Teaching understanding or algorithms?

I recently read Schoenfeld’s “When Good Teaching Leads to Bad Results: The Disasters of “Well Taught” Mathematic Courses”. 

This paper reports the results of a year-long intensive study of the teaching and learning that took place in a tenth-grade geometry class, which will be called the target class. The class took place during the 1983-84 academic year in a highly regarded suburban school district in upstate New York. The study included periodic observations of the target class and of eleven other mathematics classes, interviews with students and teachers, and questionnaire analyses of students' perspectives regarding the nature of mathematics. The target class was observed at least once a week, and was videotaped periodically for subsequent detailed analysis. Two weeks of instruction near the end of the course, dealing with locus and construction problems in geometry, were videotaped in their entirety. The analyses focused both on the mathematics that was learned, and on what the students learned about the mathematics -- including how and when they would use, or fail to use, the mathematics that they had studied.

This paper explored the difference between performing mathematical algorithms and truly understanding the underlying fundamental ideas of mathematics.  Due to the structure of the study, they were able to quantify certain classroom behaviors; time spent in questioning, active learning time, amount of praise, and amount of feedback.  The study also included the type of grouping, the size of grouping and so on.  Originally, they defined “learning” as how well the students performed on achievement tests.  I say originally, because through the article it illustrates how these tests fail in significant ways to measure subject matter understanding.

The article also suggests that providing students with repetitive routine exercises that can be solved out of context and no significance provided, actually causes the subject matter to seem frivolous to students.  This monotonous work actually deprives the students of the opportunity to apply their learning in a context that is meaningful to them.

Brown and Burton (1978) developed a diagnostic test that could predict, about 50% of the time, the incorrect answers that a particular student would obtain to a subtraction problem -- before the student worked the problem!  Teachers are still currently doing this, when we provide students with “distractors” on a multiple choice exam.  If we already know what mistakes students are going to make, before they make them, we need to start changing our instructional models.

The predominant model of current instruction is based on what Romberg and Carpenter (1985) calls the “absorption theory of learning”: “The traditional classroom focuses on competition, management, and group aptitudes; the mathematics taught is assumed to be a fixed body of knowledge, and it is taught under the assumption that learners absorb what has been covered” (p. 26) This view is essentially implying that the “good” math teacher has multiple methods of covering the same outcome.  Through these multiple methods the students will eventually “get it”.  Unfortunately, what most math teachers fail to realize, is that through these multiple methods we are forcing the students to “get” something entirely different; resistance to change.

Math teachers need to stop testing students on algorithms and more on the understanding of math concepts.  Below is an example of two problems Werthemier (1959) gave to various elementary school students.

Many of the students, who were deemed as high achievers, added the terms in the numerator and then performed the indicated division.  They followed the conventional order of operations of BEDMAS, or to some PEDMAS.  An idea which should not be taught to students out of context, which I have wrote about here.  Even though these students calculated the correct answer, I would argue that these students do not demonstrate any depth of understanding of mathematics.  To truly understand the underlying substance students should recognize that repeated addition is equivalent to multiplication and division is the inverse of multiplication. 

This example truly illustrates that being able to perform the appropriate algorithmic procedures, does not necessarily indicate any depth of understanding.  Also, the sad truth; virtually all standardized exams for arithmetic competency focuses primarily on algorithmic mastery, and not deep understanding of the math concepts. 

This is the first part of the article summarized, more to come….

Using post-it notes and a flip cam to answer math questions

 Does the price of engagement and real learning have to be in the $1 000s?

The price of a flip cam - $159. 99

The price of a pack of post it notes - $5. 00

The price of class time – 84 minutes

The price of true engagement – Priceless

The above items are all that was required to have students complete a statistics question with meaning and creativity, which is illustrated by the video.

What was the task?

In class we were investigating the average shots on goal per game in an NHL game.  The class was put into groups of 3 or 4 students and given various questions.

With this project I tried to create meaning to the questions, and not just give students meaningless numbers.

Students worked in groups and then had to create stop videos using post-it notes and illustrating how they solved each question. 

What truly made me smile….

At the end of class, my students asked me what I was going to do with the videos.  One student even asked if we could upload them onto YouTube.  I informed them that I would compile them all, remove the names, and attach some music.  The uproar that occurred was amazing.  Students were upset that I was going to remove the names!  Here are the comments:

“I am proud of this and want to show it off!”

“Can I get a copy of our video to upload on my USB drive?”

“Don’t delete our names, I want my name on in it”

Students were truly proud of their work.  In my 4 years of teaching, I have yet to witness a student be proud of the worksheet he/she completed, or the meaningless task he/she accomplished.  I am not implying this has never occurred, but I have not witnessed it yet in my own classes. 

I still remember back to my first year teaching when a student informed me, “Worksheets suck and I told my teacher that.  He still gives them to me.  I now hate [the subject]”.    What amazed me by this comment was that the student started to hate the course and not the teacher.

The most eye opening experience for me was when a student told me, “The way you have asked me to do math was what made it difficult”.  I had to truly leave my ego aside and embrace this comment.  Maybe I was the one who made the class difficult!   This comment was from 3 years ago, and in a class with the same lesson plan every day:

1)      Review homework

2)      Give students hand-outs to follow while I write on the board questions of increasing difficulty.

3)      Show and complete a word problem

4)      Hand out worksheet

5)      Give Pg. XX questions 1-XX odd for students who finish early

Since then, I have burned this lesson plan alive!

Many people have asked me, “why did you start changing the way you teach?”, and my true answer is “Because I don’t want to hear another comment like that from a student”

 I also have to say thanks to Geoff, or twitter: @emergentmath for recommending I look at the site:

http://emergentmath.wordpress.com/2011/02/11/post-it-notes-animation/

BEDMAS last along with Lottery Winners.

I would like to address a convention in mathematics that needs to change; BEDMAS.  For those who are unfamiliar with this it stands for: Brackets, Exponents, Division, Multiplication, Addition, Subtraction.  This is how I was taught order of operations.  However, division and multiplication actually are completed from left to right, and the same goes for addition and subtraction.  Therefore, we really need students to memorize also BEDMAS, BEMDAS, BEDMSA, depending on the order of the operations.

Also there could be operations inside brackets, so in actuality all that must be memorized is B(BEDMAS)EDMAS, B(BEMDAS)EDMAS, B(BEDMSA)EDMAS, …

I hope we are all confused as I am.

I have had a conversation recently where I was told, “If we don’t focus on order of operations then 3 + 2 * 10 could be 23 or 50, and there can only be one right answer”.   Now without a context 3+2*10 is 23.  However, students need to know WHY to put brackets, and when to add sometimes before we multiply. As for 2 examples:

If you are taking a cab which has a base charge of $3, and $2/km, then it would cost you $23 to take the cab 10 km.

If you are planning a party for 10 people and want to supply cookies, which costs $2/person, and cake, which costs $3/person, the total cost would be $50.

I understand that we need a procedure to solve questions that have no meaning, but for students who are just learning a new concept there should always be meaning in their learning.

This idea of memorization through acronyms can been in many subjects:

HOMES – Great Lakes

Mrs. Vasquez Eats Many Juniper Seeds Until Nurished – Planets.

I could go on and on.  I am not saying knowing the order of things is unimportant but should not be more important than the meaning and application of the knowledge memorized.  For my own students, I would rather they understand why the order of the planets are important for the equilibrium of our solar system than just the order of the planets.The sad truth is that the only time students are going to see a question, as 3+2*10, is when they need to answer a skill testing question for a prize.  Are we teaching critical thinkers, or lottery winners?

Exams last part 2

Why do we give exams?

After asking many teachers the top three answers that have been given are:

1)      “To assess, and find out actually what the students know” Rebuttal to this
2)      “If we don’t test it, the students won’t want to learn it”
3)      “Hold teachers accountable for their teaching`

Rebuttal to 2:

First, we need to understand that there is difference between learning and achievement.  For more click here. 

Second, if a student asks you, “Why do I have to learn this?” and your first or only response is “for the test”, then you are actually destroying any possible engagement.  People need to understand that learners don’t ask for the application to challenge the teacher, but actually want to understand the meaning behind the concept. 

If there truly is NO real life application then I would first advise you to contact those in charge of your mandated outcomes and ask them why you need to teach the specific outcome.  In the defense of the government, if they don’t know there is a problem, how can we expect them to find a solution?

Assuming that the outcome does have real life application, we should be focusing on the relevance and not the mindless repetition of the outcome.  Contrary to some popular belief, students do crave knowledge, but they need to be shown the “why” just as often as the “how”. 

For some outcomes this is an easy task, while for others I understand this can be quite difficult.  I, however, do believe that no matter how challenging it might be to show the “why”, the learning that will occur because of it, will make the journey worth taking. 

An exam should not be the reason anything is taught in a class.  “Teaching to the test” should be the equivalent of swearing in a classroom; something that should NEVER be done or even entertained.    I read the following, and became sick to my stomach!

Everything that has to do with the test has been given such a high priority, that there is no priority any more but that … The bottom line question comes down to, "Well, what’s going to help them do better on the test?" And if it’s not going to help them do better on the test, well, we don’t have time for that right now (Wright, 2002, p.10).

I would hope, that most agree, that the above statement is not one that teachers should be making.  If you believe, however, that a test is the only way students will learn, you are on your way to making the statement above.  I strongly encourage educators to allow students to find significance in given tasks, and you will start to see that your test no longer becomes the reason students want to learn.

Justifications first, shortcuts second.

“When you divide by a fraction, you multiply by the reciprocal.”  This is a mathematical truth, but when I asked my students to explain why I was amazed by all the tricks.  Answers ranged from “you kiss and flip” to “you can’t divide fractions.”  The latter bothered my mathematical core greatly.

After hearing these “math tricks”, I decided to write about my interpretation on why you multiply by the reciprocal and how I explained it to my students.

I started with a simple question, “8 pies were baked and cut into thirds, how many pieces are there?”  My students answered 24 quickly.  We then had a brief discussion on why when you divide a number greater than 1 by a number between 0 and 1, the answer increases. 

We then looked at the thought process of the above question. “8 divided by 1/3, is actually the equivalent of multiplying 8 by 3”.  I still, however, was not happy about the explanation.  Lucy then shouted out “each pie will create 3 pieces, and since there are 8 pies we can multiply 3 by 8”.  I smiled.

Now the tough question, “If you made 8 pies, and everyone you invited over ate 2/3 of a pie, how many people could you invite such that all the pies were eaten?”  Students realized that the work was 8 divided by 2/3, but were having troubles explaining how to do this in words.  After 5 minutes of struggling, the light bulbs started to come on. 

Paraphrased here is the explanation:

First we need to figure out how many thirds are in 8 pies.  We multiply 8 by 3 (using the same logic from above) and the product is 24. However this time, each person needs 2 pieces, therefore we now divide 24 by 2, and the answer is 12. 

In the end, 8 divided by 2/3 is the same as 8 times 3, divided by 2. 

Did my students know this already? Yes, but none of them could explain why. 

Through my own experiences, I have realized the mathematical shortcuts, tricks and magic need to stop, while rationalizations, explanations, and justifications need to begin.

Meaning first, videos second.

Usually on the last day before Christmas, a christmas movie or activity is completed or "busywork."  This year I decided, instead of watching a movie, why not let the students create one?  Today, I tried an activity that I read about on a blog.  My students were put in groups of 3-4 and given 11 questions of review.  They were then given the following instructions with a flipcam (mini video camera).

Directions

Figure out the solution to three different problems on paper and check to make sure it is correct. Check your solution with the answer key.

Plan out a 2 minute or less presentation. See Presentation below.

Write out the section/problem number(s) on your whiteboard or desk.

Write out the question with any key information on your whiteboard or desk.

Write out part of the problem’s solution on your whiteboard or desk.

Remember each person will do some explaining in the video.

Make sure that your writing is big enough to see through the video.

Get a “FLIP” and read the instructions about “The FLIP”. See below.

Once you are finished with your presentation, return the “FLIP” and begin working on the rest of the homework assignment. Make sure you put a note-card with your names and section/problem into the black bag of the FLIP.

Presentation: 2 minutes or less (redo your video if it is over 2 minutes)

VIDEO

1. Introduce yourselves: first names only.

2. Read the section/problem number(s) from the writing on your whiteboard or desk.

3. Read the question and any key information from the writing on your whiteboard or desk.

4. Explain the solution that you have written out on your whiteboard or desk so far.

5. Finish the problem by actually writing in front of the “flip” while explaining the solution.

6. Thank the audience for their time and to have a good day.

The students started to collaborate and solve the questions.  Surprisingly, on the last day before Christmas, students were engaged on meaningful tasks.  Below are some of the videos that were created.  I published one of them from my ESL student explaining in English how to do math.

ESL Student talking math

Another student discussion elimination.

I cannot find the original blog were this activity was created, if you do know the author please comment so I can give him/her the recognition he/she deserves.



Learning first, achievement second, homework still dead last.

I have received many responses to my "no homework" policy, so I decided to blog more about it.  In my classes, I do not assign daily required work.  I have, however, given my students assignments to complete out of class time, but these are not the traditional math assignments.

The myth about daily repetitive work is that this actually increases student mastery of a concept.  Many people believe the saying “Practice makes perfect”.  This saying, recently, has evolved to “perfect practice makes perfect”.  I would agree to the second statement when referring to a physical skill, such as shooting a basketball.  To master a physical skill, our body needs to mimic the correct actions multiple times.  The question that homework in a classroom does not address is “How does one practice understanding?”

Psychologist, Nate Kornell, completed a study that showed that intensive immersion is not the best way to master a particular concept.  Nate found that college students and adults of retirement age were better able to distinguish the painting styles of 12 unfamiliar artists after viewing mixed collections (assortments, including works from all 12) than after viewing a dozen works from one artist, all together, then moving on to the next painter.  Nate then deduced:

“What seems to be happening in this case is that the brain is picking up deeper patterns when seeing assortments of paintings; it’s picking up what’s similar and what’s different about them,”

Unfortunately, most classes immerse students into one concept; assign them multiple questions of the same concept, then move on to the next concept.  The research states that this is not the most effective way if we want students to retain mastery.

According to Cooper, homework increases student achievement (Even a formula has been created, 10 min per grad level)  This is a self-fulfilling prophecy.  First, educators need to realize that student achievement and student learning are not the same idea.  When looking at Cooper'sstudies, he shows that test scores increase due to the homework assigned.  Unfortunately, assigning repetitive work for students, or giving them loads of questions before an exam is the equivalent of cramming for an exam.

Cognitive scientists do not deny that cramming will lead to a better grade on an exam (thus increase student achievement), however this knowledge is quickly forgotten (student learning has decreased).  We, as educators, need to realize we are here to increase student learning first and student achievement second. 

Educators first, entertainers second.

"If students are having fun then they will learn more."  This was a comment I heard and decided to look more into this idea.  Most people understand the meaning of the words "engagement", "entertainment", and "meaning", but do people understand that engagement and meaning does not imply entertainment?

When I look back to a couple of years ago, I would start my class with a funny YouTube video or a funny website.  I believed that these videos and sites convinced students that what I was about to teach was worth learning.  Unfortunatly, I was being decieved by this notion.

Students want and need to have fun, however not at the expense of learning.  I believe true student engagement can be fun, but it doesn't have to be.  Recently, I have asked some of my students which of the two enviornments would they prefer:

Option 1: A class where you laugh and joke, watch funny videos, but do not understand the concepts being presented

Option 2: A class that is very structured where no jokes are allowed, but you can completely apply the knowledge and concepts you have learned.

Most either said option 2 or a mix between the two classes.  Not one student picked option 1.  As educators we need to realize we are not doing our students or ourselves a favour by showing non-relevant material in class.  The question then is “How do we engage students on learning and allow them to also have fun?”  The answer is simple; create activities that relate to the outcomes not to entertainment. 

To introduce non-permissible values in math, I showed an example on how 1 could equal 2.  I then asked the students to find the mistake.

The mistake is at one step you have to divide by 0 which is breaking the law in the world of math.  Was this task as fun as showing a video where a kid runs into a stop sign? Absolutely not, however this task was more meaningful to the students.  As teachers, we are not entertainers but educators, and sometimes we need to be reminded of this.

Meaning first, homework second

I have stopped assigning daily required homework to my students.  Over the last 4 years of my teaching career, I assigned daily homework, and at the start of each class I would "check", or assess, the completion of their work.  When I started to think about it, what I was doing could be considered malpractice. 

The students would open their work booklets to the assigned page and I would walk up and down the rows and either give the students a 100% or a 0%.  It is ludicrous to call this true assessment.  I was grading their work ethic more than their actual knowledge of math.  Almost every student's mark was being either inflated or deflated due to their work ethic.

I have had the discussion that daily homework teaches good work habits and/or develops positive character traits.  After reading many articles and research I have yet to find one piece of evidence that supports this claim.  Another argument is that homework "gives students more time to master a topic or skill".  I have read reports from researcher Richard C. Anderson that claims "the actual learning that is occurring depends strictly on time spent learning the concept".  However, when Anderson completed further research he found that this claim also turns out to be false.

A colleague of mine used the example of reading to illustrate the need for daily homework.  Anderson found that when children are taught to read by focusing on the meaning of the text (as opposed to strictly memorizing the phonetic sounds of the words), then the learning completed by the reader does not depend on the amount of instructional time.  His research also carried over to math, which showed that the more time spent on completing math facts only increased achievement if the achievement was based on low level thinking and strictly recall as opposed to problem solving.  The truth is that when creativity and higher level thinking is involved, the more hours spent are least likely to produce better outcomes.

Another colleague used the idea of sports to prove that homework is a necessity.  Of course it makes sense that if you practice a certain athletic skill the correct way you will improve in that area.  However, using sports to promote homework in class is using petition principii, or more commonly known as “begging the question”.  A proposition which requires proof is assumed without proof; we are assuming that an intellectual skill and an athletic skill can be classified in the same category.

The majority of people that I have encountered that are supporters of daily required homework fail to look at the tasks from the students’ point of view.  Most “drill and practice” assignments actually do the contrary to students’ learning, and actually “drill and kill” any interest in the subject area.   Also, when students are struggling with a concept, asking them to complete questions on this concept will become frustrating for them and still no actual learning will occur.  I have realized that I need to stop treating my students with the notion that “if I give them more to do, then they will know more”.

In my classes, I challenge students in meaningful contexts and provide them with questions that are similar to the ones in class.  I do not require my students to complete these questions, I do not grade these questions, and I do not force my students to do work that is not important to them.  The meaning of the math is what I put as a priority in my class, and home work as second.