solving and its teaching is necessary. The cognition of preservice teachers can be better understood by placing them in the background of in-service teachers’ perceptions. Issues such as whether their perceptions have something in common, what the differences are, how to narrow the gap, how to further optimize their perceptions, which perceptions can be optimized before they are employed, and which can only be optimized afterward are not only significant for the training of preservice teachers but also help the continuing training of in-service ones.
Mathematical problem solving teaching in China is relatively successful, and Chinese students perform very well in international mathematics competitions, such as the International Mathematical Olympiad ( Xiong and Jiang, 2021 ). A study of 495 Chinese preservice mathematics teachers showed that their beliefs about mathematics teaching are most correlated with their inquiry-based teaching practices ( Yang et al., 2020 ). However, as Cai and Nie (2007) pointed out, mathematical problem solving research in China has been much more content- and experience-based than cognitive- and empirical-based. Empirical studies of problem solving related to preservice mathematics are scarce. Therefore, China’s experience is worth examining. The present study aims to answer the following questions:
1. What do preservice mathematics teachers know about mathematical problem solving and its teaching?
2. What is the difference between preservice and in-service mathematics teachers’ cognition of mathematical problem solving and its teaching?
Establishing a model and related tools to study preservice teachers’ understanding of mathematical problem solving has important implications for future work in mathematics education and teacher education.
Literature review
Mathematical problem solving was once a hot research issue in mathematics education ( Lester, 1994 ; Schoenfeld, 2007 ). The content covered in the literature review below includes mathematical problem solving, preservice mathematics teachers, and problem solving for preservice mathematics teachers. This section expounds on the background and starting point of this study from these three aspects.
Mathematical problem solving
Problems generally refer to stimulating situations that cannot be responded to with ready-made responses and require that specific barriers be overcome between the given information and the goal. Mathematical problems can generally be divided into problems for construction and problems to prove ( Polya, 1945 ). Schoenfeld (1985) pointed out that mathematical problems are usually divided into two categories: the common practice problem and the problem that students must experience before solving.
George Polya, the founder of the mathematical problem solving theory, described problem solving as follows ( Polya, 1962 , p. v): “Solving a problem means finding a way out of a difficulty, a way around an obstacle, attaining an aim which was not immediately attainable.” In general, solving problems that require effort and exploration improves one’s thinking skills. Polya’s problem solving table provides a perspective of the problem solving process from four stages: Understanding the Problem, Devising a Plan, Carrying out the Plan , and Looking Back ( Rosiyanti et al., 2021 ). What Polya proposed was a heuristic approach, and his theories of mathematical problem solving were far-reaching.
Lester (1994) systematically reviewed the research on mathematical problem solving in terms of research content and methods from 1970 to 1984. He summarized the current state of problem solving research and recommended future studies. Schoenfeld (2013) proposed a decision theory based on his four-factor framework for mathematical problem solving. Research on mathematical problem solving is vibrant. The connotation and requirements of mathematical problem solving ability are also constantly changing. There have been many research results on the technology of mathematical problem solving, creativity in mathematical problem solving, and emotion and aesthetics in mathematical problem solving ( Amado et al., 2018 ). In terms of research methods, the research methods of mathematical problem solving include differential analysis, thinking aloud, correlation analysis, and teaching experiments ( Bao and Zhou, 2009 ).
Problem solving is a significant learning activity, but the quality of problem solving teaching needs to be improved. As Lester and Charles (1992) once pointed out, research on mathematical problem solving provides little specific information on problem solving learning. The role of the teacher in teaching is neglected, and little attention is paid to what happens in real classrooms. The research focuses on the individual, on the theory but not on the class, which is the shortcoming of current problem solving teaching research. Lester (2013) reflected on the study of mathematical problem solving teaching and made four assertions: (1) We need to rethink what mathematical problems and problem solving are. (2) We know very little about how to improve students’ metacognition through problem solving. (3) Mathematics teachers need not be problem solving experts but must be serious problem solving students. (4) Mathematical problem solving is not always a high-level cognitive activity.
It is worth mentioning that problem solving is a core activity in Chinese mathematics teaching and learning. Mathematical problem solving activities in China have the following characteristics: (1) high level of reasoning, often involving multi-step and complex formal mathematical reasoning; (2) high comprehensive knowledge, with general mathematical problems involving multiple knowledge points; (3) high operation requirements, including high symbolic calculus ability; (4) simple background, in which more emphasis is placed on the connections within mathematics; and (5) presence of various solutions and high problem solving skills. The problem solving characteristics mentioned above are affected by examinations and courses on the one hand and classroom teachings on the other, such as emphasizing “Bianshi” teaching, question-type training, and reduction methods ( Bao, 2017 ).
In conclusion, despite the many research results, we still know very little about how to develop the metacognitive abilities of students through the teaching of mathematical problem solving. The role of teachers in problem solving learning has been neglected. To improve the problem solving ability of students, we must pay attention to the part of teachers. The problem solving teaching ability of mathematics teachers is also crucial. In addition, some empirical research supported by exact facts and data is necessary to study mathematical problem solving teaching.
Preservice mathematics teachers
Increasing attention is being paid to the training of preservice mathematics teachers. Preservice teachers refer to those who want to become teachers. In this study, preservice mathematics teachers are “quasi” mathematics teachers, including undergraduate students in Mathematics education and master’s students in curriculum and teaching theory or mathematics teaching ( Tong and Yang, 2018 ; Jiang and Jiang, 2022 ). The research on preservice mathematics teachers consists of studies on the current situation, curriculum and instruction, skill training, and teaching knowledge research. Many researchers have studied the mathematical knowledge and related beliefs of preservice mathematics teachers as well as their curriculum and teaching knowledge ( Pamuk and Peker, 2009 ; Prescott et al., 2013 ; Dede and Karakus, 2014 ; Paolucci, 2015 ; Lutovac, 2019 ). Some researchers agree that preservice mathematics teachers must be better prepared for future mathematics teaching ( Land et al., 2015 ; Lau, 2021 ).
In many countries, preservice mathematics teachers are taught mathematics and mathematics pedagogy in the mathematics department and education department of their universities. The training of preservice mathematics teachers requires interdisciplinary cooperation ( Beers and Davidson, 2009 ; Hodge, 2011 ; Goos and Bennison, 2018 ). Some researchers pointed out that the teaching skills of preservice mathematics teachers need to be strengthened ( Özgen and Alkan, 2014 ; Land et al., 2015 ; Gokalp, 2016 ). Many researchers ( Llinares and Valls, 2009 ; Özmantar et al., 2010 ; Baki et al., 2011 ; Hechter et al., 2012 ; Caniglia and Meadows, 2018 ; Saralar et al., 2018 ) are also concerned about the ability of preservice mathematics teachers to use Information and Communication Technology (ICT).
Compared to their Western counterparts, Chinese preservice mathematics teachers are more familiar with traditional mathematics thinking and teaching and are not competent in TPACK ( Xiang and Ning, 2014 ). After more than 100 years of development, China’s formal preservice teacher education has formed some unique models and characteristics ( Tong and Yang, 2018 ). The training program for preservice mathematics teachers in China has two distinct features ( Li et al., 2008 ). The first is that it lays a solid mathematical foundation for normal students to have a higher mathematical literacy. The second is that it pays attention to the review and research of elementary mathematics because everyone believes that a deep understanding of elementary mathematics and solid problem solving ability are fundamental to becoming qualified middle school mathematics teachers. Under the test-oriented education tradition, qualified teachers must have high problem solving skills.
However, many questions about preservice mathematics teachers have not been effectively addressed. The study of how to train preservice mathematics teachers efficiently remains an essential topic in education worldwide.
Problem solving of preservice mathematics teacher
Although some studies do reveal the characteristics of preservice mathematics teachers in problem solving, such studies are not rich enough. Preservice mathematics teachers have many deficiencies in problem solving. In terms of problem solving skills, most preservice mathematics teachers have low problem solving skills ( Özgen and Alkan, 2012 ). For instance, regarding questioning skills, preservice mathematics teachers commit seven types of errors in asking questions about fractional splitting ( Işık and Kar, 2012 ). Some preservice mathematics teachers have difficulty asking questions about everyday life, fitting into the school curriculum at a given educational level, and posing questions that students can self-correct ( Mallart et al., 2018 ). Preservice mathematics teachers can employ problem solving strategies and problem solving, but their use of different techniques is limited ( Avcu and Avcu, 2010 ). Likewise, they have difficulty expressing operations in the mathematical language ( Özdemir and Çelik, 2021 ). Thus, they need a better understanding of problem solving and its teaching through teaching practice.
Preservice mathematics teachers have difficulty choosing specific mathematical problems, and it is rather difficult for them to find unconventional mathematical problem situations ( Temur, 2012 ). Therefore, many problems in problem solving teaching for preservice mathematics teachers need to be discussed in depth. Zsoldos-Marchis (2015) studied the effectiveness of collaborative problem solving in changing the attitudes of preservice elementary teachers toward mathematics. They found that students who used cooperative learning methods had statistically significant positive changes in their enjoyment of mathematics. The students improved their belief in the usefulness of mathematics, preferring to solve unconventional problems. Other researchers explored the learning process of preservice mathematics teachers taking part in a middle school mathematics methods curriculum, noting the need for further research on such programs ( Gómez, 2009 ). Some researchers have tried to paint a picture of problem solving in Chinese mathematics education, revealing the knowledge of Chinese preservice mathematics teachers in problem solving and teaching ( Cai and Nie, 2007 ), but the research is far from extensive. To summarize, research on problem solving and problem solving teaching for preservice mathematics teachers is necessary. Evidence from classrooms, students, and frontline teachers is significant.
Materials and methods
Research methods.
A study in Turkey used a survey research method to understand how Turkish preservice primary school mathematics teachers perceive problem solving in mathematics education ( Kayan Fadlelmula and Cakiroglu, 2008 ). The current study adopted the questionnaire survey method as it can be used to investigate the knowledge of Chinese mathematics teachers on mathematical problem solving and its teaching. First, the researchers designed a preliminary questionnaire based on the literature and then examined the problem solving and teaching knowledge of in-service mathematics teachers through the online teaching and research community (WeChat and QQ) to develop research tools and establish norms. The online survey instrument is WENJUANXING (a widely used platform for publishing online questionnaires in China). The researchers presented links to the questionnaires through WeChat and QQ, and in-service mathematics teachers willing to participate in the survey could click the links to fill out the questionnaires. The teachers were fully informed of the purpose of the study, and filling out the online questionnaire was voluntary. Data generated by in-service teachers served as the norm and could be used to develop and validate the validity of research tools. Structural modeling was used to deal with the data. After the research tools were formed, representative preservice mathematics teachers were selected to conduct a questionnaire survey. According to research ethics requirements, the questionnaire survey was conducted only after the preservice mathematics teachers signed the informed consent.
Participants
Chinese mathematics teachers attach great importance to problem solving, must solve many problems, and regard problem solving as one of the most critical teaching tasks ( Xiong and Jiang, 2021 ). This study is part of a more extensive study. In the larger study, the researchers explored how to design curriculum instruction to facilitate the development of problem solving instructional skills among preservice mathematics teachers. Therefore, it is essential to specify the sample for the study. A total of 199 in-service mathematics teachers effectively participated in the online survey, most of whom came from high schools of good quality in Guangxi, China. Therefore, they could serve as representatives of excellent teachers in the province. As they joined the online mathematics teaching and research group independently, these teachers could be considered as having a high interest in mathematics education. They already have some teaching experience and all have been engaged in the teaching of mathematical problem solving. In addition, 127 mathematics competition coaches from all over China also completed the questionnaire online. Chinese mathematics competition coaches have high problem solving skills and problem solving teaching skills.
The participants in this study came from a local key normal university in China. A total of 26 full-time first-year graduate students, 2 males and 24 females, were selected. Two majored in mathematics curriculum and teaching theory and 24 majored in mathematics teaching. They are preservice mathematics teachers and will all be engaged in mathematics teaching after graduation. Most of them have studied mathematics courses such as Mathematical Analysis, Advanced Algebra, Modern Algebra, Functions of Real Variables, Functions of Complex Variables, and Topology at the undergraduate level. They have studied practical courses such as Mathematics Instructional Design and Teaching Skills Training and have experience in micro-teaching and educational practice. However, they are still students and do not have enough teaching experience yet.
Questionnaire surveys are one of the most popular data-gathering methods in the social sciences. This study demonstrates the development and use of questionnaires according to the purpose of the study. The basic process of this study was as follows: conduct literature review→design research methods→conduct online surveys on in-service mathematics teachers→design survey tools based on online surveys→conduct surveys on preservice mathematics teachers→collect and analyze data→report results. The study first investigated the problem solving and teaching cognition of in-service mathematics teachers. The researchers developed valid questionnaires based on literature and data on in-service teachers. Then they contacted participants to ask if they would like to participate. With the participants’ consent, the researchers committed to protecting their privacy, distributed the electronic questionnaires to them, and technically assisted them in completing the questionnaires. After the participants completed the questionnaires, the researchers retrieved the data. Once the data of preservice mathematics teachers were collected, they were compared with the data of in-service mathematics teachers.
Data collection and analysis
As mentioned earlier in section “Methods,” this study collected data through the Internet and surveyed preservice mathematics teachers by using questionnaires. Therefore, the research data can reveal the knowledge of in-service and preservice teachers about mathematical problem solving and its teaching. After the original network data were collected, the data were preprocessed to obtain valid data. The researchers carried out structural equation modeling based on the valid data and then designed a survey tool for preservice mathematics teachers. Descriptive statistics were carried out on the data of both in-service and preservice mathematics teachers, and the similarities and differences between the teachers were compared. The researchers then performed inferential statistics to arrive at more general conclusions. This study used SPSS 22.0 and AMOS 22.0 to process the data. AMOS 22.0 was used to build a structural equation model for understanding mathematical problem solving and its teaching. The maximum likelihood method was used to estimate the model.
Research tools
Beliefs are part of the affective domain of an individual that influences the learning process ( Manderfeld and Siller, 2019 ). Understanding young students’ emotional factors and beliefs about mathematics is a complex task ( Giaconi et al., 2016 ), as is understanding teachers’ perceptions of mathematical problem solving and its teaching. Many research frameworks have been built on mathematical beliefs ( Hannula, 2011 , 2012 ). The teaching beliefs of mathematics teachers refer to their orientations toward teaching mathematics, which involve perspectives regarding instructional activities, the cognitive processes of students, and the purpose of mathematics ( Wang et al., 2022 ). Currently, the transmissive and constructive taxonomy of teaching beliefs is commonly used in existing studies and international assessments, such as TEDS-M ( Blömeke and Kaiser, 2014 ). Unfortunately, the framework for teachers’ perception of problem solving and its teaching is relatively rare.
In this research, understanding mathematical problem solving and its teaching refers to the knowledge and essential viewpoints about mathematical problem solving and its teaching. It is the overall reflection of individual minds on mathematical problem solving and teaching. This study characterizes preservice mathematics teachers’ cognition of mathematical problem solving and its teaching from three aspects: overall impression of mathematical problem solving, specialized knowledge, and teaching perspective. In particular, impression is the perceptual image of problem solving in the individual’s mind, specialized knowledge refers to the individual’s rational understanding of problem solving, and teaching perspective is the attitude and orientation of problem solving teaching. The preparation of measurement items is mainly based on students’ typical mathematical beliefs and Polya’s theories on mathematical problem solving and teaching ( Polya, 1945 ). It is refined and synthesized according to research needs.
Designing a questionnaire means creating valid and reliable questions that address the research objectives, placing them in proper order, and selecting an appropriate administration method. The design of the corresponding measurement items is mainly based on the five-factor cognitive framework of Schoenfeld and is refined and synthesized according to research needs ( Schoenfeld, 2016 ). For example, the five items s8–s12 in the measurement tool correspond to Schoenfeld’s five-factor cognitive framework for mathematical problem solving (the coding and content of the item will be mentioned below). Typical mathematical beliefs held by some of the students mentioned by Schoenfeld above are shown in Table 1 , and these beliefs are imperfect.
Table 1. Typical student beliefs about the nature of mathematics ( Schoenfeld, 1992 ).
Take the teaching viewpoint of mathematical problem solving as an example. Two items are set up to reflect the orientation of preservice mathematics teachers to the teaching of mathematical problem solving. The two items are “Problem solving requires independent thinking; try not to let students discuss with one another” and “Teaching students to solve problems is to tell students the solution.” Since problem solving requires independent thinking, cooperative learning promotes problem solving (teaching). Teaching students to solve problems introduces students to problem solving and inspires them to think. The formulation of the above two items is not considered the correct understanding.
Each item is set on a five-point Likert scale: strongly disagree, disagree, neutral, agree, strongly agree. There are positive and negative items. The score is based on the correctness of the options. Each option is assigned the order of 1, 2, 3, 4, 5, and each option of the reverse item is given the order of 5, 4, 3, 2, 1. The options of the positive item are assigned as 1, 2, 3, 4, and 5 in turn, and the options of the negative item are designated as 5, 4, 3, 2, and 1. The higher the score, the higher the understanding of mathematical problem solving and teaching. Of course, the so-called correct refers to the point of view with a specific basis, in line with the primary trend.
The entire measurement questionnaire initially included 20 measurement items, and after expert judgment removed 8 duplicate items, the final questionnaire consisted of 12 items. The minimum score of the questionnaire is 12 points and the maximum is 60 points. Considering the characteristics of the five-point-option Likert scale, to better analyze the data, the cognitive level is divided into four grades according to the score, of which 48–60 is graded A, 36–47 is graded B, 24–35 is graded C, and 12–23 is graded D.
According to the score characteristics, the researchers set B and above as the qualification level to present results clearly, with A set as an excellent level. Table 2 shows the number of measurement items and the coding of the three secondary indicators of understanding mathematical problem solving and its teaching.
Table 2. Problem solving and its teaching cognition measurement framework.
The items of the entire questionnaire are as follows:
• s1. Problems in problem solving refer to the exercises in textbooks, which generally have conventional and mechanized solutions.
A. Strongly disagree B. Disagree C. Neutral D. Agree E. Strongly agree
• s2. Problem solving is calculating the result according to the conventional algorithm.
• s3. The four problem solving stages are understanding the problem, making a plan, implementing the plan, and reviewing. After getting the correct answer, you don’t have to spend too much time on the retrospective stage.
• s4. Problem solving is a skill that has little meaning after a student graduates.
• s5. Problem solving requires independent thinking; try not to let students discuss with one another.
• s6. Teaching students to solve problems is to teach students ready-made solutions.
• s7. Problem solving generally requires mobilizing mathematical thinking methods through exploration, association, and reasoning.
• s8. A good problem solver must have a good knowledge structure.
• s9. Emotional attitudes have an important impact on problem solving.
• s10. Excellent problem solvers can constantly adjust their thinking to solve problems.
• s11. Good problem solvers use heuristics well and experiment with different strategies.
• s12. Excellent problem solvers have a good sense of experience.
Reliability and validity
The cognitive questionnaire for mathematical problem solving and teaching is based on literature, and its content validity has been discussed above. A total of 203 questionnaires were returned, of which 199 were valid questionnaires. Since the questionnaire has 12 items, the ratio of the number of valid questionnaires returned to the number of items in the questionnaire exceeds 10:1, which can ensure the validity of the model significance test ( Jiang et al., 2022 ).
The Cronbach’s alpha coefficient of the questionnaire is 0.770. The internal consistency is good because the questionnaire has only 12 items. The KMO value of the questionnaire is 0.798, which makes it suitable for factor analysis. From Table 3 , the average score for the test teachers (199 mathematics teachers) is 47.73; they are in the B class but very close to the A class. Their intermediate perception of problem solving and teaching is above passable and very close to good. Their minimum score is 33 (C), a failing knowledge level. Their highest score is 60, with a standard deviation of 5.356. The upper quartile is 51, the median is 48, and the lower quartile is 45. The perceptions of problem solving and teaching of half of the teachers reached an excellent level, and the understanding of problem solving and teaching of more than 75% of the teachers was above the qualified group.
Table 3. Basic statistics for the recognition of in-service teachers.
Figure 1 illustrates the frequency distribution of scores, which is approximately an inverted bell. The distribution of scores reflects the characteristics of a normal distribution, and most scored between 43–53. Although the views of in-service mathematics teachers are not necessarily correct, their knowledge of problem solving and its teaching is considered representative. Their views can represent mathematics teachers’ relatively proper understanding of problem solving and its teaching. The data of the test teachers provide a reference standard for problem solving and their teaching knowledge level, which can be compared with the subsequent analysis of preservice mathematics teachers.
Figure 1. Bar graph of in-service teacher scores.
The chi-square value of the original model after modification is 45.185, and the significance probability value is p = 0.589 > 0.05. The model is in good agreement with the sample data. Among them, RMR value is 0.027 < 0.050, GFI value is 0.966 > 0.900, AGFI value is 0.944 > 0.900, PGFI value is 0.594 > 0.500, and NFI value, RFI value, IFI value, TLI value and CFI value are all greater than 0.900. Therefore, the model is adapted.
Since factor loadings in the range of 0.30–0.40 are considered to meet the minimum requirements for explanatory structure ( Hair et al., 2009 ), as shown in Figure 2 , most factor loadings in the model are higher than 0.50. In addition, only four factor loading values are lower than 0.50 but higher than 0.40. Although factor loading values greater than 0.50 are generally considered to be of practical significance, the factor loading values of this model all reached the acceptable minimum requirements.
Figure 2. Cognitive model of problem solving and its teaching.
In the above model, the correlation between errors has practical significance because there is a correlation between the corresponding items. Routine and mechanics in item s1 have the same meaning as the regular algorithm in item s2, though the former emphasizes what a problem is. By contrast, the latter emphasizes problem solving, but the two are related. Items s3 and s4 are mainly related in that these two viewpoints are refuted by Polya, the founder of mathematical problem solving theory. The two viewpoints often appear simultaneously and are widely known. The correlation between items s9 and s12 is mainly reflected in their emphasis on non-intellectual factors.
In this study, the questionnaire survey data of 127 problem solving experts were added to the original data of 199 in-service primary and secondary school teachers. These data were obtained through questionnaires initiated by the national middle school mathematics competition coaches on WeChat and QQ groups. This study conducted a confirmatory factor analysis on the questionnaire based on these 326 data.
The results showed that the Cronbach’s alpha coefficient of the questionnaire is 0.757, and the reliability of the questionnaire is acceptable. The KMO value of the questionnaire is 0.810, and the questionnaire has good structural validity. The chi-square value of model fit obtained by maximum likelihood estimation is 63.277, and with the significance probability value p = 0.069 > 0.05, the model works the sample data. On the indicators of model adaptation, RMR value is 0.027 < 0.050, GFI value is 0.969 > 0.900, AGFI value is 0.949 > 0.900, PGFI value is 0.596 > 0.500, and NFI value (0.947), RFI value (0.928), IFI value (0.987), TLI value (0.982), and CFI value (0.987) are all greater than 0.900. Thus, all values meet the standard of model fitting.
Except for two factor loading values in the model lower than 0.5 (0.42 and 0.45), the other factor loading values are not lower than 0.5, thus meeting the minimum model adaptation requirements. Overall, the questionnaire has acceptable reliability, content validity, and construct validity and can be used to investigate the level of problem solving and teaching knowledge among preservice mathematics teachers.
The above 199 in-service primary and secondary school teachers represent general primary and secondary school mathematics teachers as well. Their data provide a reference standard for mathematical problem solving and teaching cognition for this study.
The single-sample Kolmogorov–Smirnov normality test of the scores of each item and the total score of the 199 middle school teachers in the test are shown in Table 4 . The distribution of the scores of each item and the total score of the test teachers does not obey the normal distribution. The subsequent test of the score should use a non-parametric test.
Table 4. One-sample Kolmogorov–Smirnov normality test.
Table 5 presents the cognition of the in-service teachers. The following “more than half” (coded as D) means the proportion is in the interval [50%, 60%), “majority” (coded as C) indicates the ratio is in the interval [60%, 75%), “most” (coded as B) means the proportion is in the range [75%, 90%), and “almost all” (coded as A) represents the proportion is in the interval [90%, 100%).
Table 5. Cognition of in-service teachers.
For items s1, s2, s3, s5, and s6, the majority of the test teachers reached the correct understanding level; for items s4, s9, and s12, most of the test teachers reached the right understanding level; and for items s7, s8, s10, and s11, almost all teachers tested reached the level of proper cognition.
Descriptive analysis
Twenty-six questionnaires were sent out in this study and 26 were returned. Therefore, all 26 were valid questionnaires. Recovered data were coded and analyzed using SPSS22. The Cronbach’s alpha coefficient of the questionnaire is 0.741, and considering that the questionnaire only had 12 items, the reliability of the questionnaire is acceptable. The average score of the participants is 48.08 points. The intermediate of the participants’ cognition of mathematical problem solving and teaching reached a reasonable level. The lowest score is 32, highest is 54, standard deviation is 4.30, upper quartile is 51, median is 48, and lower quartile is 47. Half of the participants understood mathematical problem solving and teaching well. More than 75% of the participants have an understanding above the qualified level and close to the excellent level. One participant has a cognition score of 32, which is not in the qualifying group.
It can be seen from Table 6 that most preservice mathematics teachers have a correct understanding of items s1, s2, s3, s5, s6, s8, and s12. Almost all preservice mathematics teachers have reached the right understanding level for s4, s7, s9, s10, and s11.
Table 6. Cognition of preservice mathematics teachers.
The specific options for preservice mathematics teachers are shown in Table 7 , which presents their views on each item. Most participants favor collaborative problem solving and believe that teaching students to solve problems is not just about giving them solutions. Most participants felt that common practice problems in textbooks have a different meaning than mathematical problems and that problem solving is not the process of practicing regular algorithms. They attach great importance to the role of problem solving retrospectives and believe that retrospectives are a crucial stage of problem solving. They think that a good knowledge structure is an essential foundation for problem solving and that proper problem solving exercises are necessary. Almost all participants believe that the problem solving process requires mathematical thinking and logical methods and that strategies are crucial in the mathematical problem solving process. They pay attention to emotion and attitude in problem solving, affirm the critical role of metacognitive monitoring and regulation in problem solving, and believe that problem solving skills still have value after graduation.
Table 7. Percentage of specific options for preservice mathematics teachers (%).
Difference analysis
The skewness of preservice mathematics teachers (−2.19) was higher than that of in-service mathematics teachers (−0.23), and their scores were skewed to the left. Their average score (48.08) was slightly higher than that of in-service mathematics teachers (47.73). Their overall awareness of mathematical problem solving and teaching was higher than that of in-service mathematics teachers. The maximum score of the preservice mathematics teachers (54) was lower than that of the in-service mathematics teachers (60), and their minimum score (32) was lower than that of the test teachers (33). Still, their scores were less volatile (standard deviation: 4.30 < 5.36). Preservice mathematics teachers (51.00, 48.00) were the same as in-service mathematics teachers (51.00, 48.00) in the upper quartile and median scores. However, in the lower quartile, their scores (47.00) were slightly higher than those of in-service mathematics teachers (45.00). Preservice mathematics teachers had a slightly better distribution of scores.
In the scores and total scores of s1, s2, s3, s4, and s9, preservice mathematics teachers were higher than in-service mathematics teachers. They scored lower than in-service mathematics teachers on other items. The scores of each item of preservice and in-service mathematics teachers were concentrated around 4 points. The scores of each item of in-service mathematics teachers were concentrated explicitly in 2, 3, 4, and 5. By contrast, the scores of each item of preservice mathematics teachers were concentrated explicitly in 3, 4, and 5 points. It can also be seen that the distribution of preservice mathematics teachers’ scores was good.
The distribution of preservice mathematics teachers’ scores did not meet the normality requirement (Kolmogorov–Smirnov normality test, p = 0.00 < 0.05). To describe the differences in cognitive level between the subjects and the test teachers in more detail, a non-parametric independent sample test (Mann–Whitney U rank-sum test) was used to analyze the issues. The score distribution of the teacher was tested, and the results are shown in Table 8 . Asymptotic significance (two-tailed) is shown in the table at a significance level of 0.05.
Table 8. Non-parametric independent samples tests (Mann–Whitney U test).
There was no statistically significant difference in total scores between preservice and in-service mathematics teachers ( p = 0.316 > 0.05). Their scores for s1, s2, s3, s5, s6, s8, s9, and s12 were not significantly different ( p = 0.484, 0.066, 0.101, 0.790, 0.857, 0.414, 0.412, 0.211). There were significant differences in their scores for s4, s7, s10, and s11 ( p = 0.030, 0.019, 0.040, 0.008).
On item s4, “Problem solving is a skill of little significance after students graduate,” preservice teachers scored significantly higher than did in-service teachers. They were more convinced that problem solving skills still work after graduation.
On item s7, “Problem solving generally requires mobilizing mathematical thinking methods to go through the process of exploration, association, and reasoning,” the scores of preservice teachers were significantly lower than those of in-service teachers, and preservice teachers failed to fully realize that mathematical problem solving is multi-dimensional mathematical thinking that requires the participation in exploration and inquiry. On item s10, “Excellent problem solvers can constantly adjust their thinking to solve problems,” preservice teachers scored significantly lower than did in-service teachers. They did not fully recognize the metacognitive monitoring and regulating effect of mathematical problem solving. On item s11, “Excellent problem solvers use heuristics well and experiment with different strategies,” preservice teachers scored significantly lower than did in-service teachers on their awareness of the use of heuristics and strategies in problem solving.
There was no significant difference between preservice and in-service teachers in their knowledge of mathematical problem solving and its teaching. Still, preservice teachers scored slightly higher on average. Preservice teachers are more aware that problem solving skills are still valid after graduation. At the same time, they know that they need to participate in mathematical thinking, actively explore, mobilize corresponding strategies, and apply metacognitive monitoring and adjustment in mathematical problem solving and its teaching.
First, this study builds a research framework, develops research tools based on existing literature, and tests the reliability and validity of the research framework and research tools. The researchers then use the developed research tools to investigate the mathematical problem solving and teaching perceptions of preservice mathematics teachers and compare the survey results with those of in-service teachers. Finally, the researchers discuss the data results.
Preservice mathematics teachers recognize that problem solving skills are transferable and remain helpful after they graduate. They believe that solving problems requires a process of exploration and effort and fully affirm the importance of emotional attitude in problem solving. They think good problem solvers can constantly adjust their thinking and direction to solve problems as well as use heuristics and other strategies. Problem solving training for preservice mathematics teachers is the core content of mathematics teacher education ( dos Santos Morais, 2020 ). This idea may benefit the good mathematical problem solving and teaching knowledge of preservice teachers. Preservice teachers have positive beliefs about solving math problems, and their views are consistent with the current movement for reform in mathematics education ( Kayan Fadlelmula and Cakiroglu, 2008 ). At the same time, the lack of awareness displayed by in-service teachers relative to preservice mathematics teachers means they need on-the-job training to avoid getting lost in their teaching practice. Some research results point out that some in-service teachers have an insufficient understanding of mathematics teaching, which negatively impact their mathematics teaching ( Setoromo and Hadebe-Ndlovu, 2020 ). The behaviors and attitudes of teachers toward teaching and learning and their knowledge base are the results of the influence of on-the-job training ( Ramatlapana, 2009 ). The participation of in-service teachers in training can facilitate their communication with their peers, help them obtain new information, and update their understanding ( Izci and Göktas, 2017 ). Teacher educators can improve the effectiveness of in-service teacher training by teaching content knowledge orientation and stimulating collaboration among teachers ( Selter et al., 2015 ).
Preservice mathematics teachers also understand the meaning of problems and problem solving, the importance of problem solving review, knowledge structure, and practical problem experience. They recognize the importance of independent thinking for problem solving but ignore the value of collaborative problem solving. Compared with in-service teachers, some preservice teachers think that teaching students to solve problems equates to telling students ready-made solutions. However, as Polya advises mathematics teachers, the best way for students to learn mathematics is to discover it independently ( Polya, 1945 ). Preservice teachers must focus more on the initiative and creativity of students and put them at the center ( Polya, 2002b ). Many teachers are usually in the early stages of their careers, teaching in ways that are not consistent with their beliefs about teaching ( Beswick, 2012 ). If they want to connect belief to practice, then they need to think about practice ( Schoenfeld, 2003 ). They can change their beliefs about mathematics and its teaching by focusing on how students learn and think about mathematics ( Hough et al., 2006 ). Teachers obtain knowledge by reflecting on the goal accomplishment, learning process, and thinking approach of the students; the efficiency of media; and the recommendation of experts ( Sudejamnong et al., 2014 ). Hence, preservice teachers need to acquire further knowledge in teaching practice. It is not enough that they know what to do but do not know why. Preservice mathematics teachers need to understand the nature of educational philosophy.
Research indicates that many preservice teachers show anxiety about teaching mathematics ( Steele et al., 2013 ) and that teacher training programs have little effect on their beliefs ( Dede and Karakus, 2014 ). However, the results of the current study show no significant difference between preservice and in-service mathematics teachers in their mathematical problem solving and teaching knowledge. Preservice teachers are more confident of the transfer value of problem solving ability, while in-service teachers are more confident that problem solving requires exploration and continuous thinking. According to Polya (2002), the founder of the problem solving theory, teaching students to solve problems introduces them to thinking. The problem solving process is a constant thinking process. Preservice teachers may need to experience more problem solving practices and problem solving teaching practices to appreciate this concept more deeply. They should have the opportunity in their studies to solve (and pose) appropriate problems of similar—sometimes also more demanding—types they use as a teacher later on at school ( Zimmermann, 2016 ). The education and professional development of preservice mathematics teachers span diverse backgrounds, each of which legitimizes different perspectives on mathematics teaching ( Ramdhany et al., 2018 ). Learning by experience is of essential importance in everyday life and the academic field ( Fritzlar, 2016 ). Preservice mathematics teachers need to learn in practice and summarize their experiences independently.
The cognitive framework for mathematical problem solving and its teaching proposed in this paper is valid. Its reliability and validity have been tested and can be used for research. The findings provide a framework for studying preservice mathematics teachers’ understanding of problem solving and its teaching. The research results confirm that preservice mathematics teachers understand mathematical problem solving and teaching well. Furthermore, knowing the similarities and differences in understanding between preservice and post-service teachers will help train preservice mathematics teachers better. The findings of this study confirm that the education of preservice mathematics teachers in China, especially at the postgraduate level, is effective in the perception of problem solving teaching ( Li et al., 2008 ). At the same time, the education of preservice teachers should strengthen teaching practices to increase their experience in dealing with practical teaching problems. Every study has its limitations, and this research is no exception. Specifically, the research framework is too simple, the items do not involve an understanding of ICT, the number of items in the questionnaire is too small, and the sample is from only one provincial key normal university. Therefore, the conclusions of this study are not generally applicable. Scientific and careful sampling can minimize the bias caused by selection ( Walters, 2021 ). In addition, teachers with different levels of professional knowledge have different pedagogical focuses on the functions and beliefs of mathematics and demonstrate different teaching methods ( Zhang, 2022 ). Drawing implications for educational practice from the comparison of preservice and in-service mathematics teacher in this study is not straightforward. Future studies should design a richer and more in-depth research framework, design survey tools with higher reliability and validity, and adopt more scientific sampling methods.
Compared with in-service mathematics teachers, preservice mathematics teachers pursuing postgraduate studies have a good understanding of mathematical problem solving and its teaching. Preservice and in-service teachers share much of the same cognition toward mathematical problem solving and its teaching. From the situation of 26 preservice mathematics teachers, it is believed that the training of China’s postgraduate-level preservice mathematics teachers has been successful. The difference is that preservice teachers acquire this knowledge through school learning while in-service teachers form more understanding through teaching practice. Moreover, the post-employment knowledge of in-service teachers may cover some of the correct knowledge when they were preservice teachers, which means that on-the-job training throughout their entire career is essential. Preservice mathematics teachers develop well in the cognition that can be established through theoretical study but are insufficient in the knowledge that can only be found through practice. Therefore, more practical opportunities must be created for them. These opportunities to practice include educational traineeships and internships but should not be limited to this, extending toward, for example, consideration of the learning opportunities ICT provides.
Data availability statement
The original contributions presented in this study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.
Ethics statement
The studies involving human participants were reviewed and approved by Human Subject Protection Committee of East China Normal University. The patients/participants provided their written informed consent to participate in this study.
Author contributions
PJ was the primary author of this research. YZ reviewed the manuscript and made suggestions for revision. YJ analyzed and processed the data for this research. BX gave the overall guidance for this research. All authors contributed to the article and approved the submitted version.
This study was mainly funded by a grant from the Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, 18dz2271000. In addition, Hunan Provincial Teaching Reform Research Project (HNJG-2022-0051) supported the cost of proofreading services in this study.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Keywords : pre-service teacher, perceptions, problem solving, mathematics teaching, in-service teacher
Citation: Jiang P, Zhang Y, Jiang Y and Xiong B (2022) Preservice mathematics teachers’ perceptions of mathematical problem solving and its teaching: A case from China. Front. Psychol. 13:998586. doi: 10.3389/fpsyg.2022.998586
Received: 20 July 2022; Accepted: 18 October 2022; Published: 03 November 2022.
Reviewed by:
Copyright © 2022 Jiang, Zhang, Jiang and Xiong. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Peijie Jiang, [email protected] ; Bin Xiong, [email protected]
This article is part of the Research Topic
Psychological Studies in the Teaching, Learning and Assessment of Mathematics
IMAGES
VIDEO
COMMENTS
Which statement about the teaching through problem solving approach is most accurate? Use of complex and non-algorithmic thinking. Selecting problem solving
2) Which statement about the teaching through problem solving approach is most accurate? A) Requires a four-step approach to problem solving.
Which statement about the teaching through problem solving approach is most accurate? A) It is closely associated with teaching for problem
2) Which statement about the teachingthroughproblem solving approach is most accurate?A) Requires a four-step approach to problem solving.
Chapter 3: Teaching Through Problem Solving Which statement best reflects the approach of teachingforproblem solving? It usually involves students
In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students.
Which statement about the teaching through problem solving approach is most ... Which is the most accurate statement regarding posing a worthwhile problem?
Shifts have been made from the teacher talking most of time to. ... To teaching about and through problem solving... identify the shifts that are occurring
Teaching students to systematically use a visual representation to solve word ... to solve word problems are more likely to solve the problems accurately.
Preservice mathematics teachers' accurate understanding of mathematical problem solving and its teaching is key to the performance of their