TEACHER WORKSHOP ACTIVITY - TOWERS FOUR HIGH
HOW TO BUILD MATHEMATICAL THINKING
Based on the longest study ever conducted of mathematics teaching in primary school and beyond, this workshop provides and models everything that teachers need to help teach mathematical problem solving for primary school years 2 - 6.
It models how teachers can transfer this activity direct into your own classroom with minimal effort and maximum effect.
The workshop is comprised of a short video and related, hands-on activities.
In this workshop, students engage in solving a combinations problem.
This problem is from a branch of discrete mathematics called combinatorics- which is usually taught in high school or college as part of probability.
REMINDER: Students will naturally develop their own strategies. Your role is to facilitate by questioning, not by teaching.
Teachers should not need to suggest strategies or provide answers.
Source: http://www.learner.org/workshops/pupmath/support/pupm1.pdf
Video 1. The Rutgers (SUNJ) Study - The Development of Mathematical Thinking
PROBLEM SOLVING or SOLVING PROBLEMS?
Is there a difference between the skills required for 'problem solving', compared with those commonly taught for 'solving problems'? Fern Hunt thinks so (view first 4 - 5 minutes).
TOWERS FOUR-HIGH ACTIVITY
The students started with, and repeatedly encountered the so-called Tower Problem, one variant of which is presented as a challenge in Workshop 1: Materials
Each group/participant is given a total of 100 Unifix (or similar) stacking cubes, comprised of two colours (50 cubes of each colour = total 100 cubes). Although not essential, sharing and discussing solutions will be much easier if the sets of colours handed to each group contain contrasting colours of “light cubes” and “dark cubes”.
HOW TO BUILD MATHEMATICAL THINKING - Towers - Year 4
Video 2. The Rutgers (SUNJ) Study - Towers Year4 - Mathematical Thinking
This video investigates the deeper benefits in the context of the Towers-of-four problem for a group of year 4 students.
INTRODUCTION
Based on the longest study ever conducted of mathematics teaching in primary school and beyond, this workshop provides and models everything that teachers need to help teach mathematical problem solving for primary school years 2 - 6.
It models how teachers can transfer this activity direct into your own classroom with minimal effort and maximum effect.
The workshop is comprised of a short video and related, hands-on activities.
In this workshop, students engage in solving a combinations problem.
This problem is from a branch of discrete mathematics called combinatorics- which is usually taught in high school or college as part of probability.
REMINDER: Students will naturally develop their own strategies. Your role is to facilitate by questioning, not by teaching.
Teachers should not need to suggest strategies or provide answers.
Source http://www.learner.org/workshops/pupmath/support/pupm1.pdf
BACKGROUND
A course initially designed as a professional development unit for teachers, grew into a long-term study of the mathematical development of students. Small groups of children begin to tackle combination problems such as 'Towers' and 'Shirts & Pants'. Students then progress to more complicated problems, such as 'Pizzas', 'Cups, Bowls, and Plates' and 'Towers of Hanoi'.
The initial 'Towers' activity, investigates how many different four-cube-high towers can be made by selecting from Unifix blocks of two colours. Although this problem would be at home in any high school or undergraduate probability class. The difference is that the students are given only objects, not mathematical formulas or explanations, and asked to make a prediction and then justify, prove and explain their results. Students jump on the problem with enthusiasm. Teachers/facilitators purposefully step back to “see what happens.”
Archival and contemporary videos show how the students worked toward a solution over time by devising their own way of representing and solving the problems.
TEACHER REVIEW
Take notes of what the children do; reflect—in writing—about your own role in the activity; collect and bring all of the children’s written work to the next session to share.
After seeing the teacher/students repeating the same, or similar, problems multiple times over and extended period. What can we say about the changes in the students’ methods over time?
After watching an overview of the Rutgers’ study. In what ways are the students in the study similar to your students? How are they different?
How do these activities relate to authentic, 21st Century earning?
There's a not-so-obvious connection between towers and genetics, and it's the following:
When you toss a coin, you get a sequence of heads and tails. When you look at DNA, there's a sequence of not two things – heads and tails – but four letters – As, Gs, Cs and Ts. And there are little chemical scissors, called restriction enzymes which cut DNA whenever they see particular patterns. And they're an enormously useful tool in modern molecular biology. And instead of asking the question, “How long until I see a head-tail-head?” – you can ask, “How big will the chunks be when I use a restriction enzyme which cuts whenever it sees G-A-A-G, for example? How long will those chunks be?” 1)
EXAMPLE ACTIVITY
The following activities are mostly variations on a theme (combinatronics activities). They make use of simple, low-cost resources most commonly found in school.
The Tower Problem
The students started with, and repeatedly encountered the so-called Tower Problem, one variant of which is presented as a challenge in Workshop 1:
Materials
Each group/participant is given a total of 100 Unifix (or similar) stacking cubes, comprised of two colours (50 cubes of each colour = total 100 cubes). Although not essential, sharing and discussing solutions will be much easier if the sets of colours handed to each group contain contrasting colours of “light cubes” and “dark cubes”.
'Problems'
You have two colours of Unifix® (or similar) cubes/blocks available with which to build towers.
Solve the problem below for yourself and develop a way to convince others that you have found all possible towers - that there are no more and no less:
GETTING STARTED
- Using only two colours of cubes, make as many different-looking towers as possible, each exactly four cubes high.
- Each tower must always point up, with the little knob on top.
- Write down your prediction of the number of towers you can build which are different.
- Draw a a circle around the number of towers that you have predicted.
- Start building your towers and the find a way to convince yourself and others (prove) that you have found all possible towers four cubes high and that they are all different.
WHEN YOU HAVE FINISHED
- Write down the number of towers you can build which are different.
- Draw a square around the number of towers that you found.
Open-ended versus Explicit Activities
How can these kind of open-ended activities ever help students succeed in solving the 'real' mathematical problems that they will face later in primary, high school and beyond?*
*It's an hour-long video summarising 12 years study. Feel free to spend a few minutes at start, middle and (most importantly) the last three or four minutes, through to the end.
At the end of each workshop, ask students to create new 'towers ten high', where each of the ten blocks in a tower has the same colour! This will make it easy for the next group to get started when using the cubes.
Record the date and the title 'Towers 4 high' on a clean page in your journal (write your name at the top of the page if you are sharing or not using your own journal).
You will need a total of 100 Unifix (or similar) Cubes in two colours (50 blocks of each colour) to build towers.
Your task is to make as many different-looking towers as possible, each exactly four cubes high.
A tower always points up, with the little knob on top.
Find a way to convince yourself and others (prove) that you have found all possible towers four cubes high and that you have no duplicates.
Write down your prediction of the number of towers you can build which have no duplicates.
Keep a record of your solution, including your justifications/proof, to share during the next workshop (Workshop 2).
After you have completed your solution for towers four cubes high, predict (without building the towers) the number of possible towers:
- Three cubes high and
- Five cubes high.
Why do you think your predictions are correct?
- What did you used to think?
- What do you think now?
RESOURCES + REFERENCES
- Introduction: https://youtu.be/Vj2qtZlCXRI (3:10)
- Part 1: 2J Getting Started https://youtu.be/oaO9JL-hbGo (4:00)
- …or… https://youtu.be/xdlgwVEDmB8 (3:48)
- Part 2: https://youtu.be/ECg6jNWIqBs (7:11)
- Towers Adapted for Y4: https://youtu.be/0nmzudyhE8o (5min )
- Review: https://youtu.be/Vgbo91z1u6U (7:00)
ANALYSIS - Source:
OTHER:
- Maths isn't hard - it's a language - https://www.youtube.com/watch?v=V6yixyiJcos