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.

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!

Making Math Formulas Tactile

The credit for this unusual idea goes to Math Giraffe and the Tactile Formula Project . Here’s how it played out in my 9th grade Algebra I classroom.

The challenge? Making Exponential Formulas out of concrete images

I believe the lesson is applicable with any formula and any level of student. Try it yourself… how well do you understand the formulas you teach?

The Lesson

I started by telling the students about ‘this cool idea I saw on the internet’.

The idea is to create formulas out of three dimensional objects, like candy or pennies, or legos, or cubes, or squares that suggest or seem to define the elements in a formula. I didn’t have the luxury of time to collect objects, so I told my students to use drawings of items that they planned to use to make the elements.

To get the creative juices flowing, I shared the images from the Math Giraffe blogpost, and started the conversation about what I saw in the images. The students quickly started noticing and interpreting what they saw in the images, and questioning why students used some of the images in the examples. Some students began interjecting their own ideas for images they would have used. This really primed the pump, so to speak, for the formulas we were going to interpret.

Images from Brigid’s Math Giraffe blog post.

Brigid, of Math Giraffe, used the classic slope-intercept formula,

y = mx + b

well known to my students. The task for my students was to do this with the multiple versions of the exponential formula:

I wrote all of the different forms of exponential formulas on the board, and let each student choose the formula they wanted to use for their visual interpretation. What followed was a deeply enriching -and eye-opening – mathematical and conceptual conversation among all of my students!

Note: We used exponential formulas, but any formula would work with this activity.

The challenge for my students was using pictures to relay the action contained in the formula elements vs just using a picture that looked like an item in the formula. An example of this was incorrectly using bananas for parentheses, because, as the student explained, “they looked like parentheses.”

This student was just a little shaky on choosing an item that revealed the inclusive nature of the parentheses. Another student didn’t use any pictures at all. He wrote out his formula using a color for each letter, basically color-coding the identity of each element in the formula with an elaborate system for understanding each part:

Did they miss the point? Absolutely not! Both were much more able to explain when and how to use the formula after this task.

This task got my students thinking more deeply about the formula itself, than just trying to memorize when to use it!

Interestingly, the exponential growth element was mostly identified as time, with one child drawing a clock face whose hands illustrated the exponent variable.

I noticed that after this exercise, students were beginning to talk about the formulas by identifying individual elements and their actions. When explaining why representational items were chosen, students were recalling examples from word and story problems from our practice work.

Research (source articles noted below) shows the value of connecting the abstract ideas of math to concrete items and has been proven to help with retaining the material. In this lesson, connecting physical objects (or in this case, drawing pictures of physical objects) allowed my students to make clear connections to what the formula was representing. More than that, this exercise functioned as a valuable informal assessment for me about what my students really understood about the elements and actions of the formulas we were learning. I look forward to using this task again!

Asking the right questions of ourselves…

We question our students to elicit and engage, to push their sensemaking, to activate prior knowledge, and to get them thinking about their thinking. But do we question ourselves and our pedagogy with the same focus?

Table taken from National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all.

In the rush to use new ideas, incorporate technology, or just ensure our students are doing math things from the minute they walk through the door, are we giving enough thought to why we are choosing an activity? What is that activity suppose to achieve? Has it earned the time it will take, not only for the students to complete, but for the grading and feedback? In other words, will it move these students further towards the goal?

Learning Targets are Not Just For Kids

My colleagues and I have been asked to share learning targets with our students; to write the daily lesson or goal on the board and go over it, so students know what successful learning looks like.

As you think about what your students will be doing, do you ask yourself just how the activity will provide the experience you want for the student? What will it tell you or your student about their learning? How will it move them closer to understanding? How will it engage the learner to produce the desired result? What will the failure of this activity tell you? Tell your student?

For the math classroom, skill builds upon skill. Knowledge is formed by understanding how the old can be reshaped or used to fit into the new. It’s like reading, but with numbers. In learning to read, we begin to decode shapes that are letters, that make sounds, that can be arranged into words (patterns of letters), that are then arranged into larger groups of sentences to convey information.

In mathematics, we learn our numbers, which instead of sounds convey amounts. At first, these amounts are concrete, as we count fingers or toes or toys or blocks or cheerios. At some point the numbers begin to represent the amount. We put these numbers together into groups that define patterns, and we put these patterns into sentences that convey information. Every time we learn a new concept, we can place that building block in context and do more than we could. A concept in math means we’ve identified a relationship, a cause and effect, a reasoning about how one action effects an outcome. When we understand the connection, we can extrapolate, interpret, compare, contrast, synthesize, and create. All of those things that we say we want our students to be able to do, no matter the subject.

Is this what your materials and lessons are teaching?

Every teacher reading this has bemoaned the lack of time we have in the classroom. How we spend that time often forces us to cut our activities, explorations, and conversations about the material. We have to hit the ‘most important’ standards, or the ‘big ideas’. Yet we know that knowledge is built by exploration, by focusing on a problem or situation, by playing with ideas. But how often have you asked ‘how will this activity increase understanding of the concept?’ ‘What is it teaching?’ ‘How does it allow showing the learning, or mastery?’ ‘Will it allow for connections to what has already been done, or to what is to come?

Is the idea or activity sticky?

By sticky I mean will this be something the student will think about longer than the activity itself. Will a student come in a day or a week or a month later and say, ‘I’ve been puzzling about this idea and I think I finally get it.’ And yes… I believe all of our ideas can be sticky – not necessarily for everybody at the same time. If we are truly giving each part and moment of our lesson thoughtful care, there will be more sticky moments than not. Those moments are what build interest, knowledge, and understanding. One of the best examples of this is the Four Fours activity. My students worked on that for over a week!

How do we get there?

1. Put yourself in your students’ shoes.

Think about what is happening to them daily. When they walk into your class, where are they coming from? Have they had time to process their last class? (Probably not.) Are they looking forward to math or dreading it? Did they do their homework – or even understand it? Is your class a relief, a chore, or an interesting, thought provoking space in their day? As I write this, I see the faces of my students, and can easily see the few that truly look forward to this class – but there are occasions where my lessons have let them down, too!

What do they need from you in that moment, to get their mind off of what has happened up to the moment they walk through your door?

For a start, pretend you are a student and walk into your classroom. Pick up that starter. Take it to a desk and try your own lesson. Where is your student brain? How does it make you feel? Does it do you want it to do? Loosen them up? Assess yesterday’s lesson? Review a skill they need in the main lesson? How will you check the outcome? This shouldn’t be a ‘take up and grade’ – it’s very name implies short, sweet, and to the point. What you want to accomplish must guide what you do. What you do sets the tone for the rest of the class. If the starter isn’t working for you, it isn’t working for them. Stop. Just stop doing what doesn’t work.

One teacher I know

has a great routine. She has trained the kids to pick up a starter (half page, every day) on their way in. Some fill it out. Some don’t. She goes over each problem, quickly working it on the board. She asks a few questions about the numbers or the process. Usually she has to get the class’ attention, many are off task. The kids who knew how to do this are already zoned out. The ones who don’t know how either copy her work down exactly, or don’t write anything. She does this quickly, and in her mind this is circling back around as a review for weak skills or concepts. She is practicing good classroom management by getting kids in their seats and working. She has trained them that math is boring.

During one week, she gave the same material as a starter and as a quiz to check for Learning. To her frustration, it did not result in increased knowledge for those who hadn’t already learned it (and I suspect it was extremely boring for the handful who did!) This is a 15-20 minute activity every day. How could she change this activity to get the desired outcome, i.e. strengthening this skill?

A process is a pattern of activity. A concept is the explanation or reasoning for why we do the process. Teaching a concept should lead to the process. In the interest of time, students often learn the process. Then it’s practice, homework, and a test. Are you teaching process or concept? Are you reviewing process or concept? Are you practicing process or concept? Concept is harder, takes more time and doesn’t work well on a worksheet. It is much more interesting, however. Concept is sticky.

You do not have to reinvent the wheel!

No time to write those magnificent lessons? Have I got a tip for you! You do not have to do this alone! Lessons and resources are out there and so many are FREE. Check out these links (courtesy of Matt Vaudrey and the #MTBoS:

Not only will you find good lessons, you will find teachers who are constantly looking for better ways to share this wonderful world of mathematics!

Here are a few more questions for you to consider, (and which I will be grappling with while planning my next classes):

3. What is my lesson intended to do? How do my materials (problem set, delivery, class activity and structure, timeframe, sensemaking, etc) support this goal?

4. Where are my kids likely to fail? What can I do beforehand to support the weak spots (Starter idea!)?

5. What does the learning of the concept look like? What do I do for those that ‘get it?’ What do I do for those that need more? How will I know (formative assessment). If they don’t start, WHY not? If they don’t finish, WHY not? Do they really know how to do it? Is homework appropriate- i.e. will this truly extend the learning?

6. Does my lesson connect this idea to what they already know? Does it give them a peek into a future idea?

7. When/how will I give them time to process what they’ve done?

8. When will I revisit? How will I revisit? (Yes! Plan for this!)

I leave you with this:

The moment you can really know a student has internalized a concept/learning target is the moment you hear/see them sharing what they’ve learned with another student. Plan for that, too.

Change the mindset from ‘finished’ to ‘learning’….

In this continuing exploration of Academy high school practices and their unique position to affect change, the idea of changing classes every day, every hour or more, is examined.

Currently, the Academy school model is some type of block schedule. Students take an assortment of courses in order to meet a specified list of credits towards graduation. Throughout the day, students get an hour to 90 minutes of a subject, bells ring, brains shift, and the activity is repeated over and over, until the final bell. Is this really the best way to learn? Is it the way we learn to tie our shoes? Did mom give us an hour of this and an hour of that as we learned to make cookies, or learn that different sized pans would only hold so much, before spilling the contents across the floor? Did a timer go off somewhere as mom yelled that it was time to go learn something else?

I believe there is a better way. We learn by experiences: examination of a situation, trying a solution, evaluating the result, trying again, evaluating, reflecting, etc. and throughout the process, storing the experience for future reference. There is also an element of sharing, talking over results with another person who has experienced, or is experiencing the same thing. With my students, however, I usually get one offering a solution and the other copying it down, in order to simply get finished. No learning, because the goal isn’t to learn. The goal we’ve set is for them to get ‘finished.’

Change the mindset from ‘finished’ to ‘learned’ or ‘learning’ and time becomes irrelevant. The goal is learning, so students don’t leave (or they return each day) until they’ve mastered the concept.

The idea that students are given a list of items to accomplish at the beginning of their high school career, and that they can finish them in any order, and in whatever time frame they are able, is a concept whose time may have come.

What if high school were as exciting and looked forward to as coming of age? Shouldn’t this be a time of discovery and promise, instead of dreaded and scary? To give our students the responsibility of their own learning, I believe they also need to know what that learning entails, so they are properly equipped to run with the responsibility!

I think it is time we return to trusting our kids with more, sooner. The movement to protect kids has pushed personal responsibility to the back burner. We are so afraid of letting kids fail, we do things for them, which creates a child that doesn’t need to step up and take responsibility. In addition, it creates a child that views failure as a disgrace, meaning that child will do everything in their power to avoid failure. I’ve observed students who work harder to avoid failing – often by cheating or straight up avoidance – than they would to engage in the learning! In their defense, the classroom can be a real snoozer if there is nothing engaging or relevant going on.

‘I believe every child can learn’ is the new mantra.

What I don’t see is the belief that every child can take control of their learning; that we can trust natural curiosity to take them places our planned lectures never could or will. There has to be a freedom on the part of the teacher to ‘hang on for the ride’, as the student forges ahead.

It’s a little like taking a horseback ride.

Every student has a mentor, advisor, or adult that holds them accountable. The entry to high school is planned in a community of parent, student, teachers. The goals for the learning are spelled out. The reins are placed in the child’s hands, with guidance. The adults are the spotters, close in the first few rides, backing off as the student gets more familiar with the process and expectations. The student isn’t being pulled along the path to the destination; the student is choosing the way, enjoying the ride, the pace, the view. The destination arrives naturally, and perhaps a little differently than first envisioned.

This idea does require a bit of a structural change. Instead of a class schedule, a school could end up with every student starting in the same place! Big room needed! The solutions could involve everything from an orientation style of instruction with starting points, to letting students come up with possible solutions that would allow them to get in the knowledge- and learning time- they need.

The Checklist

So what does this checklist look like?   Is it the same for every student? Is it modifiable, like college programs? What’s required? What’s negotiable?

Is it a list of standards, or more a list of abilities, attitudes, or problem solving? Is it a bit of both? How do we assess the learning? Who assesses the learning and levels of achievement?

This process requires collaboration among student, parent, and faculty… but it also requires a commitment on the part of teachers and school leaders to stick by the rules: student choice, student struggle (a critical component), and a plan to reward  success AND failure, because there will be both, if we’ve done it right. It’s time to do this right, and to return a love and excitement for learning back to our children!

Close Reading in math; and the “after” math…

Literacy is everywhere, including math class. And I don’t mean those clever (and not so clever) word problems.

I used a simple story book, predictions, and paired discussions for my 9th grade Algebra I -ers. I passed out colored pencils for annotation, and paper for predictions.

We started with the brief prelude, a paragraph, about a ‘happy-go-lucky young man’ who meets an old man who tells the young man,’I have a gift for you.’

There were a few other clues in the paragraph, so after asking the kids to read and annotate the paragraph individually, a volunteer read the paragraph aloud. With no further discussion, I asked each student to write down their prediction of what the story was about. Then I asked them to share their predictions – and their reasons why they thought that – with a partner. Then they would listen to their partner’s prediction and decide if they agreed or disagreed.

I modeled having a conversation…

To remind them of how to have a meaningful conversation, I modeled listening, responding with questions about what was said, and commenting on the information. We talked about how conversations were about listening and responding, not a contest of spouting information. Everyone got to practice, with a little (okay, a LOT of,) prompting!

We turned to page 2. My students were now eager to read further (we had predictions to fulfill!), and since the book’s format was laid out in sections of two to four sentences, with lots of images, I was able to let all have a turn. We took the four sections on pages two and three rather slowly, and we pulled apart the actions of the old man (he gave the boy two seeds), and the young man (he cooked one seed and planted the other seed). We talked about the passage of time – winter, when they had met, and the boy planted the seed; spring as the ‘sturdy’ plant appeared from the ground; summer, when the plant produced two flowers; and fall, when two fruits became evident. Each one of my questions caused them to return to the text for details.

Before we turned to page four, I asked my students about their predictions, and whether they had changed their predictions based on this new information. The conversation among the students turned to the details they just reviewed: how long it took to grow the seeds, how one seed could feed someone for a whole year, and why couldn’t he get food where he lived. New predictions came from the ashes of the old, with several students predicting that the plant would grow huge, up to the sky, and the boy would climb it. This was an obvious scaffold onto a familiar story. They told me the key word ‘sturdy’ meant that it would be strong and big. We turned the page.

New predictions came from the ashes of the old…

As each student read a section, we learned that the plant the young man had grown produced two more seeds from the fruit. He cooked and ate one and planted the other. We noticed from the picture that he put a hammock in a tree. Apparently he was planning on staying put. I pointed out that this was something that usually happened when people planted, instead of hunting for food. Another scaffold, as they agreed with me that he would need a place to sleep while the plants grew.

Again, we turned the page. This time, the readers learned, Jack (that was the boy’s name) had managed to grow another plant from the seed, which yielded two fruits, which yielded two seeds, one which was dutifully cooked and eaten and one planted in the ground. No changes here, but we noticed in the drawings that Jack seemed to be getting fat. Also, the plant wasn’t getting any bigger, which some of the boys seemed disappointed with. I think they were still waiting for the plant to grow up to the sky. One boy persisted in asking why Jack just didn’t move to a place where he could buy some food. We made more predictions, adjusting our expectations based on what we’d read. I asked them if their mental picture of the story, and of Jack, was changing. One of the girls agreed, and then we turned the page.

Their mental picture of the story, and of Jack, was changing…

The next two pages supported the story line. Jack continued for two years to cook and eat a seed and plant the other. The plant never got any bigger and never produced any more than two flowers, two fruit, and two seeds. It looked as though Jack would be living out his life, year by year, cooking, eating and planting seeds. I had the students make predictions anyway, and talk about what they thought Jack should do, given the circumstances. Some of the children thought Jack was stuck in a rut. On to the next page…

My readers were still eager to read. They continued, aloud, to devour the slim text (it was great to get a window into their abilities to read and decode the text – you know, for those word problems we will get to someday!) It was momentarily exciting to hear that Jack was as bored with his existence as some of my students! He said, and I am paraphrasing here, ‘if I always do what I have always done, I will never get anything different than what I’ve gotten.’ (Here I looked at my math students to see if they had taken anything of a personal nature from this comment. It appeared they hadn’t felt a connection.)

‘What,’ I asked, ‘Do you think he is going to do?’

‘He’s going to plant both seeds,’ spouted one girl. ‘But what will he eat while he waits for the plant to grow?’ I asked. They hadn’t thought about that, and in thinking about it, several students were seriously confused about how many seeds he could plant if he ate one, and he only had two to start with. (This may point to a reason that so many students struggle with adding negatives and positives.)

A serious discussion ensued as to what he was giving up if he didn’t eat the seed. Would it be worth it? We talked about how sacrifice is sometimes needed to affect change. One of the kids said he would be cranky because he was hungry – the voice of experience talking? The kids had gotten into the spirit of the lesson and were ready with their predictions. The Jack and the Beanstalk contingent were ever hopeful. There was still one boy who wanted to know why he didn’t just move to a place with food and forget about the seeds. We turned the page.

The story continued with Jack explaining that he decided to go hungry  so he could plant two seeds. To assist the verbal process, and give structure to the next round of predictions, I drew a table on the board and labeled the two columns ‘seeds planted’ and ‘seeds produced’. I then filled in the first line. Under ‘seeds planted’ I wrote 1, and under seeds produced, I put 2. I asked the kids to review the current situation: is this correct so far? After some discussion of my column titles, the students agreed. I asked them to fill in the next line. Now that Jack has decided to go hungry for a year, how many seeds will he plant? The students agreed he would plant both seeds.

I wrote a 2 on the second line under ‘seeds planted’.

‘So what goes under seeds produced?’ was my next lead in, ‘and why?’

Another look at the text produced the facts: two seeds produced two plants, each with two seeds. How many different ways could we count this, and still get answers varying from 2 to 6?!?

More fingers, more math. One girl supported her position loudly by mentioning multiplication; two plants times two seeds would be four. Several students had made the same conclusion by different routes, but getting the kids to share their explanations that they had discussed with each other was the hard part. The confidence that they can be right is so difficult to encourage! The student who mentioned multiplication was encouraged as another student agreed with her. A 4 was written in; satisfaction all around.

So, how many seeds will he plant next? And ‘why?’

‘Four’ was the immediate answer, until a single student voice reminded us that he had to eat. So began another discussion about how to calculate the number of seeds, and how many will he eat, and how many will he have to plant? Some students felt he was going to eat two of them. (We had doubled the amount of seeds grown – we must double the amount he could eat!)

We returned to the initial instructions on page 1. A student read the evidence and told everyone definitively that only one seed would feed Jack for a whole year. By this point, I was listening to these children teaching each other how to support their facts, correct their misconceptions, and expose and correct mathematical errors. Every child was involved in this discussion – every child!

One seed eaten, three seeds planted was the ultimate conclusion. (If I rendered their discussion here, you would be as bored as we were with three years of Jack eating a seed and planting a seed!) The chart was filled in, and we were ready for the next question: if he plants three seeds, how many seeds will he have to plant the next year?

While the whole process was a conversational struggle, before I left them to this final prediction, I pointed out the table and had them write it in with their predictions. We had just begun defining a function the day before, so I asked them if they thought this was a function? I was rewarded with a student identifying the input and output terms of seeds planted and seeds grown, as x and y. Several students then made the next connection to the fact that none of the inputs we had so far would repeat, so it must be a function.

Without giving away any more of the story, I will tell you that they were able to successfully calculate the answer. We stopped here as I had them make a final prediction about the rest of the story. We will revisit Jack and his adventures in the coming unit.

Here is the ‘after’ math:

Our school lit coach came by two days later and interviewed some of the students that had participated in the close reading. Here are some of their comments (I stayed across the room, out of their line of sight. I was curious to hear their honest responses.)

• “I liked it because it was different from the way we usually do math.”
• “Why can’t we do math like this all the time?”
• “I felt like I really understood the story.”
• “I remember more about how many fruits Jack planted.”
• “The table helped me understand.”
• “I think it will help me in my reading in other classes.”

‘I think it will help me in my reading in other classes.’

Wow. Now, that’s some kind of ‘after’ math!

Editor’s note: the standards for this lesson had to do with identifying functions; recognizing a function in multiple forms (such as tables), identifying functions from contextualized settings; the literary standards were identifying supporting information  and using contextual  clues to support mathematical arguments (a Mathematical Practice, as well). Additionally, the conversation supported mathematical reasoning, practicing vocabulary, and reasoning aloud. The constant predictions were embraced by the students, who stayed involved for what amounted to about 60 minutes of close reading. They had to know each outcome! No one read ahead, which surprised me a little. They really wanted to maintain the suspense! I can’t wait to revisit Jack and the rest of his story!

For those who notice the reading level, I wanted to use a text that would provide a low floor, and that would allow me a high ceiling – the actual math is writing and calculating an exponential equation. This was a great text for my ninth graders! The text didn’t cognitively get in the way of the activity.

To read Jack’s full story, visit Anno’s Magic Seeds, by Mitsumasa Anno. For more of her series of math books click here.

Knowledge Machines are here; How will you use them?

There was a time when school was about learning the three R’s: reading, writing and ‘rithmatic. Sounds like the beginning of a long ago time story, doesn’t it?

After reading this 1993 article from Wired, I realized that Papert’s ‘Knowledge Machines’ are, in fact, here.

New Year’s will be in August, this year.

If you are a teacher, that is.

On August 8, hallways and rooms will fill with the wriggling eager bodies of their parents’ best! I can hardly wait! During the summer, I’ve been loading up on great ideas, reading about effective teachers, discovering new tech and new resources, and creating learning plans that will put them into practice!

I am teaching my students Algebra I this year; 9th graders, some returning 10th, and I want them to feel the excitement, the sizzle that I feel with math. This is a new year, a new crop of children, a new chance for me to share what I love- math – with children who never fail to delight me (and challenge me, worry me, turn my hair gray, and, well, you get the idea- but that’s another post!)

The year I have planned, this year, will be different. This will be the year that every student tests proficient on the EOC, aka Georgia Milestones. My lessons will start with Wonder/notice, there will be lots of student conversation, with roles for small group work, and conversation starter posters on the wall! My class will be fully engaged, will actually complete their assignments, will receive thoughtful feedback, and grades that really show how well they’ve mastered standards. I’ll make all the calls, on time, to the parents.

My IEPs will have clear goals, my re-evals will be works of art! I’ll handle my discipline issues with skill and compassion. This year, I’ll have strong closure routines, include literacy in every lesson, hold awesome number talks, and have nimble responses to my formative instruction.

This year, my room will be organized. I’ll have study centers, whiteboard walls, standing desks, and engaged, curious students! This year – well, this year will be everything I was hoping last year would be…
So, you see, teachers really do celebrate New Year’s in August!

Murder Mystery Solved with Trig!

Dateline: April 14, 2016

The murder of Maria, whose body was conveniently found at right angles to Leg Streets A and B, has been solved! Investigators found the weapon across the river, apparently thrown there by her assailant while he was running down Leg Street A in an attempt to escape. A quick thinking officer (who had majored in math at the police academy) was able to calculate an angle measure for the angle made by the throw from the perp and the street leading to the victim.  Another savvy investigator was able to determine the distance from the suspect to the location of the attack.

With the mathematical evidence in hand, investigators were able to triangulate a conviction. Math teachers everywhere weighed in, saying it has the proportionate ability to change the way investigators do business!

Dimensions of the prisoner’s defense will be released at a later date. Film at eleven.

Okay, so I don’t really have film (we forgot to assign the job of reporter!) What I do have are a room full of kids who can now set up the proper proportions for trig problems!

Here’s how the crime went down:

Scene 1: Before the murder, I handed six students a few props:

Each student had to use the prop to arrange themselves into a triangle. The other students watching were, um, helping. (that’s what they called it!)

A short q&a followed:

Me: Okay leg A, are you opposite or adjacent to angle b?

Hapless Student holding Leg A sign: “I’m opposite, um, no, I’m right next to him (indicating student holding the angle b sign)! What does adjacent mean, again?”

We were able to sort out the definitions, and the students holding the leg signs got pretty good at determining whether they were “opposite” legs or “adjacent”  legs. A big moment came as students noticed that they could be opposite OR adjacent. More importantly, they were able to articulate WHY the status would change.

More importantly, they were able to articulate WHY the status would change.

Scene 2: The next six students were given the cards. This time, I stood back and let the first group help position the players. A little skirmish ensued as Leg A and Leg B were being positioned. After a brief discussion about whether or not leg locations could be interchangeable (did Leg A have to go in the same place as the first triangle?), it was decided that as long as a leg were placed on each side of the 90 degree angle, it didn’t matter what we called them.

The opposite and adjacent discussion began again. It was fun watching students correct these new players, or making them guess by giving them tantalizing clues!

(If you ever want to know what you look like teaching, give your students the reins. Mimicry is not dead!)

Scene 3: With everyone up to speed on definitions, the murder could now commence! Maria was positioned. Ryan was immediately suspect, as we put the crime scene tape in his hand and instructed him to escape a bit down the hall. The “weapon” was given another piece of crime scene tape and told to take off in the opposite direction. The “hypotenuse” was asked how far the “perp” had thrown the weapon. We stretched the crime scene tape from the suspect to the weapon location. It was at this moment that I heard several students say “Hey, we made a triangle.”

It was at this moment that I heard several students say “Hey, we made a triangle.”

(Scary, I know, right?)

After a bit more discussion, the students determined that we needed an angle and we needed the distance from the body to the suspect to set up a proportion to solve for the distance. Two students were dispatched with the piece of crime scene tape that had been held between the victim and the suspect (Leg A, for those of you following along). Twelve inch square floor tiles assisted in the crime scene measurement. I used my oversized protractor to come up with the angle measure, and we were ready to set up some proportions!

Back inside the room, our eager detectives checked their trig proportion info sheet (yes! They used their NOTES!) and settled on cosine, adjacent and hypotenuse. I stood back and watched them argue over who was going to set up the problem, exactly how to set it up, and how to enter the information into the calculator. Then I watched them convince one another which answer was correct.

Concrete to representational to modeling AND peer tutoring…I love it! I would say that a murder wasn’t the only thing that got solved today!

We’d like you to take over this class, six weeks into second semester….

Six weeks into second semester, I was asked to become the teacher for a group of Algebra II students.

“It isn’t that I don’t like math. Learning takes time in math, and I don’t always get the time it takes to really understand it.”*

How many more of our students feel this way, but instead of telling us with words, they distract, joke, sleep, or skip class:

…Math is such an interesting subject that can be “explored” in so many different ways, however, in school here I don’t really get to learn it to a point where I say yeah this is what I know, I fully understand it. We move on from topic to topic so quickly that the process of me creating links is interrupted and I practice only for the test in order to get high grades.

Taking Time Learning Math:A Student’s Perspective by Evan Weinberg

Would I want to come to my class?
This question haunts me. What are my kids seeing, feeling, thinking? Why does this kid come, but stay totally uninvolved? Why does this child talk, constantly, but about anything but math? Where did curiosity go? Is my class a class I would look forward to?

My personal enjoyment of math comes from the struggle with ideas and the satisfaction I get from my connection of and understanding of the relationships among those ideas. It’s like a huge puzzle that will take the rest of my lifetime to fully understand. The student’s comments in Evan Weinberg’s post resonated with what I see happening with my students. They are not learning math so much as preparing for a test about math.

They are not learning math so much as preparing for a test about math.

The current situation of ‘learn how to do this; learn how to do that’ mentality is slowwwwly changing over to ‘understand why this is so; why does this relationship work’ exploration. It will need a shift in how we teach, letting kids struggle and connect ideas (we must facilitate this exploration, but not down some tightly designed path), and changing our view of grades and mastery. I can’t say I don’t have the answer- I am working on an answer that works for me and for my students. And I’m sure I am not the only one teacher who has found the path that is taking them closer to the ideal.

This post grew out of my response to Evan’s column. His response,

“I completely agree that this is a shift, and it is ongoing. Clearly, despite the changes I’ve made to the way I teach, students still get the sense that the test is the important part, which means there is still a great deal of improvement yet to be made!”

*Taking Time Learning Math:A Student’s Perspective by Evan Weinberg