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]]>Ensure multiple opportunities for assessment, provide immediate feedback to students, create a guideline for completing rigorous mathematics independently.
A checklist assessment is an alternative to a summative assessment that provides feedback to students as they complete their assessments. Students are provided a cover sheet which includes the standards and skills they must demonstrate to meet the learning criteria. Using multiple versions of an assessment, students must demonstrate understanding (in a portfolio, on a whiteboard, on an assessment paper). When done, students return to the teacher with the assessment in hand, the teacher can then review and provide feedback instantly about the evidence shown on the paper. Usually, teachers can return the quiz to the student to make improvements and resubmit.
Persistence is key for both teachers and students. Teachers must insist that students contribute high-quality work, and students must persist in advancing thinking and improving their submissions. When students have completed enough marks on the checklist, the student will have demonstrated understanding.
Increasing student staus within group work, increase engagement of all learners, problem-solving, independent learners
Students are provided with clue cards that in some way inform the group of the activity, students are only allowed to look at their card and need to communicate its meaning with others in the group. Students can read the card, but ultimately, each student is responsible that their information is included in the final product.
A helpful tool to ensure this is happening is to remind groups before submitting their product that everyone needs to check that the information on their card is being implemented correctly. This may be an opportunity for a Huddle or I-Spy as a strategy to support more students. Tim Erickson has published a few books that have a lot of these types of problems. Here’s an example.
Increase access, help get a group unstuck without reducing rigor, suggest alternate methods, listen to an idea.
As an alternative to giving new directions, a group huddle can allow groups to continue working while also communicating a consistent message to each group. Have groups send a nominee to get the information to bring back. Stating with a huddle also gives a clear start, where one group member must take ownership of the task to share the guidelines with teammates.
Additionally, Group Huddles help groups get unstuck. For example: Bringing the task manager into a huddle the teacher may suggest a question to discuss to change the direction of the overall group conversations. Or a huddle could allow a student from one group to share information with other groups and those students can build on the first group’s thinking. For this situation, it’s important that students can build off the information from the huddle, otherwise, they are not taking full advantage of the task.
Support groupwork without reducing cognitive demand, share ideas, revising thinking.
If a team is stuck on part of a task, but another group is moved on, have a member of the “stuck” group to another team to spy on the other group’s work. The “Spy” should not talk with other groups and should watch until they understand how to get their own group unstuck. Spy’s then reports the idea back to their own group and moves forward. Low Status students make great Spy’s so they can bring back critical information to their group
Alternatively, during a task requiring a lot of justification, the teacher places a “Hint Sheet” in a location around the room. The group’s job is to reason through the solution and The Spy is the only person who can see the hints and may not bring the paper back. The Spy must report the ideas to the team clearly and may return to the hint sheet as many times is needed throughout the activity. Teachers should be vigilant to ensure this strategy is used appropriately.
Differentiate instruction, provide just in time feedback to students/groups, rehearse procedural tasks.
Levels Tasks are an opportunity for students to practice skills learned and increase rigor over time. Provide students with a half-sheet of the task. Then, when the group believes they have correctly answered and all in the group can explain, have the group manager ask for teacher review. When groups correctly answer and justify their reasoning, award them with another “Level” and repeat the process. Some students view levels as a challenge and make the activity a competition.
When making Levels, put the key learning in Level 1 or Level 2 and supplement in the higher levels (pre-teaching). The goal is that all students in the class will demonstrate mastery of the beginning levels and groups who move quickly will access higher cognitive demand or more challenging content. Not all students need to get to all levels, but everyone needs to get through ________. Do not tell students about the minimum level (otherwise, this task will backfire), rather frame the task as a challenge…”I wonder how many levels your group will get”.
Depending on the activity, students usually need about 5 or 6 Levels prepared.
Students use error analysis to address common errors in thinking. Increase the amount of justification for procedures.
The premise of a “Math Hospital” is that students are the doctors and they are operating on “sick” problems. All problems come into the hospital sick and need to be:
This could be combined with other strategies including Jigsaw or Select & Sequence. Provide students with ample time to address all three of these ideas. Students may benefit from a reporting template to help organize ideas.
Assessment practice, updating/revising thinking, fluency practice, group interactions.
Provide students with Multiple Choice forms (bubble sheets, scantron, ZipGrade form)and a multiple-choice/select-all-that-apply test that is similar to an upcoming assessment (SBA, SAT, ACT, or in-class assessment, etc.). During class, students are given the assessment to work on in groups, they must agree on answers together and as a team brings up their bubble form. Using the multiple-choice question checker (Scantron or ZipGrade form), in-class quickly grade the quiz and return to students. Then, students must recheck and provide additional information for the questions they got wrong as a group. They have at least 2 (or more depending on the need) to build mastery of the assessment questions.
Check out https://www.zipgrade.com/ if you do not have a Scantron form at your school.
Critiquing the reasoning of others, explain the reasoning, connecting mathematical concepts, compare ideas of others, understanding the logic of others.
Students are provided with only part of the task. The task has progress towards a solution, but the bottom part is missing…the ‘dog’ ate it. So, students must prepare the ending of the started work using the reasoning provided to complete the task and generate a viable argument.
Another approach to this group strategy is to provide only part of the work…the ‘rip’ is vertical versus horizontal…or take a small chunk out of the side. The missing work should be strategic to encourage group thinking and public reasoning.
Consider providing alternative procedures to common problems to help students approach a new line of reasoning.
Communicate high-quality group work behaviors, encourage collaboration, provide feedback about processing skills.
Contrary to the name of this strategy, it is not a quiz. Rather this is a piece of formative assessment for students about how they are working in a group. Simply, a participation quiz requires a few key elements.
When using a Participation Quiz, at the beginning of the task, teachers must state some “look-for” behaviors when students are working in groups (all heads in, using group roles, re-reading the task card). Then, as student engage, have a public space to write public feedback to groups. Encourage a specific group member to review the teacher notes periodically. During the task, or at the end of class, provide feedback about some strengths and areas of growth in the group processing skills observed. It’s amazing what students will pick up on without verbal reminders.
This is an opportunity to highlight good things that are happening and provide awareness for those not engaged in the ways outlined at the start of the lesson. Feedback in a participation quiz should not penalize a single student, but make them aware of the group behaviors witnessed. Think court reporter…the public notes and feedback are to document, not assign judgment.
Participation Quiz’s change the role of the teacher significantly from an instructor to a facilitator of the learning process. This shift takes some time for both teachers and students to become more familiar.
Build ideas consecutively, increase the participation of groups to a whole class discussion.
Teachers notice that different groups are having amazing conversations, but not all of the groups are talking about the same ideas. Teachers want others to hear these great conversations, so a select and sequence allows teachers to strategize to choose parts of the conversation to bring forward to the whole class. There are several approaches to this, consider reading “5 Practices for Orchestrating Productive Mathematics Discussions” by Smith & Stein for more detailed examples. Select & Sequence is just part of the 5 practices, but very effective and simple to implement. We recommend you practice all 5!
Tasks can be long and overwhelming. Sometimes it’s nice to show students a summary of this work. At a point during the task when students need a larger picture, select groups who have strong explanations of their work. Sequence these students to discuss a specific part of their work to the class. To make this more successful, prime the groups you want to talk before having them share…for example: “Sarah, when I call on your group, I want you to share out how your group came to this conclusion…team members, help Sarah get ready to share out.” Doing this for multiple groups allows for a quick and successful conversation. Encourage all students to listen to the sharing…consider your Options for Structuring Mathematical Talk.
Some mathematicians can benefit in generating models and making conclusions by examining many ways of understanding a single representation. Many examples of this exist online, here are a few that support using this option to help mathematics.
Some ideas found on this page are adaptions from work created by Jess Griffin, Karen O’Connell, William (Bill) Feeley, San Lorenzo High School Teachers, and College Preparatory Mathematics (CPM). Activities have been modified based on classroom experience by the author.
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]]>We want students to be modelers of mathematics. This task provides them with an authentic opportunity to model with mathematics using data that they are generating. Students will come to understand that the mathematics behind these tasks is very messy and models are imperfect. So, students use their understanding about the parts of exponential functions to create a model from the data.
Download PDF
Download Word Doc
Provide students about 100-200 skittles in a small cup.
This is a Level 3 task…with some modifications to the task instructions (removing the procedure, reducing the report structures) this task could easily represent a Level 4 task. In total, students must collect multiple sets of data, plot that data, make conclusions about that data (average common ratio), use the information collected and analyzed to generate an equation, graph that equation, and determine the goodness of their graph. With additional questions, students are asked to model and determine the quality of this model which requires a lot of mathematics. Teachers are encouraged to modify this task for their needs.
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]]>These examples may be downloaded and used, or provide inspiration.
Bags and Blocks Task examplesUsing the concept of a balance beam, student reason through the process of identifying the number of blocks inside a “mystery bag.” In doing so, students are practicing the reasoning of solving an equation, often complex equations using the reasoning of a balance beam.
I like to start by asking students to consider the meaning of the word “equal.” In general, students reason that equal means fair or balanced in some way. Then I show them a makeshift balance beam using buckets and a meter stick, depositing binder clips into the buckets. Eventually, I want students to determine that there are two ways to make this balanced:
Then, I throw out an idea that some binder clips have a lifting force, and suggest that I add one of those lifting clips to one side and then ask groups to determine at least two different options that they could do (knowing they have unlimited resources) to make it balanced. I will select and sequence until students say:
At this point, students have EVERYTHING they need to start the task. Some students will not understand the second part of this demonstration, that is okay for right now. Let them play around and explore.
Simple example explaining balance beams: https://youtu.be/5q9wy7G2v5U?t=35
Online interactive balance beam: http://www.learner.org/courses/learningmath/algebra/session6/part_c/index.html
My friend Nancy Ku developed two tasks that she uses in her class, she has allowed them to be shared here:
Copy and paste the items to the left of the slide…
Henri Picciotto’s MathEd.page is one of a kind and has an abundance of resources, I’ve gotten sucked into this site for hours. He invented a similar procedure to Algebra Tiles called LabGear and has published the whole book for free on his website. Learn and use Lab Gear without an investment. https://www.mathed.page/manipulatives/lab-gear.html
Algebra Tiles and Algebra Blocks are products that are valuable for students to learn more about algebra:
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]]>The post Rough Draft Talk appeared first on MathematicalTasks.org.
]]>This simple strategy relies on the work of our ELA colleagues when teaching the writing process. Expecting students to share parts of ideas is important because it sparks the ideas of other students. Additionally, expecting imperfect, rough draft responses, increases engagement and student status. Students feel good about participating in class!
When students get stuck, provide the space to talk about their thinking with a partner. Give a prompt, sentence starter or leading question to motivate the conversation. Be sure to use the words “Rough Draft” so students know your expectation that their responses are not perfect. Share within groups or as a whole class.
Please share your experiences with using “Rough Draft Talk” in the space below.
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]]>During my student teaching, my mentor teacher and I invented this rubric to assess student understanding. The generic rubric below was created to accurately assess each concept that appeared on assessments. We used the key phrase “evidence of understanding” to measure what was on the page. Scores below as 6 were rarely awarded, so there was little need to differentiate between every single integer.
I think this generalized rubric is superior to a 4 point scale. There is no need to convert the rubric score to percentage scores. It communicates mastery clearly for both students and families, and it differentiates levels of understanding.
Later, I became more accustomed to SBG and decided to modify the rubric I was using to be more equitable based on my school’s grading policies. Students who made no attempt or did not make sufficient progress were being disproportionately penalized for not attempting the work…so I adjusted my rubric.
You may question my decision to eliminate “0” through “3”…some say “If they did zero work, they get a zero in the grade book.” Unfortunately, that is not how most students think, even on a percentage scale, 4 is still failing! This is important because of the mindset of students. When students receive a failing grade, they self-assign blame and students reduce the behaviors that promote learning. So, if a student fails a single standard, they do not have an overwhelming amount of work to improve. I don’t want their perception of their grades to prevent them from learning.
Finally, in my current system of grading, we award students grades between 0.5 and 4 incrementing by 0.5 (see below). I rewrote a standard to adjust my assessment of the standards taught to reflect this new scale.
Please feel free to use these rubrics in your classroom or adjust them as needed. Comment below if you have questions or ideas about these rubrics to measure student understanding.
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]]>The post Exponents Exploration appeared first on MathematicalTasks.org.
]]>Students knowing the exponent properties is essential, more importantly, however, is knowing how they came to be. This task uses what students know about the definition of exponents as repeated multiplication and supports them in making generalizations from many example problems. Students are supported in connecting their understanding of the definition of an exponent to the properties that can be used to streamline the process. With an emphasis on finding patterns, using repeated reasoning, justifying reasoning and generalizing results, students will quickly learn the 6 most common exponent properties and be able to explain why they work!
Download: Exponents Exploration
This is a Level 3 task in that students are making connections between the exponent definition and the exponent properties. They are learning a procedure for simplifying which makes this a “Procedures with Connections” task. We are suggesting a method and supporting with a structure for how to determine these procedures, but we do not suggest a solution or a conjecture for each of the six suggested properties. Many aspects of this task maintain high cognitive demand because students will need to justify their work to generalize procedures.
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]]>Students are asked to have discussions with their partners all the time. But how many of those conversations ensured equity and structure in WHO was talking and WHEN? Students are terrible at having conversations, it’s a skill built over time. They often gravitate to their friends in the group. Alternately, they assign a status to each person in their group and vie for the one they deem is the “smartest”. Either this happens or students just give up completely.
Students want to know that there is fairness in the talking, even if they don’t want to participate. When implementing this structure, it’s important to tell students that this will ensure equity, provide a talking worthy question (academic or not) and ensure that each table member has an opportunity to talk for the same amount of time. Specifically say … if a student doesn’t want to talk, that’s OKAY! They will have their full 30 seconds where no one else should be talking. Name that sometimes this is awkward and that’s also okay. Students should see that they will have their chance to talk in this group. I’ve often done a simple A, B, C, D order.
Next, consider the diagram, there are LOTS of “pairings” that teachers can choose from to ensure certain students talk with other certain students. This may be by convenience or strategic.
Here’s a brain exercise, how many different (and what are) the orientations of groups of 2? (answers below) How can you mix up the ordering? Will you be strategic about the groupings? Why?
Especially when starting to implement group talking structures, students will not know what the expectations are or how important the structure is. This sounds crazy, but every time at the beginning of the year when any student does not follow this routine, I stop the class and clarify the importance of following this structure to ensure equity. I’ll say something like, “Someone in class was not following the structure and took someone else’s turn talking before their time was up. Let’s start this person’s turn over again so they have their allotted time. Remember, everyone has something important to say, others need ‘think-time’, this might be awkward sometimes, but please follow this routine.” After doing this a few times my students understand it’s importance.
And, why so carefully to ensure students are talking so specifically to other students? Isn’t this overkill?
Part of the human condition is to avoid working hard, students are not exempt from this natural state. Along with avoidance, many students not accustomed to this way of thinking and learning defer to other students who they believe know more about what is going on. As educators, we can not allow students to opt themselves out of the conversation. Students opt out because they:
So, what do we as teachers need to do? My solution is to afford students the opportunities to continue to engage. Some strategies to allow more students to talk because of the items listed above:
In my experience, doing these things have allowed more students to speak and feel valued in the conversations happening in their group or in class. It also reduces one or two students in class dominating the conversation which increases those student’s academic status but allows for students to opt-out of the learning.
Finally, students need to have a reason to talk in their group, structuring talk will only work if there is a conversation-worthy stimulus to talk about. There is an article titled “Options for Structuring Student Talk” that discusses ways to create talk worthy conversations.
Please let me know if I have left out any grouping:
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]]>This document is aimed at providing teachers with a tool for some options when implementing a lesson in class. Deciding which door to choose is getting easier!
See below for a few examples:
Structured Talk OptionsDownload: Structured-Talk-Options
To get students to talk, teachers need to provide an opportunity to have a discussion. So, provide a choice to consider a mathematical idea.
“A mathematician said ‘such and such’ about this idea. In your group consider if you agree or disagree with that option…group member C will start and ensure everyone talks. Prepare person D to share your group’s ideas with the class.”
Then, as students talk, decide on how these ideas will be shared with the greater class. Listen as you walk and consider your options in the second table.
“Julian, tell us about your group’s thoughts…Cassandra, could you add on to Julia’s ideas based on what your group discussed.”
This option strategically selects and sequences more than 1 student to discuss connections between math ideas in a rough draft kind of way. This MAY NOT be what is needed for your classroom at a given time, it is an option that was available during this teacher’s discussion. Maybe try another door.
After listening to a lot of groups, many students are simultaneously reaching an error that you want to make sure they are aware of. It may seem at this point that the students just aren’t getting it…then you remember you have an option. One student in a group knows there’s a wrong path, but does not know how to get unstuck.
“Alright class, we know that learning is a little messy, I want to show you an interesting thing I saw at Gabe’s Group…I wonder if your thinking is similar to Gabe’s.”
Gabe approaches the projector to show his method then you ask…
“I heard you tell your group that you didn’t think this was working…what led to that thought?”
As Gabe explains…you now have the option to try a different cycle and possibly introduce a new idea about what “your last class” tried that worked for them. Students now have the option to try a different route and the class is still a success.
Consider exercising these options very frequently in class, change it up and plan for the conversations. Student structured talk doesn’t happen on accident and the more preparation will increase the quality and the likelihood that good conversations are happening.
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]]>This task is very powerful and requires students to strategize to calculate the viability of making a stain glass window given the cost constraint. Students are provided with little guidance on how to approach this problem but are given lots of information about the problem and the constraints of the task. Teachers should be prepared to support students appropriately using group roles when using this task as it may be simple for select students to “take over” use group roles and various strategies to keep all students within the group responsible for a part of the task and ensure all can explain what is happening on their partner’s paper.
Stain Glass Task (online)This task provides a lot of information for students to sift through, while there is an outcome, there are many ways to approach this task. We believe this is a level 4 task, especially when group roles are being used since it requires relational thinking between context, geometry, and constraints without the suggestion of a procedure. Some teachers may choose to help students connect the negative space pre-activity to make connections to another activity. This would arguably decrease the cognitive demand since it suggests a procedure for calculating the area. Ultimately, students will be required to invent a procedure to solve this problem and appropriately answer the question.
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]]>The post Task Analysis Guide appeared first on MathematicalTasks.org.
]]>These 4 categories guide the work of this site. Level 1 and 2 tasks are plentiful and can be found in most math textbooks from elementary school to collegiate texts. They do have a place in a mathematics classroom, to increase the fluency for students after the concepts are learned, however, more classrooms need more Level 3 and Level 4 mathematical tasks and teachers need access to those tasks.
Task Analysis GuideNCTM publishes many articles about the Task Analysis Guide and uses it in many of their papers and research articles including “Principle’s to Action” where they summarize the research. They elude to conclusions that students who engage in high cognitive demand tasks learn more than those rich in procedures alone and that Mathematical Tasks are challenging to implement and are often transformed into less cognitively demanding tasks during instruction.
Notice in the Lower Level demands students are reciting rules or reciting procedures without much attention for why the procedure works or engaging in multiple mathematical practices or habits of mind. The High-Level demand tasks require students to use models to make connections between representations in different ways and engage in thinking that requires justification or explaining a solution, skills students need even without mathematics.
Very quickly, teachers can change the types of conversations between students in class by providing them with the opportunities to engage in tasks that are higher cognitive demand and supporting those tasks well. There are many examples of tasks that can raise the level, there are some strategies to help with that, we hope to share those in future articles on this site.
However, simply providing students with Level 3 and 4 Tasks does not increase the outcomes alone, teachers are critical in switching their role in the classroom as well. Teachers must transition from being the “holders of knowledge” in the classroom to being the facilitator of knowledge.
Without proper execution, High Cognitive Demand Tasks lose their value in the classroom and can turn quickly into something they were not intended. That is why the teacher article section of this site is so critical in providing tools for teachers to implement well and find success along the way. After teachers select tasks, tools for asking great questions and facilitating are needed. Students often want to replace the burden of thinking back onto the teachers (because they know it’s easier for them), but teachers must resist this temptation of taking over the thinking for the student which can easily happen.
Implementing tasks in a classroom is challenging, however, it is very possible. Plus, the more teachers who implement tasks well in their classrooms, the more students understand the expectations for engaging in them and then more learning happens along the way. Task implementation is for everyone, it raises the bar for all involved regardless of background experience. Tasks can also be very frustrating for new learners (both students and teachers alike), but with patience practice and rehearsal, they become more simple to implement and students will learn more.
If you would like to learn more about this work and more, take a look at NCTM’s “Principle’s to Action” book. It is a very agreeable read and support for specific learning outcomes for teachers. Highly recommended.
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