Connecting Mathematical Ideas
Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning
Heinemann Publishing ($25)
Type of Video
Mostly whole class discussion (8 out of 10), two small group interview around experience of math class (not specific mathematics content)
Length of Videos
Range from 4 to 15 minutes. majority are between 8 and 12 minutes long.
Number of Videos
Number & Quantity; Algebra; Functions; Geometry
Description of the Videos
Video 1: Border problem, 12 minutes, whole class discussion. Talk about wrong answers first, different methods for getting an answer. Lots of great talk moves turn and talk, restating, asking for understanding, having students go to board and discuss, representing student thinking, asking for referents, rephrasing, connecting equation to picture. They are edited to skip some of the conversation. Method over numerical answer.
Video 2: Border Problem 2, 12 minutes Whole class discussion. Creating a rule for the border problem. Using variables to create expressions. Functions. Introduce vocabulary when needed. Student reasoning, student discourse, defining variables.
Video 3: Convincing others 10 minutes, whole class discussion. Convincing others that your rule works for all numbers. Creating norms for discussion, reasoning, justification, evaluating the arguments of others. particular case to help you understand the general case. 2(n-1) = 2n -2
Video 4: Division of fractions 15 minutes. Making sense of division of fractions. Creating arguments and critiquing the arguments of others. Representations of division of fractions. (area model, linear model. fact family method) making sense of operations.
Video 5: Notion of proof 9 minutes What does it mean to show something is true? 2(n-1) = 2n -2 moving from specific cases to “generic cases” using specific examples to figure out why it works for all numbers, it needs to work for all numbers. You would have to explain what the variable means and why the formula works.
Video 6: Class Participation 14:24 minutes 3 minutes into it, people say they don’t want to share with the class. Teacher shares why she thinks sharing out is important. She thought reporting for your group was easier, but it actually is harder for many students. Hard to represent others’ thinking. Popsicle sticks, she explains why people think they are good, but why do people not like them. The role of right and wrong answers. Being confused is important. Trying to lower the pressure.
Video 7: Interview 1 9:17 minutes. Interview with three students about video 4 and class in general, division of fractions. They talk about what the teacher does. Talking about multiple solutions, vs just one answer and then move on. Teacher loves multiple solutions. What is the best math lesson, students talk about a good discussion where they disagree. stay on the same thing until we get it. A bad lesson - - just doing a worksheet without discussion it’s OK if you are wrong, lot of different ideas in class.
Video 8: Interview 2 4:12 minutes. Interview with three students about math class, what they like about it. Learn more by talking in groups. Teacher likes it when you make mistakes, you learn from them. Comparing teacher giving method without ever figuring why it works. Prefer "why it works" approach. You understand it better, you can figure out why you messed up. She listens and wants to hear kids’ ideas.
Video 9: Surface Area 8:22 Minutes. Whole class discussion in which students work to derive the formula for the surface area of a cylinder. Memorizing versus figuring out.
Video 10: Volume of Prisms 6:23 minutes. whole class discussion about volume of cylinder. use what we know about rectangular prisms to create formula for cylinder and other prisms. General formula, area of the base times the height.
Usefulness of the Videos
These videos in general show examples of effective discussion moves and students making arguments justifying solutions.
They also show a teacher working to establish norms around discourse and around multiple solutions, proof, and meaning making, as well as the role of wrong answers.
These videos are connected to a book that treats each video as a case study. Each chapter of the book is centered on a video, and includes background and analysis by the teacher, as well as analysis of the clip by Jo Boaler, and mathematics education researcher.
Some of the videos have transcripts connected to them on the CD.
Jo Boaler and Cath Humphreys
How have you used these videos?
What do you like or not like about these videos?
Other comments or supplimental information?