How would you finish this WODB?
How would you finish this WODB?
Which one doesn’t belong? Can you find more than one?
The first couple of weeks are always fun, if stressful. I spend the first 2 weeks introducing my students to the wealth of resources I will implement in the classroom, and today that was Graphing Stories.
My goal is to find a way for my students to create their own videos even though I can’t directly video them or their voices. They all showed interest in this project, creating something that relates and connects math instead of requiring them to calculate.
This year I introduced the topic a bit different, I am still building my classroom into a safe zone for students to think and share ideas. I told them the name of the website, and the particular video we were watching, in this case, Christoper Danielson’s How Many Ponies.
I only told my students this: that they would watch a short video clip and create a graph based on it. I immediately had some students voice their concerns, wanting me to totally structure the assignment and outline every step they needed to perform. I recognized their concern, most of my students are used to traditional math- performing algorithms in specific orders and steps. They are the student who won’t attempt any assignment without first checking in with you, wanting you at their side while they work through a problem. Many of these students are capable mathematicians, needing nothing other than the constant reassurance that they are doing the correct steps. I try to transition them away from this dependence, to become independent mathematicians so they can become confident in their knowledge and abilities to problem-solve. I reminded them the nature of what we were doing, that they needed to play with math- try things on their own without worrying about specific structures they needed to follow. I reassured them after this “practice graph” (one that we were not going to grade- I mean who grades introductory tasks and ideas? How can we expect mastery on a new concept?) I would give them some more information on the design of the website and what their task was when presented this type of problem.
After watching the video the first (and 2nd slow-motion) time, half of the students sat there and gave me the “tell me what you want” look. A few even asked “what do I do now?” My reply was simple, “Make a graph of what you just watched.” I had Students reply “I still don’t know what to do!” I would ask them to “Try something.” I reassured my students that I was not grading them on this first graph, I wanted them to play with math, try things when they were not sure, allow themselves to experience math through their experiences. Most students did attempt some sort of graph.
A few didn’t, and I didn’t want to lose them. So I asked students to finish up what they were doing so I could ask them a question. When they did I asked them, “How many of you feel Math Class is a place where you have to solve problems the way your teacher wants, quickly and with no errors?” Every student raised their hand. I followed up with the statement of “I would like to tell you that I am not the teacher that expects you to all solve problems in a particular manner, if you have a question about a problem- expect a question back. I need to ask you questions about how you think about a problem so we can work together to solve it.” I pointed to my whiteboard that still has the Four 4’s on it. Just like the Four 4’s puzzle, there are many ways to think about and approach math. I need to determine what your experiences and method is.” While I was talking I first saw a bunch of deer in headlights, students who felt like the world was just yanked out from under them. Then things shifted a little as they looked at the work we did this this week. Those students who hadn’t attempted a graph went to work.
We then talked about things I noticed while students were working: types of graphs, labels, shapes, etc. I never used a student’s name or placed their work during this time- they are not quite ready for that step yet. I saw I still need to work on making the room a safe place for them. As we discussed these things generally in class, I watched students as they compared what we talked about to their first attempt. Of course, I saw many start to erase. I quickly intervened on this, explaining that their first attempt was just a warm-up, I did not expect them to get it correct the first time they attempted it. One student told me that they expected everything they did to be corrected, and correct- that’s how Math Class was. I just replied, “Not in my Math Class.”
I then went over Graphing Stories in general, and asked them what they noticed when they first watched the video. I asked them what those things meant in their assignment, and connected what they observed to what they needed to do. We then watched Ponies again, and students were much more confident in making graphs. Although Christopher’s graph was probably one of the most complex on the site, one I wanted to work with them on. I was happy with how students progressed on this task, and I let them know that. Overall Graphing Stories was a great lesson and discussion for students, and our goal this year is to create some of our own!
I haven’t heard back from anyone (and I’m not sure I really expected to) so I’ll just relate what happened in my classroom when changed the way I asked a question. The first thing I’ll say is that changing the way you question definitely changes the way your students think- and in most cases it is for the best. Take for example this convo Christopher and I had yesterday…
I was writing an Open Middle problem, and as I often do I hit publish too quickly. This is what the first draft was…
Now, I want you to know that this was up maybe 2 minutes until I realized that I hadn’t put any real constraints on the problem. Typically with Open Middle problems you are allowed to only use a digit once, and I was modifying this for a smaller number set. OF COURSE Christopher would be online and checking my tweets because this is what happened next…
I had also realized that it wasn’t possible if you weren’t able to repeat numbers- which would be fine for the classroom environment, but not for the Open Middle format. This is what I was originally thinking of the problem so I published it instead (which caused Christopher’s reaction)
I was trying to reduce the numbers usable because I was wanting to target 3rd grade and was trying to put a smaller constraint on possible values for the boxes. Unfortunately that wasn’t working, and I had to go back to the original format. But like Christopher stated, it changed the problem and some of the thinking students needed to do.
Back to the original set of questions. The first:
Create an addition problem where 2, 3 and 4 appear somewhere in the problem or answer.
Let me remind you that I work at a juvenile center where I have mixed classrooms. There is a wide range of student ages and ability levels in one class. Most of them were VERY uncomfortable with this problem. They weren’t asked to make a direct computation, and there wasn’t any “answer” for them to work with. After their initial shock, and Mr. A’s many many times of explaining 2,3 and 4 need to appear somewhere within the problem, they produced some nice examples.
The top left response was overwhelming, with over 92% of the class producing it. It was the other responses that I really found interesting. Students used combinations of grouping 2 and 3 to create a new number (like the bottom left), and others introduced additional digits with 2, 3 and 4 to create a different sum than 9 (like the top right). I was also interested in students who created a sum equaling 234, asking students how they determined what numbers to use as addends. Many students replied that the number they used was their favorite number, which made me a bit concerned about the student on the bottom right- students always try to sneak in inappropriate things to check and see if you are paying attention. The bottom response was the one I found most interesting. This is a student who HATES math class and is working at a proficiency level that is 3 grade levels lower than her enrolled grade level. She has an IEP for Emotional/Behavior Disorders as well as Learning Disability in Mathematics. Yet she is the one who provided me with the most elegant and interesting solution. She loves patterns, and told me she wanted her numbers to be sequential, so she needed to use 5. She sat and worked for over 10 minutes trying to figure out how to create a sum of only 5. She asked me many times if she could multiply, subtract or divide- and was referred back to the wording of the problem many times. She did not give up or throw a fit however, and I believe that is because she set the parameters she was trying to achieve. Then, the light bulb went off, her face lit up and she asked me, Mr’ A,- can we use negative numbers? I referred her back to the problem and asked if it indicated that she couldn’t. She reread it and said no. So I told her she had her answer and she was immensely pleased with herself. She was in an incredible mood all day and even came into class the next day asking if anyone else had used negative numbers (the answer to which was no).
That second day, I once again asked them the “same” question, and highlighted how I made it different.
Create an addition problem where 2, 3 and 4 appear somewhere in the problem or and answer.
Even though many students found the question easy yesterday, once again the sky was falling and the math gods were against my poor innocent students. Once again I calmly took it all in stride (while laughing to myself inside because I expected this response) and verbally explained that 2, 3 and 4 needed to appear on both sides of the equal sign. They then went to work.
By changing one word, it created such a variety of thinking and solutions that I could have never elicited with the first question without very guided questions. Some students used the same thinking as the first question, but I also loved how some natural properties of mathematics was produced: place value, additive identity, reflexive property. Students introduced decimals! When does that EVER willingly happen? Once again, I was surprised by a student who took an extra step and made their sum by using only 2, 3 or 4 in all of their numbers (except for the 100 Mr. A, my brain started hurting). This has left me with many a rich discussion to take with my students as we reflect on the difference of these questions, their thinking, and their solutions.
I hope you find the same in your classroom.
Reading through a book today I saw this questions posed:
Make up an addition problem where 2, 3 and 4 are used somewhere in the problem or answer.
Ask that question of your students and see what types of responses you get.
Then twist the question slightly and ask it again (this is what I was thinking when I originally read it, so I wondered what type of varied response it would get and why).
Make up an addition problem where 2, 3 and 4 are used both in the problem and the answer.
What difference did it make? Was the change significant for teaching?
I would like to know your experiences with this, thanks for sharing.
OK, the MTBoS has really influenced the way I approach teaching and assigning problems. Coming back from Christmas Break, I needed a different way to get students thinking about math again, and found that thanks to Math = Love in the form of Gemini Puzzles.
What are Gemini Puzzles? In short, they are equality statements that are missing any mathematical symbols. Here is an example:
The thing I love about this is- I put these two on the whiteboard for students to see when they first walk in and instead of asking what we are doing today or talking about what they did last night, they started thinking about what was going on. The other great thing, I didn’t have to tell them what to do, they started playing around with the numbers to get it done.
When I asked my students what they were doing, there was a common theme:
Which is great until I ask students about the name of the problem, Gemini. I then get answers about twins and I tell them that’s relevant to the problem. They are given two equivalence statements, and they have to be twins. In other words, anything they add to one statement has to be the exact same as what they add to the second.
Then students start working in a stream of consciousness and they have this as an answer:
It doesn’t take too long for a student to say that there’s problem with this. Students are thinking “one plus one is two, times two is four”, but they are not properly showing that. Then the talk of Order of Operations hits, and students realize the first statement is only 3. By throwing in another set of symbols they find the correct solution:
By providing students with two equivalence statements that need the exact same symbols to make them true, you are giving students a reason to think about the order of operations and how they interact. I have loved listening to student talk when working on these, and I even had students proudly come up to me with different solutions to a problem. They are amazed that there can be different ways to implement mathematical operations and get the same result (see if you can find which one they discovered!)
Gemini Puzzles are a great activity for the classroom, I know I’ll be using them from now on, I hope you enjoy them as well.
I currently have a range of students from 6th to 12th grade in my classes, with various mathematical skills ranging from 4th to 12th+, and these puzzles have engaged and challenged them all. It’s a great thing to see.
When working with a student today, she noticed that:
and wondered if that pattern works for any other numbers.
So, for any integer a,b when does
Would changing the constraint on a,b change the problem? What number types would produce more values for a and b?