How would you help these students?
Thanks to Tina Cardone for the submissions. Head over to the Productive Struggle blog for further analysis and discussion of these questions.
How would you help these students?
Thanks to Tina Cardone for the submissions. Head over to the Productive Struggle blog for further analysis and discussion of these questions.
What’s the fastest way of helping this student?
Thanks to Jonathan Newman for the submission. You should click on his name to go to his blog.
You grade these tests on Sunday, and you see these kids on Monday. What does the lesson plan look like?
For more context and analysis, go check out the blog from whence these came.
Marshall ThompsonĀ is offering a substantial bounty for the correct explanation for this mistake. (He knows the truth from talking to the student.)
Who’s got it?
Compare to thisĀ post.
Are there any other mistakes that you would expect students to make on this problem?
Thanks to Kristen Fouss for the submission.
How do you teach this topic so that students don’t make this mistake? (Or do you just rail on their earlier teachers?)
Thanks toĀ Kristen FoussĀ for the submission.
In the comments: can you reconstruct each step of the student’s argument? What was he thinking?
Thanks to Kristen Fouss for the submission.
[We have a growing library of trig identities, and I’ve got like 10 more trig identities to post over the next few weeks. Seems to be a tough spot for students.]
Bob Lochel writes:
“I used this question as part of a benchmark assessment given to over 1,100 students at our high school, as preparation for the state test.Ā Only 14% of students gave the correct answer B, while 66% of students chose A as their response.Ā Iām not surprised that students would perform weakly on a domain/range question, but I am surprised that so many chose A.Ā I featured this problem as a set-up for a domain and range activity on my blog: http://mathcoachblog.wordpress.com/2013/01/22/home-on-the-range-and-the-domain/, but feel free to share it with the mathmistakes crowd.”
So, what do we think about choosing A? Any theories?
I want to share a theory on this mistake:
The student had an association between negative exponents and reciprocals and “half-powers” and square roots. As the student was parsing the problem he “fulfilled his obligation” to use that association on the number. I guess what I’m positing is that the mind works by making a connection, and then remaining in tension until that connection is used in a problem. (I’ve often had the experience of feeling as if there’s an insight that I haven’t used yet in solving a problem, and it’s like having a small weight on my back.) The mind comes to relief at the moment that the insight is used.
The student’s mind made the connection between negative powers and reciprocals and was in tension. He then used this insight at the first opportunity he saw, to relieve himself from the burden of his insight.
Some of you might disagree. For instance, you might think that the student had just memorized some rule poorly, had no conceptual understanding of powers, and gave the answer that he did.
But I think that the answer felt right because he used the fact that he knew. I’d predict that this student would be able to answer correctly.
If you think that the student just memorized a rule, is there any reason to think that a student would get a question such asĀ correct?