Math: Pen vs Pencil

Many math students are given strict instructions by their teachers to do all their work in pencil. I disagree.

The advantage of doing work in pencil is that:

  • it is easier to erase, so students are less likely to be paralyzed by “I am not sure this is correct, so I don’t dare write it down”

The disadvantages of working in pencil are that: Continue reading Math: Pen vs Pencil

What A Parent Wants From A School

As a parent, I look for two categories of attributes when choosing a school for my child:
– Ones which benefit my child directly
– Ones which benefit my child indirectly, by helping others (teachers, parents) do their jobs more effectively

Schools that satisfy more of the attributes in both categories are likely to have happier parents and more successful students.

The Administration and Teachers Should Help My Child

Directly By:

  • Being aware of history. Before the start of each school year, my child’s current teacher(s) should have reviewed all of
    – last years’ teacher comments for my child
    – my child’s transcript (all courses, all years at the school)
  • Helping my child to both pursue existing  Continue reading What A Parent Wants From A School

Short Assessment Grading: Add or Average?

Long assessments can waste precious class time unless there is much material to be assessed, but shorter assessments (with few questions) can cause small errors to have too big an impact on a student’s grade.

For example, consider the following assessment lengths where each question is worth 4 points, and the student has a total of two points subtracted from their score for errors:

# Questions Points % % Grade
1 2 / 4 50% F / F
2 6 / 8 75% D / C
3 10 / 12 83% C+/ B
4 14 / 16 88% B / B+
5 18 / 20 90% B+/ A-

The “% Grade” in the table above reflects a 7-point / 10-point per letter grade approach. A one question quiz is risky for students: they could get a failing grade for losing two points on the only question. Two question quizzes are only slightly less risky.  Only with three or more questions does this scenario start to minimize the risk of actively discouraging a student who loses several points.

Should quizzes therefore only have three or more questions? What if I don’t want the class to spend that much time on an assessment, or don’t have Continue reading Short Assessment Grading: Add or Average?

Unintended Consequences of a 0 – 100 Grading System

If a student makes four errors in the course of answering ten questions, what is an appropriate grade? Presumably, it would depend on the severity of the errors and the nature of the questions. Consider how your approach to grading might vary if students had been asked to:

– match ten vocabulary words to a word bank, or
– define each of ten words, then use each appropriately in a sentence

– complete ten 2-digit multiplication problems, or
– solve ten multi-step algebra problems, each requiring a unique sequence of steps

– answer ten questions similar to what they have seen for homework or in class, or
– answer ten questions unlike ones they have been asked before

Would you label each answer as right or wrong, then use percentage right as the grade?
Would you assign a number of points to each answer (if so, out of how many points per question)?
Would you assign a letter grade to each answer (whole letters only, or with +/-)?
What would you consider a “D” set of answers?
What would you consider an “A” set of answers?

Would your answers vary depending on whether you had created the assessment yourself, or were using someone else’s questions?

Many math/science teachers seem to use a percentage approach (based on total points earned or number correct) more often than any other, particularly when their school defines its letter grades using a 0 – 100 scale. Teachers of other subjects also use this scale often, but less so for “free-response” questions. While a percentage approach can work well for some assessments, it can have unintended consequences for others.

Similar Right/Wrong Questions

When asking a series of similar questions, such as Continue reading Unintended Consequences of a 0 – 100 Grading System

Flipped Classroom: It’s About Timely Formative Feedback

The phrase “Flipped Classroom” is appearing with increasing frequency in publications and blog postings. Yet, it seems to mean different things to different people. Many of the references I see to flipped classrooms are made by people or organizations who have a vested interest in selling goods or services, which probably affects their view of the issues.

As proposed by Salman Khan in his TED Lecture, flipping the classroom involves using internet-based video to move “lecture” out of the classroom to some other place and time of a student’s choosing. Class time can then be used for student problem solving and group work. Dan Meyer and others have critiqued aspects of Salman Khan’s approach, with some such as Michael Pershan offering constructive ideas for improvements.

Eric Mazur, a physics professor at Harvard, has also been advocating a “flipped” approach  – and for considerably longer than Salman Khan. His conception of “flipping” focuses on getting students to Continue reading Flipped Classroom: It’s About Timely Formative Feedback

Grant Wiggins on Mathematics Education

Grant Wiggins was the keynote speaker last night at the annual “Anja S. Greer Conference on Mathematics, Science and Technology” hosted by Phillips Exeter Academy in Exeter, NH. The focus of his talk was mathematics education, and the points that were noteworthy to me included the following:

Increasingly, schools and standards bodies are setting their goal for mathematics education to be the development of good problem solvers. Yet,
– few schools focus their curriculum on problem solving
– nationally, dismal percentages of students can successfully solve problems of types they have been taught to solve, let alone problems they are not familiar with
– a significant percentage of students hate their mathematics courses

We face some big questions that are challenging to answer:
– What is the problem with mathematics education today?
– What are we going to do to address it?
This is the problem that math teachers and curriculum designers must solve.

If students are to be able to solve problems of types they have not necessarily seen before, they need the ability to transfer their knowledge and skills to new domains. Yet, most of mathematics education today focuses on Continue reading Grant Wiggins on Mathematics Education

Ten Skills Every Student Should Learn

A recent eSchool News article by Meris Stansbury lists ten skills cited by its readers as being most important for today’s students to acquire:

  1. Read
  2. Type
  3. Write
  4. Communicate effectively, and with respect
  5. Question
  6. Be resourceful
  7. Be accountable
  8. Know how to learn
  9. Think critically
  10. Be happy

The list is interesting to ponder. I would not argue that any skills on the list should be dropped, however I suspect we could have endless debates about what order to list them in or how to best group them. I am happy to note that all of the skills are beneficial in studying just about any subject or discipline.

There are a few additional skills that I would advocate adding to, or being more explicit about in the above list:

Cost effective adult education: might it influence secondary education?

An article in The Washington Monthly titled “The College For-profits Should Fear” describes the founding and growth of Western Governors University. It uses an on-line model with some twists:

  • Course credits based on assessments completed. If you pass the final assessment, you get credit for the course… even if you just took the initial course assessment a few days earlier.
  • Tuition is charged per semester, not per course enrollment. This encourages students to complete as many courses per semester as they can, as it can save them money.

From what the article describes , this model seems most successful with older students – people who know what they seek, and don’t wish to waste time getting there. The WGU model is interesting for several reasons:

  • It has tuition levels that are around 40% or less that of other on-line programs, about $6,000 per year.
  • It employs full-time Mentors, who serve as a combination of guidance counselor, tutor, cheer-leader, and ombudsman for students. While they seem to provide a regular point of contact between students and the degree program, the article does not specify how many hours per week of such contact a typical student receives.
  • It uses industry-based standard assessments whenever possible as culminating assessments. The goals of the programs are therefore hopefully better aligned with the professional goals of the industries it is preparing students to enter.

The low tuition means that this model Continue reading Cost effective adult education: might it influence secondary education?

“Teacher” is an inaccurate title

People, both as children and adults, are constantly learning new things. The more actively engaged in the learning process they are, the more likely they are to learn something well and retain that knowledge. So what exactly is the person “teaching” a course doing? Their title implies that they are somehow loading knowledge into student brains. While that may fit the assumptions behind the “lecture model” of instruction, that is not the way learning works.

So what title is appropriate for people who:
– Decide on, or create a sequence of topics and tasks that engage, but do not overwhelm
– Set the stage, pique student interest, then Continue reading “Teacher” is an inaccurate title

Scheduling for Curricular Depth and Challenge

What was “the best” course you ever took? Probably one for which you had to work quite hard, one that you perceived as challenging from the outset, one for which you rose to the challenge. The course probably had a reputation as a tough course, so you probably added it to your schedule with care and made sure you did not take another really challenging course at the same time.

Major time commitments are regularly called for in schools: for musical or dramatic performances, athletic seasons, and some classes too. Could we improve the way such opportunities are scheduled so that  students can experience as many as possible each year without creating a killer workload for themselves at critical times during the year?

Challenges

What if schools offered “challenges” that lasted for either half or a full semester? Each student could be required to be enrolled in two challenges at all times. A research project, art or engineering project, dramatic or musical performance could each count as a challenge, as could a varsity sport, as could any number of academic and extra-curricular offerings. To qualify as a “challenge”, an offering would have to:

  1. Culminate in a public performance, presentation, or display of student work.
  2. Involve extensive Continue reading Scheduling for Curricular Depth and Challenge

Lecturing: There Are Better Ways ToTeach

Many widely used math textbooks seem written for a traditional “lecture-style” teacher. They can be challenging to teach from if you are trying to reduce time spent “talking at” the class.

Some of the NSF-funded mathematics texts published over the past decade make it much easier for a teacher to avoid lecture mode, but:

– from a parent’s perspective, some texts don’t seem to have much of a role for the teacher, so how can/should a teacher add obvious value (in student and parent eyes) to what is in the text?

– the lack of prominently highlighted boxes around all information needed for the test is a source of student gripes. Students need to re-learn “how to learn” when a text or teacher takes a different approach, so time and effort needs to be devoted to this at the start of the year.

– the format of each unit can begin to Continue reading Lecturing: There Are Better Ways ToTeach

Integrating Mathematics With Other Subjects

What if most activities in school asked students to “reach and defend a conclusion”?

  •  in Math, about quantitative or geometric relationships, about measurements of worldly phenomena, etc.
  • in Music, about the effect of a melody line, about a particular mix of instruments, etc.
  • in English, about effective use of language or metaphor, about storytelling techniques, etc.
  • in Visual Arts, about the effective use of color or negative space, about how a work can be interpreted, etc.
  • in History, about a set of events, about relationships between societies, etc.
  • in Physical Education, about the effects of various activities on the human body, about the effectiveness of various strategies in a sport, etc.
  • in Science about whether two measurements are related in some way, why they might be related, the consistency with which they seem related, about cause and effect, etc.

What might our schools look like under such an approach?

The Purpose of High School Mathematics

The 2011 Anja S. Greer Conference on Secondary School Mathematics at Phillips Exeter Academy provided many opportunities to hears others’ ideas about the purpose of our High School Mathematics Curriculum.  Some of the statements I noted were (with apologies that none are exact quotes, and my lack of attribution on some):

In life, not to mention just about any academic subject, students should question information they come across, then work to support or refute it using numbers as needed.

Quantitative situations can be found in poems, literature, environmental claims, social justice issues, and social service needs.  We teach mathematics so that students can decide for themselves whether the quantities involved make sense or not.  Ray Williams (St. Mark’s School, Perth, AU) presentation.

Let the students ask Continue reading The Purpose of High School Mathematics

Re-thinking Our High School Math Curriculum

Nils Ahbel of Deerfield Academy gave a thought provoking presentation at the 2011 Anja S. Greer Conference on Secondary School Mathematics (held at Phillips Exeter Academy in Exeter, NH) on the history and potential future of the American High School mathematics curriculum.  The Prezi that he used to illustrate his talk can be found here.

As I recall, his core points about the state of things today were that:

– our curriculum has remained largely unchanged for 119 years (witness the content of the textbook whose pages fill the number 8 in his prezi).

– the current goal of most high school curricula is to Continue reading Re-thinking Our High School Math Curriculum