Math Curriculum Design

Designed the education goals, instructions, assessments of math course (Counting and Probability) for Grade 10 to improve student’s self-regulation, with detailed lesson and units demonstration, emphasizing the self-regulated learning.

Overview

This course I designed is to embed self-regulation into math education for students in high school. To address self-regulation, I will basically focus on three parts: cognition and meta-cognition, motivation and emotion, strategy and action. The content is based on the unit of counting and probability. The whole course is about 10 hours, containing the in-class lecturing and after-class project. This poster will show how goals, assessments, instructions and evaluation are aligned and justified, based on theories and learner characteristics. In each part, the general description is followed by the specific design for two lessons about using permutation and combination to calculate probabilities of the event (40*2 mins).

Learners in the context:

According to the average score of PISA, (The Program for International Student Assessment) test in 2015, more than one in four students in Beijing-Shanghai-JiangsuGuangdong (China), Hong Kong (China) are top-performing students in mathematics, meaning that they can handle tasks that require the ability to formulate complex situations mathematically, using symbolic representations. It seems to be a very exciting result when I looked at the statistics. However, the problem hiding behind the proud grades is worthy of attention. For example, according to Li (2004), the Chinese way of math education basically contains two parts: learning-questioning and learning-reviewing, which indicates the leading roles of teachers instead of students. This pattern is not beneficial to foster student’s independent learning ability and thinking ability in the long run. A lot of effort has been endeavored into explore the problems that emerged in the instructional practice and tapping into the reasons behind the results. Self-regulation is one of the focuses that the researchers are paying attention to. According to Elliot’s research (2017), self-regulated learning contains the aspect of cognition and metacognition, learning strategy and actions, as well as motivations and emotional management. Some researchers look deeply into the relationship regulation and math achievement of high school students and find out that effort regulation can successfully predict student’s math achievement based on the framework of self-regulation theory (León et al., 2015). Another research (Kim et al., 2015) focuses on the role of effort-regulation in the motivation and engagement of student’s math learning and finds out that high-performance learners are different from the low-performance learners in the motivation of math learning.

 

Learner Profile

Relationship between students and teachers

Goals

Common Core State Standards for Mathematics

Conceptual and procedural knowledge :

“Use the rules of probability to compute probabilities of compound events in a uniform probability model:

        9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.”

Dispositional and Meta-level:

Self-regulation knowledge in the Common Core State Standards for Mathematics is elaborated in an inexplicit way. There is a chapter in the beginning part about mathematical practice.

        “1 Make sense of problems and persevere in solving them. (Motivation and Emotion)

        ...

        5 Use appropriate tools strategically. (Action and Strategy)

        6 Attend to precision. (Disposition)

        7 Look for and make use of structure.(Cognition and Metacognition)

        … ”

 

the Four Pillars of Learning from UNESCO

       “provide cognitive tools required to better comprehend the world and its complexities”

 

Goal Specification:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Cognitive conceptual knowledge:

       c.c.3.3.2.1. When two events are said to be independent of each other, what this means is that the probability that one event occurs in no                                                         way affects the probability of the other event occurring.

       c.c.3.3.2.2. P(E∪F) = P(E) + P(F) – P(E∩F).

       c.c.3.3.2.3. P(E∩F) = P(E) * P(F).

 

Cognitive procedural knowledge:

       c.p.7. IF the two events cannot happen at the same time, THEN A and B are mutually exclusive events. 

       c.p.8. IF the two events can happen at the same time:

              a. IF the one event will influence the probability of the other one, THEN they are not independent. 

              b. IF the one event will not influence the probability of the other one, THEN they are independent

       c.p.9. IF the two events are independent:

              a. IF the question is to calculate the probability of A event happening AND B event happening, THEN apply P(E∪F) = P(E) + P(F) – P(E∩F). 

              b. IF the question is to calculate the probability of A event happening OR B event happening, THEN apply P(E∩F) = P(E) * P(F).

​​​

Cognitive dispositional knowledge:

       c.d.1. Manage your emotion and embrace the failure.

       c.d.1.1. Tell yourself to be happy when fail to solve the problem, since you get a chance to nail it and find the misconceptions you have.                            That is a good experience that others might not have. 

       c.d.1.2. Persist in solving a problem even the process is better and tell yourself “I will make HUGE progress if I solve this problem.”

 

Meta conceptual knowledge:

       m.c.1. Maintain the personal knowledge map every day. 

       Add the new knowledge to the old map and mark the connection between knowledge components. 

 

Meta procedural knowledge:

       m.p.1. Apply the learning strategies to learn better.

       m.p.1.1. Apply the 5-step problem-solving strategy to solve the problem. 

              - Identify the problem type.

              - Contrast the new problem with the old one and find the similarity and difference. 

              - Find the “trigger” to transform the new problem into the one solved before. 

              - Self-check whether all the sub-problems are addressed one by one. 

              - Reflect on the problem-solving process and record the learning outcomes in the learning log. 

       Reflect on the problem-solving process and record the learning outcomes in the learning log.

Meta dispositional knowledge:

       m.d.1. When you are not motivated or have the negative emotion, you can detect that and hold the positive attitude to that. 

       m.d.2. Evaluate the learning process and progression in emotional management and self-regulation. 

       m.d.3. Learn better about your dispositions: risk-taking or conservative? 

 

Assessments

In class:

  • In-class Online test: to test whether the students master the basic knowledge for this class.

  • Game:

    • Domain knowledge: how to solve the problem using the formula of independent events.

    • Dispositional: Learn better about your dispositions: risk-taking or conservative?

After class:

  • Self-evaluation log form.

  • Extra practice if not doing well on the in-class practice

  • Maintain the knowledge map.

  • Play Yahtzee with parents and friends and trying to apply the strategies we taught today.

  • Think about the project, since today’s class is tightly related to one of the final projects.

For program-focused assessment:

  • Detect the misconceptions that the instructors never thought about. (Self-evaluation form.)

 

Instructions:

Social space

To provide the supportive learning environment and classroom climate for the student, two steps should be conducted: creating the climate and maintain the climate. Since this lesson is not in the beginning stage of the course, the poster will focus on maintaining the climate.

 

Physical space

  • Mobile desks and chairs for in-class activities.

  • One tablet for each student to do the in-class test. To guarantee the quality of learning, the students are only accessible to the practice system instead of any other apps (social media, personal email box) or any other irrelevant web pages.

 

Timeline for the lesson

Activity 1: Stating Learning Objectives (2 min)

Instruction: Direct instruction

Narration: “Today we are going to learn the lesson: Probability of the union of two events: independent events. What’s more, there will be activities in the second half of the class to play Yahtzee together! Excited, huh? But before that you will need to master the knowledge about what is independent events and how to calculate the probability of the two independent events. I believe you will be a better player after you learn the relevant knowledge!”

Slides: specific learning objectives listed.

Student’s interaction: Get excited.

Justification:

Evoke student’s excitement about the coming class.

Make the expectation clear.

Learner:

Students think math learning is boring.

 

Activity 2: Introducing the concept of independent events. (5 min)

Content: Given the example of coin tossing and dice throwing, ask a question and guide the students to think about the independent events.

Narration: “Now get a coin in your pocket, toss it now… What’s your result?... Now it again… What’s your result? … Do you think the result of the second one is influenced by the first one?... Yes, they are independent... But for the example of card drawing we talked the last lesson, they are not independent, they are dependent. Now could you please tell me some examples of independent events and dependent events in your life?”

Student’s interaction: They will gain the deeper understanding of independent events when comparing the examples of coin tossing and card drawing and giving examples of the daily events.

 

Use the question-driven method to scaffold student’s independent thinking ability.

 

Activity 3: Compare the independent events and mutually exclusive events. (5 min)

Instruction: Make analogy and contrast about the independent events and mutually exclusive events. Go through the process with students.

Student’s interaction:

Learn the similarity and differences between the two concepts. Students might use this way to reorganize the knowledge and organize the future knowledge using the same strategy.

 

Model the recommended thinking process for the students to imitate.

 

Activity 4: Calculate the probability of event A AND B happening. (5 min)

Activity 5: Calculate the probability of event A OR B happening. (5 min)

Activity 6: In-class quiz about the content just learned. (20 min)

The system will provide immediate feedback when the students submit one answer. But the feedback is not the direct answer but the hint for next step thinking. Since the students are applying the problem-solving strategy introduced in the previous course. The intelligent system will detect what is the misconception and provide individualized feedback to each student. More practice or homework will be generated based on the student’s in-class test.

 

Provide individualized instructions to increase learning efficiency.

Make students realize which part is “mastered” and which “needs extra work”.

 

Student’s interaction:

They might understand when they read the feedback. And they will get the practice exercise based on their performance.

Individualized feedback will help the students to realize the part they don’t understand and provide more feedback they really need.

 

Activity 7: Break for 10 minutes.

Chat with students about their feeling of the course.

Set proper and comfortable climates (e.g. belonging, listening and sharing).

 

Activity 8: “What’s your choice?” – play Yahtzee together.

Time length: 10 min

Instruction:

8.1. First, give some simple introduction about Yahtzee (The rules are already sent to students a week before the class and suggest them to play with their family and friends).

Cast a question: What’s your choice, if you get “12456”?

Give students 1 min to think through and make the decision to move to the area of “throw again” or “stick to the current”.

Ask the students to talk about their rationale behind the reason. The calculation process applies the formula introduced today.

Along the process, write down the rationale in the whiteboard about the thinking process. During the process of making decision, students will learn better about themselves: more advantageous (risk-taking) personality and conservative (steady) personality?

 

Student’s interaction:

Join in the in-class activities and enjoy the process while applying the formula learned today.

Use game to engage students and motivate students to learn math.

 

8.2. What’s your choice, if you get “11224” for the first round?

8.3. What’s your choice, if you get “22333” for the first round?

 

Activity 9: Wrap-up and homework

Time length: 5 min

Instruction (Direct instruction):

Review the content learned today about the definition of independent events, the formula to calculate the probability of the two events happening and apply the formula to solve problems in the in-class test and Yahtzee game.

Release the homework:

  • Self-evaluation log form.

  • Extra practice if not doing well on the in-class practice

  • Maintain the knowledge map.

  • Play Yahtzee with parents and friends and trying to apply the strategies we taught today.

  • Think about the project, since today’s class is tightly related to one of the final project.

  • Make students realize which part is “mastered” and which “needs extra work”.

  • Use game to engage students and motivate students to learn math.

  • Organize knowledge using some learning strategies and tools, like knowledge map.​

Research / Evaluation:

Implement

Triangulation: cognitive, observation, interpretation.

Fidelity Check of Teacher’s Practice:

  • Two teachers co-teaching this course will benefit the fidelity of teacher conducting the course and ensure that the teachers are following the purpose instructional and assessment design.

  • The in-class session can be videotaped and re-watch afterward to see whether the instruction keeps the fidelity and whether there are certain ways to improve the instruction.

  • the assessment system is bi-directional: the teachers can assess the learning outcome of students; at the same time, students can give feedback about the instructor’s teaching.

Fidelity Check of Student’s Participation:

  • First, the teachers can review the record that students have along their schooling to detect individual differences among the students

  • Second, the instructor can ask the students about their learning process. For the summative assessment, the pretest, mid-test and posttest contain the self-evaluation scale to assess student’s self-regulation.

  • More importantly, they might need to develop a rubric to assess the student’s self-regulation. After the qualitative data is collected, they can code the behavior according to the data together to ensure the reliability.

 

 

Impact

Research question: Whether the game feature and non-game feature will influence student’s motivation and domain knowledge mastering.

 

Independent variable: One class has Yahtzee game in the second lesson; the other one has another in-class practice session to solve more math problems about today's content.

 

Dependent variable: Improvement of domain learning and self-regulation.

 

Data collection and scoring:

Gaining scores of pretest and posttest about the domain knowledge.

Gaining scores of pre-survey and post-survey about self-regulation.

The learning curve generated by the system.

Student’s self-evaluation form.

System log data can show the student’s self-regulation behavior in an indirect way.

 

Hypothesis and related predictions:

The game group will have a better score in self-regulation especially in the motivation part; the non-game group will have better score in domain knowledge testing.

 

Sampling: randomly assigned.

Validity:

Internal validity – keep the learning time the same about the class

Conduct validity - The design to test both domain knowledge and self-regulation in self-report and behavior are proper way to test the improvement, which ensures the conduct validity.

The co-teaching mode is a good way to ensure reliability - With two instructors, one will step back to observe and document while the other takes the lead

For the non-game group, the game will be played in the happy hour in the last class session at the end of the semester to ensure they enjoy the fun that the other group had.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Big ideas:

Cognition and Metacognition:

c.m.1. Make the goals and expectation clear to make the task achievable.

c.m.2. Use the question-driven method to scaffold student’s independent thinking ability.

c.m.3. Provide individualized instructions to increase learning efficiency.

c.m.4. Make students realize which part is “mastered” and which “needs extra work”.

 

Motivation and Emotion:

m.e.1. Set proper and comfortable climates (e.g. belonging, listening and sharing).

m.e.2. Give the students freedom and autonomy to set the norm of the class.

m.4.3. Evoke student’s excitement about the coming class.

m.e.4. Use the game to engage students and motivate students to learn math.

 

Strategies and Action:

s.a.1. Organize knowledge using some learning strategies and tools, like knowledge map.

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