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.