1. The Basics to Teaching Mathematics
1.1. All Math Falls into Two Categories
1.1.1. Computational
1.1.1.1. Ex: factoring quadratic equations with a coefficient greater than one
1.1.1.2. It’s easy to relearn providing you have a strong grounding in math reasoning.
1.1.2. Math Reasoning
1.1.2.1. ~Definition~ The application of math processes to the world around us.
1.1.2.2. It’s harder to teach
1.1.2.3. It’s what educators want students to retain even if they don’t go into mathematical fields.
1.1.2.4. “The way we teach it in the U.S. all but ensures they won’t retain it.”
1.2. What are the Five Signs You're Teaching Math Reasoning Wrong?
1.2.1. Lack of Initiative
1.2.1.1. Your students don’t self-start
1.2.1.2. You finish your lecture and immediately five hands shoot up asking for you to re-explain everything at each student’s desk
1.2.2. Lack of Perseverance
1.2.3. Lack of Retention
1.2.3.1. You find yourself having to FULLY re-explain concepts three months later
1.2.4. Aversion to Word Problems
1.2.5. Eagerness for a Formula
1.3. What are the Five Steps to Making Your Instructional Time Student-Centered?
1.3.1. Use multimedia
1.3.2. Encourage student intuition
1.3.3. Ask the shortest question you can
1.3.4. Let students build the problem
1.3.5. Be less helpful
2. Lesson Plan Examples
2.1. World Record Balloon Dog (Algebra I)
2.1.1. ~Step 1~ -Ask: -Predict what amount of time to pop all of the balloons is too low. -Predict what amount of time to pop all of the balloons is too high.
2.1.1.1. Answers may vary
2.1.2. ~Step 2~ -Watch the Video -Ask: -In the video, the dog pops 25 balloons in 5 seconds. Estimate how long it should take the dog to pop 100 balloons.
2.1.2.1. 20 seconds
2.1.3. ~Step 3~ -Reveal that the previous world was 41.67 seconds -Ask: -Should the dog beat the previous world record? -Does your estimation make sense based on the world record? -Are you adjusting your answer now that you know the dog’s world record? Will you adjust it down or up?
2.1.3.1. ~Yes ~No ~Up
2.1.4. ~Step 4~ -Watch the Video -Ask: -What is the independent variable? -What is the dependent variable? -Given that the dog pops 25 balloons in 5 seconds, how many balloons should it pop per second? -What an equation for your estimation. -What is the slope? -Plot the equation in Desmos. -Use desmos: How many balloons should be popped after 30 seconds? (adjust domain)
2.1.4.1. ~Time in seconds ~Number of balloons popped ~5 balloons ~y=5x ~5 ~Attatched links for graph answers
2.2. Pool Bounce (Geometry)
2.2.1. ~Step 1~ -Watch the Video -Ask: -Write down predictions from video: Which letter will the ball hit for each shot?
2.2.2. ~Step 2~ -Show the Image - Inform students that the angle the ball going in is equal to the angle going out -Ask: -Which shots look real and which ones look fake?
2.2.2.1. ~Answer image attached
2.2.3. ~Step 3~ -Fill out the worksheet using protractors -Ask: -How many of your answers were different from your predictions? -Have you played pool before? Do you think that helped or hurt you when predicting? Did it have any effect?
2.2.3.1. ~Answer video attached
2.2.4. ~How Could This Lesson Be Modified to Use Technology~ -I could turn this into an extra credit opportunity using Google Forms. In theory, each step would be on a page; they couldn't go on to the next page before answering the question or be able to change their answers after moving on -Using technology means it would not have to take up instructional time
2.3. Money Duck (Statistics)
2.3.1. ~Step 1~ -Watch the Video -Pause where it shows how much money can be inside the duck. -Ask: -(Each student individually) How much would you be willing to pay for the duck, assuming you don’t know how much money is inside? -Find the highest and lowest amount.
2.3.2. ~Step 2~ -Show the image -Ask: -Which distribution(s) is/are impossible?
2.3.2.1. ~B, C, and D
2.3.3. ~Step 3~ -Show the Image -Ask: -(Using context) What is each distribution showing? -What are their expected values?
2.3.3.1. ~A. $6.50 B. $30.60 C. $21.35 D. 17.20
2.3.4. ~How Could This Lesson Be Modified to Use Technology~ -Rather than using the given distributions, you could use Google Forms to collect students’ data, making the activity feel more personal. -Using Google Forms would allow you to collect the counts for how much students are willing to pay more quickly. -After collecting data, you could make dot plots/stem-and-leaf plots/histograms/box plots, find the five number summary, the probability for students to choose each amount, create a probability distribution, and discuss shape/symmetry in relation to the distribution.
2.4. Splash or Splat (Algebra II)
2.4.1. ~Step 1~ -Present the problem
2.4.1.1. You may have seen or heard about the circus act in which someone dives off a high platform into a small tub of water. Well, the interactive circus troupe has come up with a new wrinkle on this act. They have attached the actor’s platform to one of the seats on a ferris wheel so that it sticks out horizontally perpendicular to the plane of the ferris wheel. The tub of water is on a moving cart that runs along a track parallel to the plane of the ferris wheel, and passes under the end of the platform. As the ferris wheel turns, an assistant holds the diver by the ankles. The assistant must let go at exactly the right moment so that the diver will land in the moving tub of water. If you were the diver, would you want to trust your assistant by spot judgment? A slight error, and you could get a splat instead of a splash. The diver has insisted that the circus owners hire your group. You need to figure out exactly when the assistant should let go. Your analysis will be tested carefully on a dummy before the real diver.
2.4.2. ~Step 2~ -Ask: -Make a physical model of the situation in the problem using the materials provided. -What other information do you need?
2.4.3. ~Step 3~ -Give students the rest of the information -Ask: -Think to yourselves: What measurements have I given you that you didn't think about? -Change your physical models accordingly -Add measurement labels to your model -Show the students how their physical model should look
2.4.3.1. ~The ferris wheel has a radius of 25 ft ~The center of the ferris wheel is 32.5 ft off the ground ~The wheel turns at a constant speed, making a complete turn every 20 secs and turning counterclockwise ~When the cart starts moving, it is 120 ft to the left of the base of the ferris wheel ~The cart travels at a constant speed of 7.5 ft/sec ~The water level in the cart is 4 ft above the ground ~When the cart starts moving, the diver’s platform is at the 3:00 position in it’s cycle ~Students should assume that when the cart starts moving, it is immediately going at 7.5 ft/sec ~Note: We do not need to worry about gravity and the time it takes for the diver to fall
2.4.4. ~Step 4~ -Let students attempt to solve the problem using trial and error -Show students how to work through the problem using their models (math reasoning) -Show students how to set up the equation for the problem and how to solve it
2.4.5. ~How Could This Lesson Be Modified to Use Technology~ -While there are many programs that can be used to illustrate the problem (ex: turns the problem into a video showing students the exact time when the assistant should let go), many are time-consuming if students are not familier with the program. -In the Webinar, they showed the audience how to use Sketchpad. -I attached a sped-up video of myself illustrating the situation; I used the program on my laptop, but it would be much more efficienct if used on a tablet.
3. What Does it Look Like in the Classroom?
3.1. Technology
3.1.1. How to Pick the Right Programs
3.1.1.1. When looking for a program for students, you should pick one that gives more feedback than “correct/incorrect,” and the solution.
3.1.1.1.1. The feedback should be the computer “trying to understand” why the student gave that answer, and showing the student where the answer started to be incorrect and why.
3.1.1.1.2. Example of a Bad Computer Response
3.1.1.1.3. Examples of a Good Computer Responses
3.1.1.2. Many math educational computer programs do not differentiate between an “almost right” answer and a “completely wrong” answer.
3.1.1.2.1. This kind of feedback does not account for the amount of thought in the answer; It does not show the student if they are “on the right track” or not.
3.1.1.2.2. Ex: many programs say, “Incorrect answer,” when they should say “Uh Oh! I think you might have…” or “Did you mean to put…”
3.2. Real-World Applications
3.2.1. What is IM 9–12 Math™ certified by Illustrative Mathematics®? What can we learn from their curriculum?
3.2.1.1. **~Definition~** A for-profit curriculum that, “examines the structure of a lesson through the lens of the design features of the curriculum and with a focus on the philosophy and instructional shifts.”
3.2.1.2. What is good about their program/curriculum that I can implement in my classroom or find in other programs for free?
3.2.1.2.1. The scope and order of the units create a deep understanding of mathematical concepts, build fluency with procedures, and solve mathematical problems that reflect past and future experiences.
3.2.1.2.2. Math vocabulary is essential to the curriculum; Mathematical concepts are first taught using informal language, and technical terms are introduced later on after students are able to conceptualize what the words represent.
3.2.1.2.3. Class activities allow for collaboration amoung students.
3.2.1.2.4. Most of the instructional time is student-centered.
3.2.1.2.5. Formative assessments are utilized to address what students are missing, and how to fill in those gaps
3.2.1.2.6. Teacher-facing materials support teachers to develop, refine, and reflect on instructional practices.
3.2.2. David Milch, a creator of popular TV shows, swore off creating shows set in the present day.
3.2.2.1. He believed these types of shows shaped the neural pathways to expect simple problems; he called this phenomenon, “impatience with ear resolution.”
3.2.2.2. Dan Meyer explains that the solution to many problems in sitcoms are predictable from the beginning, and because the episodes are short, it does not take long for the viewer to feel validated in their prediction. The same concept is applied Mathematics. Questions that students do not find predictable are called, "Impatient problems."
3.2.2.2.1. “It creates an impatience, for example, with irresolution. And I’m doing what I can to tell stories which engage those issues in ways that can engage the imagination so that people don’t feel threatened by it.”
3.2.2.2.2. Meyer uses textbooks in America as an example. He believes they, especially the massed produced textbooks, are functionally equivalent to turning on a sitcom for the brain.
3.2.2.3. For These Types of Questions, Ask: How do we use the information given, and apply it to each question?
3.2.2.4. Here's an Example and How to Go Through it with Your Students
3.2.2.4.1. When working with students, show how to decide which information is unnecessary. ~Ask~ What do I need to know? What steps do I need to do to get the answer? What information is given in the problem that I can use to solve the steps?
3.2.2.4.2. As the teacher, you need to build comfortability around conversation in your classroom; it can not be student-centered instruction if the students won’t discuss the problems with each other
3.2.2.4.3. ~How Can the Problem Be Improved Upon~ It would be better to have a picture of a real water take or a video of someone filling one up (sped up of course) because it makes the problem feel like it can be applicable to the students’ lives
3.2.3. What's the Purpose for Teaching Math Reasoning?
3.2.3.1. The goal is to let kids figure it out using math reasoning before giving them instruction on how to write it out.
3.2.3.2. It helps kids connect to a positive experience rather than their negative perception.
3.2.3.3. ~Why is the Traditional Approach Harming Students' Mathematical Understanding~ "Regardless of which problem-solving tactic or pedagogical approach is used, students are taught that when you add fractions, the "bottoms" must match; when you multiply, they don't. Many students hear this algorithm and simply memorize it, adding it to their ever-growing list of seemingly random rules, such as when they learned to carry the 1."
3.2.3.3.1. Traditional math learning in the U.S. uses rote memorization rather than understanding each process. This makes it much more difficult to learn more complicated concepts and apply the learning to the outside world.
3.2.3.4. Examples of Problems that Promote Development of Math Reasoning
3.2.3.4.1. "If you write 1/2x + 5 = 9, many grown adults will run screaming from the room. But say to your favorite 8-year-old, "I'm thinking of a mystery number. I can't tell you what it is, but if you cut it in half and add 5, you get 9," and even before you continue to, "What's the mystery number?" that child will be off and running trying to figure it out -- and will likely solve it. She'll realize that if she had to go up 5 to get to 9, she must have been at 4 before that. And if she had to cut the number in half to get 4, she must have started with 8, the answer. It's the exact same question as that equation, but posed as a mental-math mystery number."
3.2.3.4.2. “On the notorious fraction front, Stanford professor Stephen Boyd gives a great example: He rolls three round cakes into the classroom, with one cake cut into 10 equal slices, the next cake into eight slices, and the third into six. He tells the 25 kids in the class, "Figure out how to divide these so you all get the same amount." There are multiple ways to solve this, but the students are allowed to explore -- never mind the massive dose of motivation, since they don't get to eat cake until they find the answer!”
3.2.3.4.3. “Three-dimensional solids follow a really cool pattern discovered by Euler: The number of faces (flat sides) plus the number of vertices (pointy corners) minus the number of edges always equals 2. You could write this on the blackboard and hope students remember it. Alternatively, you could hand them a stack of glow sticks and Styrofoam balls, as we do in Bedtime Math's "Crazy 8s" after-school math club, and let the kids build 3-D shapes themselves to see how the edges, faces, and corners relate to each other. Once kids derive it, they'll never have to memorize it.”
3.2.3.4.4. “In her book What's Math Got to Do with It?, Stanford professor Jo Boaler describes a classroom lesson involving fence pieces: The teacher gives 36 equal lengths to the students and tells them to fence in the biggest possible area for their cows. Right away the kids discover a truth that even grown-ups don't grasp: A line of a certain length can outline rectangles with vastly different areas. A long skinny rectangle -- let's say 17 by 1 -- has almost no area, but the closer to the perfect 6-x-6 square, the bigger the area. Eventually the kids break out of the rectangle mode of thinking and realize they should aim to make a circle, best approximated by a 36-sided polygon. By the end of that lesson, they've learned a key geometry reality and practiced their math facts, too.”