MoreOrLessPenAndPaper

Question: Is 'less pen and paper' the real goal [of constructivist math]?

Witness the work to be assigned in Passport to Mathematics, Book 1, by McDougal Littell, 1999, page 13.

(This is going to be long because I have a point to make.)

Example 2

The list shows the whole numbers from 1 through 72.

1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72

a. Copy the list and circle all the multiples of 4. b. Color all the multiples of 6 blue. c. Describe the numbers that are circled and blue.

Solution
a. The multiples of 4 consist of the 4th, 8th, and 12th columns.
b. The multiples of 6 consist of the 6th and 12th columns.
c. The numbers that are circled and blue are in the 12th column. They are all multiples of 12. So, in general you can csay that if a nuymber is both a multiple of 4 and 6, then it is a multiple of 12.

Study Tip
The pattern shown by a list can depend on how the list is written. In Example 2, the pattern would not be as clear if you used 10 columns instead of 12.

My observation: This example is obviously a study or review of multiples and finding common multiples. It is Lesson 1.3 entitled "Making a list". (Lesson 1.1 was entitled "Looking for a Pattern") Here are the goals for this Lesson 1.3:

Goal 1: How to solve problems by making a list. Goal 2: How to use lists to help you solve problems.

(No mention of multiples in the goals. No mention of looking for patterns.)

Now my question: How much pen and paper is used in this assignment? I am aware that there is something 'fun' about seeing patterns in math. It can also be enlightening to see these patterns. But to have students copy the entire list before finding the pattern -- ?? What a laborious, time consuming task!

How much more time could students spend on finding common multiples for other pairs of numbers in the time wasted just making this list? Lots of time here that could have been used on speed drills, mental math reviews, other computations, etc.

So to answer my question: No, apparently pen and paper work isn't bad afterall. It's only bad if I use pen and paper for traditional math algorithms and drills. A "new math" discovery-type lesson can use all the pen and paper that it wants.

Also, note that Study Tip. Keep that in mind as I give the Extra Example suggest in the T.E.

Extra Example 2

Copy the list of 72 numbers in Example 2
a. Draw a square around all the multiples of 5.
c. Describe the numbers that are in squares and colored yellow.

In light of the Study Tip, does anyone find it odd that this extra example uses the same list structure?

(This list is but one type of list that students are making in this lesson.)

What would you rather your students spend their math time on making lists or actually finding common multiples and doing computations?

One more point: Did any one read the two goals? Any thoughts?

-- InterestedTeacher

Comments

Is it your impression that classes are doing HUGE amounts of paper-and-pencil-and-glue projects these days?

instructivist.com

-- CatherineJohnson - 29 Jun 2005

Projects are being pushed on us all the time. Administrators and curriculum directors want to "see" something that students are doing and making. Teachers can then display the "made" things on bulletin boards, or better yet, outside the classroom on the walls of the hall!! To the ooohs and aahhs of everyone that passes by!

Now, everyone can see what you "did"! But what did students "learn"? If you can show that students "did" something, they're bound to have learned something, right? This is shallow thinking.

What did you learn? Paper and pencil (and glue) can demonstrate what you've learned, or they can just show that you "made something", or they can do both. Hopefully, they do both, but what should be the most important? And how can it prove that you understand?

Some people will say that it's not what they've learned, or even if they've learned, but how they feel about their project that's important.

Projects take time, something that is precious in the classroom. A typical math lesson must be a balance of review and practice of recent lessons, drills, introducing the new concepts, teacher-directed practice of the new concepts, and then individual practice of the new concepts. A teacher is constantly making judgments as to where to spend that time. But it's all got to be planned so that learning takes place.

Back to the projects and this is critical: when did the learning take place? Was it when the student was being taught directly by the teacher (algorithms, strategies for getting the right answer)? Or was it after hours of constucting a project (make that a group project for more points with the higher-ups)? Sometimes the projects are so long that the whole purpose is lost for some students. There's just too much lag time between the beginning and the end, and too much vague, off the track work for some students to even figure out what they have done.

Projects with glue are important for younger students because we're working on developing fine motor skills. But I'm seeing primary-grade type paper-and-pencil-and-glue projects being done in upper grades more and more. Why?

The constructivists don't want students to learn any other way but through students' own "makings" and "doings" and "constructings". Even if it takes hours. So, one of the reasons is that projects are supposed to help students figure out and make learning "their own".

Another reason for primary-type projects done in upper grades is for the "show". Many of my projects have been just for show, and I've grieved inwardly, if these were in-class projects, because I've felt that I've cheated the students from learning twice as much through other teacher-directed means. But even then, my projects have been on material that I've already taught them.

I can't imagine risking allowing students to come up with their own "method" through a project in Math. There is too much at stake -- the future of this student rests on learning incrementally one step, one concept after another, and learning it properly, so he can successfully solve a problem. After hours, days, maybe even years of successfully solving problems, then is the time for doing an involved math project. In the mean time, lots of paper and pencil drill, lots of practice, using the algorithms over and over until it's second nature.

I'd much rather see students working with manipulatives. That's the good kind of hands-on stuff! At my school that seems to satisfy the higher-ups, and then it's back to the drill and practice. And some of it is paper and pencil. Not any glue in 5th grade math in my room.

-- InterestedTeacher - 29 Jun 2005

I don't see it so much as a split between pen/paper versus other techniques (calculators, computers, or manipulables). I see it as a change from a traditional bottom up approach to education that emphasizes mastery of basic skills, to a top down approach that emphasizes "real world" problems. In the top down approach, the student is required to construct his or her own basics, which may or may not ever happen. And, of course, the top down approach doesn't require mastery of the basics because they don't think there is any linkage between skills and understanding. They think that content changes so fast that only the skill of constructing is important.

Manipulables and calculators can be very useful in certain situations. The question is whether they are wasting time or not.

-- SteveH - 29 Jun 2005

Absolutely, use of manipulatives can be a waste of time. When concepts are already nailed down, using manipulatives might be a waste of time. Occasional use as review would be beneficial

I love to use fractional/percent manipulatives when introducing percent. Saxon has "pie pieces" (with percent and fractions written on each piece)which are good for this. I'll ask students to demonstrate that 2 tenths is equal to 1/5. Or perhaps, lay out at least 2 different ways to show 30%, or to demonstrate which is more, 3 fifths and 1 tenth or 3 fourths, using pie pieces. Once a student demonstrates mastery of this, continued use of these manipulatives for this purpose would be unnecessary.

Weaker students need more time, along with teacher directed practice, that "traditional bottom up approach" as SteveH^? so aptly describes. (I love his writings.) These students may take months, or even a year or two, making a connection between fractions and percent. Many a time, all I have to say is "Get out your 'pie pieces' and let's look." They provide a way for students to get the answer while they are learning and before understanding comes. But the repeated usage of manipulatives, or other visuals, helps to bring understanding and eventually students will interchange fractions and percent easily and accurately.

Now, a related "project" that I would consider a waste of 5th graders' time would be to glue these manipulatives on posterboards so we could put them on our hall wall to "show off" that we've learned 3 ways to make 30%! And to use class time to make this "project". No, no. Let's just work with the pieces for 5-10 minutes on our desks, and then move on to other invaluable teacher-directed work on algorithms, speeddrills. And some of it will be paper and pencil.

-- InterestedTeacher - 29 Jun 2005

Projects are being pushed on us all the time. Administrators and curriculum directors want to "see" something that students are doing and making. Teachers can then display the "made" things on bulletin boards, or better yet, outside the classroom on the walls of the hall!! To the ooohs and aahhs of everyone that passes by!

That's exactly what the Illinois Loop folks said.

My school, this year, had a hall filled with prime factor DIORAMAS.

-- CatherineJohnson - 29 Jun 2005

Projects take time, something that is precious in the classroom. A typical math lesson must be a balance of review and practice of recent lessons, drills, introducing the new concepts, teacher-directed practice of the new concepts, and then individual practice of the new concepts. A teacher is constantly making judgments as to where to spend that time. But it's all got to be planned so that learning takes place.

YES YES YES!!!!!!

This is one of the things that just makes me NUTS: the talent, experience, and flat-out briliance that go into being a really expert teacher are completely unrecognized and unacknowledged in all the constructivist rhetoric & propganda.

I find the idea that the teacher is NOT to be the 'sage on the stage' insulting to teachers.

-- CatherineJohnson - 29 Jun 2005

Projects with glue are important for younger students because we're working on developing fine motor skills.

EXACTLY.

AND EVERYONE KNOWS THIS.

It's just chronic no common sense-y out there.

-- CatherineJohnson - 29 Jun 2005

Many of my projects have been just for show, and I've grieved inwardly, if these were in-class projects, because I've felt that I've cheated the students from learning twice as much through other teacher-directed means. But even then, my projects have been on material that I've already taught them.

You know what you need? (Which you may already have, of course.)

You need the teacher-equivalent of THE CAKE MIX DOCTOR book.

It's incredible.

She has a 'Darn Good Chocolate Cake' that tastes like you spent hours making it, but is in fact a doctored-up chocolate cake mix.

But people love it, and they feel happy and flattered (as they should!) that you made it for them.

Do you have classroom crafts like that?

Things where you can get a huge bang for your buck?

-- CatherineJohnson - 29 Jun 2005

SteveH^?--fantastic comment.

That's EXACTLY right.

-- CatherineJohnson - 29 Jun 2005

I use the fraction manipulatives from Saxon Math 6/5 and find them very beneficial for teaching fractions and percent.

One important time saver: have students color the fraction pieces (both sides of paper) before they cut them out. Red for the 50%, blue for 25%, etc. Students learn the colors and since both sides are colored, it saves them time finding them. Keep them in zip baggies. They should last all year.

-- CarolynMorgan - 30 Jun 2005

Catherine, one thing comes to mind. The best bang for my bucks has been a box of craft sticks from the hobby store. I use them all the time in math. In fact, I have students keep a baggie of them in their desks, maybe half the year. They leave them there for the next class.

Uses:

• To show what happens when you combine an odd numbered set with an odd numbered set; or with an even numbered set.

• To regroup tens and ones, rubber bands hold the tens, until you need to break them apart to borrow. (This would be more for grades lower than 5th.)

• To demonstrate averaging: Pick a group of 3-5 students. Give each a bag of sticks, have them count them. Then tell them to redistribute them so that each has the same number. (Be sure to plan this ahead of time so the numbers are good.) It's really fun when you pit one team against another to see who can do this more quickly.

• Great for demonstrating dividing a large number, with or without remainders. This really shows what "the remainder" means.

• Demonstrate parallel lines, intersecting lines, perpendicular lines. Also horizontal, vertical, and oblique lines. (each student with own set of sticks at desk) (This is one of my favorites.)

My instructions: Start with a pair of horizontal parallel lines (sticks); close the area in by using a pair of parallel oblique lines.

or

Start with a pair of lines that are not parallel, but are touching; close the area in with another pair of lines that are not parallel.

or

Start with a pair of vertical lines; close in the area to make a quadrilateral that has only one pair of parallel lines.

• Show right, acute, obtuse angles (students with set of sticks at desks)

• Show specific polygons. (right triangle, oblique triangle, pentagon) (students with set of sticks)

(I love them because I can quickly look around the room and see who's struggling.)

• This year I used them to help a student who couldn't understand a problem such as: If 2y=16, then how much would 4y-1 equal? He needed to see the concrete. (And this happened the day I was being observed by my principal. I never had used the sticks for this before, but I coulnd't get him to "see" it. My principal loved it!)

And this next year I may try to use them for Bar Graphs. (I've been paying attention, Catherine!) I don't know, they may be too long, we'll see. But for students that need the visual, it may be helpful.

I've had my box of craft sticks for 13 years.

-- InterestedTeacher - 30 Jun 2005

One important time saver: have students color the fraction pieces (both sides of paper) before they cut them out. Red for the 50%, blue for 25%, etc. Students learn the colors and since both sides are colored, it saves them time finding them. Keep them in zip baggies. They should last all year.

Great idea.

I didn't even bother using the things (though Christopher could use some serious OT work on cutting paper, that's for sure).

I used my plastic fraction manipulatives (they're over on the Math Supplements page).

I used them again yesterday, as a matter of fact, and it reminded me of Carolyn's post on the CA Board of Ed study finding that fraction manipulatives are good for 7th graders.

Yesterday Christopher was, on his own, adding up fractions & finding equivalent fractions--he was just idly doing this, while waiting for me to get things organized.

Last summer I had to HAUL him through fraction lessons (which was what he'd flunked on his Unit 6 test).

This summer those fraction manipulatives are much more 'real' to him.

-- CatherineJohnson - 30 Jun 2005

Wow--fantastic advcie Carolyn & Interested Teacher.

I'm going to try to get this pulled up to the front page (I'm trying to get Andrew's home program set up today, so the only reason I'm sitting at my computer is that I'm scanning in Edmark pages.)

Or else I'll post a notice.

Still, I'd like to get some of these points highlighted.

-- CatherineJohnson - 30 Jun 2005

How are you going to do bar models with craft sticks?

That's a great idea, by the way, getting the craft sticks.

I'm going to get some for Andrew.

All the Singapore kids have them, I think.

-- CatherineJohnson - 01 Jul 2005

OK, I'm finally getting to your question on how I'm going to try to do bar models with craft sticks.

Take a problem like:

Joe had 2 times as many apples as Mary. Altogether they had 24 apples. How many apples did Joe have? (Or did Mary have?)

It takes a visual picture for most 5th graders at first, and for some, those visuals are necessary for a long time. I envision using craft sticks on this because students can begin with two different bars, one twice as long, to represent the two students. But because you can move them around, students can put the sticks in one long bar to total 24 apples and can then separate the total into the two bars again.

I don't know; it seems good as I think about it. Some children just need multiple types of manipulatives to 'see' and 'understand' problems.

-- InterestedTeacher - 04 Jul 2005

We usually draw those Saxon 'towers'. I think this will work too.

-- InterestedTeacher - 04 Jul 2005

It takes a visual picture for most 5th graders at first, and for some, those visuals are necessary for a long time. I envision using craft sticks on this because students can begin with two different bars, one twice as long, to represent the two students. But because you can move them around, students can put the sticks in one long bar to total 24 apples and can then separate the total into the two bars again.

I just had a Reading War with Andrew; I sat him down to do his Edmark pages (first time ever at home) and he slapped his head, screamed, etc.

I'm in a daze.

So...I'm having trouble visualizing this....I'll ask one quick question: I take it craft sticks come in different sizes?

Is that right? (These are popcycle sticks, right?)

Where do you get them?

(I can easily imagine using different sized sticks for bar models--)

-- CatherineJohnson - 04 Jul 2005

Also, what are 'Saxon Towers'?

-- CatherineJohnson - 04 Jul 2005

Do you know what these things are, or how they work?

-- CatherineJohnson - 04 Jul 2005

Yep, they are just 'popcycle sticks', but they are all the same size. Otherwise, it wouldn't work. Craft sticks come boxed by the 1,000 at hobby stores. Great investment. "These things" which you referenced look like them, only mine are so old they aren't colored.

Saxon towers are the upright bars that are used to sketch 'Larger-Smaller-Difference' stories. In my Saxon 6/5, that is Les. 36.

So I just continue calling them towers when we get to Les. 50 where we learn to use the upright bars(towers) to solve fraction-of-a-set type of problem, but I have them include other steps and other activity which really guarantees that students understand get the right answer. The Learning Center teachers at our school really love how we do this, because it helps those L.C. students "see" what we are doing. I wish I were good at drawing and using our printer/copier to send the drawing so I could show you what I do. I'm going to take a stab at a verbal explanation.

Students must have a colored map pencil in additon to their regular pencil. Let's pose a problem: If 3/4 of the 24 cars were red, how many red cars were there? For the problem finding 3/4 of 24, we would all draw a tower We write the fraction on the left of the tower, and the total on the right of the tower. (Every time we do it this way.) With their regular pencil, students circle the denominator. They divide the tower into that number (4) of parts. Then they also really divide the 24 by that 4, using the area to the right of the tower. The answer '6' is then written in each of the 4 parts of the tower (6 red cars, 6 red cars, 6 red cars, 6 red cars). Now they circle the numerator of the fraction with the colored map pencil (3) and lightly shade that number of sections of the tower. (Remember the number '6' is written on each section.) They can then see how many red cars are included in the shaded part of the tower.

We work on this and work on this, but students really do like it; they think it's fun. We go to the board and I let them use my colored chalk!!! They soon learn to sketch their tower, making their towers' height cover the same number of lined spaces equal to the denominator. I hope that sentence made sense!

Bright students learn quickly to solve the problem without using the 'tower', and I don't require it for them.

Of course, drawing these towers comes before students learn to change 'of' to x and multiply. We really work hard on learning this concept.

I've had students use this sketch to help them with other problems when I wasn't even expecting it.

I hope I have made this clear. If not I'll have to email you my phone number and we can chat longer.

I think I can make the craft sticks work on bar models.

-- InterestedTeacher - 05 Jul 2005

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