For assessments to be reliable and valid, educators need to work together. A great summative assessment is also one that will not just determine what students know, but will also help them continue to learn and consolidate key learnings of the standards.
What would happen if educators were made responsible for co-creating and co-analyzing the summative assessments that would be comparable among schools? This would be an action that would replace the use of a summative assessment created in isolation of other collaborators and experts, and also separate from standardized summative assessments created by outside agencies.
It seems reasonable that for co-creation of assessment to occur, educators would have to meet up regularly to talk about goals, processes, procedures, evidence and more. Educators also have a myriad of different understandings about assessment. This inevitably leads to different ways of creating and implementing assessments. If educators had the opportunity to co-create assessments, then one might surmise that the assessments would become more aligned between and among learning environments, and would create more reliability and validity across and between schools.
When summative assessments are chosen and created by outside sources, teachers do not have the say in terms of what standards will be measured — therefore they are unable to consider their own student needs and developmental readiness. In addition, when educators create their assessments in isolation, the richness of discussion and knowledge of multiple people is not utilized. This is a very important component of assessment, and one that would promote deeper learning among students.
In addition, co-created summative assessments could allow educators to decide upon the standards that they will use to evaluate learning. For instance, in terms of math, it is important to think about the difference between proficiency standards, and content standards — and does it matter when it comes to summatively assessing students? Perhaps most importantly, when educators are co-creating the assessments, there is likely an increased probability that the students themselves are able to shape the assessment in valuable ways.
Then is it important to think about having a facilitator to help train educators in this creation process? Is more training required for that? Perhaps content coaches could serve as key catalysts to help teachers analyze the content areas that are necessary for students to perform and achieve on summative assessments. A coach or trainer who can help facilitate the creation and analysis of assessments, and ensure alignment to standards of the greater institution.
Is it reasonable to bring teachers together, on a regular basis, to build high quality summative assessments? Then to meet again to collaboratively analyze the student work? All the while having access to coaching and support for creating and analyzing assessment data?
While there are no clear answers, one element is certain, that working together on assessment creation is a way to make summative assessments more reliable and valid for learners.
What does it mean for students to have a choice in math? After my last post about struggling students, it was interesting to see that the aspect of ‘choice’ was what really resonated, and was shared here in Aviva Dunsiger’s blog, and here on VoicEd Radio. This really got me thinking deeper about ‘choice’ in math.
The word ‘choice’ can be difficult in math because it is often laden with differing expectations and pre-conceived notions. Are we talking about students choosing the math problem? Choosing the type of problem? Choosing the strategy? Choosing the right numbers? The right operations? We can apply student choice in so many ways.
I wanted to investigate the aspect of choice in math. More specifically, how students come to select a mathematical strategy.
For the purposes of this blog post, when I discuss choice here, I am thinking about how we help students understand and make choices about where and when to use the most appropriate strategies. I also think that it is important to draw the distinction between helping students to execute a strategy correctly, versus selecting a strategy to implement. There are strategies that students will execute very well, and will be able to use to obtain the correct answer. However, it is the latter that I wish to focus on here, because I think it is imperative that before we help students to solve correctly, we want them to be able to select a strategy. And if students are proficient with using strategies, we want to push them even more with math problems that force them to dig deep and consider what strategies they need to use.
When we talk about selecting a strategy to use in math, we hope that students can develop along a continuum of strategies ranging from inefficient to efficient that can help them solve a math problem in a timely fashion. There are many trajectories and continua of math development, my favourite is the Lawson Continuum.
Efficiency is important. If students are unable to use a strategy efficiently, then they are more likely to choose a strategy that is NOT going to work for them. Therefore, it becomes very important that we help students not just learn the strategies, but also become efficient in them, and understanding how they help in solving different types of math problems.
Some strategies will be more efficient than others, and depending on what strategy a student chooses, this will tell us a lot about where a student is developmentally. When we know this, we can make sure that we implement the opportunities and next steps needed to continue to develop.
Sometimes, however, we will learn that a student is developmentally unable to choose. There are several variables that can affect the ability to choose an appropriate strategy.
The first issue surrounding student choice includes their ability to:
- read and understand the problem, and
- implement a strategy.
In order for students even start with being able to select a strategy, they need the ability to read and understand the information in the math problem, THEN they need to develop the ability to implement various strategies. Therefore, students might need different interventions and scaffolds to first understand the math problem, then they will need a repertoire that they could try to implement BEFORE they are able to make a selection. Their ability to implement a strategy will depend on what they have learned, practiced, and their mathematical mindset.
What can we do to build this repertoire of strategies that they can regularly practice?
Number Talks are just one way that students can learn about mathematical strategies. The power behind Number Talks lies in the visual representation of student thinking. This paired with the ability to help students listen, and engage in productive discourse helps them to make explicit connections with strategies. This modelling and practice will help students to become aware of the choices available, while at the same time allowing them to understand how other students think mathematically. Other ways to help students become aware of strategies include mini-lessons, and guided practice of strategies applied to various problem structures. (You may have great ideas yourself that I hope you will share here.)
However, some students will continue to struggle in their ability to choose an effective strategy. Math anxiety is a factor for some students that limits their ability to choose.
Math anxiety is another factor that can negatively diminish the ability to choose an appropriate strategy. We all either know first hand, and / or have witnessed the effects of anxiety in our students when it comes to math. I know that when I feel anxious, I become concerned about how others are evaluating my choices. Other intrusive thoughts may accompany anxiety, and this greatly reduces working memory for students. They may also feel like they have to choose the same strategy as their friends – or choose the one that their teacher is looking for. These types of intrusive thoughts greatly reduce the ability for a student to process the math problem at hand and make the best choices.
But perhaps choice is best discussed in terms of how strategy selection occurs at different developmental stages in math. Some students may be developmentally ready to recognize a strategy that is right for them, and educators can track this and assess student progress and decide upon the next steps to become more efficient.
However, other students will need extra support. They are our struggling students.
While many students are able to choose the most efficient strategy, others will need the strategy to be chosen for them – e.g., like the student who has a learning disability in visual-spatial/visual-motor skills who needs explicit visuals and precise strategies chosen for them for a while to help them learn the strategy to solve a particular math problem. Why not modify a math problem, or provide a visual of the strategy that a student should use, until they are able to commit the strategy to memory and decide how to use it in a problem situation on their own.
My question becomes, how do you teach strategies, and how do you help your students choose which ones to use, and use them efficiently?
Students with math difficulties often struggle with the skills surrounding addition and multiplication.
It is devastating for children who struggle with math, to be labelled as low achieving students, and score low on high-stakes assessments. Cognitive deficiencies can deeply affect the development of mathematical strategies that are involved even in simple addition and multiplication.
The Lawson Continuum is just one developmental trajectory that educators can look at to help us understand that students can use different strategies to solve the same problem. As students develop in math, and gain practice with problems, they are hopefully able to become more efficient with how they solve problems.
A neuro-typical student might start with less efficient strategies on the Lawson continuum, and over time, master more efficient strategies that help them develop mastery of the skill. The hope is that students that are developing ‘normally’, will be able to choose the most appropriate strategy for the specific problem they are working with, and will continue to be able to do this as problems get more and more difficult.
But what about the students who are not developing ‘typically’?
It is up to educators to make the changes necessary to help all children to be successful. If a student is following a different path of development than another child, it is up to us to identify that and figure out how we will help them learn.
It is important to note that not all students need complete the same math problem in the same way. If we can understand the unique needs and strengths of struggling students, and be very sensitive to the those needs and unique student experiences, we will make big differences in the lives of struggling students.
Educators can learn how students access math problems, and what stages of development a student is at, and what strategies can help move them forward. How the strategies are taught and modified, will depend on the student.
To help struggling students learn key mathematical strategies, we can:
- Adapt developmental math strategies and create iterations of strategies for students to practice with. Perhaps this includes different ways of ‘counting on’.
- Provide Choice: Encourage students to use their own strategies, or strategies that work best for them. If we are always requiring the same strategy from everyone, then some students will miss out.
- Provide feedback – focus on the task, not the student, and ask key questions that draws students to thinking about the strategy and where to go next. Help them to consider the reasonableness of their answer.
- Ask for students to orally describe the math – this helps to increase reasoning skills.
- Provide explicit instruction of a math strategy. This could be a mini-lesson for a guided group, or an individual conference.
- Practice the same strategy within different problem structures over and over again.
- Spend time discussing the problem, before solving the math. This helps students to practice and learn how to understand the math problem. Help them discover the ‘story’ in the problem. Put the literacy in the math.
- Design tasks that are just right for your struggling students. They do not have to be the exact same as everyone else’s math problems in the class. We do not need to have a one size-fits all approach where everyone always answers the same problem.
- Encourage struggling students to use a particular strategy frequently and consistently to build memory and mastery. Note that this does not mean solely memorizing a basic fact.
- Provide opportunities to practice familiar strategies with various problem structures.
- Visually model thinking for students as often as possible
- Model think-alouds for students so that they understand ways of ‘thinking’ about the math
Not all students develop at exactly the same rate – but some students will struggle greatly as they progress in elementary school. Ensuring that we still work to help these students in the ways described above will not only help them fill in gaps, but also will help them become better able to use strategies that will help them in their mathematical thinking both now and in the future.
Even if students are not developmentally ready to understand a more efficient strategy, they can still practice and master the strategies that are developmentally appropriate for them, in our efforts to ensure success.
As a math facilitator, I am often asked by educators often how they can support students with a diverse range of developmental needs. A problem of practice that I think we need to be aware of, is that a one-size-fits-all approach to teaching math, including instructional strategies, assessment and feedback processes can negatively impact learning and development of some students. While research has made great advances in instruction and assessment for students with diverse needs, I think we need to look deeper. Particularly when it comes to anxiety, and more specifically math anxiety. I don’t believe that anxiety is the same for everyone, so why would math anxiety be any different?
A lot of research exists into math anxiety already, especially when it comes to the cognitive impacts of math anxiety – for instance, we know that anxiety takes up valuable brain resources necessary for understanding the math. We also know that timed tests and drill and kill practices absolutely promote math anxiety. Jo Boaler has been part of extensive research that delves into the negative consequences of math anxiety, and also what educators can do differently in the classroom to ensure that we are not exacerbating it. Indeed, changing the way that we teach and view math goes a long way to preventing additional math anxiety.
However, what about student psychological development in math, and how anxiety mediates the perceptions of the instruction, assessment and feedback in math?
How do we scaffold the feedback and assessment in ways that impacts student learning in math?
The instruction, assessment and feedback strategies that might work for dealing with math anxiety in one child, may need to be very different than the anxiety that is presenting in another child.
How does math develop when students are developing along different trajectories of anxiety?
What does math anxiety look like in different children? How does instruction and assessment interact with a myriad of variables? What are the personal variables that each child is coming to the classroom with? Looking closer at the impact of instruction, assessment and feedback on the students, and how they perceive the strategies is important. This is in contrast to just looking at how to change math instruction so that students can cognitively improve, and looking toward developmental changes in math anxiety. The variables that research can look deeper into include socioeconomic status, gender, race/ethnicity, family dynamics, psychopathology, neurological differences and more.
I think that this could have very big implications for how we teach and facilitate math learning for elementary school students.
So I am left with questions like, do all children with math anxiety develop along the same developmental trajectories of learning? What about the child who has anxiety but still performs well, versus the child who also has dyscalculia? What if the anxiety is influenced by a separate mental health issue, a neurological difficulty, self-esteem issue, motivation issue? What if it is caused by a trigger in the surrounding environment? As such, are there different trajectories of instruction, assessment and feedback that we should be following?
Do we treat the underlying issues separately, and address math anxiety as a one-size fits all instructional approach? Or do we look at different developmental trajectories, differentiated instruction and assessment practices?
It sounds like common sense, but there appears to be little research exists that looks at how the students develop in math through their anxiety, and how instruction, assessment and feedback impacts this.
I think that as I delve into this topic more and begin to understand what supports are necessary for children who are experiencing math anxiety, a greater realm of possibilities will emerge.
I look forward to hearing thoughts,
Facilitating quality math talk in the classroom is essential to promote student learning. However, this can be very challenging.
One cornerstone of effective math talk, is the ability to listen, question, paraphrase and learn from other students. Despite the fact that teachers create classroom norms and expectations around effective communication, students may be inadvertently reinforced to listen to the teacher only.
Why does this happen?
This could happen for a myriad of reasons, like the teacher feels pressure to complete the lesson in a set time frame, or an overwhelming need to cover too much content, not wanting to embarrass a student who has just shared a misconception, perhaps we as teachers feel pressure to teach the way we were taught, or maybe we just do not yet see the importance of focusing on what the students are saying vs what we are teaching. Therefore we don’t appropriately follow through with student thinking. Regardless of the reason, it is important to recognize what our own patterns are so that we can adequately identify the steps that we need to help create a Math Talk Community.
What is a Math Talk Community?
A math talk community is one where students play an active role in listening to their peers, engage in conversation about the math, and learn to give each other effective feedback. In this type of community, the onus moves from the teacher to paraphrase what students are thinking, to the students themselves to paraphrase and reiterate what someone else has said. It continues until all students are engaged and making connections. Students are encouraged to regularly use the vocabulary, learning goals available to personalize their own strategies, create representations, and ask questions of others in meaningful ways. In a math talk community, teachers use pedagogical strategies and supports to scaffold student learning, and help students to understand that they are not on their own, have the time and space to work with new and old ideas, and not merely rely on the teacher. Strategies can include, but not limited to:
- Feedback dice or cards,
- Whole class discussions during consolidation after minds-on, and action parts of a three-part lesson
- Number Talks
- Group work opportunities
- Regular opportunities for students to paraphrase each other, ask questions
- Visible vocabulary walls paired with visuals and frequent connections to them in a variety of new math learning settings
- Reviewing and connecting regularly back to the the learning goals and success criteria
- Co-created success criteria
- Promoting effective questioning to other students and the teacher
- Visually representing student thinking on the board
- Keeping an idea going long enough to engage all students in the classroom
What happens when a Math Talk Community develops?
In a math talk community, much like a #feedbackfriendly Classroom, students learn more deeply. This is because the time, space, and opportunity are provided to get at any misconceptions. If misconceptions are not addressed, then we can run into problems because this is where students may get stuck understanding what to do to get better, begin to fake what they know, and develop fixed mindsets and negative views about their math learning.
Communication is such an important cornerstone of a math classroom. It is essential to develop math talk so that students can develop true conceptual understandings, and become flexible with strategies and ideas that can be applied to their math learning and the world around them.
What strategies do you use to cultivate math talk in your classroom?
Truth and Reconciliation and Literature in the Library Learning Commons – Canadian School Libraries Journal
— Read on journal.canadianschoollibraries.ca/truth-and-reconciliation-and-literature-in-the-library-learning-commons/
Feedback in the Library Learning Commons – Canadian School Libraries Journal
— Read on journal.canadianschoollibraries.ca/feedback-in-the-library-learning-commons/