Collaboration & Perseverance: What Do They Look Like?

I recently wrote about this year’s circumference of a cylinder lesson.

As I was looking through some pictures, I ran across these two from last year’s lesson.

What do you see in these pictures?

I was struck by what I saw: collaboration and perseverance.

What do collaboration and perseverance look like in classrooms you’ve observed? What about in your own classroom?

How do you create a culture of collaboration in your classroom?

How do you make sure your students know that we want them to learn mathematics by making sense of problems and persevering in solving them?

Thank you to all who share your classroom stories of collaboration and perseverance, so that we might add parts of those to our own classroom stories, as the journey continues.

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Solving Right Triangles

One of the NCTM Principles to Actions Mathematics Teaching Practices is to build procedural fluency from conceptual understanding. We began conceptual understanding in our lesson on Trig Ratios. So we started our Solving Right Triangles putting-it-all-together lesson with some “Find the Error” (what’s wrong with the procedure) problems from our textbook.

Students worked a few minutes alone before sharing their thoughts with their teams. What can you find right about the given work? What is wrong about it? How can you correct the given work?

We started our whole class conversation with #1. Students began to recognize how many options there were for correcting the given work. The Class Capture feature of TI-Nspire Navigator gave students the opportunity to see and compare each other’s calculations. They looked at and compared trig ratios for complementary angles D and F.

We moved to #2. What error did you find?

The triangle isn’t right. I don’t think we can use sine, cosine, and tangent for triangles that aren’t right.

Not yet … but eventually you’ll be able to.

And then BK said, “But I worked it out.”

If I haven’t learned anything else, I’ve at least learned to hear my students out. I hadn’t noticed what he’d done when students were working individually, so I didn’t know what we were about to get into. “How did you do that?”

I drew in an altitude from angle B and made two right triangles.

And so he did.

He recognized half of an equilateral triangle, so he used 30-60-90 triangle relationships to get enough information to write a ratio for tangent of 55˚.

This is what happens when students learn mathematics steeped in using the Standards for Mathematical Practice. Students practice make sense of problems and persevere in solving them. They practice look for and make use of structure. Even in what seems at first glance like simple, procedural problems.

And so the journey continues, with the practices slowly becoming habits of minds for my students’ learning, seeing glimpses of hope more often than not …

 

The Diagonals of an Isosceles Trapezoid

It was the day before the test on Polygons, and so I thought that writing a proof and then giving feedback on another team’s proof might be helpful.

Students worked alone for a few minutes, thinking about what was given and what could be implied. Then they worked with their team to talk about their ideas and to begin to plan a proof.

Some were off to a good start.

Some were obviously practicing look for and make use of structure.

Some were stuck.

I talked to several groups, listening to their plan, asking a few questions to get them unstuck.

And then I got out colored paper on which to write the team proof.

 

The clock was ticking, but I thought that surely they would be able to trade proofs with another team for feedback within a few minutes.

 

I talked to another group. They were reflecting ∆ABC about line AC.

What will be the image of ∆ABC about line AC?

The answer? ∆ACD.

Of course that is wrong. It seems so obvious that ∆ABC is not congruent to ∆ACD. And I’m also wondering how that helps us prove that AC=BD, since BD isn’t in either of those triangles. But that’s where this team of students is. I now have the opportunity to support their productive struggle, or I can stop productive struggle in its tracks by giving them my explanation.

 

My choice? Scissors. And Paper. And more time.

What happens if you reflect ∆ABD about line AC?

 

Oh! The triangles aren’t congruent.

 

So are there triangles that are congruent that can get us to the diagonals?

∆ABC is congruent to ∆BAD.

 

How do you know?

A reflection.

About what?

This pencil!

 

So what is significant about the line that the pencil is making?

It’s a line of symmetry for the trapezoid.

It goes through the midpoints.

 

(One of the team members was using dynamic geometry software to reflect ∆ABC in the midst of our conversation, but I don’t have pictures of her work.)

 

So the plan was for team to write their proofs on the colored paper and then trade with other teams for feedback. Great idea, right? So how do you proceed with 15 minutes left? Proceed as planned and let them give feedback with no whole class discussion? Or have a whole class discussion to connect student work? Because as it turned out, no two teams proved the diagonals congruent the same way. I chose the latter.

 

I asked the first team to share their work.

 

Their proof needs work. But they have a good idea.

They proved ∆AMD≅∆BMC, which makes the corresponding sides congruent, so with substitution and Segment Addition Postulate, we can show that the diagonals are congruent.

 

Next I asked the team to share who proved ∆ABC≅∆BAD using a reflection about the line that contains the midpoints of the bases. Their written proof needs work, too. But they had a good idea.

 

Another team proved ∆ACD≅∆BDC.

 

Another team constructed the perpendicular bisectors of the bases. Since the bases are parallel, a line perpendicular to one will be perpendicular to the other. I’m not sure they got to a reason that the perpendicular bisectors have to be concurrent. They could have used ∆AZD≅∆BZC to show that. Instead, they used a point Z on both of the perpendicular bisectors (they know that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment) to reason that ∆AZB and ∆DZC are isosceles & then used Segment Addition Postulate and substitution to show that the diagonals are congruent. Not perfect. But a good start.

 

NCTM’s Principles to Actions discussion on support productive struggle in learning mathematics says, “Teachers sometimes perceive student frustration or lack of immediate success as indicators that they have somehow failed their students. As a result, they jump in to ‘rescue’ students by breaking down the task and guiding students step by step through the difficulties. Although well intentioned, such ‘rescuing’ undermines the efforts of students, lowers the cognitive demand of the task, and deprives students of opportunities to engage fully in making sense of the mathematics.”

 

So while I didn’t rescue my students, we also never made it to an exemplary proof that the diagonals of an isosceles trapezoid are congruent. Did they learn something about make sense of problems and persevere in solve them? Sure. Is that enough?

 

Would it be helpful to lead off next year’s lesson with this student work? Or does that take away the productive struggle?

 

Is it just that we have to find a balance of productive struggle and what exemplary work looks like, which is easier in some lessons than others? If so, I failed at that balance during this lesson. Even so, the journey continues …

 

Growth Mindset & GRIT & SMP1

If we want our students to be mathematically proficient, and if we want mathematically proficient students to make sense of problems and persevere in solving them, how will we help them when they don’t? or won’t? or feel like they can’t?

Jill Gough and I have been working on leveled learning progressions for the Standards for Mathematical Practice. Here is the visual for SMP1.

 

I wonder how much making sense of problems and persevering in solving them has to do with the work of Carol Dweck on Mindset and Angela Duckworth on GRIT. I had the opportunity to hear Angela Duckworth speak at the AP Annual Conference a few years ago.

 

One of the ways that our students can earn Problem Solving Points in our course is to determine how much GRIT they have:

Angela Duckworth says that the key to success is GRIT. Watch her TED Talk. Then determine how much GRIT you have. Then email your instructor a reflection with a response to at least one of the following prompts:

I like …, I wish …, I wonder …, I will …

We have enjoyed reading our student reflections on GRIT.

I like this idea and I do believe in it. I believe a lot of people don’t really understand how extremely important it is though. I think a lot of people would watch the video and think “oh cool grit whatever” and not realize that that’s more than likely is what will get you hired coming out of college and that it will probably take you farther in life than anything else. I wish more people understood that. I wonder if GRIT is something you can turn off and turn on, like we know it can change but can you just decide you want to be gritty for this one thing and be gritty.

 

 

I like that Angela Duckworth and Princeton (and you too, Mrs. Wilson) are speaking out and beginning to normalize this idea that intelligence is a fixed point, that we can’t change, is all wrong. Yes, it’s true we are not all rocket scientists- but should the people with less of an initial gift for learning have any less of an education? I’ve felt that in our school and our society there are a lot of limitations, including how high you rank in standardized tests, that influence how much you are pushed and expected to succeed. However, I don’t think that people who rank lower in testing scores should be shoved aside and given just the bare minimum. If the fear of failure was not so prevalent in the school system, maybe kids would believe that they can succeed after the initial failed attempt; that not just the ‘smart’ kids will be the ones to succeed.

I wish that someone had told me about this sooner, and that we were setting the goal at something more like GRIT, not just if you get the answer faster or easier than someone else. I’ve been in the smart track my whole life so I might sound out of line, but even I know that I won’t be a mathematician or the one to find a cure for cancer. No matter how hard I try, there is reality to remember, and though I’ve had encouraging parents and many very helpful teachers, I’ve still had the idea of my failure put into place. Can I wipe away that misconception that I was hardwired with a certain capacity for greatness? Even if I do, I feel that I just wasn’t born with a lot of determination. I’m sad to say my GRIT score was only 2.7 or so.

I wonder if this idea will die away or flourish in the new minds of the next generation. Before I came to your class, I’d always had teachers who would seem to forgive our wrong answers, but never one who said that, if used in the right way, it could actually help improve our overall smartness. I wonder how I could improve my measly 2.7 GRIT to something stronger. I wonder if I’ll ever find a motive to push me through, something to fuel my resilience.

 

I will work on not giving up; what better time to muddle through than high school? Opportunity to dump homework and just watch netflix abounds, but I will make a conscious effort to improve my GRIT and become a more responsible, diligent person.

 

 

I will definitely try harder in school and in other commitments after watching this video. The grit survey site gave me a grit score of 3.88. It also stated that I have more grit than 70% of the US population. Wow! I am shocked that 70% of the US has a grit score lower than 3.88. I am not fully satisfied with that score, so I will try harder to increase my grit score.

 

 

I took the grit survey and my result was 3.25. That makes me grittier than more than 40% of the United States of America. I will work hard to persevere on any project I begin. When I do projects, it always feels like I work so hard when I start, but as I get closer to being finished with it, I don’t work as hard as I could. I need to work on having patience to see something completed. I will also work to not get so discouraged when I get something wrong or when I don’t understand something. Once I start to do some of these things, I will become more successful and grittier.

 

 

I like how Angela Duckworth developed a grit questionnaire and how she admitted that she didn’t know how to instill grit in kids. I also liked how she ended with “In other words, we need to be gritty about getting our kids grittier.”

I also took the grit survey, and got a 3.5 out of 5, which is apparently better than 50% of the US population. I don’t if I should be happy that my 70% is better than nearly 160 million people or sad for the same reason.

 

 

Does it help for us to make our students and children aware of growth vs. fixed mindsets? Does it help for us to purposefully use growth mindset and GRIT language with our students? And whether or not research shows that it helps, can’t it not hurt if we want all of the learners in our care to make sense of problems and persevere in solving them?

 

The Base Angles of an Isosceles Triangle

Our students come to us knowing that the base angles of an isosceles triangle are congruent. But they don’t know why.

CCSS-M.G-CO.C.10: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

We leave this proof as an exercise on the unit assessment, which is why there is so much setup in the exercise. (Didn’t you always love math textbooks that left proofs as exercises at the end of the section for you to do instead of actually working through the proofs during the section?) I do wonder what would happen if there were no hint. And if this were an exercise in class, of course I wouldn’t give a written hint. But since this is on the test, I am admittedly limiting the amount of productive struggle that I expect from my students.

How would you expect your geometry students to prove the base angles of an isosceles triangle are congruent? What misconceptions might your students have?

This year we got several of the traditional SAS (and SSS) proofs:

And we got a few of the rigid motion – reflection proofs:

I think we still need some work on these proofs … like explicitly stating that A lies on the perpendicular bisector of segment BC because it is the same distance from B as it is from C.

 

We got a long paragraph proof with the misconception that the two smaller triangles formed by the altitude will be 45˚-45˚-90˚, but then ending with an argument for reflecting the triangle about its altitude to show why the base angles are congruent.

We got another argument for constructing the altitude/angle bisector/perpendicular bisector/median from the vertex and using HL to show that the decomposed triangles are congruent.

And an argument I haven’t seen before using inequalities in triangles.

What we didn’t get were blank responses. Our students are learning to make sense of problems and persevere in solving them. Our students are learning to look for and make use of structure. Our students are learning to construct viable arguments and critique the reasoning of others. As the journey continues, our students are becoming the mathematically proficient students that we want them to become.

 

Proving Triangles Congruent – SAS

CCSS-M. G-CO.B.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

This standard made me realize that the textbooks I had used for a long time allowed the ASA, SAS, and SSS Triangle Congruence Theorems into our deductive system as postulates. We’ve always proved SAA and HL, but for some reason I thought the others were in the back of the book in a section of more challenging proofs of theorems. (I at least knew that the proofs weren’t left as an “exercise” for students at the end of the section on congruent triangles.)

How can we use rigid motions to show that SAS always works?

Here is one student’s suggestion.

We’ve mapped ∆ABC to ∆DEF with C to F using vector CF, and rotating ∆A’B’C’ about F using angle C’A’D will map one triangle on top of the other. But have we used the given SAS? We know that ∠B≅∠E, not that ∠C≅∠F.

Here is another student’s suggestion.

Once you’ve mapped C to F using vector CF, the student suggests rotating the new triangle 180˚ about C.

We know that using dynamic geometry software doesn’t prove our results for us.

But using dynamic geometry software does help convince us that we are proving the right thing. I cannot remember where I recently read (a Tweet? a blog post?) that students need to be convinced a statement is true before they will expend effort proving it. It takes a lot of Math Practice 3 for us to make it through explanations for why SSS, SAS, and ASA provide sufficient information for proving triangles congruent.

We can use a translation and a rotation, but we need to map ∆ABC to ∆DEF with B to E using vector BE. We know that ∆A’B’C’ is congruent to ∆ABC because a translation preserves congruence.

Then what rotation will ensure that ∠B’ maps onto ∠E?

Rotating ∆A’B’C’ about E using angle C’EF will leave E=B’=B’’. B’’C’’=EF because a rotation preserves congruence. A”B”=DE because a rotation preserves congruence, and ∠B≅∠E because a rotation preserves congruence.

If the given triangles do not have the same orientation, a reflection will be necessary, which could then be followed by a translation and/or a rotation as needed. Note: I’ve recently seen different interpretations of “orientation”. We say two figures have the same orientation if the clockwise order of the vertices is the same.

Even if the triangles do have the same orientation, a reflection or sequence of reflections can be used.

Since EF=EC’, E is on the perpendicular bisector of C’F. Reflecting ∆A’B’C’ about the perpendicular bisector of segment C’F will leave E=B’=B’’ since a point on the line of reflection will be its own image.

Since ∠DEF≅∠C’EA’ and EA’=ED, A’ and D will also have to coincide after the reflection about the perpendicular bisector of C’F.

Thus, ∆ABC≅∆A’B’C’≅∆DEF.

Thinking through the proofs of SSS and SAS make our traditional congruent triangle problems look like a waste of time.

Can we show that the two triangles are congruent?

Students look at this and immediately see that one triangle is a rotation of the other 180˚ about the midpoint of segment AC.

 

Students look at this problem and see the same rigid motion to prove congruence.

 

And another, except that this time, someone initially suggested a reflection about segment AC.

 

Can we recover showing congruence from the initial reflection?

Of course … another reflection about the perpendicular bisector of segment AC shows the given triangles congruent.

And then we think about why that works.

As the journey continues, I am grateful for standards that push my students and me to think outside our comfort zone, giving all of us the opportunity to make sense of problems and persevere in solving them.

—

Resources:

From Illustrative Mathematics, Why does SAS work?

Usiskin, Peressini, Marchisotto, Stanley. Mathematics for High School Teachers: An Advanced Perspective, Pearson 2003.

 

Productive Struggle: The Law of Sines

NCTM’s Principles to Actions suggests eight Mathematics Teaching Practices for teachers. One of them is to support productive struggle in learning mathematics. The executive summary states: “Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.”

What does productive struggle look like? What does it sound like?

I saw a glimpse of what productive struggle looks like yesterday. I get to share a room with a teacher (who happens to be a former student of mine), and so I listen with one ear when I’m in the room working at my desk during her Precalculus class. The lesson was on the Law of Sines, but Trisha didn’t tell the students from the beginning that was the learning goal. Instead, the students focused on the math practice make sense of problems and persevere in solving them.

She presented a situation. And the students made assumptions and asked questions.

One I remember hearing was “I guess we can’t just use a measuring tape?”

Then she asked them to solve the problem.

And so they did. These students didn’t balk at the task. They all worked. They didn’t even talk very much at first … you could hear them thinking in the silence that encompassed the room. That’s when I looked over and realized that I was seeing productive struggle in action. Productive struggle isn’t always quiet, but it definitely started that way for these students. Eventually, students listened to Ain’t No (River Wide) Enough while they worked.

When solving the non-right triangle without knowing the Law of Sines, the students used another Math Practice – look for and make use of structure – to draw auxiliary lines. Some drew an altitude for the given triangle to decompose it into two right triangles. Some composed the given triangle into a right triangle.

 

Trisha collected evidence of what students could do using a Quick Poll.

So if we are given one side length and two angle measures of a triangle, is there a faster way to get to the other side?

More productive struggle … the numbers are now gone, students are reasoning abstractly to make a generalization.

And they did.

And they derived the Law of Sines in the meantime.

How often do we give our students a chance to engage in productive struggle? In how many classrooms is the Law of Sines just given to students to use, devoid of giving students the opportunity to “grapple with mathematical ideas and relationships”?

When I discussed what I saw with Trisha, she noted that last year, only a few of the students in her class successfully solved the triangle prior to learning about the Law of Sines. This year, all of them tried and most of them succeeded. These are the students with whom we started CCSS Geometry year before last. These are the students who have been learning high school math with a focus on the Math Practices. These are students who are becoming the mathematically proficient students that we want them to be. Because we are letting them. As the journey continues, we are learning to leave the front of the classroom behind so that we can support productive struggle in learning mathematics.

 

Visual: SMP-1 Make sense of problems and persevere #LL2LU

What if we display learning progressions in our learning space to show a pathway for learners? After Jill Gough (Experiments in Learning by Doing) and I published SMP-1: Make sense of problems and persevere #LL2LU, Jill wondered how we might display this learning progression in classrooms. Dabbling with doodling, she drafted this poster for classroom use. Many thanks to Sam Gough for immediate feedback and encouragement during the doodling process.

I wonder how each of my teammates will use this with student-learners. I am curious to know student-learner reaction, feedback, and comments. If you have feedback, I would appreciate having it too.

What if we are deliberate in our coaching to encourage learners to self-assess, question, and stretch?

[Cross-posted on Experiments in Learning by Doing]

 

SMP1: Make Sense of Problems and Persevere #LL2LU

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

I can make sense of problems and persevere in solving them.  (CCSS.MATH.PRACTICE.MP1)

But…What if I think I can’t? What if I’m stuck? What if I feel lost, confused, or discouraged?

How might we offer a pathway for success? What if we provide cues to guide learners and inspire interrogative self-talk?

 

Level 4:

I can find a second or third solution and describe how the pathways to these solutions relate.

Level 3:

I can make sense of problems and persevere in solving them.

Level 2:

I can ask questions to clarify the problem, and I can keep working when things aren’t going well and try again.

Level 1:

I can show at least one attempt to investigate or solve the task.

 

In Struggle for Smarts? How Eastern and Western Cultures Tackle Learning, Dr. Jim Stigler, UCLA, talks about a study giving first grade American and Japanese students an impossible math problem to solve. The American students worked on average for less than 30 seconds; the Japanese students had to be stopped from working on the problem after an hour when the session was over.

How may we bridge the difference in our cultures to build persistence to solve problems in our students?

NCTM’s recent publication, Principles to Action, in the Mathematics Teaching Practices, calls us to support productive struggle in learning mathematics. How do we encourage our students to keep struggling when they encounter a challenging task? They are accustomed to giving up when they can’t solve a problem immediately and quickly. How do we change the practice of how our students learn mathematics?

 

[Cross posted on Experiments in Learning by Doing]

 

 

A Tank of Water

Towards the end of last semester, we worked on a task that Tom Reardon shares as The Great Applied Problem.

What is your question? (Note that this lesson was before I read Michael Pershan’s post considering how we ask students about what we can explore mathematically.)

why are we always looking from the top and never the side      1

other than circles and rectangles what are the cross sections of shapes?         1

are the 2 bases the same size          1

what are the sizes of the different cross sections?            1

i wonder how to find the volume of a partially filled cylinder in which the base of the liquid is a sector     1

how many cubic units of water are in the shape currenty?        1

is the water filling up or flowing out. what is the length of each cord as it fills 1

I wonder if there is a relationship between the diameter of the base and the change in dimensions of the rectangle…    1

the areas of the bases created by the water         1

What is the total volume of the tank          1

what is the smallest rectangle possible to be a cross section?     1

at what rate does the volume change        1

is it filling up or draining     1

How much water is in the cylinder?           1

how much water will the cylinder hold      1

how much water would it take to fill the tank       1

I wonder what the volume is of the water in the tube.    1

how much water will it hold            1

will the water flow through the straw        1

what is the volme ofthe water         1

what is the volume of the water in the cylinder?  1

does the circumference change      1

how would the shape change as the cylinder shift           1

whats the ratio of the volume of water to the cylinder    1

how much more water do you need to fill it up    1

How much water can the tank hold?          1

how will the volume change if the water increases          1

What are the measures of its radius and horizontal height, or of either if the volume is given?        1

 

We settled on how much water is in the tank.

What is the least amount of information you need to answer the question?

Teams worked together to make a list of the measurements they wanted to use for their calculations. Very few teams wanted the same information. Some differences were minute, such as one wanting the radius and another wanting the diameter. Or one wanting the depth of the water and another wanting the distance between the center and the chord. Or one wanting the radius and another the length of the chord. Some differences were bigger, such as one wanting the ratio of water to total volume.

What information was I willing to give them?

Thanks to a spreadsheet included in Tom’s problem, I had plenty of measurements from which to choose to give students. But I hadn’t thought through whether I was willing to give the ratio of water to total volume instead of the length of the radius. I ended up not giving the ratio. But I did give some teams the length of the chord instead of the radius.

As I watched students work, I noticed that they had the opportunity to look for and make use of structure. Dylan Wiliam talks about asking students questions that push their thinking forward and probe their understanding. This task did just that.

What misconceptions would students have about this figure?

We talked about what’s there that’s not there. What do you see that isn’t pictured?

A semicircle

What else do you see that isn’t pictured?

A diameter

What else do you see that isn’t pictured?

A radius that forms a right triangle with half of the chord.

What else do you see that isn’t pictured?

An isosceles triangle formed by two radii and the chord.

What else do you see that isn’t pictured?

A sector.

What else do you see that isn’t pictured?

The region formed by decomposing the sector into a triangle and a segment of a circle.

Is that same region a semicircle?
It’s not. But that is what a lot of students thought when they first looked.

We are learning to look deeper to make sense of the structure before we jump into calculating.

One student reflected on working through this task in class. She also talks about using the practice make sense of problems and persevere in solving them: “trying to find what we needed to know”, “tried different ways to find the area”, “drew diagrams”, “made a plan”, “discussed different approaches”. She also talks about the math practice construct viable arguments and critique the reasoning of others: “We all decided what we wanted to know to figure out how much water was in the tank. And then we tried to explain our reasoning to the class. We all discussed what we wanted to know then decided together what we really needed to know.”

I’ve used this task for several years, but I’ve never introduced it like this before. Previously, I’ve asked my students only to calculate.

And so the journey to be less helpful continues …