Simon Gregg recently posted « Making Sense » on his blog. Its the story about what happened when he asked his students the following:

ALMOST ALL of his students responded with 30 as the answer. Surprising? Unique to just these students?

**Maybe… **

**or maybe not.**

Robert Kaplinsky‘s « How Old is the Shepherd » video demonstrates that Simon’s students aren’t unique in giving a numerical response to this type of question.

Simon’s hypothesis as to why the students responded the way they did, if I understand correctly, is that it is related to conformity (see his aformentioned post find out about « the famous Asch conformity experiment »- famous maybe, but I didn’t know what it was!). Students see others answering 30 and decide to go with it- even if they’re not sure.

I’m going to throw something else out there- an alternate hypothesis as to why students responded the way they did.

Students are used to being in « answer getting mode » when faced with a word problem.

They see numbers and, in the eternal words of Eminem, students « don’t just stand there, (they) operate! ». Their brain shuts off and they do stuff with the numbers wether it makes sense within the context of the problem or not.

Evidently, I’m speaking in general terms. However, the following images found in *Powerful Problem Solving * by Max Ray-Riek is one of the reasons I think my hypothesis tends towards being true.

(I believe the images are originally from « What’s All the Fuss About Metacognition? » by Alan H. Schnoenfeld, University of California, Berkley.)

The first image is the analysis of problem solving actions in relation to time of a « novice ». The second is the same analysis but for an « expert ». Notice any differences? The novice reads and jumps right into « Explore ». The expert on the other hand spends some time analyzing the problem, planning, implementing, verifying then GOES BACK TO ANALYZING based on the results he/she obtained before planning, implementing and verifying again.

Novice –> no analyzing.

Expert –> LOTS of analyzing

Those little triangles found in the analysis of the expert? He/she stopped to ask him/herself « What am I doing? Why am I doing what I am doing? Does it make sense? ».

I think many students, especially « struggling » students believe that people that are good problem solvers **just know** the answer and they **know it quickly**. We see that it is actually the opposite.

So what do we do about this?

I think we should show students these two images in order to dispel the misconceptions that « you have to just know the answer » or that speed is better when problem solving. We need to get them thinking about the importance of analyzing the problem. In saying this, I don’t believe that checklists are an answer (i.e.- Read the problem twice; Circle the important words and numbers; Highlight the question; Verify your solution, etc.) I think that checklists tend to keep students in a procedural mode and less in a thinking mode. I want my students to be thinking all the time! Secondly, getting students to *Notice and Wonder *is a way to encourage them to be in analyzing mode. Even better would be to take away the question in the problem in order to transform it into a situation. Then do the noticing and wondering. After they have noticed all that their is to notice (and they will get better at this over time), they will be in a better place to answer almost any question that we can throw their way. They will better understand the situation, the relationships between the numbers and the context of the problem.

With this in mind, I’ve started conducting what I am calling an « experiment ». About two years ago, my former colleague Céline recorded the audio of 20 grade 4 students responding to the « How Old is the Shepherd? » problem à la Robert K.- except that there were 25 sheep being garded by 5 dogs instead of 125 sheep. This was done in French as I work in a French School board in Ontario, Canada. Here were the results

**12****responded with « 30 »****9 responded with an answer based on reasoning**(i.e.- « the question doesn’t make sense »; « 50 because farmers are usually older but not too old ».)

Recently, I’ve started to record the responses of current grade 4 students. The difference this time around; however, is that I am presenting them with the situation « **There are 25 sheep. They are being guarded by 5 dogs.** » and I am asking them « **What do you notice about this situation?**« . Only after they have stopped their « noticings » do I present them the situation again with the added question « **How old is the shepherd? »**. Here are some of the noticings from the students (before the « actual » question is asked):

- There are 25 sheep.
- There are 5 dogs.
- 1 dog is guarding 5 sheep.
- It’s at a farm.
- They must be in some sort of a pen.
- That’s a bad idea. The dogs shouldn’t be near the sheep- they will eat them.
- I have a dog at home

As for the responses to « How old is the shepherd? »? Here’s a sample:

- « I can’t answer the question because there is nothing in the problem that talks about his age. »
- « The question doesn’t make sense. »
- « 35. I added the number of sheep and the number of dogs. »
- « Around 40. Farmers are usually middle-aged. Not 60 and not 20… so I would say around 40. »
- « I don’t know because it doesn’t say his age. »
- « The situation doesn’t go with the question. »

Out of 13 responses so far, 3 were something along the lines of 25 + 5 while the other 10 responses were based on reasoning (i.e.- « the question doesn’t make sense »).

Without the « notice » part? 12 out of 20 responded with « 30 ». With the « notice » part? 3 out of 13 responded with « 30 ». Scientific journal worthy results? Not so much. Compelling? Maybe. But for sure interesting to say the least. I think it points to the power and importance of getting students to analyze problems before « doing stuff » with the numbers. Huh. It’s like the folks at the Math Forum (or is is « the folks formerly known as the Math Forum à la Prince »?) know what they are talking about.

So that is my journey so far in the « Students just start calculating with the numbers they see and it doesn’t even make sense! » world.

I’m wondering where people are in in this same journey?Thoughts?Questions? Comments?

Yes please.

Interesting. I took the opposite direction and gave my son the question (how old is the teacher?) with the « data » (class has 25 students, 5 tables) hidden. After considering the question for a bit, I revealed the data and asked for his thoughts. Immediate response is that the information given doesn’t relate to the question, so he doesn’t know. Just one data point, but I’ll see if I can find an environment to test on some more kids.

What does « explore » mean in the two diagrams showing stages of problem solving? Normally, I wouldn’t think that exploring is bad, but that seems the case when comparing novices with experts. Alternatively, it could imply that cases requiring exploration (truly novel situations, cases contrary to established norms in teh field) may be situations where novices would outperform experts.

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Hi Joshua,

I find the « question first » approach interesting; however, what I see as the common piece between what you tried and what described is the time given for the student (child) to analyze and consider the information before being asked to « find an answer ». In essence, we are forcing them to slow down- which I think is a good thing.

I believe « explore » means that the person is doing stuff- sort of like trial and error, or just trial. I think exploring is good as long as there is always self-questionning attached to it such as « What were the results of what I just tried, How does this relate to the context of the problem, What did I learn that informs my next trial? ».

Thanks for your thoughts!

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I really like how you are using notice and wonder as a sort of antidote that jump starts their critical thinking and helps them avoid being a mindless math robot. Any opportunity for these types of conversations in a classroom should be jumped on. Thanks for sharing!

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Salut Pierre !

I will write in English so everyone can understand… I really appreciate your post and your reflexion! Last year, when Céline presented us (Réseau des CP) with the videos, a collegue of mine went straight to a Grade 5 classroom to test it out. Most of the kids responded as the students in the video by actually telling her what the age of the farmer was. The interesting part was the conversation she had with the students afterwards. I was fascinated by what she found out by talking to the students and I’m sharing with you a few of the responses. Some students said that they knew that the question made no sense but they felt they had to do something and give an answer. Another student said that he too knew the question didn’t make any sense so he took the 2 numbers, did the 4 operations just to make sure the teacher would at least find one of them right. Lastly, the one that intrigued me the most…some students answered the question knowing it made no sense BUT when asked why, they said to please the teacher. Wow! So students are doing math to please the teacher vs reasoning. That’s something to reflect on.

Thanks again for sharing your ideas! It’s always a pleasure to read you 😉

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Salut Julie!

« Pleasing the teacher vs reasoning » is something to consider- especially as to why that is and what we can do about it. Pas facile! J’aimerais en discuter de vive voix quand on aura la chance…

Merci pour ajouter à la réflexion!

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