Common Core Aligned Lessons: Rich Tasks, or Same Stuff, just re-aligned?

I got an email from Achieve just this week inviting teachers to start testing the lessons on the Achieve the Core website.

I was so excited! All of this with one search tool! I clicked through: There were lessons for every standard! I clicked again: For every grade! I quickly clicked on the lesson promising to teach students to find the zeroes of quadratic equations (that’s such an important concept and I was looking to expand on the rich task we used in the workshop). And then… Well, does anybody remember the sound the record player made when the needle would slip and slide across the vinyl??

Since attending Common Core Standards training this summer, where we learned how to implement rich tasks for conceptual learning, and learning about Dan Meyer’s work, I have been interested in sources of these strong lessons and in modifying many of my existing lessons. The Stanford class ‘How To Learn Maths’ cemented my desire to get even better at teaching math through numeracy, rich visualizing, and good questions to get students thinking about what the numbers they are using really represent.

Here is what I found:
My click on the quadratic lesson connected me to the website Share My Lesson. The lesson plan was beautifully written: standards, number of days, list of materials (hmmm, a graphing calculator, but not graph paper?), the detailed notes handout- one for each day of the two day lesson (fill in the blank), and even a group ‘discovery activity’. Further down, there is a chart with a column of expected student answers/misconceptions, etc. that looked interesting, (in fact, that was the best part) and another section with a three column ‘prior knowledge, current knowledge, future knowledge’ chart (although prior and current knowledge would seem to be the same thing, but current knowledge is apparently what they are supposed to learn in the lesson; which makes that future knowledge in my book!)

There isn’t any instruction on the formative assessment, although perhaps the teacher will make sure the blanks on the notes are correctly filled in…

I fast-forwarded to the instructions for the lesson: graph (using the calculator) four given quadratic equations and identify the zeros. Hmmm. How are they supposed to know this? I checked the prior knowledge column on the lesson plan. Nope. Nothing about zeros. The current knowledge column (remember: the goal of the lesson) was that the student ‘would be able to’ find the zeros of the factors of the quadratic. Factors? But we didn’t factor anything. Oh, wait, it says here that factoring is the next lesson! STOP!!!

Where is the rich task? Where is the productive struggle? Where are the mathematical practices?

This great lesson, common core aligned and all, appears to be more of the ‘feed kids details and have them take notes’. Even the group activity, having them find the differences in the graphs isn’t creating the conceptual understanding of what they are doing, what the graph represents…. Can you feel my frustration here?!

We (teachers) are going to have to undergo a shift in thinking about what good lessons look like. It is going to require kicking out textbooks and no longer training students how to get good at multiple choice tests. This is a paradigm shift. Yet, here is the Achieve the Core website leading teachers to more of the same dry, lecture heavy, notes and memorization-filled stuff!

In the interest of good reporting, I went to two other lessons, one for sixth grade on fractions, and one for eighth grade algebra. They were similarly structured.

If you are interested in what this national resource of lessons is offering – and interested in helping improve the lessons – then here is your chance (you might even win stuff!):

Participate in the Common Core Challenge:
1. Watch short videos of master teachers while using the CCSS Instructional Practice Guides
2. Apply what you saw to a lesson of your own
3. Tell them about your experience
About the winning stuff, the email said, and I quote, “As a participant, you’ll be eligible to win great prizes while helping to continuously improve tools designed to support teachers like you.”
The Common Core Challenge was developed as part of the Common Core Teacher Institute held on October 6th at NBC News’ Education Nation 2013.

This is your chance to give them feedback on these lessons. Let’s give our teachers every chance to succeed in the classroom, because this is the only way our kids will succeed. That success will transfer to a confidence that we won’t need standardized tests to see!

Seriously, though, sharing good lessons so that we don’t have to create our own and giving feedback to make good lessons better will allow us to improve what is happening with students across all disciplines, across all schools and across all SES.

I have found the most wonderful group of teachers and resources and community through the MTBoS site and through Twitter ( I am @the30thvoice), so I know my teaching is going to get better and my students are going to be challenged.

How a students starts is out of our control, but how a student finishes is in large part due to how he/she is taught. Be that teacher for your students!


Sixth Grade, Fractions, and Fair-shares

I have been assigned a room full of bright six-graders. They are successful; achieving high proficient scores – they are students who are “on the bubble” of scoring at the advanced level. A review of their work reveals the problem: Operations with fractions.

My goal is to help them create a stronger conceptual understanding of fractions, in order to allow them to be more successful in their fraction interactions!

The initial lesson involves some baby steps, in order to formatively assess the understandings about equal sharing, naming shares and rebuilding wholes. Materials will include the students themselves and a fair share box (see attached image- thanks to Jere Confrey).

I tell them they are going to gather in teams and ask them to come up with ways that we can divide the class so that each team has an equal number of students, WITHOUT COUNTING FIRST. At this point I am not looking for anything but suggestions on grouping.

I will use thinking questioning to get them started, clarifying q’s to help them express ideas and repetition: students repeating what others have said. The students will then be asked to critique, analyse and decide how best to group.

They will use a method they decided upon to create the teams. We may need time for trial and error here, as well as how to divide ‘leftovers’ when the class does not divide evenly!

Once they have successfully created even teams, I will pose questions (I will use the family feud style of questioning, so every student gets a chance to answer, can ask for help, can challenge another group’s answer, etc)

First question: Have we divided the teams evenly? Are they ‘fair’? (There may be discussion about this, especially since students sometimes want ‘smarter’ kids in their group!)

We will have to talk about ‘leftovers’ and how to divide them. (Options: let the students adjust groups until we have even amounts, or give the extra students jobs before the exercise begins so we don’t have to divide any bodies!)

Q: (work on in groups) How would you describe your team in relation to the rest of the student teams in the room? ( this is where I will go around and listen to what the students think about their relationship to the other groups. I want to find out about their level of expertise with the language of groups- I anticipate that I will get anything from ‘one group of five students out of 20’ to ‘1/4 of the whole class’.

I will have the students share their comments and ideas. I will then let the students give each other feedback on the reasonableness, accuracy, and will take the opportunity to guide the vocabulary for the discussion, by restating, and asking students to restate.

At this point, I expect to have identified any conceptions about fair sharing and language of sharing the students have.

Now we will fill in the chart:
How many students in each team?
How many teams?
What was the total number of students in the class
So what can we say about each team?

I will have the students tie their comments to the chart, walking through the naming conventions they use. If there is not a connection to fractions in the discussion, I will shape my questions: ex. Do they see another way we could name each group in relation to the whole?

Once we have a fraction on the table, we can start rebuilding. (At this point, I would have them go back to their seats.)

I would flip the table at this point, to show the connection to the fractions they usually see. I want them to make the connection from part to whole.

The next step will be to reconstruct our groups: if one group is 1/nth of the whole class, how will we talk about two groups in relation to one? What about the whole class in relation to one group? (n here represents how many groups a class actually comes up with, but this could also be the goal, to start with two teams, three teams, four teams, and have the students introduced to the mathematical extension of ‘n’!)

After this discussion, I would reinforce the problem solving with the penguin problem, (how do we share 30 snowballs among three penguins?) using small groups. At this point, I will be checking in with each group to hear what they are saying and to reinforce the ideas of sharing, naming and reconstructing. I will be listening for understanding.

All of this should only take a fraction of the lesson time. I am asking the students to think about fractions in a primary way, to connect the more elaborate work of operating with fractions. The students will be encouraged to use the language of fractions, wholes and reconstructing as they work with grade level problems involving fractions. Students will be encouraged to break down fractions or reconstruct them to assist in solving problems, with drawings, with manipulatives, or using the fair share table. Then they could begin to operate with the fractions, understanding how to combine groups, and in further lessons, what happens when groups are multiplied and divided.

I see my understanding of how children instinctively do this work informing my ability to identify students who have “lost” or buried the skills and to assist in reconstruction of this knowledge to enhance and strengthen their future conceptual understanding.


Standards Based Grading. Hmmmm.


Standards Based Grading. Hmmmm.

I want to know how to use this particular beastie to my and my students’ advantage. What do I know about it?

(I sent out some tweets. #SBG and #MTBoS to find out more!)

It is based on the common core standards for my subject/grade level.
I’ve seen these as sets of “I can…” Statements- is there a list somewhere, or does each teacher create his/her own?

(I sent out more tweets: @TarheelMommy95 @algebrainiac1 @JustinAion @WHSRowe were a few of those who generously shared what resources, including two references I definitely want to read: Marzano’ Formative Assessment and Standards Based Grading and Designing and Teaching Learning Goals & Objectives.

A score is given for each “I Can…” that reflects the student’s level of achievement of the standard, usually 0 to 4 or 5, depending on the teacher.
There are some great rubrics in the CCSS materials that could help me here- maybe there are teachers who have also simplified the rubrics for their students and parents – to help them begin the process of steering their own learning. (I feel another tweet coming on! I also remember a blog…. Mrs. Aitoro’s that might help me here. I will ask her to share…

I will want to have some kind of tracking system. That means that I will need to connect each task to the appropriate standard/skill. I will need to be able to identify what mastery is for that skill (oh yeah, the “I can…” statements and the rubric statements will help me there!)

How does the student exhibit improvement? I’ve seen a blog that explains the grade book set up – one grade per standard, it gets changed if the student exhibits improvement, but doesn’t get lowered. I like the idea of two tests with 100% success allows the student to not have to test that standard again. Can I assume mastery after two times? Will the rubric for mastery be clear enough? Will I run the risk of the student forgetting? Does mastery indicate memory or true conceptual understanding?

Thankfully, because of Task #2 and #MTBoS (thanks Justin @j_lanier and the rest of the MTBoS team!) I know I am not the only one out here asking these questions and I don’t have to re-create the wheel for my particular hamster cage! I really want to do this. And now I know I can!

So let’s say a television is falling on your head. Will a bigger TV kill you faster?

You are standing on the sidewalk. Somebody yells, “watch out!” And you look up and realize a television is hurtling towards your head. You have 2 seconds to move out of the way. What floor was the television dropped from?

This is a reverse of the classic egg drop problem, which asks the student to figure out how long it will take an object to fall from a specified height. The formula for gravity and time is usually provided. We are going to turn this lesson on it’s head, literally!

The lesson is designed for two 45 min lesson periods. I’ll give the standards and practices at the end (or maybe I’ll let you tell me which ones this lesson hits!)

The lesson starts with an exploration of gravity and a ‘where does the formula come from’, and moves to the exploration of the above scenario – quickly, before somebody gets smacked on the head by a TV! (The title of the blog is the question that kicks off the gravity part!)

Start with the question: So let’s say a television is falling on your head. Will a bigger TV kill you faster?

Have the students make conjectures in their own minds. Have the silent thumbs up when they are ready. Pair students up and have them share their conjectures, giving reasons that the other person can articulate. Go around the groups and ask what they decided: what the question was asking, whether a bigger tv would fall faster than a smaller one, give the reason(s) they felt that was true or false.

If your facility offers a one or two story drop (football stadium announcer stairs?) conduct this next experiment empirically.
Or, use this great video called Misconceptions about Falling Objects.”

First, watch only the first part of the video, where the interviewer is asking random people whether a heavier ball will fall faster or the same as a lighter weight ball.

Stop the video before they get to the proof part. Survey your students. See how many agree with the heavier is faster theory. Ask them why. Have them explain their thought process.

At this point, you can take them outside, let them try it themselves with two different weight items, or you can play the rest of the video, or both.
If you get to do the outside drop, bring a stopwatch. Set up some students to drop and some students to time the drop and determine how fast the ball drops. Have the students estimate the height (if you can, measure it ahead of time, so you can work the math first) and come back to the class.

The task ahead of the students is two- fold:
1. To understand that items fall at the same rate, no matter the weight, and to perhaps extend to the connected idea that gravity is a pull that has nothing to do with weight – weight has to do with density of matter. (If you have a wonderful science teacher, maybe she will work on a connected gravity lesson!)
2. Develop an understanding of the parts of the formula used in so many of their quadratic one-variable math problems, as they develop their speed of gravity based on the tests, or in discussion from the information from the video.

I will tell you that the video gives the formula in meters per second. The formula in the Alg II Word problems are generally given in feet per second so take some time here to work the students through a conversion process, so they will see the variations as different ways to write the same formula.

Rest or break here. Debrief the students, let them write out their understanding of the question asked in the title- will a bigger TV kill you faster? See how many students changed or enlarged their views. Have them articulate what changed, or if they thought objects would fall at the same rate, how did what they did support their initial thought.

Pick up the next day with the formulas. Let the students (whole group) come up with thoughts as to how they are different and why one formula might be more appropriate than another.

Now, pose the second scenario:
you are standing on the sidewalk beside a multi-story building. Above your head, someone has knocked a window air-conditioner out of the window. If you don’t move it will land on your head. You have 2 seconds to move. How high up does that window have to be to give you time to get out of the way? If you double the height, does that double the amount of time you have to move?

Have the students each think about the problem individually and observe their ideas. You may need to ask some questions of those students who can’t get started, to help them find an entry point. Ask them if they could draw the problem, or have them list things they will need to know (rate of gravity) to solve the problem. (Working the problem looking for height requires re-arranging the formula, which is why I wanted the students to really understand the parts and where the numbers come from.)

The next step is group discussion. In groups of 3 or 4, have them share their ideas and work together to discuss how they feel they can solve the problem, discuss which version of the formula will be appropriate to use, and details, like the average height of floors in a multi story building.

Encourage through questions some hypotheses, maybe a drop from the second floor. At this point there will be some discussion about how to mathematize the answer. One of the recurring problems for students deals with the square root solutions required to solve quadratics. Students should have already worked through the various ways to solve quadratics, so you will want to listen to the conversations surrounding these issues.

Some students will be able to solve for the square and some will not. Have the student who does solve the square explain his/her understanding of the process. If students in each group come up with differing ideas, let them question, critique, convince, until they can agree on some solution. After all, their life depends on knowing how fast they must move!

As you listen, decide which groups to call on for sequencing purposes. Bring the group together and let each group talk about what information they used to solve the height issue. As students speak, if someone realizes their mistake, let them correct it, then iterate the thought process – why did they think the mistake was reasonable?

Continue to ask questions to guide the students through the ideas, asking students to reiterate what they heard another student saying; asking a student to clarify a point, to dig deeper into the thought process.

They should, of course, come up with two answers, positive and negative. So the next question is which answer makes sense and why? What does the negative answer represent? What would have to happen for the negative answer to make sense?

The students can practice their skills by asking them the same problem, only transferring the location to Mars. If the television dropped from the same height on Mars, would they have more time or less?

Don’t forget to ask about doubling the distance. Let the students confirm whether or not that is true and why.

This is a good time to pull the students together and have students restate the ideas they’ve heard. Watch and listen for evidence of a change in student understanding. Are they making the same errors or different ones?

To cement the knowledge, have this similar problem ready: you are in a boat at the foot of a cliff. You look up and see a man tumble straight down. Develop some different scenarios for the height of the cliff and the time it takes him to fall.

Okay, I won’t make you guess the standards and practices- but if you see any I missed, let me know!
The CCSS task that is directly addressed here is Algebra II: Quadratic Equations in one variable (standard A.REI.4)
The activity supports all eight mathematical practice standards. The various individual, small group and whole group tasks give plenty of opportunity for formative assessment of math skill levels and understandings. The Mars thing – have either an internet access to look up the gravity on Mars, or have it ready and let the students who get done early work on the Mars formula for the rest of the class to use. They can do a mini present on how they got it.

There are sure to be multiple paths students took to get the answers. Let whole group discussion offer opportunities for students to share how they thought about solving. Put the paths on the board and let students compare the paths – they will find ways to add some of these new paths to their own toolbox if skills!

As always – feel free to use, modify, comment, and question. Thanks!


Be Less Helpful

“It is not as important that managers have succeeded with the problem as it is for them to have wrestled with it and developed the skills and intuition for how to meet the challenge successfully the next time around” The Innovator’s Solution: Creating and Sustaining Successful Growth Clayton M. Christensen, Michael E. Raynor The above quote applies to hiring good people to help businesses grow and succeed. I could change the word ‘manager’ to ‘student’ and define exactly what the career ready student must look like! Good companies know they will never find a perfect experience match, but instead will look for skills that allow the transfer of abilities. In math having the students actually work at solving math challenges will give them those transferable skills – and the confidence to use them! What does this process look like during an actual lesson? The initial introduction to this problem-solving task ‘stuff’ might present a challenge for both you and your classroom. It did for mine. I was following the pattern of ‘tell students the goal of the lesson’ (i.e. factor polynomials); run through the lesson with questions and discovery (felt like pulling teeth!), work problems with them; give them practice. My frustration was in the students’ comprehension- I could tell with my questioning that they didn’t ‘get it.’ ” Why don’t you just tell us?” one student even said. There was no desire to look for possible solutions. they just wanted me to give them the answers and accused me of not teaching them. The frustration on all sides stopped the learning process. I was asking them to do something they didn’t have the tools to do. I failed to train them in the method I wanted them to use! I’m here to keep you from making the same mistakes I did. Lay the groundwork first with mathematical thinking. Then get out of the way and be less helpful! Be Less Helpful I first heard this phrase while watching a TED talk by Dan Meyer. The genius of his approach, letting students look at a situation and decide how to solve it, was breathtakingly simple. It pinged deeply against what I was beginning to learn about in the common core ‘key task’ ideas. It fit in with my own ‘questioning and discovery’ method. I had to know more. What I found was a community (blogs and on Twitter!) of teachers who are committed to teaching their students through challenging tasks, and giving the students lessons as the child decides they need the skill to solve the assignment!” The difference? Nobody asks, “When am I ever going to use this?” They are putting their hands out for the teaching. They are engaged and interested because they are in charge of the process. The key to success is in choosing tasks that are going to teach the standards you want them to learn. Here is a good checklist for the tasks you want to use: 1. Identify the mathematical goals for the task: what standards will the students experience as they solve this task? 2. Identify how prior knowledge will be scaffolded. 3. Identify how students will demonstrate that the mathematical goals have been met. 4. Work the task in order to anticipate possible solution paths; ensure a variety of representations and/or strategies. 5. Identify common misconceptions. In other words- everything you always do for a lesson! Here is the difference: you are not going to tell them how to solve the task, or what methods or formulas to use. you are not going to remind them of where they have already seen the material or tell them how to start thinking about the task. You may or may not provide an illustration- the following task requires they visualize the triangle themselves. This is not the time for remediation lessons for kids missing skills. Let them struggle. As much as you want to give in and tell them what to do, don’t. Ramp up the questions to help the student find the entry point that fits their skill level. No, they may not get as far as the rest of the class today, but they will be further along than when they started! The task allows students to explore, investigate, and make sense of mathematical ideas on their own. Let it provide personal challenge and productive disequilibrium, too. Be less helpful! Here is an example of a key task: Teaching the Converse of the Pythagorean Theorem Students are told to envision a triangle, sides a,b,c, where side c is a specific given length. The task is to use various lengths for sides a, b and determine what effect the lengths have on angle C. (You may specify that sides a, b must be shorter than C, as an introductory exploration, but that is all.) As you go around the room, looking on as students individually engage the task, remember that you are not allowed to tell anyone where to begin. You should be ready to ask prompting questions (prepared beforehand) to give students ideas about entry points if they can’t get started. Have advancing ideas for students who get done quickly, (ask them to come up with a different method to do what they did, or ask them what happens when a,b are longer and shorter, or longer and longer). During the group work, listen for everyone sharing. Require that students justify and defend their work to the group. Be prepared with clarifying questions. If a student is changing their work because of another student, ask them to tell you what changed their mind- this articulation of ideas is critical. This is a good time to decide on selection and sequencing for the whole group discussion. Your goal here is to assess their learning and advance them toward the mathematical goals using questions (to prompt, to clarify, to restate). The whole group discussion is the opportunity for summarizing the learning from the groups. Encourage every student (I utilize Accountable Talk) to participate, either by sharing ideas, or restating comments from others. This is the place to make connections among solution paths- let the students make the connections (remember, we are being less helpful!) Don’t forget to tie in what they have done to the vocabulary. In the converse lesson, this is the place to tie the mathematics the students have used into one of the three versions of the converse theorem, (and to the standards goals for the lesson). It is important to have another problem or two that require similar (but not exact) engagement for ‘setting’ the skills they just used, and expanding on what they just did. Don’t forget this step. I believe in having students talk about what they have discovered- not to me, but to another student, or in a journal. This would be yet another step in formative assessment for learning. Or, I could have just asked the students to use the Pythagorean Formula, given the measurements for sides a,b,c and asked the whether angle C is 90 degrees. What do you think?