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Lesson and Assessment Design – #T3Learns

What are we intentional about in our planning, process, and implementation?

  • Are the learning targets clear and explicit?
  • What are important check points and questions to guide the community to know if learning is occurring?
  • Is there a plan for actions needed when we learn we must pivot?

On Saturday, a small cadre of T3 Instructors gathered to learn together, to explore learning progressions, and to dive deeper in understanding of the Standards for Mathematical Practice.

The pitch:

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Jill and I fleshed out the essential learning in more detail:

  • I can design lessons anchored in CCSS or NGSS.
    • I can design a lesson incorporating national standards, an interactive TI-Nspire document, a learning progression, and a formative assessment plan.
    • I can anticipate Standards for Mathematical Practice that learners will employ during this lesson.
  • I can design a learning progression for a skill, competency, or process.
    • I can use student-friendly language when writing “I can…” statements.
    • I can design a leveled assessment for students based on a learning progression.
  • I can collaborate with colleagues to design and refine lessons and assessments.
    • I can calibrate learning progressions with CCSS and/or NGSS.
    • I can calibrate learning progressions with colleagues by giving and receiving growth mindset oriented feedback, i.e. I can offer actionable feedback to colleagues using I like… I wonder… what if…
    • I can refine my learning progressions and assessments using feedback from colleagues.

The first morning session offered our friends and colleagues an opportunity to experience a low-floor-high-ceiling task from Jo Boaler combined with a SMP learning progression. After the break, we transitioned to explore the Standards for Mathematical Practice in community. The afternoon session’s challenge was to redesign a lesson to incorporate the design components experienced in the morning session.

Don’t miss the tweets from this session.

Here are snippets of the feedback:

I came expecting…

  • To learn about good pedagogy and experience in real time examples of the same. To improve my own skills with lesson design and good pedagogy.
  • Actually, I came expecting a great workshop. I was not disappointed. I came expecting that there would be more focus using the TI-Nspire technology (directly). However, the structure and design was like none other…challenging at first…but then stimulating!
  • to learn how to be more deliberate in creating lessons. Both for the students I mentor and for T3 workshops.
  • I came expecting to deepen my knowledge of lesson design and assessment and to be challenged to incorporate more of this type of teaching into my classes.

I have gotten…

  • so much more than I anticipated. I learned how to begin writing clear “I can” statements. I also have been enriched by those around me. Picking the brains of others has always been a win!
  • More than I bargained. The PD was more of an institute. It seemed to have break-out sessions where I could learn through collaboration, participation, and then challenging direct instruction, … and more!
  • a clear mind map of the process involved in designing lessons. A clarification of what learning progressions are. Modeling skills for when I present trainings. Strengthening my understanding of the 8 math practices.
  • a better idea of a learning progression within a single goal. I think I had not really thought about progressions within a single lesson before. Thanks for opening my eyes to applying it to individual lesson goals.

I still need (or want)…

  • To keep practicing to gain a higher level of expertise and comfort with good lesson design. Seeing how seamlessly these high quality practices can be integrated into lessons inspires me to delve into the resources provided and learn more about them. I appreciate the opportunity to stay connected as I continue to learn.
  • days like this where I can collaborate and get feedback on activities that will improve my teaching and delivery of professional development
  • I want to get better at writing the “I can” statements that are specific to a lesson.
  • I want to keep learning about the use of the five practices and formative assessment.

We want to see more collaborative productive struggle, pathways for success, opportunities for self- and formative assessment, productive conversation to learn. and more.

And so the journey continues…

[Cross-posted on Experiments in Learning by Doing]

 
2 Comments

Posted by on October 19, 2014 in Professional Development, Uncategorized

 

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Systems of Linear Equations

We are working on Systems of Linear Equations in our Algebra 1 class.

The first day, our lesson goals were the following:

Level 4: I can create a system of linear equations to solve a problem.

Level 3: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

We used the Mathematics Assessment Project formative assessment lesson Solving Linear Equations in Two Variables to introduce systems.

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Then we asked students to draw a sketch of a system of equations with one solution, a sketch of a system of equations with no solution, and a sketch of a system of equations with many solutions.

We used the Math Nspired activity How Many Solutions to a System to let students explore necessary and sufficient conditions for the equations in a system with one, no, or many solution(s). We used a question from the activity so that students could solidify their thinking about the relationship between the equations for different solutions.

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We sent the questions as Quick Polls so that we could formatively assess their thinking and correct misconceptions.

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We like using this type of Quick Poll (y=, Include a Graph Preview) while students are learning. The poll graphs what students type so that they can check their results and modify the equation when they do not get the expected result.

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(Note: on the second day, we asked again for students to create a second equation to make a system with one, no, and many solution(s), but the given equation is in standard form.)

 

On Day 2 of Systems, the learning goals are slightly different.

Level 4: I can solve a system of linear equations in more than one way to verify my solution.

Level 3: I can create a system of linear equations to solve a problem.

Level 2: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

We started with a task from Dan Meyer’s Makeover series, Internet Plans. Part of this makeover was a complete overhaul of the context. Students write down a number between 1 and 25. They view this flyer (from Frank Nochese).

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And then they decide which gym membership to choose, using the number they wrote down as the number of months they plan to work out.

130708_2lo

When I saw the C’s and B’s on the number line on the blog post, I actually thought that the image was beautiful – a clear indication that one plan is better than the other two for a while, and then another plan is better than the other two. This image absolutely leads to the question of which one becomes better when – and thinking about why plan A is never the better deal.

Our students started calculating. Several asked what $1 down meant before they could get a calculation. Then they shared their work on our class number line.

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We were dumbfounded at the results. Really. It was hard to know where to start. We were prepared to help student develop the question with correct calculations. We hadn’t thought about what we would do when students couldn’t figure out which plan was the better deal for a specific, given number of months.

What would you do next if this is how your students answered?

 

We started with the lone C at 4 months, hoping someone would claim it and share his calculations. He did: I divided $199 by 12 months and then multiplied by 4 to get the cost for 4 months.

 

Another student said he couldn’t do that because you had to pay $199 for 12 months no matter how many months you actually used it.

 

Another student said that A was the better deal because they were planning to get every other month free by working out more than 24 days each month. We discussed how realistic it is to work out at a gym 24 days a month.

 

We did eventually look at a graphical representation of each plan. And talked about what assumptions had been made using these graphs.

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Can you tell which graph is which plan?

Can you write the equations of the lines?

In the first class we never made it to actually calculating the point(s) of intersection (class was shorter because of a pep rally), but other classes did calculate the point(s) of intersection.

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It’s not a waste of time to think about Plan C costing $199 for 12 months.

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But then we really should have created a piecewise function to represent the cost for 13-24 months as well.

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The number lines weren’t much better in subsequent classes.

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So maybe the students got to practice model with mathematics, just a little, even though we have a long way to go before students will be able to say, “I can create a system of linear equations to solve a problem”.

The teachers were definitely reminded of how much work we have to do this year, as the journey continues …

 

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SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 2)

We want every learner in our care to be able to say

I can reason abstractly and quantitatively.
(CCSS.MATH.PRACTICE.MP2)

SMP2

But…What if I think I can’t? What if I have no idea how to contextualize and decontextualize a situation? How might we offer a pathway for success?

We have studied this practice for a while, making sense of what it means for students to contextualize and decontextualize when solving a problem.

Students reason abstractly and quantitatively when solving problems with area and volume. Calculus students reason abstractly and quantitatively when solving related rates problems. In what other types of problem do the units help you not only reason about the given quantities but make sense of the computations involved?

What about these problems from The Official SAT Study Guide, The College Board and Educational Testing Service, 2009. How would your students solve them? How would you help students who are struggling with the problems solve them?

There are g gallons of paint available to paint a house. After n gallons have been used, then, in terms of g and n, what percent of the pain has not been used?

SMP2 SAT1

A salesperson’s commission is k percent of the selling price of a car. Which of the following represents the commission, in dollar, on 2 cars that sold for $14,000 each?

SMP2 SAT2

In our previous post, SMP-2 Reason Abstractly and Quantitatively #LL2LU (Take 1), we offered a pathway to I can reason abstractly and quantitatively. What if we offer a second pathway for reasoning abstractly and quantitatively?

 

Level 4:

I can create multiple coherent representations of a task by detailing solution pathways, and I can show connections between representations. 

Level 3:

I can create a coherent representation of the task at hand by detailing a solution pathway that includes a beginning, middle, and end.  

Beginning:

I can identify and connect the units involved using an equation, graph, or table.

Middle:

I can attend to and document the meaning of quantities throughout the problem-solving process.

End:

I can contextualize a solution to make sense of the quantity and the relationship in the task and to offer a conclusion. 

Level 2:

I can periodically stop and check to see if numbers, variables, and units make sense while I am working mathematically to solve a task.

Level 1:

I can decontextualize a task to represent it symbolically as an expression, equation, table, or graph, and I can make sense of quantities and their relationships in problem situations.

 

What evidence of contextualizing and decontextualizing do you see in the work below?

Screen Shot 2014-09-24 at 8.43.43 AM Screen Shot 2014-09-24 at 8.46.55 AM Screen Shot 2014-09-24 at 8.43.55 AM 

 

[Cross-posted on Experiments in Learning by Doing]

 

 

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SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 1)

We want every learner in our care to be able to say

I can reason abstractly and quantitatively.
(CCSS.MATH.PRACTICE.MP2)

SMP2

I wonder what happens along the learning journey and in schooling. Very young learners of mathematics can answer verbal story problems with ease and struggle to translate these stories into symbols. They use images and pictures to demonstrate understanding, and they answer the questions in complete sentences.

If I have 4 toy cars and you have 5 toy cars, how many cars do we have together?

If I have 17 quarters and give you 10 of them, how many quarters will I have left?

Somewhere, word problems become difficult, stressful, and challenging, but should they? Are we so concerned with the mechanics and the symbols that we’ve lost meaning and purpose? What if every unit/week/day started with a problem or story – math in context? If learners need a mini-lesson on a skill, could we offer it when they have a need-to-know?

Suppose we work on a couple of Standards of Mathematical Practice at the same time.  What if we offer our learners a task, Running Laps (4.NF) or Laptop Battery Charge 2 (S-ID, F-IF) from Illustrative Math, before teaching fractions or linear functions, respectively? What if we make two learning progressions visible? What if we work on making sense of problems and persevering in solving them as we work on reasoning abstractly and quantitatively. (Hat tip to Kato Nims (@katonims129) for this idea and its implementation for Running Laps.)

 

Level 4:

I can connect abstract and quantitative reasoning using graphs, tables, and equations, and I can explain their connectedness within the context of the task.

Level 3:

I can reason abstractly and quantitatively.

Level 2:

I can represent the problem situation mathematically, and I can attend to the meaning, including units, of the quantities, in addition to how to compute them.

Level 1:

I can define variables and constants in a problem situation and connect the appropriate units to each.

 

You could see how we might need to focus on making sense of the problem and persevering in solving it. Do we have faith in our learners to persevere? We know they are learning to reason abstractly and quantitatively. Are we willing to use learning progressions as formative assessment early and see if, when, where, and why our learners struggle?

Daily we are awed by the questions our learners pose when they have a learning progression to offer guidance through a learning pathway. How might we level up ourselves? What if we ask first?

Send the message: you can do it; we can help.

 

[Cross-posted on Experiments in Learning by Doing]

 
4 Comments

Posted by on September 28, 2014 in Standards for Mathematical Practice

 

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Origami Regular Octagon

We folded a square piece of paper as described in the Illustrative Mathematics task, Origami Regular Octagon. I didn’t want students to know ahead of time that they were creating an octagon, so I changed the wording a bit. We folded (and refolded … luckily, there was not a 1-1 correspondence between paper squares and students). Students worked individually to write down a few observations and then we talked all together.

Screen Shot 2014-09-21 at 8.47.08 PM

It’s an octagon.

There are 8 equal sides.

There are 8 equal angles.

It’s a regular octagon (this is the first year my students have come to me knowing what it means for a polygon to be regular).

How do you know there are 8 equal sides and 8 equal angles?

Because we folded it that way.

How do you know there are 8 equal sides and 8 equal angles?

Because one side is a reflection of its opposite side about the line that we folded.

What is the significance of the lines that you folded?

They are lines of symmetry.

There are 8 of them.

The opposite sides are parallel.

How can you tell?

This took a while. Maybe longer than it needed to.

Another student raised his hand.

I figured out that the sum of the angles in the octagon is 540˚.

(I don’t have the sum of the interior angles of an octagon memorized since I can calculate it, but I did know that 540˚ was too small.)

How did you get that?
I made an octagon and measured the angle. Then I multiplied by 8.

On your handheld?
Yes.

Okay. Let’s see what you have. I made him Live Presenter.

He showed us the angles he measured that were 67.5˚.

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It might help if we can see the sides of your angles. Will you use the segment tool to draw them?

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Other students argued that we needed to double 540 to get the sum of the angles in the octagon, 1080˚.

What else do you notice?

Triangles.

Congruent triangles.

Right triangles.

Students noticed different numbers of triangles.

And they recognized that we knew about congruence because of reflections.

Somehow we asked the question about the value of the angle (x).

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I set up a Quick Poll to collect student responses.

Almost everyone got the correct answer of 22.5˚.

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Can you tell me how you got your answer?

One student used the ¼ square with a 90˚ angle that had been bisected by the folded line to be 45˚ and the bisected again by the folded line to argue that x was 22.5˚.

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Did anyone do something different? Hands went up all around the room.

AC hasn’t talked to the whole class yet today, so I asked what she did.

I saw a circle with 360˚ and divided by 16.

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Then DC’s hand went up. 360/16 is equivalent to 180/8. I saw a line divided into 8 equal parts (or straight angle).

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Then TC showed us the isosceles triangle she used with the 62.5˚ base angles.

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Then someone else showed us the right triangle he used with the complementary acute angles.

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Before we knew it, we had spent almost an hour talking about a regular octagon. And learning math using quite a few Math Practices: construct a viable argument and critique the reasoning of others, look for and make use of structure, use appropriate tools strategically.

I’ve wondered before how much longer we will need to talk about generalizing relationships for interior and exterior angles in polygons. Today I got a glimpse of students being able to figure out those relationships by looking for and making use of structure. The only concern that remains is the length of time it would take to do that on a high stakes standardized test such as the ACT or SAT. And so the journey to do what is best for my students continues …

 
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Posted by on September 21, 2014 in Geometry, Rigid Motions, Tools of Geometry

 

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Visual: SMP-5 Use Appropriate Tools Strategically #LL2LU

We want every learner in our care to be able to say

I can use appropriate tools strategically.
(CCSS.MATH.PRACTICE.MP5)

SMP5 #LL2LU

Level 4:
I can communicate details of how the chosen tools added to the solution pathway strategy using descriptive notes, words, pictures, screen shots, etc.

Level 3:
I can use appropriate tools strategically.

Level 2:
I can use tools to make my thinking visible, and I can experiment with enough tools to display  confidence when explaining how I am using the selected tools appropriately and effectively.

Level 1:
I can recognize when a tool such as a protractor, ruler, tiles, patty paper, spreadsheet, computer algebra system, dynamic geometry software, calculator, graph, table, external resources, etc., will be helpful in making sense of a problem.

 

Suppose you are solving an equation.

Are you practicing use appropriate tools strategically if you use the numerical solve command on your graphing calculator?

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Or what about using your calculator to substitute values of x until you find a value that makes a true statement?

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Are you practicing use appropriate tools strategically if you use a computer algebra system to explain your steps?

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Or what if you use the graphing capability of your handheld?

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Consider each of the following learning goals:

I can explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution, and I can construct a viable argument to justify a solution method.  CCSS-M A-REI.A.1.

I can solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. CCSS-M A-REI.B.3.

I can explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. CCSS-M A-REI.D.11.

Does use appropriate tools strategically depend on the learner? Or the learning goal? Or the teacher? Or the availability of tools?

 

[Cross posted on Experiments in Learning by Doing]

 
2 Comments

Posted by on September 15, 2014 in Standards for Mathematical Practice

 

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SMP5: Use Appropriate Tools Strategically #LL2LU

We want every learner in our care to be able to say

I can use appropriate tools strategically.
(CCSS.MATH.PRACTICE.MP5)

SMP5

But…What if I think I can’t? What if I have no idea what are appropriate tools in the context of what we are learning, much less how to use them strategically? How might we offer a pathway for success?

Level 4:
I can communicate details of how the chosen tools added to the solution pathway strategy using descriptive notes, words, pictures, screen shots, etc.

Level 3:
I can use appropriate tools strategically.

Level 2:
I can use tools to make my thinking visible, and I can experiment with enough tools to display confidence when explaining how I am using the selected tools appropriately and effectively.

Level 1:
I can recognize when a tool such as a protractor, ruler, tiles, patty paper, spreadsheet, computer algebra system, dynamic geometry software, calculator, graph, table, external resources, etc., will be helpful in making sense of a problem.

We still might need some conversation about what it means to use appropriate tools strategically. Is it not enough to use appropriate tools? Would it help to find a common definition of strategically to use as we learn? And, is use appropriate tools strategically a personal choice or a predefined one?

Strategic

How might we expand our toolkit and experiment with enough tools to display confidence when explaining why the selected tools are appropriate and effective for the solution pathway used?  What if we practice with enough tools that we make strategic – highly important and essential to the solution pathway – choices?

What if apply we 5 Practices for Orchestrating Productive Mathematics Discussions to learn with and from the learners in our community?

  • Anticipate what learners will do and why strategies chosen will be useful in solving a task
  • Monitor work and discuss a variety of approaches to the task
  • Select students to highlight effective strategies and describe a why behind the choice
  • Sequence presentations to maximize potential to increase learning
  • Connect strategies and ideas in a way that helps improve understanding

What if we extend the idea of interacting with numbers flexibly to interacting with appropriate tools flexibly?  How many ways and with how many tools can we learn and visualize the following essential learning?

I can understand solving equations as a process of reasoning and explain the reasoning.  CCSS.MATH.CONTENT.HSA.REI.A.1

What tools might be used to learn and master the above standard?

  • How might learners use algebra tiles strategically?
  • When might paper and pencil be a good or best choice?
  • What if a learner used graphing as the tool?
  • What might we learn from using a table?
  • When is a computer algebra system (CAS) the go-to strategic choice?

Then, what are the conditions which make the use of each one of these tools appropriate and strategic?

[Cross posted on Experiments in Learning by Doing]

________________________

“The American Heritage Dictionary Entry: Strategically.” American Heritage Dictionary Entry: Strategically. N.p., n.d. Web. 08 Sept. 2014.

 
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Posted by on September 14, 2014 in Standards for Mathematical Practice

 

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