Month: January 2002

Assessment is key to planning. Knowing where your students are at allows you to plan for their next steps and build on their existing maths knowledge and skills. But what is the most effective form of assessment? And what assessment practices best fit with explicit teaching?

Martin Holt, educational consultant and academic in maths education at the Australian Catholic University (ACU) believes that both questioning and feedback play an integral role in effective maths assessment. Martin discusses this in the Mathematics Hub’s webinar, Explicit teaching in mathematics: Effective assessment. He expands on the effectiveness of using questioning to target explicit concepts and processes, and discusses ways to give feedback that is specific and actionable.

Questioning

Questioning is one of the seven components of explicit teaching, and is an effective strategy in assessing students’ understanding of maths concepts. Planning thoughtful and well-constructed questions can draw out what students know and understand, as well as highlight any misconceptions they might have.

Questioning can help you to:

• probe student thinking
• draw out the mathematical reasoning students use
• extend students’ mathematical thinking
• encourage students to reflect and synthesise
• find out how students apply their knowledge
• extend students to transfer maths strategies to other problems.

Using different types of questions allows you to gather information about what your students know. This data can then be used to inform your explicit teaching focus.

Equitable approaches to questioning

How do you question students in your classroom? How do you ensure everyone is included and willing to contribute?

There are numerous approaches you can employ in your teaching that will encourage a more equitable range of responses from across your class.

• Build a team culture where students are encouraged to contribute knowledge to help the group learn together and solve problems.
• Encourage participation by cultivating the expectation that all students contribute, not the expectation that their responses must be accurate.
• Give students the right to pass if they don’t feel comfortable answering a question.
• Ask questions that allow all students to be part of the response, such as:
  • How did you make a start with this problem?
  • How did you work it out?
  • Who solved it a similar way?
  • Who solved it differently?
  • How did you record what you found out?

Using these strategies will help to foster a classroom culture of trust, which will lead to more students in your classroom becoming actively involved. This will give all students the opportunity to share their ideas, and it will give you the opportunity to learn where each student is at in their maths learning.

Feedback

Feedback is also one of the seven components of explicit teaching. We all know the power of providing insightful feedback, but how is feedback linked to assessment?

Feedback can help students to build on what they have already learnt, and you can use this to assess their progress. Feedback is a way to explicitly outline the next steps in a student’s learning.

You can provide written feedback using sentence frames such as:

• It’s great that …
• Because you can already …
• You are now ready to …

In Martin’s webinar, he explains that feedback needs to be ‘just in time, just for me and delivered where and when it does the most good’. Digital feedback is a wonderful tool to help you do this. It can provide instant feedback to all stakeholders.

All students enter a lesson at different points, and they will exit it at different points as well. Your aim is to ensure that each student moves forward on their learning trajectory, regardless of their entry point. Providing feedback is also a great opportunity to model the accurate use of mathematical language.

Formative assessment

Formative assessment tasks will provide you with valuable data about your students’ knowledge and skills. Knowing precisely where your students are at will allow you to plan your next steps.

As part of the planning process, have a go at solving the assessment task yourself before assigning it to the students. This will give specific insight into the range of possible strategies your students might employ to solve the problem.

Having a learning intention for the task will allow your students to understand the focus of the lesson. Learning intentions are usually presented at the beginning of a lesson, but this doesn’t always need to be the case. It can also be beneficial to have students explore the overall concept first, and then make the learning intention explicit to students.

Making the success criteria explicit is also vital. Explaining to the students what they need to achieve provides a pathway for them to achieve success. This can also be done by showing students work samples, so they know what success looks like.

Using exit tickets upon completion of the task allows you to gather information about what the students know and if they have any misconceptions.

Effective assessment will support you to plan for explicit learning so you can emphasise key mathematical points and build each student’s knowledge and skills. Find out more about assessment strategies that promote explicit learning by working through our free Explicit Teaching in Maths modules.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics.

When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to The Mathematics Hub

An upside-down sea ice slab showcasing brine channels that facilitate the drainage of liquid brine and support convection along the interface.

A new applied mathematical theory could enhance our understanding of how sea ice affects global climate, potentially improving the accuracy of climate predictions.

The authors of a new paper published in the Proceedings of the Royal Society A: Mathematical and Physical Sciences, offer new insights into how heat travels through sea ice, a crucial factor in regulating Earth’s polar climate.

Dr. Noa Kraitzman, Senior Lecturer in Applied Mathematics at Macquarie University and lead author of the study, says the research addresses a key gap in current climate modeling.

“Sea ice covers about 15% of the ocean’s surface during the coldest season when it’s at its vast majority,” Dr. Kraitzman says. “It’s a thin layer that separates the atmosphere and the ocean and is responsible for heat transfer between the two.”

Sea ice acts as an insulating blanket on the ocean, reflecting sunlight and moderating heat exchange. As global temperatures rise, understanding how sea ice behaves will become increasingly important for predicting climate change.

The study focuses on the thermal conductivity of sea ice, a critical parameter used in many global climate models. The movement of liquid brine within sea ice, which can potentially increase its heat transport, was not accounted for in previous models.

Dr. Kraitzman says the unique structure of sea ice, along with its sensitive dependence on temperature and salinity, means it is challenging to measure and predict its properties, specifically its thermal conductivity.

“When you look at sea ice on a small scale, what makes it interesting is its complex structure because it’s made up of ice, air bubbles, and brine inclusions.

“As the atmosphere above the ocean becomes extremely cold, below minus 30 degrees Celsius, while the ocean water remains at about minus two degrees, this creates a large temperature difference, and the water freezes from the top down.

“As the water freezes rapidly, it pushes out the salt, creating an ice matrix of purely frozen water which captures air bubbles and pockets of very salty water, called brine inclusions, surrounded by nearly pure ice.”

These dense brine inclusions are heavier than the fresh ocean water which results in convective flow within the ice, creating big “chimneys” where liquid salt flows out.

The research builds on earlier field work by Trodahl in 1999, which first suggested that fluid flow within sea ice might enhance its thermal conductivity. Dr. Kraitzman’s team has now provided mathematical proof of this phenomenon.

“Our mathematics definitely shows that such an enhancement should be expected once convective flow within the sea ice begins,” Dr. Kraitzman says.

The model also offers a way to relate the sea ice’s thermal properties to its temperature and salt content, allowing theoretical results to be compared with measurements.Specifically, it provides a tool to be used in large-scale climate models, potentially leading to more accurate predictions of future conditions in the polar regions. Sea ice in the Arctic has been declining rapidly in recent decades. This loss of ice can lead to a feedback loop: as more dark ocean water is exposed, it absorbs more sunlight, leading to further warming and ice loss.The loss of sea ice can affect weather patterns, ocean circulation, and marine ecosystems far beyond the polar regions.

Dr. Kraitzman says understanding the thermal conductivity of sea ice is important for predicting its future.

The researchers note that while their model provides a theoretical framework, more experimental work is needed to integrate these findings into large-scale climate models. The study was conducted by mathematicians from Macquarie University in Australia, the University of Utah and Dartmouth College, New Hampshire in the U.S.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article to be given Macquarie University