Course:MATH TAAP/2014/discussion

Discussion notes from meetings

February 6th - Conclusion

February 3rd - Self-directed Learning

It's brontosaurus.

Pòlya heuristics seem to follow pretty much these self learning steps.

Secretly put in the goals into the problems. It is generally a tough sell to persuade students to incorporate these strategies. Dedicate more time to method versus content (especially 110, not much place in the other courses).

Teaching by examples might promote this.

Problematic node

Students often try nothing or a single thing, then are stuck, unwilling to change their strategies (assessement of the task/planning/reflection). The first step from knowing nothing to knowing something.

No reflection after exams no change in behaviour (solutions are underused). Not mixing up practice.

How to assist transition

Going through a lot of worked problems (make these worked examples available).

Work explicitly on non-content skills (110)

Give randomized content practice opportunities. ________________

What kinds of reflective practices are promoted by our current calculus courses?

We give back exams and assignments and try to encourage students to look over them. Is it productive to offer extra credit to students if they redo their exam and make up their mistakes? It might be if we are encouraging students not only to redo the problems, but reflect on their incorrect work.

From Figure 7.1 on page 193, which node or link is the most problematic for the students?

Students have trouble even at the first node, but the most problematic is probably planning. If we can get students to the reflection level, this would be wonderful! Many students are used to just "doing" the math, with no idea of why they are doing it and what comes next. This is not a cycle that we have to only move forward in; we can move forward and back.

How can we assist the transition over this problematic link?

"Meta-scaffolding" could get them used to the process of planning during problem solving. By "meta-scaffolding", we would ask them to first assess, then consider what they do and do not know, and then plan what they need to do in order to solve the problem, all before they start the actual computations. If we break the process down, ask them mindful questions along the way, and encourage a gradual process, the process of planning might sink in over time.
Get the students to explicitly perform some of these meta-cognitive skills will help them in the planning stage. We as instructors should set a good example in showing we work through problems with a plan in mind. Posting solutions that are not "pefect" could also aid students in understanding that the path to a solution isn't always smooth. Also, when working through examples in class, we should ask students what the plan for the problem is.

January 30th - Online activities -- Social Identity & Course Climate

Ideas for online activities :

Social Identity & Course Climate How does the social identity development of students relate to first year calculus courses?

Intellectual development was interesting
In mathematics there are true statements; different from other fields where there are not true statements.
Even if we don't think that we've directly offended someone, we don't know if we have or not.
Parabolic curve example: say throw a ball vs. a missle's projectile path.

Are there any common practices in the UBC math department that may marginalize (intentionally or unintentionally) certain groups of students?

We interact in a certain way, but we can't interact with a student in the same environment.
A lot of people put mathematics on this weird pedastool and view it as a raw measure of intelligence.
Getting a bad mark in an english class isn't as big of a hit on one's ego as it is in a math class.
Don't make a point of your effort to make the enviorment comfortable for a particular demographic. It defeats the purpose.

Are there any aspects of the syllabus that may marginalize (intentionally or unintentionally) certain groups of students?

Writing on an exam "three halves" rather than 3/2 might throw off some students who are ESL.
Do we have a bias on people who haven't taken calculus in a bit of time? A semester versus a year?
We need to let them know in advance that they need to be proficient in the past material.
Computer/internet access? Is this a bias? We require Webwork and piazza, but do we know every student has consistent internet access?

January 27th - Online activities -- Practice & Feedback

Not giving feedback too quickly or too often: robbing them of an opportunity to attempt to fix the mistake themselves. Recall the portion of Socratic method which speaks about constructive vs nonconstructive feedback.

Webwork has immediacy.

Homework does not have enough repetition, across different assignments. The practice portion is not well established.

If students are cramming for exams and passing, then forgetting the material 2 weeks later, are they learning? Why not have open-book exams for applied sciences, to have them more conceptual and less formulaic. Perhaps the policy would be to have a sheet they can take in to the exam.
- But homework is already concenptual-based, and open book - and they're still not learning the concepts. What does this imply?

Perhaps assignments should ask "what does this mean?" for various symbols in a question to check understanding.

Students might only look at grade, and ignore the feedback. For exams, there is frequently not enough feedback.

Strategies to convince students to reflect on their work and read feedback: withhold solutions, and assign the exam as a homework to correct; have a policy for remarking where students must write an explanation about why they deserve more marks; exam wrappers

About Practice & Feedback

How well do current assessments in math courses facilitate learning? To what extent does the feedback on these assessments help students develop mastery of the material? How does the current assessment organization promote integration of the received feedback? How could we improve that?

Assessments are frequent and well-targeted at specific performance goals. The timeliness of feedback is good. Unfortunately, feedback is often poorly targeted and summative in nature. Moreover, few opportunities to utilize feedback in practice are given. We rarely assess performance on the same concepts twice. Webwork allows multiple attempts, but provides poor feedback. Only the few students who ask about webwork questions to the instructor (or, a lesser extent, their peers) have the opportunity to receive useful feedback and incorporate it into their practice. Encouraging more students to ask about webwork questions will lead to them receiving more useful feedback and to them incorporating the feedback. Written assignments offer feedback that is too delayed and is typically summative rather than formative. Quizzes have fewer questions to mark, so there is more time for formative feedback. Asking the marker to give more detailed comments seems worthwhile. Clickers give real time group feedback and demonstrating the correct solution gives formative feedback.

January 23rd - Developing Mastery

What is expert blindness or the expert blind spot?

Unconscious competence. Experts lack concious awareness of what skills and knowledge they are utilizing. Difficult to realize that you need to teach things you don't even know you are using. Even if you are aware that you have an expert blindspot, it is very difficult to see what is in it. Experience with what students ask about can help you see what's in your blindspot. Not all student questions are useful for that though. (Aside: In general, not every student questions are about things that confuse all students. They are often specific to individual students.)

Do the students' level of mastery at the end of first-year calculus match that of the expectations of the instructors? If not, at which stage of mastery (component skill, integration, application) do students get stuck?

No, mainly because of unrealistic expectations. First, as an instructor, I think it is easy to assume that students who pass the course have essentially mastered the content. This is not the case. 60% mastery is not really mastery. Ability to apply the concepts will likely be weak because they have not practiced that. Ability to even integrate concepts will likely be weak, on average, due to insufficient practice. Mastery of components will be less disappointing. Components skills are computing limits, derivatives, and integrals using the simple rules and formulas, stating maximization procedures, statting IVT, stating product rule, etc. Integration of components would take place in solving a max/min problem, computing a CDF, or using trig substitution, using the product rule. True applications are virtually nonexistent in typical first-year calculus. Integration of components is where students get stuck, mainly.

What strategies could we use to relieve this bottleneck?

Text suggests 3 strategies:

1.Give students practice to increase fluency (i.e., to make component skills automatic). This allow students to focus cognitive energy on integration of components.

2. Temporarily constrain the scope of the task. Start of with problems that require very little integration (minimizes cognitive load). Then gradually step up the complexity.

2^{prime}. Writing things down or using a problem solving rubric can also help reduce cognitive load.

3. Include integration explicitly in your performance criteria. So students can know to focus on integration.

______________________

Expert blindness is when an expert is so good at something but does not realize that it is a skill that needs to be learnt (and then the world is doomed).

Do levels of mastery (expected and actual) match?

No, they should have a better intuition about the concepts (what is the derivative used for). Stuck in the application step, or integration (solve the quadratic in some problem, can't start the problem). Problems taking out techniques(integration, application).Guessing what will happen (integration, application).

How to resolve this bottleneck?

Emphasize on the specific concepts used in examples. Point out the need to master some concepts that will be used later (scaffolding). Give them challenges (you need to be able to do this underwater). Stress the importance of doing this automatically. List the component skills explicitly. Mention when a concept arises in a problem, proof or example. Giving hints where the students get stuck. Reword the problem in hints (linear approximation → tangent line). Discussions on what to do when solving a problem (four-step planning). Gradually ramp up the difficulty of problems. Use the easy problem to show how to solve the hard ones.

January 20th - Motivation Factors

Student’s motivation:

instrumental value (pass a required course, achieve a good grade to enhance future prospects)
attainment value (satisfaction from getting the right answer)
students may not believe they can achieve success, on the other hand students may expect a good grade with little work
student’s performance in past math classes influences motivation

Instructor’s Goals:

would like students to obtain necessary skills and tools that will serve them in the future
attain some understanding and appreciating for mathematics as a discipline
develop good study/work habits
mathematical communication; better writing skills

How can we reconcile:

authentic examples
reward what the instructor values
good communication of expectations

Students' Goals:

They value the grade
Self-esteem
Don’t want to be embarrassed by proving they don’t know something
They value being prepared for the next course.
Getting into Sauder
Social aspects

Instructor's Goals

Importance of hard work.
Try to make them work hard.
Teach diligence: assigning hard problems doesn’t necessarily prove that diligence is important. Want to convey not to give up.
Have the average be good, or higher than other sections (competitions with the other sections).
Wanting to see them succeed on the exam.
Have them prepared for their coming courses; see that they know the stuff they need.
Give them tools they can use in their future, whether mathematics, social, or confidence.
Mainly, wanting them to learn the tools, have fun, and have them do well on the exam.
Goal is to get students to find something that they find worthwhile.

Ways to reconcile

scaffolding certain topics.

Related rates: -draw a picture -what do you have -what do you need -what are the rates of change? -what equations can you derive from the picture? Give them a lot of structure at first and remove it bit by bit.

January 16th - Knowledge organization

Vanessa and Laurent

I think we encourage a very strict web that they need for exams. We don’t allow for the students to go outside the web that they should know for the exam. So, as soon as we ask a question that requires them to make a different connection, they are completely stuck. i.e. they can only do organization they have seen previously.

They don’t make a link between one theorem they are learning at the beginning of class to one theorem they learn at the end of class.

We should try to bring other types of mathematics into our problems.

When you are writing some theorem you could ask what previous concept that was saw last month did we use here?

Trap them into situations that look like one theorem that they use, but you remove the parameters.

In a lot of classes, especially more advanced ones, a prof says the objective is to get to “THE THEOREM”. It all goes back to the sequential organization; we need to get to THIS result. We put too much importance in the end result, which may be important, but it’s only at the end because it relies on all the previous stuff.

Make them create the web rather than us. This will help us recognize where there are gaps in the knowledge.

Ask them to build a web throughout the class. Start it at the beginning and build it as a project.

Alastair & Kyle

"What is their organisation like?"

"A pretty weak one"

"Table of contents of a text book"

Algorithmically (flow chart), and Chronologically.

"Changes to be made"

Make heirachry/connections more explicit.

Could you use "Wikipedia" as a model on how to organise calculus concepts.
Get students to make mindmaps/wiki with a particular map- through a particular lens. EG "When" "How" "What are the properties of?" "How does this connect to the real world?"

Give a student a SINGLE node, and get them to look at ALL the connections. EG- if you ask a student to tell you everything about "discontinuous functions", examples, tests etc.

Summarise at appropriate times.

"Mixed" assignment question that cover multiple topics.

Rob and Matt

More mileage out of hard (old) homework problems
Students starting thinking when the instructor stops talking
Short in-between lecture problems (increase retention/connection)
Homework problems that emphasize/require connections
Student directed end of class summary

January 13th - Prior knowledge in first year calculus

Matt

Common misconceptions in calculus courses

Goal is to gain proficiency at performing specific tasks, i.e, to be programmed with an algortihm
The answer is more important than understanding
The instructor's role is to program the student with algorithms.

Ideas to address these

Communicate through evaluation that process is the goal above answers.
Computer simulations/programming in class to emphasize understanding the process
Have students write a psuedo code assignment. An demonstrate their code in class.

Alastair

Common misconceptions in calculus courses

Freshman's Dream: All functions are linear (like x -> x^2)
Sandwhich fractions: 3/4/2
sin^{-1}(x) = 1/sin(x)
y^{-2} = \sqrt{y} (= y^{1/2})
Fraction mistakes

Ideas to address these

Explicity point out these errors using numerical examples, after an exam perhaps.

Kyle

Examples of prior knowledge hindering learning for calculus students

Insufficient or inaccurate knowledge of arithmetic and algebra
Insufficient or inaccurate knowledge of physics or physical intuition
Lack of activation of accurate knowledge about physics

Ideas to address these

Review at start of course, including exercises. Review in context throughout course, possibly with exercises.
Use physical examples. Review the physical principles that underlie the example.
Use examples that rely on less knowledge of physics. Give students explicit **everyday** physical examples
Use assignments to do the above.

Misconceptions that undergrads bring into calculus courses.

Functions - "What is a function?" Abstractly defined functions (eg. floor function, piecewise functions) difference between equations and functions. "Every day" functions (eg. like the area function for an integral)

Calculus/Mathematics is about numbers when in fact its about functions/ideas.

Many methods to solving a problem vs one method only.

Math has one answer but many ways to get to the answer

Math is complete (ie there are no other problems to solve). All problems that can be solved have been solved.

Math is devoid of logic. (Of course math is all about logic)

Basic algebra skills (eg. sin(x+y) = sin(x) + sin(y)). Distributive property and when to apply it.

Exponential rules (and logarithms rules)

Solving x^2 - 4 = 0 gives two answers (ie forget the plus minus).

How we can fix these misconceptions

Assign more conceptual questions. Eg. Mean value theorem: Horse Race example: Start and end at the same time but at some point they went at the same speed. What about 3 horses?

To help with functions; perhaps explaining functions via pictures. There are more functions than just poly, trigs, etc.

Eg. Meat Machine. Toaster vs Slot machine. Helps to relate new knowledge with previously learned knowledge.

Logic things: Know the sun exists but some days you can't see it.

Algebra skills: Try to show more steps in class.