Thank you for this thoughtful and helpful essay ]]>

Understanding mathematical functions is one of the core elements in the field of mathematics , which makes people to enjoy mathematics .A student who get the concept of functions can easily use and aplreciate for example derivatives and integration which in turn can apply in physics such as calculating power,velocity.

I appreciate this research.

I suggest if this article can addressed in several universities . As it can make difference . ]]>

1) By now I consider a major factor in student’s misunderstandings and inability to do certain types of problems to be their inability to interpret what I call Mathematical English in a sufficiently precise way. I believe we have to devote conscious attention to teaching them this new language. It is impossible, I believe, to do math properly without knowing this language.

2) Specifically, I do not believe we are in general forcing students to develop rigorous skills in deductive logic. When I say this, I do not necessarily mean skills in writing proofs. I do mean, when doing the types of problems you describe as well as computation problems, to use proper logical deductions. Specifically, clear understandings of when something has to be verified for *all* cases or just one. And the difference between an implication and its converse.

I believe 1) and 2) above might play a role in your Function Understanding Thought

3) In my experience many students have a poor understanding of functions, which then leads to many errors. I see this too often even in courses such as multivariable calculus. I believe we assume that students have already internalized a correct understanding of functions from what they learn in precalculus and calculus. My observation is that in fact these courses do not really test this carefully and students are able to do well in those courses without ever really developing a precise understanding of what a function is. Usually students don’t even believe it’s important to know what a function is.

I almost always ask students at the beginning of a math course, any course, what a function is. I see nervous and bewildered reactions, and the few students who are willing to venture an answer usually say one of the statements you list in Function Understanding Thought 2. But it is clear to me, especially by the rigid wording they use, that they are simply parroting what they were told or read and that what they say does not imply a *working* knowledge of what a function is. I believe that we try too hard to give such precise but opaque definitions, stated in Mathematical English, assuming that the students will take our cue and use them properly. My belief is that we should find a way to give students a *working* knowledge of what a function is and in fact avoid such formal statements. I like your diagrams, but they’re only a start towards knowing how to *use* functions.

4) By the way, I *like* the question about i^n, because it does expose the misunderstandings students have but we don’t always see. Simple but “tricky” problems like this give us an opportunity to spot these gaps and work with students to fix them. I want more problems like this. One of my favorite precalculus problems, taken from the infamous Hughes-Hallett text is: Find the exact values of arcsin (sin 4) and sin(arcsin 4). Walking a student through this, using the Socratic method, can be quite effective in diagnosing and advancing a student’s understanding of what a function is.

]]>I think teaching of mathematics is generally improving. It’s more thoughtful and some research on teaching and learning is impacting classrooms. Better materials are being created and disseminated. There are many layers and nuances that public discussions and K-12 testing regimes don’t usually capture and still way too much ranting and blaming.

]]>https://web.ma.utexas.edu/users/mks/ProbStatGradTeach/ProbStatGradTeachHome.html

Handouts from a course for a master’s program for secondary math teachers. Many of these will be good resources for teachers of statistics. Several of the students in the class said they planned to use the ideas in the handout Logarithms and Means to introduce logarithms in their own classes. (There are some external links for the course at https://web.ma.utexas.edu/users/mks/396C08/Mf396C08home.html)

https://web.ma.utexas.edu/users/mks/CommonMistakes2014/commonmistakeshome2014.html

These notes are from a Continuing Education Course that was specifically for teachers (although some of the students in the course were statistics teachers), but provides good background/enrichment for statistics teachers.

https://web.ma.utexas.edu/users/mks/M358KInstr/M358KInstructorMaterials.html

Instructor materials and tips for a course initially developed to give prospective secondary math teachers background to teach AP statistics.

https://web.ma.utexas.edu/users/mks/360M05/360M05home.html

Materials for a course on problem solving for prospective math teachers.