How can I stop teaching calculating and start learning to do maths? How can inquiry based mathematics and argumentation make the student a more active learner?
From observations and data collected from testing, it was identified that students struggled when faced with mathematical word problems, there was a lack of knowing what the question was asking and what mathematical operation and strategy could be applied. Observations also showed that students struggled to actively discuss and provide a mathematical explanation using the context of the problem, not just the numbers. When assessing learning we need to look beyond improvement in mathematical scores to how students engage with mathematics and the relational aspects with their peers.(Hunter and Anthony, 2011). Often in our testing we do not gage what the students are thinking and they may be on the right path but give a wrong answer because of time constraints or simple number mistakes, there may also be a lack of motivation as the question may be uninspiring.
In discussions with my inquiry buddy we both felt that children were often too reliant on the teacher for modelling strategies, which obviously has its place but is centered around learning mathematics rather than doing mathematics. We also agreed that the children needed to be active in developing mathematical problems/inquiry so they understood the process and required solutions and were motivated.
What am I trying to achieve?
To engage the students in becoming active learners to have them 'doing maths', to be able to discuss, justify and explain their problem solving strategies and challenge the thinking of others.
How are we going to achieve this?
To engage the students in becoming active learners to have them 'doing maths', to be able to discuss, justify and explain their problem solving strategies and challenge the thinking of others.
How are we going to achieve this?
From research it became evident that the practices and opportunities found in inquiry based mathematics may engage students in explanation and justification and understanding of mathematical problems.
The document Learning to “friendly argue” in a community of mathematical inquiry by Roberta Hunter, Glenda Anthony June 2011. It is argued that effective pedagogy in inquiry based mathematics allows students opportunities not only to share their ideas in a mathematical community but also to analyze and evaluate the thinking of other members. They state inquiry based mathematics requires students to:
The document Learning to “friendly argue” in a community of mathematical inquiry by Roberta Hunter, Glenda Anthony June 2011. It is argued that effective pedagogy in inquiry based mathematics allows students opportunities not only to share their ideas in a mathematical community but also to analyze and evaluate the thinking of other members. They state inquiry based mathematics requires students to:
• provide mathematical reasons/justification for their thinking, rather than just a description of their thinking
• as a group, explore two or more ways to explain a strategy solution
• analyse explanations and construct ways to revise, extend and elaborate on sections others might not
understand
• predict questions that could be asked by peers, and prepare mathematical responses
• ask questions that clarify an explanation (What do you mean by? What did you do in that bit? Can you show
us what you mean by...? Can you draw a picture of what you are thinking?)
• agree on the construction of one or more solution strategies that all members can understand and explain
• work together to check, explain and re-explain in different ways the group explanation.
From this one of the first objectives was to discuss, explain and model how and why to ask questions of each other, to engage in 'mathematical argument'. As we had been working on fractions I gave the students a fraction problem to go and solve in pairs and to then feedback their findings to others, explaining and justifying, and for others to question, using what we had developed.
The pairs worked together and used diagrams and discussion to solve the problems. However,in some pairs one person did most of the work and the other passively listened. When explaining to the larger group there was some discussion of different ways of solving, but most had the same answers so didn't engage in too much 'argument'. There was also little discussion around different strategies. From this I needed to develop an inquiry which challenged them to' mathematically argue' and which involved the use of different strategies, number knowledge, place value and basic facts.
The pairs worked together and used diagrams and discussion to solve the problems. However,in some pairs one person did most of the work and the other passively listened. When explaining to the larger group there was some discussion of different ways of solving, but most had the same answers so didn't engage in too much 'argument'. There was also little discussion around different strategies. From this I needed to develop an inquiry which challenged them to' mathematically argue' and which involved the use of different strategies, number knowledge, place value and basic facts.
There has been considerable attention within research literature to creating inquiry-based learning environments in which students not only express their own ideas but also use mathematical reasoning to challenge those of their peers and the teacher.
Anthony, G. & Hunter, R. Journal of Urban Mathematics Education July 2011, Vol. 4, No. 1, pp. 98–119
©JUME. http://education.gsu.edu/JUME
We next decided to create an inquiry around milk usage in the school, the first step was to develop our inquiry questions.
We recorded them on the IWB where we could return to them and change or update.
We next decided to create an inquiry around milk usage in the school, the first step was to develop our inquiry questions.
The children worked in pairs to solve the problems. They were motivated by collecting the data and as a group discussed how to collect it and who, they agreed to split up and go around the classrooms and then to share their findings. In pairs they decided how they would present their answers, they chose mainly to use bar graphs to show consumption. One pair used a pie chart which opened up a debate of how and why we display data and the best way.
When it came to working out the consumption of milk per litre some struggled as they were unsure about milliliters and litres. However, working this out required multiplication and division and questioning others in the group, which was a positive step towards 'mathematical argument.
A pair discuss their mathematical reasons for working out consumption.
IMG 0183 from neil carrington on Vimeo.
Buddy observations and discussions.
During observation my buddy commented that the children were motivated by what they were doing and were able to explain their inquiry. However, she did comment that in some of the pairings there seemed to be a dominant person who often spoke up or seemed to be doing most of the thinking and working. From this I felt I needed to have the pairs feedback to me as well as the group so I could aim questions at them which involved both answering. My Buddy also questioned; why I had chosen these particular students, how much input I had given and did they choose the inquiry topic. I had chosen them because this was a trial and the literacy around inquiry based mathematics had indicated that students needed to take ownership and have the skills to partake. These students were also identified as having the skills, independence and knew what strategies they could use (had received, modelling and scaffolding), but often struggled to apply them and were often not motivated by teacher developed problems, although they had shown they liked to be challenged. In answer to other questions, I had chosen the topic, but the students had developed the learning task and questions. As for my input I was acting more as a facilitator and only gave input when needed. However, as discussed above after observation I had to have regular questioning, feedback and feed-forward meets to assure all were on task and using the mathematical inquiry process and justifying their answers. After a further observation my buddy commented that all the students were engaged and working on their inquiry and that they could justify what they were doing and explain their answers. Her feedback also highlighted that the students were autonomous in choosing what and how they answered questions, presenting their findings and use of technology. During this feedback I was asked how the students could use Inquiry based mathematics beyond the school and involve their whanau. From this we have started another Inquiry based around a bedroom makeover which requires work at home and communications with their whanau. This inquiry is also more individual based, but as part of it they will share and justify their problem solving, answers and methods to others.
Student feedback on Inquiry based mathematics
IMG 0184 from neil carrington on Vimeo.
Bedroom makeover
IMG 0202 from neil carrington on Vimeo.
PMI of Inquiry
The pairs used different strategies to work out the daily, weekly and yearly consumption. We discussed the use of multiplication and addition and the pairs tried to justify their choice a calculator was allowed to work out the yearly consumption. This conversation was useful as it indicated the need to move some to multiplicative thinking.
A pair discuss their mathematical reasons for working out consumption.
IMG 0183 from neil carrington on Vimeo.
Buddy observations and discussions.
During observation my buddy commented that the children were motivated by what they were doing and were able to explain their inquiry. However, she did comment that in some of the pairings there seemed to be a dominant person who often spoke up or seemed to be doing most of the thinking and working. From this I felt I needed to have the pairs feedback to me as well as the group so I could aim questions at them which involved both answering. My Buddy also questioned; why I had chosen these particular students, how much input I had given and did they choose the inquiry topic. I had chosen them because this was a trial and the literacy around inquiry based mathematics had indicated that students needed to take ownership and have the skills to partake. These students were also identified as having the skills, independence and knew what strategies they could use (had received, modelling and scaffolding), but often struggled to apply them and were often not motivated by teacher developed problems, although they had shown they liked to be challenged. In answer to other questions, I had chosen the topic, but the students had developed the learning task and questions. As for my input I was acting more as a facilitator and only gave input when needed. However, as discussed above after observation I had to have regular questioning, feedback and feed-forward meets to assure all were on task and using the mathematical inquiry process and justifying their answers. After a further observation my buddy commented that all the students were engaged and working on their inquiry and that they could justify what they were doing and explain their answers. Her feedback also highlighted that the students were autonomous in choosing what and how they answered questions, presenting their findings and use of technology. During this feedback I was asked how the students could use Inquiry based mathematics beyond the school and involve their whanau. From this we have started another Inquiry based around a bedroom makeover which requires work at home and communications with their whanau. This inquiry is also more individual based, but as part of it they will share and justify their problem solving, answers and methods to others.
Student feedback on Inquiry based mathematics
IMG 0184 from neil carrington on Vimeo.
Bedroom makeover
IMG 0202 from neil carrington on Vimeo.
PMI of Inquiry
Positive
|
Minus
|
Interesting
|
Students’ motivation and engagement.
Participation and collaboration.
For many students, learning and becoming confident to construct,
present and question mathematically.
Thinking multiplicative as well as additive.
More active learners and able to realize what it
is to do mathematics as well as learning mathematics.
Justifying and ‘arguing’ more.
Poses a challenge.
Forces the use of number knowledge, place value
and mix of strategies.
|
For some students, learning and becoming confident to construct,
present and question mathematical explanations is a lengthy process, which
requires the teacher to continually press students to provide conceptual mathematical
explanations.
For some students the expected mathematical practices, such as
constructing representations, making arguments, and explaining their
thinking, may be difficult or make them feel uncomfortable. Some found the
thought of arguing difficult and some went quite if proved wrong.
Within groups some children doing most of work and others
passive.
|
Not sure where the Inquiry will go and what the
outcome will be.
Could be an age issue, some children may be too
young to do what Anthony & Hunter describe above about Inquiry based
mathematics.
Wanted
to use ICT to display findings/data.
|
What next?
Continue and develop inquiries that motivate and engage and require mathematical understanding. To use it with other students and try at other levels. More learning at home so class time is spent reflecting, justifying and where next. To gradually build on and extend; my expectations of the students to engage in justification and mathematical argumentation and the student being the total active learner and the teacher more the facilitator . To use Explain Everything to develop and record inquiries. To develop some problem solving questions to use as pre inquiry data and some for post based around choosing mathematical operation and strategies.
Just watched this on Inquiry based maths and the presenter really shows how collaboration can really engage and work.
Continue and develop inquiries that motivate and engage and require mathematical understanding. To use it with other students and try at other levels. More learning at home so class time is spent reflecting, justifying and where next. To gradually build on and extend; my expectations of the students to engage in justification and mathematical argumentation and the student being the total active learner and the teacher more the facilitator . To use Explain Everything to develop and record inquiries. To develop some problem solving questions to use as pre inquiry data and some for post based around choosing mathematical operation and strategies.
After staff meetings, PD with Bruce Moody, this inquiry and PACT trial, have realized the need to use the different strands as away of 'doing mathematics' and to enhance and confirm the use of number.
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