CoL Inquiry - Using Evidence to Guide Practice - Part Five

My math practice this year has made considerable changes; personally I like I've grown in confidence with my practice, the research, the conversations (with colleagues and students), reflections, and all of it has left me feeling more capable as a teacher, and it has translated into students who feel much more confident and are enjoying math. One of the pieces of qualitative data I gathered at the start of my inquiry was their voice; how were they feeling about maths; here's what they said.

I will be reissuing this same questionnaire this week, and I am super excited to see the results; in speaking with my students they are feeling more capable, and demonstrate this on a regular basis in class.

In terms of data the previous blog post has what we have done so far, eAsttle test provided me next steps, and Gloss testing is being done - as I write this I am nearly finished with my inquiry group, and even without another Gloss Test from the start of the year (reminding myself how good they are, and that I need to do more of them)  I can feel their conceptual understanding has improved. In terms of quantitative results, I have now tested 6 of the students in my inquiry group. Each of them has reached Stage 6 in Addition/Subtraction, most, if not all have reached Stage 5+ in Multiplication and division. 

This still means they are working towards achieving at the level that they should be, however, they have all shifted upwards; their OTJs had them placed at Year 4, some early, some late, but definitely below the expected level. A single test is not indicative of their progress, but I can't help but be excited to see this acceleration.

Cook Islands Summit

In October I was privileged to be a presenter at the 2024 Cook Island Teacher's Summit, and I'll start of by saying WOW. What an awesome experience, and opportunity; to be able to connect and share ideas with a wonderful group of teachers is something that I will always treasure. It is opportunities like this that make me so thankful to be a part of the Manaiakalani family.

I thought it would be cool to share with you what I presented, and share some of the awesome PLD that I got a chance to be a part of;

I'll get mine out the way first. My presentation focused on my inquiry question for 2024 - How can I improve my student's conceptual understanding of mathematical concepts?

This question was one that built on my MIT '23 project of improving students comprehension of word problems, and came about because I realised that vocabulary and comprehension was only a small part of what my students were finding difficult; they also needed support in their retention of basic facts, and number knowledge. Some of them needed additional support through the use of manipulatives to make connections between abstract equations, and the real world applications of them.

This was where T-Shaped Mathematics came from; an adaptation of a learning model that we use in our literacy planning; the idea that in order to improve conceptual understanding of a topic our students must be offered a range of learning opportunities, exercises, and strategies that best suit them. By offering this wide range we can ensure that students can find the one that works best for them.

I have built my mathematics program this year on the foundation of the CPA approach, as well as elements of traditional, and reformist pedagogy; rote learning through the regular practice of multiplication and division, and addition/subtraction through repeatable exercises as warm ups; a range of questions which required the use of manipulatives, both physical and digital, and fun and engaging games that we can play in math that foster collaboration, and learning from their peers. The aim of this was, in simple terms, to do the mahi, and then get the treats. Following this principal has been instrumental in the improved engagement, and results I have seen this year.

Examples of how this is done can be seen in the slides below; 

The best part of the Cook Island Summit, aside from the amazing location, weather, people, and places we got to see, was the opportunity to learn from expert teachers, and to spend the time talking, and sharing ideas with them;

I was fortunate enough to be able to attend some awesome workshops; the first was a workshop entitled Beyond the Reef; presented by Pelu Leaupepetele, the principal of Kedgley Intermediate. This workshop was focused, primarily on leadership in schools, but carried with it some critically important messages, particularly for teachers of primarily Pasifika schools, of vulnerability, honesty, and openness, and how relationships between the leadership, teachers, and students can create a school environment where students and teachers open up, and can share and be vulnerable with each other, and in doing so, can build trusting relationships which foster a school which is healthy, and happy.

The second workshop is one I am very thankful I was able to attend, it was extremely popular, and with good reason. The workshop was titled Ruapekapeka, and the title, which I knew to be a famous battle in the New Zealand Wars, was so much more. Presented by Ian McGee and Tess Whelan from Taylormade Media we were introduced to matautanga.co.nz, an absolutely incredible resource. The ins and out of which I could spend all day writing about, however I simply implore you to explore their resource. It is amazing, and absolutely free. I will 1000% be using this as the foundation for my Aotearoa's History unit starting off 2025.

The first day done and dusted we all gathered together to experience Captain Tama's tour. What an awesome day - I cannot empasise enough just how beautiful the Cook Islands are, and we were gifted an incredible, and hilarious trip out to a small island, maybe 10 minutes by boat from Rarotonga. We were kept fed, and entertained by the wonderful Captains and spent a lush couple of hours swimming, sunbathing, and nearly being eaten by an overly friendly Giant Trevally. 

I will share with you more details about Day 2 of the Summit in another post, because there is simply too much to read in one go.

CoL Inquiry - Part Four (Testing Reflection T2)

 Last time we spoke I mentioned that we would be running an eAsttle Maths test. I always try to remember that tests are a diagnostic; a way of identifying areas of strengths, and weaknesses, and gives us, as teachers, a really good opportunity to reassess what has worked, and what we need to improve on, not a reflection of how good we are as teachers, or how well my students learn. It is merely a snapshot, and even if it feels discouraging, it is not the whole story,

In this round of testing I have a mixed bag; for context, at this stage (Term 2) students at year 6 who are achieving at the norm achieve a 3P in their eAsttle tests.

The testing results for the students in my inquiry focus group (names excluded) are as follows; 

- 3 scored a 2B (more than 2 years below norm)

- 3 achieved a 2P (2 years below norm)

-  2 achieved a 2A (1.5 - 2 years below norm)

- one student achieved 3B (0.5 - 1 year below norm)

- one student achieved 3P. (at the norm)

It can be difficult when trying to align the results to see progress; eAsttle, PAT and Gloss all have their own scales, and scoring systems which can make it difficult to get a clear picture of progress. One of the things I will be doing in Term 1 of next year is being consistent with the testing to ensure I have a clearer picture of their progress as the year goes own. This will be alongside the school wide testing that we are required to do (PAT).

I will also be conducting Gloss tests, and will share my findings as Term 3 goes on.

 

CoL - Part Three (CPA Approach)

 Teaching Maths was the thing I was most intimidated by when I made the move to primary teaching. I am an English major, and trained as a teacher of English. My learning journey with math was an interesting one. I did well when I was at in primary; I was good at following the "algorithms". I was lucky enough to live in and attend school in Chile, and was learning long division by 8 years old, but I think algebra was where the problems started. Dealing with unknown variables, and identifying patterns by applying formulas was very confusing. "What happened to 8 + 9 is 17, carry the one...?!" . It seemed that everyone who did well was in on a secret that I wasn't privy to. Their brains were able to unlock the mysteries of maths, and I felt like I was locked out.

I made a promise to myself, that I was going to do what I could to ensure that no student in my class was going to feel as lost, and confused as I was. I didn't know how best to do that, and 4 years after becoming a primary teacher I am still figuring it out, however. I think I have found something that has changed my view on maths, and I hope can help my own students to learn maths by giving them the keys that will help them unlock the "mysteries" of maths.

I found myself trying to read as much as I could about conceptual understandings; I felt like if my students could build a strong conceptual understanding in my next unit, that it would allow them to apply sttrategies to a wider range of problems, including word or picture problems which require inference, and comprehension on top of mathematical problem solving.

Identifying what my students problem is has been a multi-fold approach; I've looked at their results in PAT Maths, Gloss, and through my observations in class. My original hypothesis was focused around the fact that they needed support with their vocabulary, and that that was a major limiting factor in their ability to solve problems, like the ones in PAT tests, and activities in the Caxton textbooks. What I have observed and noticed is that a lot of them do not have much confidence with the overall concepts. If the question is easy to understand, even if the mathematical operation is challenging, they can apply their strategies to it. 

What I would like to do know is to continue supporting their vocabulary, but also to improve their confidence and self efficacy when it comes to solving the wide variety of problem types that they will encounter. 

The CPA approach was one that resonated with me early on; developed by psychologist Jerome Bruner it's focus is on building conceptual understanding of maths by working through stages; concrete, pictorial, and abstract.

The Concrete element involves the use of materials; for my students this was particularly instrumental in helping them develop their conceptual understanding of multiplication and division. Being able to move blocks, or beans, or money around, and physically manipulate these objects meant that strategies like grouping, repeated addition, or subtraction had practical meaning. I think this approach was the most pivotal, as I could see them responding, and understanding. Those "ah ha!" moments that we cherish so much as teachers were really evident in this stage.

We then moved on to the Pictorial stage, in which students could see visual representations of the problems; this worked really well with arrays, and grouping as well because they could see the objects they were being asked to apply their strategies to. 

The Abstract stage being the final stage of the approach is one that really threw me for a loop the first time I read about it; I grew up doing maths with symbols from a very early age, and it hadn't clicked to me that these symbols and digits were abstract representations. Once I started to think more carefully about when I gave my students these abstract questions it became apparent that my sequencing had been askew.

The other aspect to this was based on an interesting article that was shared with me from a couple of different people, and obviously it resonated with them, so I delved into it. The original article was an Op-Ed published by the University of Auckland, titled "What maths teaching could and should learn from cognitive science" - it spoke on the fact that these "exploration" type of maths lessons showed no discernible benefit to student's learning. Following up was the article that was referenced in the Op-Ed which was a paper titled "Why Minimal Guidance during Instruction Does Not Work: an Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching".

This was again, pretty eye opening; it spoke about the fact that simply allowing students to explore concepts with little or no scaffolding or support in the form of explicit teaching, meant essentially that we were relying on students to learn simply by exposure or osmosis. There needed to be concrete, explicit teaching of these concepts if they were going to be able to learn, retain, and apply them to the wide range of problems they will be faced with.

We have recently come to the end of a 5 week unit, beginning with multiplication, moving on to division, and then to applying strategies, and will be conducting an eAsttle test on them in the next week, I'm excited to see if the results reflect the progress they have made in group work; knowing they can do it with me, it's time to see if they can apply these concepts with the training wheels off, so to speak. 

I'll see you guys in the next instalment!

References:

Bourtzinakou, E. (2023). Developing Mathematical Reasoning; The role of the CPA model in students’ progress from standard to Reasoning and Problem-Solving questions. https://www.et-foundation.co.uk/wp-content/uploads/2023/02/The-role-of-a-CPA-model-in-developing-mathematical-reasoning_Gateshead-College-CfEM-action-research-report-2021-22_compressed.pdf

Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why Minimal Guidance during Instruction Does Not Work: an Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. Educational Psychologist, 41(2), 75–86. https://doi.org/10.1207/s15326985ep4102_1

Main, P. (2021). Concrete pictorial abstract approaches in the classroom. www.structural-learning.com. https://www.structural-learning.com/post/concrete-pictorial-abstract-approaches-in-the-classroom

N.A What maths teaching could and should learn from cognitive science - The University of Auckland. (n.d.). Www.auckland.ac.nz. https://www.auckland.ac.nz/en/news/2024/04/10/What-maths-teaching-could-should-learn-from-cognitive-science.html

CoL Inquiry - Part Two (Diagnostic Tool)

 After deciding that I was going to create a diagnostic tool to determine my student's understanding of different mathematical concepts it was now time to create the tool.

I wanted the tool to function as a standardised way of checking if they were able to understand what the different strands of mathematics were asking of them, and not assess whether or not they could solve the problems.

In this question, I wanted to know if they would be able to match the term, to the definition, to the example. 

In this one, they needed to be able to order the sequence of problem-solving steps.

I ran the test with a sample group of students; I didn't use my target group for the prototype testing - my rationale for this was to try and get a sampling from a range of different abilities, reading levels, and attitudes towards maths. The prototype was tested by 11 different students, and the results were a little disappointing - not because I felt like they hadn't succeeded, but because I felt that the test itself was too confusing. In the reflections with students they were unsure what they were being tested on, and that some of the questions were confusing, even for the extension students.

This leaves me at a bit of a crossroads, is the test tricky because I have made the questions and the tasks too confusing, or is it an indication that this test actually challenges their ability to comprehend the maths concepts, and it's actually revealing holes in their understanding?

I am going to gather feedback from my colleagues, adapt the test using Google Forms to see if the more rigid structure can work as a scaffold them and remove ambiguity. Hopefully the results will show me there is merit in continuing to pursue this diagnostic tool.

In the mean time, I am continuing to explore the CPA Approach to my maths teaching. Check out the next post to learn more about CPA, and how it has been working out in Room 8.

CoL Inquiry - Part One

It's been a couple of weeks since my students sat their PAT Math test, and so I've spent a lot of time reflecting on their results. At the start of each year we use these tests to help us to understand our students better; to see which areas of maths they succeed at, and which ones they are struggling with. 

The target group for my inquiry cycle will be a group named Storm; students in this group have OTJ's which place them around Level 2 of the NZC and they averaged a 3.3 Stanine in PAT testing. 

There are 10 students in this group, and with a group this large there is a wide variety of self-efficacy, interest, and enjoyment when it comes to maths, so, to get a better understanding of who they are as Maths learners I conducted a Student Voice survey; of the 10, only 3 felt that they were "good at math", the others did not know (good news is, none of them thought they were not good). 

They were pretty divided on how they felt about maths; a couple of them used only positive words to describe maths, words like "fun" "interesting" or "exciting", but what was interesting was that almost half of them used the word "stressful" to describe it, including some of those students who also said it was fun, and interesting.

I found this dichotomy super interesting, because, to me, it mirrors my own feelings towards maths. As a student when I encounter concepts that I do not understand, it is stressful. It's like trying to solve a puzzle, but half of the pieces are from another puzzle, no matter how I try to apply my learning and try to make the puzzle pieces fit, they just won't; however, if I find the other pieces, if I strengthen my understanding of a concept, or a formula, or a strategy, suddenly the pieces start to fit, and that is what lead me to the most recent development in my inquiry.

Do my students have a well-rounded understanding of the concepts and vocabulary that we cover? 

Do my students understand that addition is calculating the combination of 2 or more numbers, or that multiplication is the repeated addition of the same number, for example. How do we expect them to apply strategies to concepts that they are not fully aware of? How can they be adaptable, and flexible with their application if they do not first understand what they are being asked?

Without trying to cover the wide spectrum of concepts in the math curriculum I chose to focus on the two areas of maths in which my students achieved the lowest results; Number Strategies, and Algebra, and to work out where their understanding of the concepts within them are.

It is easy to assess their ability to solve problems in these areas, we teach a strategy, we give them a problem, and either they solve it correctly, or they do not. Testing their understanding of the concept is proving to be a little harder. After discussion with my colleagues, and conducting research of my own I could not find anything that fit my needs, GloSS testing where students explain their strategies, and eAsttle tests gave me some insight, but not an umbrella view of what they understand these concepts, and words to actually mean.

What I would like to do is build a diagnostic test, or interview that can assess whether or not they understand what a concept like "addition" is, rather than test their ability to add together numbers.

I'm working on putting together a prototype at the moment, so check back in to see what I've got soon!

MIT 2023 - The End of the Journey

As my father always told me  "All good things must come to an end", and this MIT journey for 2023 has finally come to an end. It is a bittersweet goodbye. I walk away from this project with a tremendous sense of pride in what we have accomplished together. MIT 2023 has been one of the most significant highlights of my year and has opened up so many opportunities, discussions, conversations, reflections, and friendships that have improved my practice and my outlook on teaching; the value of the MIT project goes far beyond the tools we have created, it is an opportunity to take part in a community of learning that is equal parts fun, inspiring, and motivating.

MathVentures is just the beginning; starting my investigation into math vocabulary has inspired my next inquiry cycle; investigating how we can continue to help our students comprehend the word problems they encounter in maths, and how we can support our students to build a robust math vocabulary that will help them to better understand how maths is applied in more real-world contexts.

Check out MathVentures here!

 In a way, the end of this journey opens up a door to a whole new set of opportunities to learn and develop my practice, and in turn, improve the outcomes for my akonga. As lifelong learners I think it is so critical for us to continue to pursue these avenues as they open up to us, and while MIT has come to an end this year, it is the start of a whole new journey.

I would like to first acknowledge my fellow MIT alumni for 2023; Crystal, Michelle, Essie, Jayne, and Maiken. You guys have been a blessing to work on this project with. Without the support and encouragement from you guys, I don't know if my project would have reached the stages that it is at now. Thank you so, so much. I am so lucky to have been a part of such a cool crew, and I won't forget our adventures any time soon! That boat trip to Russell is easily one of my favourite memories. 

Secondly, I would like to acknowledge the tremendous work put in by Matt Goodwin, Dorothy Burt, Justine Todd, and Jenny Oxley; the knowledge, experience, and feedback they have given us along our journey have been invaluable, and their contributions to our projects cannot be understated.

I cannot recommend this program highly enough, if you are thinking of applying, please do. You will gain invaluable insights, relationships, and confidence in your own practice that is hard to come by elsewhere.

Kia ora rawa atu,

Gabe