MathNotations

Example I
40% of the the Freshman Calculus class at Turing University withdrew. If 240 students left, how many were in the class to start?

Solution without explanation or discussion:

0.4x = 240 ⇒ x = 600


Example II
40% of the the Freshman Calculus class at Turing University withdrew. If 240 students were left, how many were in the class to start?

Solution without explanation or discussion:

0.6x = 240 ⇒ x = 400


Thinking that the issues in the problems above are more language-dependent than based on learning key mathematics principles or effective methods? I would expect that many would say that using the word "left" in both problems was unnecessarily devious and that clearer language should be used to demonstrate the mathematics here. Perhaps, but when I taught these types of problems I would frequently juxtapose these types of questions and intentionally use such ambiguous language to generate discussion - creating disequilibrium so to speak. If nothing else, the students may become more critical readers! Further, the idea of using similar but contrasting questions is an important heuristic IMO.

Even though I've been a strong advocate for a standardized math curriculum across the grades, I fully understand that the methods used to present this curriculum are even more crucial. Instructional methods and strategies are often unpopular topics because they seem to infringe on individual teacher's style and creativity. BUT we also know that some methods are simply more effective than others in reaching the maximum number of students (who are actually listening and participating!). I firmly believe there are some basic pedagogical principles of teaching math, most of which are already known to and being used by experienced teachers.

Percent word problems are easy for a few and confusing to many because of the wide variety of different types.

Here are brief descriptions of some methods I've developed and used in nearly four decades in the classroom.

I. (See diagram at top of page)
The Pie Chart builds a strong visual model to represent the relationships between the parts and the whole and the "whole equals 100%" concept. How many of you use this or a similar model ? Please share! There's more to teaching this than drawing a picture but some students have told me that the image stays longer in their brain. I learn differently myself but I came to learn the importance of Multiple Representations to reach the maximum number of students.

II. "IS OVER OF" vs. "OF MEANS TIMES"
The latter is generally more powerful once the student is in Prealgebra but, of course, the word "OF" does not appear in every percent so many different variations must be given to students and practiced practiced practiced practiced over time. The first method can be modified as a shortcut in my opinion to find a missing percent and that may be its greatest value. However many middle schoolers use proportions for solving ALL percent problems. I personally do NOT recommend this!

Well, I could expound on each of these methods ad nauseam and bore most of you, but I think I will stop here and open the dialg for anyone who has strong emotions about teaching/learning per cents...

Posted by Dave Marain at 6:52 AM 9 comments

Labels: heuristics, instructional strategies, middle school, pedagogy, percent, percent word problem, SAT strategies, SAT-type problems

Have you ever inserted batteries in a device only to find that it didn't work? You reverse the batteries and try again, but no luck. You can't find the polarity diagram to guide you and you're dealing with 3 or 4 batteries and all the possible combinations! Well, that just happened to me as I was inserting 3 'C' batteries into a new emergency lantern I just purchased. There was no guide that I could see. I knew there were 8 possibilities but it was late and my patience quickly ran out. I tried it again the following morning, shone my small LED light on it and saw the barely visible diagram.

After seeing the lantern finally operate, I realized I should have used a methodical approach -- practice what I preach!! Then I thought that this might be a natural application of the Multiplication Principle one could use in the classroom. Of course, it would work nicely if you happened to have the identical lantern but you might have some of these in the building or at home which take 2 or more batteries. IMO, there's something very real and exciting about solving a math problem and seeing the solution confirmed by having "the light go on!" I'll avoid commenting on the obvious symbolism of that quoted phrase...

Instructional/Pedagogical Considerations

(1) I would start with a small flashlight requiring only one battery to set up the problem. For this simplest case, students should be encouraged to describe the correct placement in their own words and on paper.

(2) Would you have several flashlights/lanterns available, one for each group of 2-4 students or would you demonstrate the problem with one device and call on students to suggest a placement of the batteries? Needless to say, if you allow students to work with their own flashlights, they will look for the polarity diagram so you will need to cover those somehow. That is problematic!

(3) Do you believe most middle school students (if the polarity diagram is not visible) will randomly dump in the batteries to get the light to go on and be the first to do so? Is it a good idea to let them do it their way before developing a methodical approach? Again, if a student or group solves the problem, it is important to have them write their solution before describing it to the class. If there is more than one battery compartment, students should realize realize the need to label the compartments such as A, B, C , ... Once they reach 3 or more batteries, they should recognize that a more structured methodical approach is needed so that one doesn't repeat the same battery placement or miss one. One would hope!

(4) Is it a drawback that the experiment will probably end (i.e., the light goes on) before exhausting all possible combinations? How would we motivate students to make an organized list or devise a methodical approach if the light goes on after the first or second placement of the batteries?

(5) I usually model these kinds of problems using the so-called "slot" method. Label the compartments A, B, ... for example and make a "slot" for each. For two compartments we have

A B
_ _

Under each slot, I list the possibilities, e.g., (+) end UP or DOWN (depending on the device, other words may be more appropriate). Here I would only concern myself with labeling the (+) end, the one with the small round protruding nub. For this problem I would write the number (2) on each slot since there are only TWO ways for each battery to be placed. Note the use of (..). In general, above each slot I would write the number of possibilities. For two compartments (or two batteries), the students would therefore write (2) (2). They know the answer is 4 but some will think we are adding rather than multiplying. Ask the class which operation they believe will always work. How would you express your questions or explanation to move students toward the multiplication model? The precise language we use is of critical importance and we usually only learn this by experimentation. If one way of expressing it doesn't seem to click with some students, we try another until we refine it or see the need for several ways of phrasing it. This is the true challenge of teaching IMO. We can plan all of this carefully ahead of time, but we don't know what the effect is until we go "live" (or have experienced it many times!).

Perhaps you've already used a similar application in the classroom - please share with us how you implemented it. Circuit diagrams in electronics also lend themselves nicely to this approach. Typically, I've used 2, 3 or more different coins to demonstrate the principle but the batteries seem to be a more natural example, although I see advantages and disadvantages to both. At least with the batteries, students should not question the issue of whether "order counts!"

I could say much more about developing the Multiplication Principle in the classroom, but I would rather hear from my readers.
If you've used other models to demo this key principle, let us know...


REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

Updates (Pls Read!!)
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.

Posted by Dave Marain at 9:53 AM 8 comments

Labels: combinatorial math, instructional strategies, middle school, multiplication principle, pedagogy

A simple warmup for Grades 3-12?? Can one problem really be appropriate at many levels?

Would 3rd, 4th or 5th graders guess the obvious answers 0,0 or 2,2 provided they understand the meaning of the terms sum and product? Do youngsters immediately assume the two numbers are different? Children at that age are thinking of whole numbers, however, what if you allowed them to try 3 and 1.5 (with or without the calculator)?

For middle schoolers: After they 'guess' the obvious integer answers, what if you were to ask them: "If one of the numbers is 3, what would the other number be?" If one of the numbers were 4? 5? -1? -2? Is algebra necessary for them to "guess" the other number? Would a calculator be appropriate for this investigation? Would they begin to realize there are infinitely many solutions? What if you asked them to explain why neither number could be 1...

For Algebra students: If one of the numbers is 3, they should be able to solve the equation:
3+x = 3x; they can repeat this for other values including negatives as well. They should be able to explain algebraically why neither number could be 1. Let them run with this as far as their curiosity takes them!

For Algebra 2, Precalculus and beyond: See previous ideas. Should they be expected to solve the equation x+y = xy for y obtaining the rational function y = x/(x-1)? Analysis of this function and investigation of its graph may open new vistas for this 'innocent' problem about sums and products. Does this function really make it clear why 0,0 and 2,2 are the only integer solutions?

The original question is well-known. At any level, I would recommend that they be allowed to explore and make conjectures before more formal analysis. High schoolers enjoy coming up with 0,0 and 2,2 as much as 8 year olds! Modifying it and asking probing questions as students mature mathematically is the challenge for all of us. Have fun with this 3rd grade question!

Posted by Dave Marain at 8:33 AM 0 comments

Labels: arithmetic sequence, investigations, pedagogy, warmup

Now that our family health crisis has abated (my daughter is doing well), I guess it's time to jump back in with both feet. A math program leader in a district with which I am consulting, asked for my opinion on an important issue of curriculum and instruction.

How much time should middle school teachers spend on the traditional vertical algorithm for adding and subtracting mixed numerals vs. converting to improper fractions immediately?

I assumed that both methods are still commonly taught with about equal time given to each, but I wasn't all that sure about how that was across the country. This is where I need the help of my informed readers.

First, my thoughts. From a practical perspective of those who utilize fractions in their occupation, I would guess that the mixed numeral form is most commonly employed. Whether it's the carpenter taking measurements to see how many board feet of wood must be ordered (or for precise measurement to the nearest sixteenth of an inch) or someone following a recipe in the kitchen, I can't imagine that converting to improper fractions would be their first choice. On the other hand when I personally need to add fractions in a math problem, I usually use improper. I took an informal survey of one of the groups I'm working with and the majority stated they were taught both methods and some preferred working with mixed numerals and others said it's more complicated that way.

Are the number of steps roughly the same?

Mixed Numerals Algorithm for Subtraction:
(1) Convert the proper fractions to common denominator form.
(2) If needed, regroup, i.e., "borrow" 1 from the whole number part of the larger mixed number, convert the 1 into common denominator form and combine this with the other fraction (of course students are shown short-cuts for this which they blithely and mechanically follow without much thought).
(3) Subtract the whole numbers and the proper fractions.
(4) If the resulting fraction is improper convert it and add the whole number part to the previous result.

Improper Fraction Algorithm:
(1) Convert each mixed numeral to an improper fraction by the traditional algorithm (again blithely and mechanically without much thought).
(2) Determine a common denominator (or the lcd) and convert each fraction.
(3) Subtract the fractions.
(4) Convert the answer to mixed numeral form by the traditional division algorithm.

Now I may have combined steps or there are oversights but essentially they appear to be roughly the same number of steps. However, the difficulty or complexity level of the steps
may not be equivalent.

I also feel that the mixed numeral form requires somewhat more conceptual understanding even if the child does it routinely. It may also prepare the youngster for working with algebraic expressions like A + B/C, but that's debatable. Further there seems to me to be a strong connection between the Mixed Numerals Algorithm and adding and subtracting denominate numbers. For example:
Subtract
15 hr 37 min
9 hr 46 min
I doubt that we would encourage students to convert both to minutes first, subtract, then convert back to hrs and min. I could be wrong there!

I feel there are arguments on both sides here. My instinct is that both need to be taught but it's not clear to me how much time should be spent on each method. Certainly some youngsters could handle both with facility while some would struggle mightily with at least one of these methods.

Further, I suspect there are some youngsters who convert mixed numerals to improper fractions procedurally without full conceptual understanding that a mixed numeral is an addition problem!

Your experiences and thoughts...

Posted by Dave Marain at 4:44 PM 13 comments

Labels: fractions, improper fractions, middle school, mixed numerals, pedagogy

The English language has many confusing phrases but "as many as" IMO has blighted the youth of many an algebra student. Perhaps you think I'm exaggerating this? At the beginning of the school year, write the phrase in the title of this post on the board and have your PreAlgebra/Algebra I (or higher) students write one of the two equations on their paper. Give them only a few seconds, then compile the results. Let us know if the vast majority choose the correct equation. Of course, the outcome depends on the group and many other factors but if we have enough data it might prove interesting. I'm basing this on many years of questioning students. Perhaps I am the only one who has experienced this phenomenon!

The abstraction of algebra is difficult enough for some youngsters. Students who are new to our language have particular difficulty with idiomatic phrases but those born here also seem to struggle with the verbal parts of word problems - that's completely obvious to any algebra teacher of course. If only we could remove the words from a word problem!

Certainly teaching vocabulary and math terminology is an essential part of what we do as instructors. We should also hold students accountable for this vocabulary by assessing it directly.

In this post, I'm inviting readers to share some of the coping mechanisms and pedagogical strategies they use in the classroom to help students survive phrases like "as many as." What phrases seem to cause the most confusion among your students? How about "x is four less than y?"

Here is my initial offering. Let me know if you do something similar or if you feel this might be helpful (or if you vehemently disagree!).

KEY STEP: First decide from the wording of the problem if there are more girls or more boys. In fact, this should have been my original question -- not the equations! It is critical for students to be able to translate the verbal expression into a comparative relationship: Which is the larger quantity? Number of boys or number of girls? Hopefully, most youngsters would interpret the original problem to imply that there are more girls than boys. Hopefully! Ask this question first (metacognitively, students need to learn to ask themselves questions like this when they are reading).

NEXT STEP: Now the issue is where to place the "2" in the equation. Based on the key step above, we know that the number of girls is the larger quantity. Ask them why 2G = B would be incorrect.

Better alternative for some:
We all know that those who have difficulty handling abstraction benefit from concretization, i.e., using numerical values:

Have them write both possibilities:
B = 2G and G = 2B
Now have them substitute values for G and B that make sense for the original problem, say
G = 12, B = 6. Some struggle with this!
By substituting (students like the phrase "plug in") these values into both equations, they should see that 6 = 2⋅12 does not make sense. The correct equation should become apparent. Should...
Of course, most youngsters need to practice many of these before they reach comfort level.

Your thoughts, suggestions, anecdotal evidence???

Posted by Dave Marain at 7:19 AM 16 comments

Labels: algebra, algebra sense, pedagogy, verbal phrases, word problems

The following is the first in a series of strategies for teaching concepts that often prove difficult for many students from middle school on. These are not based on carefully controlled research studies following clinical methodology for a dissertation. They are based on 30+ years of learning how to do it better!! I suspect that's why we refer to the practiceof teaching. Our readers are encouraged to share their own favorite methods that have been helpful to their students or to themselves. These ideas are intended only as suggestions. Each teacher will, of course, bring her/his own ideas and style to bear on the lesson.

Most of you know the classic algebra word problem type that has appeared frequently on standardized tests and math contests:

THE BIG QUESTION
Jack averaged 40 mi/hr going to school and 60 mi/hr returning from school over the same route. What was his average speed in mi/hr for the round trip?

Since there has been a decrease over the past 25 years in the number of word problems to which our students are exposed, some youngsters may not get to see one of these until reviewing for SATs or in their physics class.

From watching how students approach this type of question, I'm getting a sense that we need to introduce the basic concepts earlier on in middle school, which I am sure already occurs in some programs. In planning to teach methods of solving these kinds of problems, I usually tried to return to basic principles of math pedagogy - keep it simple and start with concrete numerical exercises that built on prior knowledge. What does all this jargon mean?


Start with a review of averages, then move on to combined averages before attempting to explain the round-trip rate problem!

[Concerned that such development will take too much time? There won't be enough time to review homework and provide enough practice for the homework assignment? My supervisors never threatened to fire me if a lesson lasted for more than one day and if, heaven forbid, I did not assign homework that first evening! Some ideas just cannot be rushed.]

Suggested Question #1:
Jack had a 70 avg on some tests and a 90 average on some other tests. Can his overall average be determined?
More specifically: When do you think 80 will be the correct answer? When will it not?

Comment:
Question 1 is intended to provoke thought and encourage an intuitive response, not a calculated answer!

Suggested Question #2:
Jack had a 70 average on his first 4 tests and a 90 average on his next 6 tests. What was his overall average for the 10 tests?

Comments
Note that I am suggesting beginning with problems to which middle school students may better be able to relate than a rate-time-distance question. The first question above is fundamental in developing the concept of the original rate problem.

These questions should help many students focus on the essential idea that we need to know how many are in each sub-group!

Since most students connect average to dividing a TOTAL by some quantity, they should feel comfortable in solving the average grade question as follows:

(TOTAL PTS)/(TOTAL NUMBER OF TESTS) to arrive at an average of 82.

BUT DON'T STOP THERE! Stress the UNITS of this result to build the rate concept:

Posted by Dave Marain at 7:28 AM 9 comments

Labels: average rates, averages, instructional strategies, Instructional Strategy Series, pedagogy

One of the reasons I began this blog was to share the collective wisdom of experienced math teachers as a benefit to the novice. Well, here I am 18 months into MathNotations and I don't believe this has yet been specifically addressed. I expect the comments or follow-up posts to be even more beneficial than what I'm writing below.

Here's what I'm asking my readers --
In this post, I will begin enumerating one or two instructional components which I believe should be an integral part of most (math) lessons. Since I have strong antipathy towards jargon, I will try to avoid technical phrases like 'set', 'hook', although closure is ok.

Note that I put math in (..) to emphasize the point that I regard many of these suggestions as integral to effective lessons in general!

Note: These lesson components should be independent of teacher style, makeup of the class, content, etc.


Background
I do know that newbies often feel overwhelmed by all of the differing expectations coming from their immediate supervisor, colleagues, principal, other administrators, courses of study/syllabi, district technology initiatives, state standards, state standards, NCTM Standards/Curriculum Focal Points, standardized test specs -- just to name a few! I haven't even mentioned what they learned from their methods classes, the influence of their math teachers in their formative years, advice from just about everybody. When all is said and done, it seems that the number one concern on the part of most evaluators in the beginning is classroom management, effective delivery of content being number two. Of course, evidence of content knowledge becomes of greater importance if there is an immediate supervisor who has math certification.

How does one navigate through this morass without losing one's mind? Prioritize! Less really is more! Rather than attempt to build the perfect lesson to please the observer, be guided by what you know will lead to demonstrable evidence of learning. Yes, planning is critical. I will comment on that further.

Here then is just the beginning of what I expect to be an extended discussion and one which I am considering publishing as a pamphlet. Please adhere to the Creative Commons License in the sidebar if reproducing any of this.

DISCLAIMER
I am stating unequivocally that these are my own personal ideas of what makes an effective math lesson. I do not want anyone to say that I am telling anyone how to teach!

Each of you out there will have your own list, although I'd be surprised if there wasn't considerable overlap. The order of course will vary. These are the principles by which I was guided both as a classroom teacher and as a supervisor. At the beginning of the year, I would meet with teachers to discuss what I was looking for in the lesson. For clinical observations, I would also have a preconference to discuss specifics. This was particularly of critical importance before observing the non-tenured teacher.

THE BEGINNING
1) Class Opener - Critical first 5 minutes - Establishment of Routines

a) Allow students to socialize/decompress for a couple of minutes as they enter, but let them know what is expected of them; close door at late bell. Establish iron-clad routines for students to follow if they arrive after that - stick to it!

b) Math Warmup/Problem of the Day already on the board or projected on a screen using the overhead or PowerPoint (or Word) from the computer; the warmup can be used to review prerequisite skills for the upcoming lesson, SAT review, an opportunity for students to practice their communication (e.g., writing) skills in math, etc.

c) Answers to some or all of the homework exercises can be written on the board or projected on a screen from overhead or computer. Virtually every publisher of current texts provides ready-made transparencies both for WarmUps and answers to homework, not to mention PowerPoint presentations for every lesson! Some educators object to displaying answers like this as it invites students to quickly copy these on their paper. You may want to have selected answers displayed rather than all. There is no foolproof method here, so use your own judgment. The important thing is to busily engage students from the outset. While students are working on their warmup problem, the teacher is circulating, checking homework and engaging students. This personal interaction with students means so much (e.g., Lily, I saw you in the play on Thu night -awesome!).

Ok, folks, this is just a beginning...
Please contribute your suggestions!

Posted by Dave Marain at 2:10 PM 5 comments

Labels: instructional strategies, pedagogy

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is 9ドル.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

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Did you overlook our Mystery Mathematician for May? I've received two correct responses thus far, but submissions can still be emailed until the 15th of the month. Don't forget to include the info requested in a previous post.



If (A+3) ÷ (B+5) ≥ 10 and B ≥ 7,
what is the least possible value of A?
DISCUSSION

The use of multiple representations of a concept or procedure in mathematics is highly recommended by NCTM and other math education experts. Also known as the Rule of Four, it suggests that instructors use some or all of the following, when introducing a new concept. This requires careful planning and considerable thought on the part of the teacher. Over time and with experience, it will flow. However, it does help to see many models of this heuristic for geometry, algebra, etc.

The Rule of Four suggests that a concept be presented
(a) Using natural language (words)
(b) Numerically (concrete examples, 'plugging in', use of data tables, etc.)
(c) Visually (e.g., using graphs, charts, concrete models)
(d) Symbolically (algebraical mode)

From my experience, many students will approach the problem at the top by ignoring the inequalities and simply plug in 7 for B. They've learned that this strategy usually works on standardized tests. It is our role as educators to challenge them to think more deeply. Create disequilibrium by provoking them with a question like,
"But to make a fraction small, don't you need to make the denominator as large as possible?" Of course this statement does not apply to this problem, but I'll wager that it would cause some to reconsider their initial answer!

Do you think that most students would quickly recognize that the relationship between A and B can be described by a linear inequality, which can be then be approached both algebraically and graphically? Do you think I need strong medication for asking you that question!

To deepen their understanding, one could ask:
How would you have to change the above problem so that one could ask for the greatest possible value of A?

I plan on posting further examples of the Rule of Four. I am aware that I have not fully demonstrated this technique for the problem above. I'm only hinting at it. More will likely come out in the comments...

Posted by Dave Marain at 10:58 PM 4 comments

Labels: algebra 2, linear inequality, multiple representation, pedagogy, precalculus, Rule of Four

NOTE: PLS READ THE COMMENTS FOR A DETAILED DISCUSSION OF THIS PROBLEM, WHICH SHOULD PROVE QUITE CHALLENGING FOR MOST MIDDLER SCHOOLERS.

The previous activity I posted regarding integers that have exactly four factors might lead to some interesting discussion regarding a general description of such numbers. All of these kinds of problems could be handled by simply giving students the general rule for determining the number of factors of any positive integer. I have given this well-known number theoretic formulation in earlier posts, so I won't review that at this time. However, there is a greater benefit to be derived from having students investigate these relationships. The following activity should be adapted to meet the needs of your students.

Some educators react to these kinds of deeper investigations with reactions like:
(a) I have a curriculum to cover. I don't have time for this.
(b) Unless this kind of question appears on state testing, it's simply not practical for me to do this.
(c) My students are just not ready for this kind of thinking.
(d) Dave, you're out of the classroom now, so you're forgetting the realities of most classrooms. Some students don't know their basic facts and you want me to do higher-order thinking! Gee, Dave, are you forgetting we have classified children mainstreamed in our classes? Get real!
(e) Dave, stop suggesting HOW we should teach and just give us the problem. You're trying to impose your style on others - it doesn't work - we each bring our own style to a lesson.

[Comment: I have strong reactions to some of the above, but then I'd be arguing with myself! I'll respond to some of these in the comments section or devote an entire post to these critical issues if my readers decide to respond to this.]

I'm certainly not suggesting that these kinds of explorations should BE the curriculum. There must be a balance between these problem-centered approaches and skills development. I am suggesting there needs to be some time devoted to deeper cognitive processes to foster mathematical development. The following investigation is far from one inch deep! I may continue it later but I'm hoping some will suggest extensions, make comments or report back how it played out in real classrooms (also how it was adapted/revised).


INVESTIGATION/READER CHALLENGE

Part I
1. The number 6 has 4 factors: 1,2,3,6 (or in paired form: 1,6;2,3).
Suggested Questions:
If we double the number 6, what do you think will happen to the number of factors? Will it increase or stay the same? Will the number of factors also double?
Mathematicians, like scientists, make conjectures or educated guesses, but not wild guesses! We need some evidence or data on which to base our conjectures. With your partner, fill in (and possibly extend) the following table, then formulate your conjecture using correct mathematical language:
Positive Integer.................Number of Factors
6...........................................4
12.........................................____
24........................................____
48.......................................____

Do you think we have enough data to make a conjecture or should we continue the table? Record your observations and then state your conjecture or 'rule'.

Teacher Tip: Depending on the maturity of the group and their experience with these kinds of formulations, you may want to start them off with a prompt:
If we double a positive integer, the number of factors _______________.

Many students will be convinced they have found a mathematical rule that will always work. That's one of the objectives of this investigation: To help them understand that
(a) Pattern recognition does not a rule prove!
(b) The conjecture is based on starting from the number 6. There is no basis for assuming that their 'rule' will be valid if we start from a different positive integer!

Part II
This time have students start from a different integer: 18
Make a table similar to the one above, again doubling the integer in the left column.
Again, record your observations and then state your conjecture or 'rule'.

Suggested Questions:
Do you think there is a more general rule that covers all cases or are there simply different rules for different integers? If you were going to investigate this further, what other kinds of starting positive integers would you try?

Teacher Tip: Asking many questions stimulates student thinking and leads to more questions and deeper thought processes on their part. A free interchange for a couple of minutes is invaluable here to have students come to see that mathematical research requires persistence and an attitude of inquiry. As Ms. Fribble from the Magic School Bus would say: ASK QUESTIONS! (or something like this!).

Part III - Start from an odd integer this time: 15
Part IV: To be continued...

Posted by Dave Marain at 7:20 AM 7 comments

Labels: factors, investigations, middle school, number theory, pedagogy

The following was my comment on Joanne's stimulating discussion on 'Teachers wonder about direct instruction.'

Although my primary focus is currently on WHAT we teach rather than HOW, I must strongly endorse Mr. Strauss’ reasoned and thoughtful comments. Good teachers have always blended successful methods of the past with the best of what is currently known about the different ways that children learn. No single style can possibly meet the needs of our more and more diverse learners we encounter every day. There seems to be considerable confusion about the technical meaning of DI as developed by Mr. Engelmann. One would need to thoroughly study his rationale and approach to make an informed judgment and I suspect many are responding to the ‘label’ rather than its substance just as many react to ‘discovery learning’ as if it is a method to be used all the time. Effective math lessons I’ve observed for the past 10 years included the essential components of instructional/learning theory:
1. Motivated the lesson (a ‘hook’)
2. Articulation of the objectives of the lesson (what students will know and/or be able to do at the end of the lesson) - this must be carefully thought out during planning and conveyed clearly.
2. Connected current learning to prior learning
3. Reviewed the necessary prerequisite skills for success
4. Provided clear explanations both orally and in writing (on board, on handout or in an electronic presentation)
5. Maximized student involvement via questioning, promoting of dialogue or an activity
6. Assessed what was actually learned (e.g.,responses to questions or requiring students to complete a specific task).
When you remove all the labels, Joanne, it comes down to this: How do we know that the objectives of the lesson were achieved? When I am transmitting parcels of information directly to students, I am still engaging their minds by asking many many questions of different taxonomies to check for their understanding as well as checking if they are still conscious! When I propose a challenging problem and give them a few minutes to work on it in small groups, I am still monitoring their progress carefully and asking guiding questions.
If DI includes all of these components and allows children to explore at times and tackle unstructured open-ended questions for which there is no clear blueprint for solution, then I applaud DI and I guess I’ve been using it all along. If ‘Discovery Learning’ includes all of these components, then I guess I’ve been using it all along and I applaud that too.
Again, as Larry so ably expressed it, good teachers FIND A WAY that works for most of their students most of the time. There will always be some in the class who are not able to grasp the material for a myriad of reasons, often having nothing to do with the child’s ability. Rather than continue this general debate, perhaps we should be looking at REAL examples of effective teaching and then we can applaud these efforts and use them as models for the rest of us, rather than debate the category into which the lesson falls. Oh well, this will never happen, because real examples and pictures would obviate all of the rhetoric and we’d have nothing to blog about!

Posted by Dave Marain at 7:44 AM 2 comments

Labels: direct instruction, discovery learning, math instruction, pedagogy