Engineering 'aha' Moments in Algebra:
Reflection on 6 Day PD Program
Mornington Peninsula ASISTM Cluster
Victoria
Sample student and teacher work below was presented during this six day PD from MC professional learning program. The course was presented in 3 sets of two days with about a term between each set. These examples are testimony to the value of investing serious money in teachers through quality professional development executed over time.
Participants in the course were:
 Mt. Eliza Secondary College  Jodi Wilson, Maria Antoniou
 Mornington Secondary College  Bev Platt, (Robert Althorp)
 Rosebud Secondary College  Greg Muir, Enoka Ramanyake & Greg Lee
Following the sixth day of the course the organiser, Stephen Daly, Mt. Eliza Secondary College reported:
 The verbal feedback that I have received from our school has been fantastic.

Negotiating the 5 most important outcomes of the course. See below. 
Sandwich Challenges
Between the sets of two days participants were expected to trial material from the course and prepare a report to illustrate what they did and what students learned. These Sandwich Challenges are a characteristic of Six Day MOTM Courses. As always, teachers responded to the challenges with enthusiasm and professionalism. Here are some examples:
Bev chose Task 10 Find My Pattern as the starting point for her algebra trial and built her teaching craft around the suggestions in its partner Maths300 lesson.
Bev's presentation folder.




Course Outline
Days 1 & 2
Background
Algebra Makes Sense
 Patterns & Generalisation
 Substitution & Solution
 Pairs & Pictures
 Positive & Negative
Rectangle Algebra
Connected Learning
Days 3 & 4
Refresh & Report
Investigating Tasks
 3 Lives of a Task
Task Based Unit Plans
 Modelling, Invitation, Skill Practice
 Replacement Unit & other Structures
Research Support
Student Publishing
Assessment Opportunities
Mathematics Makes Sense
Days 5 & 6
Refresh & Report
More on Unit Plans & Structures
A Working Mathematically Curriculum
Algebra Surprises
Leading The Way

This includes going outside and giving students large size number cards with the challenge of arranging themselves in a pattern. After some simpler examples the students are expected to find their way through a pattern that involves adding one term and subtracting the next. This devious approach to developing patterns was originally developed by Year 4 students and led to the task above.

As the cameo explains, solving these patterns can lead to plotting pairs of points. For instance, given this set of numbers  11, 13, 13, 15, 15, 17, 19  if you were told they made a pattern with 13 as the first number and 19 as the last, the first challenge would be to Find My Pattern.
After a while, through the application of various strategies, you might perceive a 2/+4 pattern and sort the numbers as 13, 11, 15, 13, 17, 15, 19. Because these numbers are now in sequence each one can be assigned a position in the sequence and the following set of ordered pairs develops:
(1, 13) (2, 11) (3, 15) (4, 13) (5, 17) (6, 15) (7, 19)
and graphing these pairs produces a picture that at first site may be surprising, but then adds to the explanation.
 Can you work backwards from the sample graph to create a Find My Pattern Challenge.

Find My Pattern then suggests students are invited to round off the investigation by creating their own pattern challenges for others. Based on an idea found in another source, Bev's students put a lot of work into creating pattern challenges such as those shown here.
Jodi based her first classroom trial around Task 130, Protons & Antiprotons, and its partner Maths300 lesson which provides a stimulus newspaper article and software. She captures the essence of the trial in this PowerPoint Presentation which includes linked student work. These links may not work from within the web mounted version of the show, so they are linked here:
Thoughts on Matter / Proton & Antiproton Grids / Student Report
In the second round of trialing, Maria and Jodi planned an algebra unit, based around Maths With Attitude Pattern & Algebra Years 7/8, and Maria recorded their experiences in a scrapbook.
The unit Maria & Jodi created included the use of Investigation Journals and an assessment task. The assessment task required students to develop their own task cards (with answers).

Read below then return and click here to see inside the scrap book.



When we were making the connection between Protons and Antiprotons and Positive and Negative numbers, a few students asked the question: Does 34 mean 3+^{}4 or 3^{+}4? After completing both questions with Protons and Antiprotons students realised that there was no difference. 
Students were given an assessment rubric directly linked to VELS (Victorian government prescribed assessment criteria).
See more of Maria's scrapbook by clicking on the cover photo above and using the spacebar or arrow keys to move through the photos.
For a full screen view, right click to download the PDF slide show then run it from within your computer. 

Asked to comment on the value of this three week unit, their evidence of learning and to make other comments that might help their colleagues, they wrote:
Jodi
The three week unit was very much worthwhile. It made me restructure the way I teach and has benefitted the students in the way they approach tasks. The evidence for our success is our Algebra Topic Plan. Students are writing about their maths and their writing has improved as we have done more investigations. I would like more time to plan and execute units like this.
Maria
The three week unit was worthwhile because it gave the class the opportunity to get away from the text book and an opportunity to work on handson tasks. Our evidence of success is in the photos we took, anecdotal comments from the kids such as "We like this stuff.", the work in student Investigation Journals, the results of the 'create your own task card' assignment. It is important when running a unit like this to be familiar with the resources you are using and to gather all the materials you need in advance.

Greg and Enoka developed their lesson sequence from Task 178, Match Triangles, which had been explored in the first workshop guided by its partner Maths300 lesson. Greg worked with Year 8 and wrote about his trial under the title Edited Confessions of an Algebra Teacher, which is reproduced here.
The aims of the series of lessons were to:
 solve the 'Match Triangles' problem and apply this knowledge to related situations
 introduce students to algebra through a series of hands on investigations
 work in teams
 consider problems and create a formula to solve them


Starting Point
An initial discussion of what students did through their holiday break and what type of work their parents did was led in the direction of one student whose father was a builder. Through this the construction of houses was investigated and the use of trusses between floors. He pointed out the trusses in the covered way between two school buildings which were triangles.
A conversation about how they were made ensued and the cost of making the triangles for the trusses was considered. Students came to the board and drew what they though the trusses looked like and the most cost effective method was decided using an initial triangle and then adding to it. This led us into the task.
Exploration
The students organised themselves into groups of 3 or 4 and were given icypole sticks to construct a row of triangles. A question was posed: What if you wanted to make many more triangles, how could you work out how many icypole sticks would you need without actually making the triangles?
Students sat in groups and wrote down different formulas and tested them by making the triangles. As a whole group they wrote their formulas on the board and discussed what was similar. They were all about the same, eg: 2 x triangles + 1 = icypole sticks.
After this they were asked to make different shapes, come up with formulas for them, then test their formulas. All results were recorded on the board and ordered pairs were made for triangles, squares, pentagons and hexagons. These were entered onto an Excel spreadsheet and then graphed.
They were asked if there was a shorter way and they came up with letters to represent the words, ie: 2xT+1=i. This was a chance to explain that the 'x' sign wasn't needed and the students accepted 2T+1=i. Students were now asked to find the number of icypole sticks for 15 triangles. They came up with answers based on the formula and were sent away to make the number of triangles and check their answers.
Questions were set by the students for a written report. The groups answered the questions and the graphs were submitted as part of the report.
Reactions
 A group of girls came running across the room very excited with several formulas relating to different shapes and they said,
We've found a pattern with the shapes. If you have a shape you multiply the number of shapes you want by one less than the sides of the shape and then add one. Just like the triangles.
Through questioning, we found their formula worked for all the shapes they had made.
 Three boys who usually work like a frozen polar bear actually finished all required work and sought help out of class to make sure that they were on the right track.
 One boy who was a reluctant participant made two graphs and compared them saying that the graphs were steeper as the number of sides went up. So the Squares graph was steeper than the Triangles and so on.


After these handson lessons were finished the students completed some exercises out of the text book and the statement was made that a lot of the questions were like the icypole stick questions and they could understand the questions a lot more now.
Enoka was working with Year 7 in the first month of the year  their first year in secondary school. Her icypole stick patterns developed into a project about building 'row houses' given the costs per wall and roof. Enoka's PowerPoint Presentation provides background, including trialing alternative forms of assessment.
[Note: This is a 9.5Mb file and might take a while to load. Alternatively you can right click, save it to your computer and open it from there.]
These work samples suggest the students' enthusiasm for algebra and the breadth and quality of their presentations.
Folded papers attached to the model record details of the pattern.
Greg explored several investigations during the course and rounded off with a 4 week Space & Logic based around Maths With Attitude Space & Logic Years 7/8.
The first investigation that 'did his head in' was ThirtyOne and an Investigation Guide he developed has been added to the Task Cameo. Click the image to reveal the Cameo.
During the Space & Logic unit, Greg was exploring Maths300 Lesson 48, String Shapes, and read that Lesson 51 Hunting For Stars could be used an extension. Doing this confirms work on polygons developed in String Shapes and integrates with pattern work of the type developed in ThirtyOne.


The investigation begins outside or in a gymnasium with a number of students sitting in a circle and passing a ball of wool to each other according to given rules. The trailing end of the ball marks out a polygon or star. Hence the name. The lesson continues inside at tables as the students explore more circles and passing rules on a prepared sheet, then is extended further by companion software. This photo slide show gives an idea of student involvement. (Spacebar or arrow keys will change the slides.) Greg has also contributed:
String Shapes begins outside with each group of 6  8 students sharing a large loop of string and a set of Polygon Pages. One student is Recorder. Students have to create a large string version of the polygon on the page and find ways to test the accuracy of their model. Inside, class discussion begins to uncover measurements and features that are the properties of particular shapes. Greg added to the lesson by asking each group to take apart their polygon booklet and find out as much as they could about the shapes on the pages they received. This collected wisdom was then turned in as a group report. More detailed discussion followed as a class before the students were issued with a new booklet each and challenged to complete at least 4 pages with as many properties as possible about their chosen shapes. Many students chose to complete the whole booklet. The text book work on polygons and properties was a breeze after this investigation.
This photo slide show shows highlights from the lesson sequence. (Spacebar or arrow keys will change the slides.)
Another component of the MWA unit is students using handson tasks as an invitation to work more independently as a mathematician.
Evaluation comments were gathered from teachers at every stage of the course and included these reviews:
My 'aha' moments have been:
 The activity River Crossing  realising there was a basic pattern which can be further explored as graphing, simultaneous equations, forming equations, transforming.
 Allow students to expand their ideas into different areas and not necessarily follow the lesson plan.
 Solving the 'kangaroo' puzzle/ Listening & viewing other people's presentations/ Working with group members to 'solve' puzzles.
 To allow students to think outside the square/ Look at different tasks and resources available from the Task Centre/ Big picture broken into smaller tasks and guides.
 Looking at different ways to approach different types of lessons and topic starters. Getting more involved in the practical applications.
 Realising that there is a better way for kids to understand mathematics than using the traditional approach.
During the final day of the course participants constructed a group negotiated text in response to the challenge to list, in order, the most important course outcomes. They were agreed to be:
 Enhanced the enthusiasm for and enjoyment of teaching mathematics in a fun and interesting environment.
 Experienced a better way for students to learn  working like a mathematician.
 Changed the way I conduct my classes  alternative ideas and directions.
 Enhanced learning of teachers with the tools and resources of Maths300 and Maths With Attitude.
 Provided opportunity to share ideas and work with others in the same direction to extend and plan units of work.
Perhaps even more importantly all participants were willing to consider taking a leadership role in local or district or state professional development programs.
