Do you think teaching math causes you to over think??? Almost like “going insane “…NaNa Dunn from a post on Visual Math’s FB page

Oh yeah! I used to dream about my lessons… over and over during the night. I’d wake exhausted. Here’s an example of what I do now…

When you realize the students can’t ‘see’ the thing you are teaching them

My kiddos are learning to write quadratic equations. This requires them seeing patterns of growth. The first lesson in this series was to help them see the growth in a pattern of cubes. I used a YouCubed lesson that involved coloring each growth step. Just that. This exposed several issues. The lesson with numbers was the next day, and I was pleased by the intensity of their interest, but here’s why I think that happened: I had prepared them for what they were looking for.

Color each growth step. Image from Stairs to Squares Gr3-5, YouCubed.org

Quotes taken from the original FB post…

Beth Hanna McManus writes “My 8th graders… totally missed the concept that the altitude to the base of the triangle goes through the vertex and hits the base at a 90 degree angle. I explained this in mathematical terms, in layman’s terms, using words like straight to the base, “shortest distance”, had them draw them on the white boards (checking and helping each drawing) multiple times, had them label triangles. When it came time for homework, no one knew what to do. I flipped out and spent the rest of the day in a dither wondering what is wrong with me that I can’t communicate with these kids…”

When I plan a learning experience, I look for the base skill/image/idea a child needs to be able to have to participate in a lesson. It may be a simple warmup – for the triangle problem above, after realizing they didn’t get it, I might have them draw the altitude, pointing out the way the angle looks, and how it starts in the vertex, in several different triangles. Just that. No math, just letting them learn to see what they are looking for.

Starla Adams writes, “Yes, yes, and yes….says the teacher who is out of strategies to teach long division after Friday. I actually thought I was going a little bit insane for an entire hour.”

Teaching long division is challenging. I’ve worked with 9th graders who do not understand what dividing does. If they don’t have an understanding of partitioning and regrouping, long division is just a nonsensical set of steps that they must follow – and memorize. A warmup using manipulatives (coins buttons beads) to set the stage for dividing into groups would help them understand what they are looking for. Then long division can be taught as a routine to get there.

Another common student skill gap with division is poor factoring skills. A warmup (or preview lesson the day before) with a factoring math problem string like this video from Pam Harris, might help strengthen their math fact fluency.

While working on any new skill that requires factoring, try giving students a factor chart. They won’t be using all their working memory on remembering number facts, but on learning the process of the task at hand.

Learning plans should empower students…

I realized that my 9th graders didn’t yet know that they had the power to create their own understanding. They were waiting for me to tell them what to do. Danielle Love and Kay Butler point out ways to shift the heavy lifting (and learning!) to the students!

As teachers, we spend lots of time creating learning plans. Many of us already know what misconceptions kids have, and what errors are going to get made. So lets plan ahead to expose -and remediate and preview- so that these issues don’t cause student failure during the learning of new material. These little discovery sessions and warmups are critical to building understanding and are often worth every minute of time we spend on them!

By all means, overthink and go crazy, in a most productive way! Math teachers, you rock!! I would love to hear how other teachers prepare for these misconceptions and gaps!

(And no, I don’t have those recurring math dreams nearly as often anymore!😂)

Thanks to Shana McKay of Scaffolded Math and Science, and and this really interesting thread on her fantastic FB Visual Math!

Let’s quit calling them 21st Century Skills; These babies are useful in any century!!!


Dan Meyer has struck again:

“I spent a year working on Dandy Candies with around 1,000 educators… In my year with Dandy Candies, there was one question that none of us solved, even in a crowd that included mathematics professors and Presidential teaching awardees. So now I’ll put that question to you.” Dan Meyer’s full post

What is it about a challenge like that?!? Of course I had to follow the thread! 

I read the over 100 comments as writers posed solutions, wrecked solutions posed by others, and even wrecked their own solutions! I watched as they systematically used the faults in their solutions as a springboard to better – but apparently still breakable – solutions.

I also heard a ghost of an admission that there may not be a single solution,

as timteachesmath writes, “Which broken algorithm is best so far? An algorithm that fails for ‘720’ but works for 95% of really composite numbers less than 720 might be better than one that works for ‘720’ but only works for 80% of really composite numbers less than 720.”

There is a Lesson here!!!!

I teach algenra 1 and geometry; that means 9th and 10th graders. I want to challenge them with Dan’s problem.

It’s simple, right? We are just talking about a box of CANDY!

I can see you now, shaking your head in disbelief: 9th and 10th graders able to frame an answer to a problem that even 1000 math teachers couldn’t solve.

Not only that, I can give this lesson to both algebra AND geometry!

Here is my explanation of the sequence of activities that would make the most sense to their budding understandings of math:

Essential Understanding: The best packaging involves the least surface area.

  1. The least surface area results from the tightest (closest) configuration of a cube’s side lengths.
  2. The surface area is a result of the combined areas of the six sides of the candy box.
  3. To find the minimum surface area for any number of candies, check for the following conditions: a) if the number is prime: 1, 1, the prime; b) a perfect cube: root squared times six; c) numbers with three primes: use the three primes; d) numbers with four or more primes: Multiply groups of the prime factors back together to find three products. These three products will be the three factors that will be the measurements of the box.
  4. Calculate the surface area from the measurements of the box.
  5. The box with the least surface area will have the factors that are closest to each other. It is possible for two of the factors to be the same number.

720 is a great example for (d):

Prime factors of 720 are 2, 2, 2, 2, 3, 3, 5

While they can be multiplied back together to create numerous factors, not all sets of three factors will give us minimal surface area.

Some of the sets of three that can be created are:

4, 4, and 45;

8, 6, and 15;

10, 6, 12;

And so on, until we get the multiples 8, 9, 10;

Checking for the optimal area involves a handshake (multiplying) among each of the three numbers – 8 times 9, 8 times 10, and 9 times 10, adding the products together and multiplying by 2.

Does anybody else see the individual lessons embedded in this process? This one problem is incredibly rich!

It’s not the solution, it’s the building of understanding!

The interactive process of doing this by hand is a wonderful opportunity to teach finding primes (6n-1), (6n+1). Students might also feel the need to learn how to find prime factors (and learning that all numbers are products of primes!). The question would arise about the geometry of area vs surface area. (Think of the manipulatives! I wonder of my kids would feel silly stacking cubes of jello!!!)

We also wouldn’t be able to ignore the eminently practical side of saving the planet through minimal packaging – not to mention the extension of how many candies we should pre-package for the best shipping (i.e, how many boxes can fit into a bigger box? Can we afford to package odd sizes and still keep our costs low enough to generate profit and sales?) (ooh! I can teach my kids to design boxes – quadratics, anyone?) Here we could also lead the class into the sales curve (parabolas – more quadratics! I’m in Heaven!)

By Jove! I think I figured out why Algebra and Geometry finally got together! They complete each other!!!

And I love the fact that once my students come to this understanding of the problem, they could begin to write a viable solution, either in algorithm or in code. Or maybe their understanding leads them to the conclusion that a single algorithm isn’t possible – did somebody just whisper the word “proof”? (You did just think that – you know you did!)

Just think of the STEM project ideas this activity could generate…

As many of Dan’s commenters pointed out, this is tedious by hand. But the truth of the matter is – they knew how to begin solving the problem by seeking to understand the problem to be solved! These are the skills our children need to learn. These are the lessons we need to teach. Let’s quit calling them 21st Century Skills; these skills really are useful for any age, anywhere. I’m living proof! (I’ve made it this far on those skills, haven’t I? LOL!)

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!

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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!

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