Often we adults engage students with closed-ended questions and then consider this as having a conversation with the student. I witnessed this first hand in a high school consumer math course I co-taught. The adults sat with the students the first day after December break for a conversation about their break. The questions were consisted of and were similar to the following. “Did you have fun?” “Did you eat a lot?” For some, like my son, this is appropriate. For many others, we are offering low hanging fruit that does little to move them forward.
Ask open-ended questions that prompt the student to engage in critical thinking such as analyzing and evaluating – below, courtesy of Jessica Shabatura. Work this into IEPs and 504 to have teachers implement this. For example, I asked the students what they liked about break. Then I asked why they liked it. Here is an example of me questioning my son, who does not have a disability, when he was maybe 4.
Slope is one of the most important topics covered in high school algebra yet it is one of the least understood concepts. I have two observations about this. First, slope is often introduced with the formula and not as a rate of change. Second, students intuitively understand slope as rate of change conceptually when presented in a relevant, real life context. The challenge is compounded when slope is presented with the y-intercept.
In the photo I present slope and y-intercept in a context students can understand (money is their most intuitive prior knowledge). The highlighting makes it easier for them to see the context, specifically the variables. I have the students work on this handout and I circulate and ask questions.
Here’s a typical exchange – working through problems 11, 12:
Me: “Look at the table, what’s changing?”
Student: “the cost”
Me: “How much is it changing?”
Me: “20 what?”
Student: “20 cost”
Me: “What are you counting when you talk about cost?”
Me: “So the price is going up 20 what?”
Me: Show me this on the yellow” (student knows from before to write +$20)
Me: “What else is changing?”
Me: “By how much”
Student: “1 people…person”
Me: “write that on the green”
Me: “Now do this same thing on the graph. Where do you start?” (they put their pencil on the y-intercept
Me: “What do you do next?” (they typically know to move over and up)
Me: “Use green to highlight the over” (they highlight)
Me: “How much did you go over?”
Student: “1…1 person”
Me: “Now what?” (Student goes up.)
Me: “Highlight that in yellow.” (They highlight.)
Me: “How much did it go up?”
Student: “2…20…20 dollars”
Me: “What is a rate?” (I make them look at their notes until they say something about divide or fraction or point to a rate)
Me: “So what is the rate of change?”
Student: “$20 and 1 person”
Me: “Look at the problem at the top. What is the 20?”
Student: “$20 per person.”
I point out that you can find this rate or slope in the equation, the table and in the graph.
This is a page from an Elmo book in which a cutout figure of Elmo can be inserted into various settings such as a bakery. In lieu of reading the book I asked my 4 1/2 year old son questions about each setting, e.g. “what is Elmo now?”
The questioning approach I used was to ask open-ended questions and follow up or leading questions, e.g. “how do you know he’s a baker?” This approach works at all ages but is probably more common at earlier ages. What I have found is that students progress through school learning that math questions are typically right or wrong with little critical thinking. Students are afraid to answer questions because it’s all or nothing. On another post I address how we can shape critical thinking and this questioning is another approach.
I have a video showing this questioning of my son on youtube.