To see my previous posts, reflecting on sessions 1-4, click on the below links...
Session 1: http://goo.gl/zGhmxD
Session 2-3: http://goo.gl/2pjIQR
Session 4: http://goo.gl/w61q9o
Session 5 of How to Learn Math by @joboaler via Stanford University's online platform (class.stanford.edu) is called 'Conceptual Learning, Part 1: Number Sense'.
The session is possibly the most interesting yet due to the amount of classroom practice you get to see via the videos that are posted in the session. The session begins by calling on recent research to suggest that students' foundational knowledge of mathematics is what determines how successful they are in their future mathematics. Now, I've always been a believer that maths is like a set of building blocks and without the basics you don't get very far; you need a base level in order to build upon.
This is what this session was about. The session looked out how, at the basic level, students count, count on, have knowledge of number bonds or use 'number sense', the ability to break down and move around parts of numbers in order to make arithmetic easier. For example, when adding 7 and 18 you could add 18 and 2 to make 20 and then add on the remaining 5 (from the 7) to make 25. This was one of the examples I was given.
This was when the session got really great...
I was asked to then watch a few teachers going through some 'Number Talks' or 'Math Talks'. These classroom observations were fantastic in showing teachers' methods in finding out students ways of working out multiplications, additions and thinking of basic number questions. The clips showed a class of high school/undergrad students answering questions like 18 x 5, 12 x 15 and 25 x 29.
In each of the clips I was asked to note down what the teacher was doing and the 'teacher moves' they were using in the 'Number Talk'. These alone were really useful in thinking of ways to pull out answers from students, cover mistakes that crop up and get students to really think about how they're explaining their answers. I particularly found it interesting how one teacher used leading questions to drag out clarification from students as to what they were thinking. Lead ins like 'because...' and 'you knew that...' helped to get students to think of ways of explaining their previous thoughts.
What I also liked was that the teachers visualised the problems for students to give an added representation of the problems given. These were then linked to algebra and the distributive/associative laws.
A tip I picked up during the videos was that when a new idea or question was posed the teacher would get students to discuss with one another what they thought, rather than just waiting for someone to respond, as was stated - 'when ideas are complicated or new, sharing ideas can help u clarify our own thinking'.
The best thing about the 'Number Talks' is that it allows you and your peers to see the number of different ways of looking at a problem. It allows you to discuss common misconceptions and cover mistakes (learning from them in the process). For example with the first 18 x 5 question you could:
halve 18 to make 9, multiply this by 5 to give 45 and then double it to get 90
do 10 x 5 and 8 x 5 to get 50 and 40 and then add to give 90
do 20 x 5 to give 100 and then subtract 2 x 5 to give 90
you could visualise the problem in your head as being a multiplication problem set out in 'columns', going through what you carry over at each step and then coming to your answer
you could split the 18 into 6 and 3 and the 9 into 3 and 3 and then multiply these numbers together
you could draw a rectangle with length 18 and width 9, split it up as you feel best and work out individual areas before adding together
and so on and so on.
The beauty with this open approach to seeing the thinking involved is that you don't automatically see ALL possibilities, just the one that you perhaps prefer or know best. So, by getting all answers from a class you get to see other people's thinking and then can approach a new problem with an additional perspective.
I kept hearing phrases like 'number sentence' and 'friendly numbers'. These may be terms they use in the USA more often they we do in the UK, or perhaps they're used in the primary setting more than secondary but I can't say I've come across them myself, until now!
For clarity, a number sentence is a way of working out a problem, so for the problem where students were given a 'dot card' and asked how many dots were on it they were asked to say how they approached the problem. One student said they saw one row of 3, then a row of 2 and then another row of 3 and a final row of 2. Their number sentence would then be 3 + 2 + 3 + 2 = 10.
A 'friendly number' is a number that is 'nicer' to count with, like 10 and 5 and students try to break larger numbers down to these 'friendly numbers' to make the addition/multiplication/division etc easier to do. So for example 16-13 could be 10-10 and then 6-3 to give 3, rather than counting backwards, which requires a more difficult skill.
I continue to really enjoy the course:
it's making me think about the types of tasks I want to focus more heavily on this year
it's getting me to think about the language I use in class
it's getting me to think about the questions I pose in class
it's getting me to think about the messages my classroom can give students
and ultimately it's getting me to think more about how my students learn maths.
The videos are fantastic, the resources and references you can read through the online platform/download are great. I like the peer feedback facility and the short tasks that you are asked to do on there. If anyone hasn't started this already I suggest you sign up - there's plenty of time before the expiry of the course at the end of September.
I'm already getting intrigued about the student version of the course that will be coming out and whether this will be in the same format (online, free, through stanford.edu) and how best to get my students on board with it and signed up! Hopefully more details will come available in due course...?