This is a series about “developing the question” in math class.
Curmudgeon has taught math and science for thirty years and runs the Math Arguments 180 blog, an indispensable source of interesting prompts and questions.
Here are three images he’s posted in the last month:
In nine classes out of ten, you’ll find a teacher ask her students to calculate the area of those shapes. Maybe Curmudgeon would ask his/her students to calculate their area also. That’s a fine question. But Curmudgeon does an excellent job developing the question of calculating area by first asking:
- What is an easy question we could ask about the shape? A medium difficulty question? A hard question?
- What is the best way to find the area of the shape?
- What combinations of addition or subtraction of figures could you use to find the area?
Each question develops the next question. Earlier questions are informal and amorphous. Later questions are formal and well-defined. They all develop the main question of calculating area. They all make it easier for students to answer the main question of calculating area and they make that main question more interesting also.
This technique runs back to my workshop participant’s advice that “you can always add but you can’t subtract.” Once you tell your students your question, you can’t ask “What questions do you have?” Once you tell students what information matters, you can’t ask them “What information matters here?” Once you tell them to calculate area, it becomes very difficult to ask them, “What shapes combined to make this shape?”
Tomorrow: Why Graphing Stories does a pretty lousy job of developing the question.
Preparation: If the main question is “sketch this real world relationship,” what are ways we could develop that question?
Kyle PearceAugust 14, 2014 - 4:26 am -
Until reading this post, I was super geeked to use Graphing Stories to help introduce the topic of sketching distance/time graphs this upcoming semester. I’m very interested to see what suggestions you’ll have that could allow for something very visual like Graphing Stories to better develop the question.
I just re-watched “Time” to see if I could target some of what hurts the development of the question and I can see the “you can always add but you can’t subtract” situation taking place. Too much “adding” has happened including:
– Giving the students the two variables that will be related.
– Identifying the independent and dependent variables by placing them on their respective axis’
– Even the appropriate scale has been introduced
In order to improve this particular graphing story, I might suggest that the video be chopped up into your 3 act format:
– Show the video in real time without a timer.
– Students discuss what questions they have.
– Narrow down on a question as a group (I’ll assume, the question is narrowed down to a d-t relationship for example)
– Discuss independent/dependent variables and why they think so.
– Make a prediction by creating a sketch of the d-t graph.
– Share out. Possibly group solutions in a Bansho-style manner. This is can be done very efficiently with technology.
– What other information do you need to make your d-t graph more accurate?
– Show the timer.
– Possibly show the distance the second hand moves each second.
– Students create a second, more accurate d-t graph.
– Share out student work, discuss.
– Introduce the graph with a scale.
– Show the d-t graph being created as an overlay to the original clock video.
What would this look like to you guys?
When I get closer to this portion of my course, I’ll make it a point to chop these videos up into 3 act chunks and share back if anyone else feels it could be beneficial. Interested to see what other views their are on how these could be modified to help better develop the question.
Kevin MartzAugust 14, 2014 - 4:57 am -
I see similarities between “Curmudgeon’s” approach and my approach to questioning. Resist the temptation to lead students to the intended answer. They may get there on their own AND you can learn as you work towards an answer.
MeagenAugust 14, 2014 - 5:25 am -
Devoted reader with a request: What about developing questions for non-calculation analysis of quantitative info? I work in adult education and am currently revamping social studies curriculum. My goals are to help compare differing treatments of the same topic, and integrate quantitative and qualitative data when analyzing arguments. All I saw on Curmudgeon’s blog were arguments about calculation, not using data to argue about real world topics. I love your real world orientation, so would love an example of developing questions to use the calculated data to build arguments about, say, civics. Or better yet, developing questions that promote teachers to make decisions on the quant data they receive on students!
Howard PhillipsAugust 14, 2014 - 6:39 am -
I checked out the second one, “Height”, and only if the graph is of the height of the image on the picture is there any sense in “the answer”. In reality, and any kid who has swung on a rope knows, you don’t get higher on the far side than the height you started at. This video raises so many questions about the relationship between picture and reality that it could provide hours of discussion in the right hands.
John PalkovicAugust 14, 2014 - 7:55 am -
Nice examples. This looks like textbook learner-centered instruction to me. It meshes with the ideas of Piaget and the social constructivists (Santrock, J., Educational Psychology, chapter 12). Here’s a quote from Santrock, from a math student in a class with a learner-centered approach: “He’s not going to give you the answer. He’s going to make you think.”
I have perused Curmudgeon’s blog, I don’t see where the 2nd figure is posted by him.
Brian MillerAugust 14, 2014 - 9:58 am -
“What is the best way to find the area of the shape?”
The word “best” seems like a high entry point – It could work, but it would be important that the students had already bought into the idea that finding the “best” is a process that begins with finding all of them, and then analyzing the strengths and weaknesses of each.
Dan MeyerAugust 14, 2014 - 2:40 pm -
Feel free to skip my next post! A+ work.
@Meagen, happy to help if I can. I’m struggling with your question, though. Most arguments around quantitative data involve some kind of calculation, whether that’s a calculation of a median, a confidence interval, a linear regression, etc. Without the calculation, we’re just looking at a disorderly pile of numbers. Can you elaborate?
Christine LenghausAugust 14, 2014 - 4:43 pm -
Thanks Dan – I read your post before going into class this morning (currently on financial maths eg simple interest and compound interest) so rather than telling them a few formulas – started a conversation about money – if I had $5000 I could put it under my bed for a year (and its buying power would decrease) or … then we had a discussion on what lending our money to the bank would entail and how much I would get back – so needed to link getting paid for lending money (interest rate), how long I was going to lend them my money (time), how much I had to lend (principal) and how much they would pay me. Then I said but I don’t write all these words what could I shorten them to and how do they relate to each other. A far better start!
CurmudgeonAugust 14, 2014 - 5:45 pm -
@meagan said “All I saw on Curmudgeon’s blog were arguments about calculation, not using data to argue about real world topics.”
That’s true. I started it in January and it has been mostly focused around my own classes and trying to present, sort, and share things that I might use as starters. This coming year, I’ll be teaching the first dedicated statistics class my school has ever had, so I would like to feature some of what you ask for. I’m hoping however, that teachers such as yourself have some data sets and questions to get me started? Always happy to have contributions that I don’t have to filch from Dan, Fawn, Andrew, etc ….
Simon TerrellAugust 14, 2014 - 7:34 pm -
I think perhaps a good second question as a follow up to the “amorphous, general ” questions might be;
“Can we find the area?”
Let kids have some time to solve the area of one of the shapes, then present those ideas and see if the class can pick out the merits of each solution.
The addition/subtraction method may be one of those solutions. Before asking students that third question, perhaps it might be more effective to see if a student came up with that naturally and ask if the class can figure out how/why this method worked.
Allow the class to adopt that method or nudge the conversation in that direction.
MeagenAugust 19, 2014 - 11:17 am -
Dan & Curmudgeon:
Here’s a sample lesson of what I’m trying to describe in non-traditional high school level Social Studies.
I introduce students to both text and quantitative data from groups espousing two different interpretations of the Second Amendment. Together we answer their questions about the material to aid comprehension. Then I ask them individually to evaluate which position they would support, and identify evidence in defense of their position. No calculation necessarily required, but higher order thinking about real world quantitative information expected. I am ATTEMPTING to verbally walk them through a similar process to what they will see on the GED Social Studies Test Extended Response. The test provides multiple pieces of content on an enduring civics issue, and test takers must develop an argument in their response, citing specific evidence.
Particularly when it comes to diving into the graphs, I think I could better development my questions, if you need another example for “Needs improvement.” I look forward to your constructive criticism.