Category Archives: Standards for Mathematical Practice

SMP6: Attend to Precision #LL2LU

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

I can attend to precision.


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But what if I can’t attend to precision yet? What if I need help? How might we make a pathway for success?


Level 4:
I can distinguish between necessary and sufficient conditions for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.


How many times have you seen a misused equals sign? Or mathematical statements that are fragments?

A student was writing the equation of a tangent line to linearize a curve at the point (2,-4).

He had written


And then he wrote:

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He absolutely knows what he means: y=-4+3(x-2).

But that’s not what he wrote.


Which student responses show attention to precision for the domain and range of y=(x-3)2+4? Are there others that you and your students would accept?

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How often do our students notice that we model attend to precision? How often to our students notice when we don’t model attend to precision?

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Attend to precision isn’t just about numerical precision. Attend to precision is also about the language that we use to communicate mathematically: the distance between a point and a line isn’t just “straight” – it’s the length of the segment that is perpendicular from the point to the line. (How many times have you told your Euclidean geometry students “all lines are straight”?)

But it’s also about learning to communicate mathematically together – and not just expecting students to read and record the correct vocabulary from a textbook.

[Cross posted on Experiments in Learning by Doing]


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Visual: SMP-8 Look for and Express Regularity in Repeated Reasoning #LL2LU

Many students would struggle much less in school if, before we presented new material for them to learn, we took the time to help them acquire background knowledge and skills that will help them learn. (Jackson, 18 pag.)

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

I can look for and express regularity in repeated reasoning.


But…what if I can’t? What if I have no idea what to look for, notice, take note of, or attempt to generalize?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

Are we teaching for a solution, or are we teaching strategy to express patterns? What if we facilitate experiences where both are considered essential to learn?

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked.  (Rothstein and Santana, 151 pag.)

What if we collaboratively plan questions that guide learners to think, notice, and question for themselves?

What do you notice? What changes? What stays the same?

Indeed, sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning. (Wiliam, 104 pag.)

How might we design for, expect, and offer feedback on procedural fluency and conceptual understanding?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

[Cross posted on Experiments in Learning by Doing]

Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series) (Pages 18-19). Association for Supervision & Curriculum Development. Kindle Edition.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan (2011-05-01). Embedded Formative Assessment (Kindle Locations 2679-2681). Ingram Distribution. Kindle Edition.


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SMP8: Look for and Express Regularity in Repeated Reasoning #LL2LU

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

I can look for and express regularity in repeated reasoning.


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But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4

I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3

I can look for and express regularity in repeated reasoning.

Level 2

I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1

I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.


What do you notice? What changes? What stays the same?

We use a CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning.

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What do we need to factor for the result to be (x-4)(x+4)?

What do we need to factor for the result to be (x-9)(x+9)?

What will the result be if we factor x²-121?

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What will the result be if we factor x²-a²?

We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1? (And then connect the result to the graph of y=x²+1.)


What happens if we factor over the set of real numbers?

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Or over the set of complex numbers?

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What about expanding the square of a binomial?

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What changes? What stays the same? What will the result be if we expand (x+5)²?

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Or (x+a)²?

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Or (x-a)²?

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What about expanding the cube of a binomial?

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Or expanding (x+1)^n, or (x+y)^n?

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What if we are looking at powers of i?

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We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?

[Cross-posted on Experiments in Learning by Doing]


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Which One Doesn’t Belong?

You’ve seen “which one is different” before.

(I first remember seeing this particular question from John Bament at a T3 session in 2014, although he might have gotten it from somewhere else. He sent it to the participants as a Quick Poll and showed us our quite varied results.)

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You’ve seen “Odd One Out” before.

These two images come from the Mathematics Assessment Project formative assessment lesson on Comparing Investments.

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I observed this lesson in a classroom a few weeks ago. It didn’t bother students that more than one answer can be correct, and they naturally explained why they chose what they did without the teacher even having to prompt them with “How did you get that?” or “Why?”

My coworker and I introduced Christopher Danielson’s Which One Doesn’t Belong to our beginning K-2 teachers recently. They began to think immediately about how they could do something similar with language as well as math. (And they were thrilled to learn something in PD that they could immediately take back to their classrooms.)

When I recently learned about Mary’s Which One Doesn’t Belong site, I decided to spend some time on it during our recent Math PLC meeting.

We started with a page from Christopher’s shape book. Our assistant principal (former history teacher) was thrilled to be able to immediately participate in our discussion. (How many of our students feel the same when we offer them low-floor, high-ceiling tasks?)

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We did a number WODB (one teacher fist-pumped another assistant principal when they figured out that 9 didn’t belong since the sum of its digits isn’t 7). Thanks, Pam!

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Then we moved to Rachel Fruin’s geometry Which One Doesn’t Belong. Our history teacher-turned assistant principal was still able to participate. She didn’t have the same vocabulary that the rest of the math teachers in our department had when stating why one doesn’t belong, but she learned some math vocabulary and we learned to see the images through different eyes during our shared experience.

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We ended our PLC with Hunter Patton’s Graphs & Equations 7.

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I recently heard that one measure of the success of professional development is whether the teacher’s practice changes as a result of what was learned. (Another part to this would of course be how long the teacher’s practice changes … one lesson? A few lessons? Or permanent change in lessons?) So I was thrilled to notice that the teacher with whom I share a room gave her precalculus students a WODB to try at the end of their opener later that day.

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They were studying rational functions. Which one doesn’t belong?

Before I knew it, students were in different corners of the room based on their initial responses.

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They shared thoughts with each other before sharing with the whole class.

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I tried the geometry WODB with my geometry students yesterday. I asked them to send me their response so that I could decide whether moving to one of the four corners of the room would be worthwhile. I asked bottom left to gather, bottom right to gather, and then top left & top right to gather. Why doesn’t your choice belong?

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Now work on your mathematical flexibility. Instead of being satisfied with one way to answer, find multiple responses.

Find a reason that each one doesn’t belong, and let me know when you do by selecting that choice on the new Quick Poll (now multiple response).

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Now sorted by individual responses so I can see which students need support:

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I’ve offered problem solving points for students who create their own WODB, and I look forward to seeing the results. Thank you, Mary, for creating a place for us to share and learn together … for creating a site that our teachers were able to immediately incorporate into their own learning and their students’ learning.


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The Circumference of a Circle

Thanks to Andrew Stadel’s CMC-South session, we started our lesson this year with a focus on construct viable arguments and critique the reasoning of others.

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Create an argument for comparing the height and circumference of the bottle.

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Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility. (Thanks to Andrew for this idea in particular. I’ve had students partner with others with opposing arguments on many occasions; I had not thought about the importance of partnering with others with the same argument to make your argument stronger. In the session I attended, we shared our argument with someone who answered like we did at least twice.)

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

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Uh-oh. There were apparently some pretty good convincers from the height < circumference argument. I thought fast about what to do next. I didn’t want to immediately call on someone right or wrong to share her argument with the class – I wasn’t ready for the individual/partner thinking to stop.

So without resolving the first solution, I showed another picture.

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And sent another Quick Poll.

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Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility.

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

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I could go with a whole class discussion based on these results.

CK reflected on this task in a Math Practices journal: “My first instinct was to say, ‘yes, the height is greater than the circumference’, because just looking at the can gave me the impression that the circumference was not very much. Then I was told to prove my argument, so I drew a diagram. …” (I think it’s interesting that CK chose to reflect on SMP1, make sense of problems and persevere in solving them, for this task, even though I emphasized SMP3, construct viable arguments and critique the reasoning of others, in class. The practices complement each other so well.)

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We went on to think a little more about pi, using some data that students had measured at home and submitted via a Google doc and some data through the automatic data capture feature of TI-Nspire.

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Based on feedback from students, I think this will be the last year for our What is Pi? lesson in its current form. We are getting students in high school who have learned math with the standard: CCSS-M.7.G.B.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. And so our students are now coming to us with some understanding of the formulas for the area and circumference of a circle, unlike before.

I’ve recently learned that several of my geometry students wish that we weren’t learning the geometry the way that we are. They like their previous math classes better because they didn’t have to always think about why.

We are trying to change the habits and practice of how students learn mathematics. Focusing on the Standards for Mathematical Practice has required me to think through and plan learning episodes differently than before. Focusing on the Standards for Mathematical Practice requires my students to interact in those learning episodes differently, even though some don’t prefer to. And so the journey continues …


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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?


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Why I Still Believe in CCSS

I’ve written before about why I believe in CCSS, so I won’t repeat those stories here.

I want to point you towards a few resources that might be helpful to understand why many educators are convinced that we must do something on a large scale about making mathematics less about following procedures and getting answers than it is about making sense of concepts and developing sound reasoning as to why something works.

How many of you have ever said or heard someone say, “I can’t read.”

How many of you have ever said or heard someone say, “I can’t do math.”

How many of you put fractions on your “top 5 things learned in 1st grade” list and as the “best thing about 1st grade”?


As a parent, the first recommendation I have for you is to read Mindset by Carol Dweck. Reading Mindset has changed the way I talk with my daughters about what they are learning, what mistakes they make, and what successes they have.

Christopher Danielson has written 5 reasons not to share that Common Core worksheet on Facebook. If you want to know more about Talking Math with Your Kids, read his blog by that same name. Better yet, subscribe to it. In particular, you might be interested in his post on Dots!

Look at the Standards for Mathematical Practice. I’ve written lots of posts about using them with my students to learn math.

If you want to know more about research evidence for needing these standards, Jo Boaler succinctly discusses Why Students in the US Need Common Core Math.

Sign up for How to Learn Math: For Students from Stanford University. It is a free online course that is currently open and runs through September 15, 2015. You work at your own pace through the six sessions. My daughters and I have been taking it together this semester. The conversations that we’ve had about mindset and learning math have been helpful in how they now react to learning something that isn’t easy from the beginning. We’ve also had several of our students take the course. Their reflections provide evidence that their attitude about learning math has been impacted positively. I can’t imagine that you wouldn’t want the same for your own children.

“I will no longer base my intelligence on the fact that I get an answer right because when I make a mistake my brain is growing which means I’m actually learning from my mistake.“

“I like that the course points out that everyone has the ability to be good at math and that people do not require speed to be a ‘math person.’ However, I wonder if that the only reason people think that they are bad at math is because they can’t process the information as fast as other people. In the future, I will try to push myself to leave my comfort zone and feel that it is ok to make mistakes.”

“I actually enjoyed some of the material I learned throughout this course. It gave me a new perspective on learning math. One of the best things I have learned is that everyone has the potential to learn math. No one is just born as a math genius. This boosts my overall confidence about my math ability. It makes me want to work harder so that I can get better at math. It also gives me hope towards all of my classes knowing that you can do anything if you just put in the effort. Another thing I learned is that not succeeding can be a good thing. Your brain will learn from its mistakes. If you always win at what you’re doing there is no point in doing it anymore right? You will not want to push yourself to do better because you will think you have already succeeded, when you are actually nowhere close. I liked learning all of these new things. I actually wish I would have known them earlier in my life. I could have put them towards my attitude on learning and even more specifically on learning math.“

Finally, read the recently released paper Fluency without Fear and be sure that the administrators and teachers at your children’s schools read it as well.

We are better together than we are alone, and we can make a difference in how our students learn and understand math.


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.


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?


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?


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.  


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


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


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?

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[Cross-posted on Experiments in Learning by Doing]


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Posted by on September 29, 2014 in Standards for Mathematical Practice


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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.


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]


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


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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.


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]


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


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