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.

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.