Carl's Teaching Blog

A place to talk about teaching and learning

Author: Carl Oliver (Page 1 of 32)

Stepping Towards Algebraic Thinking #NCTMRegionals 2017

Thanks for your interest in my talk at the 2017 Orlando Regional. The talk is about my use of patterns, which I’ve blogged about here a few times.

Here are my slides: Link to slides

If you want to do more with patterns make sure you check out visualpatterns.org

Here is an interesting prompt that came up in our discussion.

Voyage of The Creole – A Mini-Project for Linear Equations

If you are interested in teaching a captivating historical mini-project with linear equations that can be easily adjusted to fit your unit plan. This project is centered around the historical account of a successful slave rebellion that took place onboard slave ship The Creole in 1841. Students make decisions about where to steer the commandeered ship so that 100+ slaves can find freedom. Students use different linear representations and can also find the solution of a system of two linear equations.

Voyage of the Creole (google drive version)

Download (PDF, 232KB)

Map of Americas from eduplace.com

Let’s walk through the different parts of this project.

Introduction and Bonus Problem

The project introduces the story’s hero Madison Washington, a slave working on a plantation in Virginia, in this first part. This section is important for making sure the students understand the scenario, and there is also an interesting open-ended problem. The problem that asks students to find the number of Quarters and Dimes in a bag when given their total value, and the total number.

Having a problem at this point begs the question, “Why are you starting the project with an open-ended problem that isn’t necessary for the rest of the project?” Beginning the project with an open-ended problem is something that can help students think about some of the ideas that will come up later in the project. Having students work on the open-ended problem can give you a chance to ease students into creative thinking, using mathematical practices, and making connections. Students can work on the problem in a variety of ways, and it might be valuable for students to begin the remainder of the project feeling confident about their ability to forge their own mathematical path. This kind of problem is one that has been presented in other ways, and you can find ideas in Dan Meyer’s post, and in the comments, from when he wrote about these kind of problems. Finally, it is also something that can be omitted if there are fears the project is going long.

Part II, III, & IV

In this section of the project the main character Madison is taken as a captive on to The Creole. Within a few days he and the other captives take control of the ship and Madison must decide where to sail the ship. For the captives to avoid returning to slavery, they will need to find a place that isn’t in the United States, and then calculate the amount of time it would take to get there.

For students, this part of the project involves measuring with a ruler on the attached map and perhaps some prior knowledge of world geography. Students will want to take the ship to places all over the map and I’ve found it’s good to let them try any of them. Students who choose to travel a really long distance won’t succeed, and that’s OK. They will use and demonstrate the same mathematics as the people who pick safe locations, and add more variety. So encourage them to explore. I typically say that they can’t pick the same location as their neighbor for the sake of variety and to check against plagiarism.

Students will also have to convert the millimeters on the chart to miles. Depending on your students, this conversion may be something you want to scaffold, perhaps with a mini-lesson about changing units. If this is too much support for your students, try removing the box with the conversion ratio.

Students will then have to use a table to make an equation for the ship’s progress. This will require them to think through how to find the speed of the ship, or the slope of the equation, and also the y-intercept. The y-intercept is negative, which may be confusing for kids. The idea that the boat would cover less ground on its first day makes sense and there are a lot things kids could say (i.e. maybe the boat started off slow, maybe they took some time to load on the first day, maybe they went the wrong way first and turned around). This is a good opportunity for the students to make sense of the situation and compare the model to the situation, not merely write down the negative number because that’s what it is. Look for students to write something on the lines below.

The final question of this section asks students to recall the distance they measured, and use the equation to determine how many days it would take them to figure it out. Be on the look out for students who want to continue the table from above and keep filling it in until they get the distance.

Part V

In this section of the project, students use a linear function to see if The Creole can outrun the faster Navy ship with only a 5 day headstart. The story says that the Navy Boat travels at a rate of 42 miles per day, and they have the urge to make an equation similar to the one from Part IV. The Navy ship could be behind The Creole, by 5 days, so that means this equation needs a constant term. Students often get confused on what amount this should be, which is five days of travel. Understanding why the sign of this coefficient has to be negative is an important things for students to understand and explain as well. Once they have this equation, students can use this equation to first, see if the navy ship could beat them to their destination. If the numbers work out right, student who chose very far destinations will find that the Navy will beat them to their destination. This naturally leads to the next question should be: “where would the two ships meet?”

Part VI

In the last part of the project, students need to find where the ships would meet if they were to keep travelling according to the two equations that have been created. Sometimes students have already figured out 1 or more methods to solve a system of equations, but if that is not the case this is still a good question for kids to explore. At this point in the project the students understand the two equations, and what they represent. It would be a great time for these students to play with the idea of finding the solution of a system of equations and would set the stage for exploration in future units. If this project is meant to be an assessment of linear equations, then the students have already shown their abilities with regards to that in the previous sections. Students who lack practice with an algebraic approach can try using a table, or an organized guess and check, and in the process learn why an algebraic approach would save so much time! So this part of the project is valuable for any students who were able to complete all the work up to this point.

For all students at the end, there is the question of whether it would make sense to keep travelling towards their same destination. Yes, a Navy ship might not be prescient enough to follow the students all the way to their destination, but the idea is to make the best decision. This section helps the Madison Washington decide the best choice and offers another opportunity to see if students are making sense of the situation.

Alternate For Parts IV and V

In writing this up, it seems like it make a lot of sense to make the equations solve for the distance remaining in the journey, not the distance traveled up this point in the journey. This would be cool because then students would each have a different value for their equations and would make things more unique. I haven’t actually taught this version, so I made this a separate google doc with the slightly re-worded questions about those equations.

Conclusion

As the project ends you want to tell what actually happened to the ship. This video does a really good job of highlighting the themes.

Project Notes

This is the project that I used to do a long time ago in a school that was far, far away. I think there a lot of things that I could do differently with the project if I were to have more time, but I don’t teach this unit anymore, so I haven’t had the chance. I would love to hear any ideas you have about the project so please let me know your thoughts in the comments!

This project could lead to a interdisciplinary exploration of other topics around slavery from that era. Many topics could be tied into this project, so it may be good to think about whether your history teacher, or your english teacher, or you want to include additional information on topics like the fugitive slave act, the slave trade, the 3/5ths provision of the constitution, and more.

The information is pretty historically accurate. However, In October of 1841 when the ship left the port, there was still slavery in many parts of the world, but not entirely. I haven’t done all of the research to know whether the slaves would have been granted their freedom if the Creole theoretically landed on any country that a student might choose. Allowing students to choose ANY country that isn’t the US and expecting freedom might not reflect what actually would have happened if the Creole landed there.

This project also has A LOT of text. This may pose problems for ELL, or students with learning disabilities, or just kids who don’t like reading. The idea was that having a story could be as engaging, perhaps as engaging as a video, and hopefully that story can drive students to follow through with the project. It can be really easy to throw the story out as you modify this for your population. Definitely make the adjustments, but please keep enough of the story to help your students emphasize with Madison’s plight.

As always, if you use this, please let me know what you think in the comments!

Clog: Back on the VNPS horse

We haven’t used VNPS since the first class where some kids send like they weren’t having their needs met. We have done some journal prompts and those were particularly insightful. One student wrote a really thoughtful response about how she liked doing “the problem that we worked on in groups on whiteboards.” She detailed how talking about her thinking with the group helped her and seeing the ideas on the board really helped her understand why her initial thinking didn’t lead to the right answer. These details came straight from the heart, as I didn’t even ask for any of this. After reading that response how could you not decided to give VNPS another try.

What were the real issues that arose last time? One issue was some students felt that being in groups triggered their anxiety. Given our population doesn’t always mean students have time to build community, I can’t guarantee I can just run enough ice breakers to make everyone comfortable with each other. This time I have kids the options to work independently, but they have to be able to share out. Another student also had trouble working on a prompt that I only delivered via spoken words. To help with how they receive information, to I decided to give them the task on paper. Aside from that I kept everything the same.

When class started I decided to set up the VNPS I noticed a few things. Everyone was approaching the possible with more seriousness and focus. Unfortunately, some students who had the paper were just sitting down and doing the problem on their own instead of talking with their group. Instead of having a prompt for all students, next time I’ll have to project the prompt on the whiteboard, or tape one to each whiteboard so as to have students look at it visually. Taking the problem may help demarcate where each group is supposed to be. The need for separation arise after one of my whiteboards disappeared when I was on paternity leave. Two of the groups were crammed together on the front whiteboard and they somehow merged to become a mega-group while I was across the room. The mega-group ended up with a lot less participation from all students. Next time I need to either have clearer boundaries, students writing on the windows, or someone needs to return my friggin whiteboard!

Clog: First Day VNPS Anxiety

Today was the first day I got to teach a class this year! I’ve been coming in to school daily since mid-August 15th, so there was quite a bit of anticipation for actual teaching. The class that I’m teaching is called Problem Solving. This class was largely developed out of the Crossing the River with Dogs book by another teacher but has been, and continues to be, heavily adapted. The first section that I plan to teach in this 8 week course is Draw a diagram. Specifically, the Farmer Ben problem. It’s where you have to count up the number of cows and ducks that Farmer Ben could have if he had 22 animals which have 56 pairs of legs.

Before we got into a problem, we started in a circle. My plan is to do a circle kind of sporadically through the class, in order to build community and support my school’s restorative justice initiative. Once we sat in the circle I asked kids to think of a problem they solved and then choose one of three questions out of a hat:

  • What helped you understand the problem?
  • What was the process you took to solve it?
  • What were other ways it could have been solved?
  • How did other people help you solve it?

In retrospect, I probably should have started with something fun like “what are you watching on Netflix.” Instead we heard everyone listing all of the problems in their life on the first day of class. A few people harped on their personal flaws, or biggest mistakes, or some other horrible things that happened to them, and how they worked their way out of it. It wasn’t a mood lifter, probably going to have to tweak that for next time.

Next we counted off into groups of 3 and started doing VNPS using the Farmer Ben Problem. I wanted kids to think and to be a little uncomfortable, and that definitely happened. There were a few things that left me wondering whether I was setting kids up for failure.

One of Peter Liljedahl’s shifts for creating a thinking classroom was about reading the problem out loud instead of putting it up in words. Hopefully it will underscore the importance of paying attention and listening. Immediately after I finished reading it people asked me to read it again. After reading it the second time one student asked me to read it again and he had a look on his face like I just asked him to walk across fire. “I read it once, that’s why you need to pay attention,” I said, ushering him towards his white board, “you have to ask your group mates.” A few minutes later it was clear that he was standing on the side of his group mates while the other two members of groups were barrelling towards a solution. I checked in with the group and those two students knew just what they were doing, but that first student was silent and making a face that said “I have no idea what’s going on.”

Meanwhile, another student was making a face that said “I suffer from anxiety. Working independently on math problems was my sanctuary. Now my heart can’t stop beating, and I just want to go do the problem on my own.” This left me a little stumped. The student and I spoke after class, but I struggled to help this student connect the group work we are doing, to any real world that she has imagined. If anyone has any ideas of how to tweak the VNPS for these two students, let me know in the comments.

My first ever implementation of VNPS went very well for the majority of my class. Kids were talking and thinking together with a familiarity and comfort usually reserved for weeks into the cycle.  Some groups didn’t want to stop working after the time ran out. It is definitely something I am going to have to keep doing and keep tweaking. Next time I do it, I might use a 3-act of some kind that could have us do Act 1 in the circle, and Act 2 on the whiteboards.

Talking Math, College, and the Hard Work of Preparation with a ‘white buffalo’

Our amazing and wonderful College and Career Office took over our Friday PD to in the name of the City’s ‘College Access for All’ Initiative. Along side the powerpoint slides, and the brainstorming was a really powerful group of guest speakers…our former students! They arranged a panel of 6 recent graduates and asked them about their successes and failures in the post-secondary world, how we helped, and also what we could have done better. 2 students were doing well in college, 2 others were hitting bumps in their path through college, while 2 others weren’t in college at all. We were all understandably proud of each of the students as they described how we helped them with their current life. The students who were not in college pointed to the way that the school prepared them for adult responsibility through our internship program. Those in college were grateful for their experiences as well.

As the third student begins to give his remarks the art teacher leans over to me with pride and says “A black male in college!” “I know,” I replied, “it’s like seeing a white buffalo!” There are many more black males in college than there are white buffalo, but they are few enough that each one is sacred. The persistence rates at Michigan State when I attended were shockingly low, when something like 1 in 5 black male students, or less were actually making their way to senior year. It was a great source of personal pride for me to persist and finish in 4 years and I was quite proud to see Roger persisting as well.

Everyone was filled with pride as all of our students talked about the positives of their experiences. The mood changed when the graduates told us about their struggles. When it came time for students to talk about how unprepared they were in college this student’s comments sucked all the air out of the room. “I was very unprepared as a math student. You guys really need to make students aware that math in college is no joke. I got past the placement exam, but I failed calculus twice.” Roger spoke with the intensity of someone who was fighting for his life. “You guys really need to not be so nice to kids and hold them accountable when the aren’t doing what they are supposed to be doing. Once they get in to college they are going to be unprepared, and those professors won’t give them any breaks.” As I write this I know that I’m not capturing how deep his words were. All the staff erupted in applause afterwards. The applause served as a commitment for us as a school to continue to hold students more accountable in order to make them better prepared for post-secondary success.

The struggles of college readiness

It is difficult to prepare students for challenging college math in a transfer school where students are struggling to just finish their high school requirements. Typically students arrive having finished over 2 years of high school, typically needing between only 2 to 3 semesters of math. That isn’t enough time to teach much math. This student only needed 1 semester of math credit when he arrived. While students only take a handful of math classes with us, we have to try and provide nearly all of the math classes that students could possible take in order to plug the holes in their transcript. This means we need to provide algebra, geometry, trigonometry, statistics, and math electives. This student took geometry and an algebra class designed to help students place out of remedial math. So we have a small window to meet students math needs, and our course offerings are thinly dispersed along the spectrum. We struggle to offer more challenging courses that go beyond the typical high school requirements.

How are we going to prepare students to pass calculus in college?

If students are going to be more prepared in college, we need to convince them to take more math, above and beyond what they need, because we know they are not prepared without it. Telling kids who are already over age that they need to take more math classesSounds as fun as feeding my daughter vegetables.

So how do we go about meeting this student’s call to action? After the talk I pulled Roger aside and asked “If you rewind the clock back to you when you were in high school, would you have taken an extra math class that would have been more rigorous in order to prepare you for college?” He was honest and said no. He would have been too busy socializing across the street in the park to want to do that. Back at school two years later, his current self desperately wished his younger self had that additional class as an option. Instead Roger’s past self took the bare minimum needed to graduate. While it is important that we give students what they need to graduate high school, it is important that we also get them prepared for life after college. That would mean we would have to get students who don’t want to do challenging math, like Roger, to do challenging math. That was the charge he left me with he left to talk with other staff. “You have to figure out how to get kids like me to take that class.”

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