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About MDM4U1
MDM4U1 is a grade 12 university preparation Math course, "Mathematics of Data Management," which focuses on the study of the use of data (or information) to make decisions in situations that involve uncertainty. It, therefore, addresses two closely related concepts: probability and statistics. Probability refers to the likelihood, or chance, that an event occurs. In contrast, statistics refers to the science and art of learning from data, which consists of collecting, analyzing, interpreting, and presenting data, and then using probability as a guide to help us make an inference about the population from the sample set of data.
The Origin of Probability
Probability was first used in games of chance or gambling with dice and cards. These games started as early as 3500 B.C. in Egypt, where ancient Egyptians played a board game called Senet using counters.
Later, Gerolamo Cardano wrote the book Liber de ludo aleae (Book on Games of Chance) in 1525 and published it posthumously in 1663. Other well-known mathematicians who began the development of probability theory in the mid-1600s were Blaise Pascal and Pierre de Fermat. In 1662, the book La Logique ou l'Art de Penser (in Latin, Ars cogitandi) was published anonymously in Paris and talked about how to deal with making decisions under uncertainty. In 1713, the book Ars Conjectandi (Latin for "The Art of Conjecturing") on combinatorics and mathematical probability, written by Jacob Bernoulli, was published posthumously by his nephew, Niklaus Bernoulli.
Figure 1: Ancient Egyptians' Senet (the world's oldest known board game) in around 3500 B.C.
The Practice of Statistics
The practice of statistics became popular when the merchant, John Graunt, published Natural and Political Observations Made upon the Bills of Mortality related to statistics and applied probability in the mid-1600s in the city of London, England. Probability was then used to make inferences about the population data.
Data Management in General
In general, to practice data management in situations under uncertain conditions, students would first need to identify a question that they want to investigate. Then, they would need to design and create a statistical experiment or a biased-free survey and then define its scope (i.e., who is their target audience). After that, they would need to select a sampling method (e.g., simple random sampling, systematic sampling, stratified sampling, cluster sampling, etc.) and collect sample data from the population. Next, they would analyze the collected data (e.g., calculating the sample mean and standard deviation) and estimate the population statistic(s) (e.g., the confidence interval of the population mean or true proportion) from the sample statistic(s) using probability theory. In the end, they would make a general conclusion or inference to their question and present their findings.
Figure 2: Bills of Mortality in London, England in 1664
Summary of the Practice of Data Management:
Here are the units of learning in the MDM4U1 course:
Unit 1: Introduction to Probability
Unit 2: Permutations
Unit 3: Combinations
Unit 4: Probability Distributions for Discrete Variables
Unit 5: Organization of Data for Analysis
Unit 6: One-Variable Data Analysis
Unit 7: Probability Distributions for Continuous Variables
Unit 8: Two-Variable Data Analysis
In some units of learning in this MDM4U1 course, students will learn how to use Google Sheets (or other spreadsheet programs) to simulate a random experiment with measurable outcomes/properties. In this type of random experiment, the occurrences of outcomes involve chances and cannot be predicted with certainty. Examples of such random experiments are the flipping of a fair coin, the tossing of two unloaded dice, and the spinning of a spinner.
Students will also learn how to use Google Sheets to calculate the sum of entries in a row or along a diagonal in Pascal's triangle and to perform statistical analysis (e.g., calculating mean, median, mode, first and third quartiles, and interquartile range, estimating a line-of-best-fit, and creating a residual plot) on a given set of data that can be downloaded from the website of Statistics Canada or other sources of data. Furthermore, the students will get hands-on experience in designing a bias-free survey using Google Forms, selecting a sampling method, collecting data based on the selected sampling method, analyzing the data, and presenting the results.
Panoramic Pictures of Mr. Ho's Classroom
Student Feedback
Mr. Ho's grade 12 Math students said: "Data is the best" and "We will ace data!"
G12 Feedback from Data Students 1.movG12 Feedback from Data Students 2.movResults of a Survey on Mr. Ho's Teaching Performance
Mr. Ho's MDM4U1 students (i.e., Audrey Senthivel-Davidson, Sophia Faraone, Bowen Zhu, Jayden Breslow-Bardell, and Owen Sabol) conducted a survey that asked a sample of his students from his morning and afternoon MDM4U1 classes how they felt about Mr. Ho's teaching style (whiteboards and thinking classroom) and assessment practice in semester 1 of the 2023-24 school year. The sampling method that they used was stratified sampling with two strata, one consisting of students in the morning class and the other consisting of students in the afternoon class. The results are summarized in the following graphs:
Student-Centred Learning: Building Thinking Classrooms
Here are the photos of Mr. Ho's students actively engaged in the in-class activities in the MDM4U1 course:
Technology-Based Learning: Micro:bits in Mr. Ho's Data Management Classes
Mr. Ho attended two professional development (PD) workshops on micro:bit technology in Semester 2 of the 2024-25 school year, offered by TDSB's Teaching & Learning in Global Competencies, and had his MDM4U1 students work on the rock-paper-scissors championship game using micro:bit technology. Here is Mr. Ho's lesson plan on this micro:bit activity.
2024-25 MicroBits in Mr Ho's MDM4U1s.mp4Technology-Based Learning: Google Sheets Activities
Here are photos of students working in groups and using Google Sheets to simulate experiments in which the occurrences of outcomes involve chances and cannot be predicted with certainty:
Problem-Based Learning (Real World Problems): COVID-19 & Drug Testing + Social Injustice
Here is a video showing Mr. Ho's students who were engaged in activities that connect data management to social injustice. This lesson began with a real-life example related to COVID-19. Mr. Ho then introduced the concept of Bayes' rule, as an enrichment topic, to his students and applied it to solve a COVID-19-related problem. Then, his students worked in groups to solve a conditional probability problem related to drug testing. After that, his students used their results to discuss how drug testing might bring social injustice to people in a visible minority group in some countries. Every student in Mr. Ho's class was engaged, and learning happened throughout the entire class.
Data Management and Social Injustice.movGame-Based Learning: Poker Games
Here are the videos showing the game-based learning in Mr. Ho's MDM4U1 classes.
Game Based Learning(Wed, Oct 25, 2023).movMr. Ho's student teacher, Mr. Eric Huang, taught the MDM4U1 students how to calculate the probabilities of different types of poker hands on Wednesday, October 25th, 2023. Everyone was having fun playing the poker games in the class and calculating probabilities.
2024-25 Game-Based Learning (MDM4U1).mp4Mr. Ho and his MDM4U1 students were having a fun time calculating probabilities while playing poker games. This is game-based learning.
Inquiry-Based Learning: Investigation of a Topic Through Designing a Survey
Here is a video showing Mr. Ho's students exploring and investigating a topic of their choice, designing their own survey using Google Forms, collecting data, analyzing the results, making a conclusion about their topic, and presenting their work in a report:
Survey Design.movKinesthetic Learning: Percentile of Your Height
Here is a video showing Mr. Ho's students measuring their heights and calculating the percentile of their heights:
Height Percentile (Wednesday, Nov 29th, 2023).movPercentile of Lives
Here is a picture taken by Mr. Ho's wife in a shopping mall and analyzed by Chess.com's Live Review app. It indicates that Mr. Ho has a 99th percentile of lives.
Consolidation Tasks
Here are photos showing Mr. Ho's MDM4U1 students working on their consolidation tasks near the end of the class.
Optional Check-Your-Understanding (CYU) Questions
Here are photos showing Mr. Ho's MDM4U1 students lining up to get their optional check-your-understanding (CYU) questions to practise at home.
Students' Work at the End of Mr. Ho's Math Fusion Classroom
Here is a video showing students' work at the end of Mr. Ho's Math Fusion Classroom:
Mr Ho’s Math Fusion - Students Work.movCulminating Project: The Origin of Probability & Its Applications - Games of Chance
Mr. Ho's MDM4U1 students worked on a culminating project at the end of the course. One requirement of this culminating project was to have them design and create a game that involves chance. They needed to calculate the theoretical and experimental probabilities for the outcomes of their games, and there needed to be at least 10 players to play their games. Here are short video clips showcasing their work.
2024-25 MDM4U1 Games of Chance (Semester 2).mp4MDM4U1 Probability Project Jan 2024.movMDM4U1 Probability Project June 2024.movPage updated
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