##### All Topics for this Forum

## Topic: "MSPnet Academy: Breaking the Cycle of Failure: From Numerical to Algebraic and Geometric Reasoning"

### Topic Posts

###
MSPnet Academy Discussion
January 11 - January 25, 2012

Too many students leave our K-16 schools numerically illiterate and wanting little to do with mathematics. This session will focus on how the Standards for Mathematical Practice can be employed and developed as students learn to reason with and about number. We will examine how daily practice with mental arithmetic can develop productive habits of mind, and enhance students understanding of the arithmetic properties and place value while building a more solid foundation for geometric and algebraic reasoning.

### Archives for this academy

### Numerical reasoning

posted by: Ruth Parker on 1/11/2012 3:03 pm

*post updated by the author 1/11/2012*

### Broadening the national dialog

posted by: Bob Kansky on 1/13/2012 8:20 am

One of your closing questions was "How might we initiate a broad national dialog on this topic?" [In responding, I interpret the "we" to include both individuals and groups -- e.g., NCTM, NCSM, AMTE.]

One obvious task will be to bring the college mathematics community on board, since the education of prospective teachers and the professional development of practicing teachers is molded by the instruction modeled in college mathematics classes. To what extent is MAA (and its Project NExT) committed to CCSSM?

A somewhat less obvious route to broadening the effort would be to engage with the science education community. The NRC Framework for K-12 Standards in Science will be used by Achieve, Inc. to develop the New Generation Science Standards. The first section of the Framework presents eight Scientific and Engineering Practices. Although there is not a one-to-one correspondence between the NRC practices and the eight CCSSM Mathematical Practices, there is considerable agreement. Both sets of practices step back from the "what" of the subject to address the "how" and 'why" of doing the the subject.

So why not broaden the CCSSM practices-implementation workforce by merging efforts with science education. Some like groups and individuals would be:

** NSTA (Francis Eberle, Alan McCormack, Harold Pratt, Rodger Bybee);

** Achieve, Inc. (Jean Slattery); and

** NSES developer Henry Heikkinen.

### Response from Alabama to Ruth Parker's Presentation

posted by: Deb Ohara on 1/13/2012 12:01 pm

Our goal is to train math and science teachers in hands-on, engaging, and inquiry-based instructional methods, which is right in line with the mathematical practices that are now included in the Common Core Standards. I was excited by the webinar presentation because this is exactly what our initiative is trying to accomplish with teachers. Many of our local math specialists have been trained in Number Talks and they are sharing those strategies in the classrooms as they support teachers.

You asked. "Are there ways that we could make this case to a broad national audience?" I wanted to mention just how many times recently that we have seen these same or related strategies and comments that support what was said in the WebEx. 1) We have participated in the Greater Birmingham Mathematics Partnership, which is another great program that promotes Number Talks. I believe Sherry Parrish is associated with that initiative. 2) Dr. Yeap Ban Har, a mathematics educator from Singapore, in a speech in Alabama, said that in Singapore, the purpose of teaching mathematics was to teach thinking and problem-solving. They do not worry about teaching algorithms. The example math problems that he presented and the strategies that their students learned were very much aligned to what was supported in the WebEx. 3) Recently our AMSTI math specialists were trained in OGAP strategies (On-going Assessment Project). This is a program out of Vermont (Vermont Mathematics Partnership) which also utilizes the very same strategies and arguments demonstrated in the WebEx. In fact, the research that has been done by those folks has been quoted as some of the best research on Learning Trajectories. 4) CPAM (Council of Presidential Awardees in Mathematics) has recently formed an affiliation with NCTM and other leading organizations in an attempt to have more influence on mathematics education policymaking.

I believe that all these initiatives should join together with the common goal of making this case to a broad national audience. We need to influence and educate politicians, policymakers, and state and local school administrators. But we must not forget the teachers in the classroom, either. My concern is on the local and state level. Because we are working directly with teachers, I see that many mathematics teachers in the schools are not given much opportunity to participate in good professional development, combined with research evidence, and they are therefore resistant to changing teaching strategies, even though what they are doing is not working well. Our initiative is trying to influence teachers to reflect on instructional practices and make some changes. We are making headway, especially considering how understaffed we are, but we certainly could use some help!

*post updated by the author 1/13/2012*

### Mental arithmetic

posted by: Richard Askey on 1/14/2012 12:34 am

Parker's comments on teaching arithmetic in the 19th century were incomplete.In the preface to "The New Arithmetic" book edited by Seymour Eaton, 1893 edition, Truman Henry Stafford wrote: "in fact, habits of correct calculation are as important in arithmetic as in reasoning: it was the fault of the earlier teaching of this subject to overlook the necessity that the pupils should reason out the steps: then followed a time in which the later necessity was recognized rather to the detriment of accuracy; and now, owing in part to the higher scientific applications of the subject, it begins to be seen that both accuracy and correct reasoning are indispensable." I would say "generality" rather than accuracy, but make the same point.

In "Primary Mathematics", the older program used in Singapore, they teach general methods for three and a half years, but teach them well with the ideas of place value and making tens used regularly. In the start of 3B, the book used in the second half of third grade, there is a short unit on mental mathematics. This seems to me to be a very reasonable way to develop arithmetic.

At ICME-10 in Copenhagen, there was a Russian display and one problem came from a 19th century book on mental mathematics.

(10^2 + 11^2 + 12^2 + 13^2 + 14^2)/365 = ?

The answer is 2, and the goal is to do this without having to do any messy calculations, where messy is not messy at all.One wants an elegant solution. I would not give this before students have learned some algebra, but all of you know enough to be able to understand a solution, so I suggest that you try and someone should post a solution. I have asked quite a few people this problem. The most rapid good solution I got was from a mathematics education professor at East China Normal. I asked her if Chinese students would be able to do this as she did. She said some, but probably not a lot.

Most mathematics educators stress the Standards of Mathematical Practice. Let me suggest that there are a lot of things in the Content Standards which should be stressed first. How to understand and teach fractions is one topic, and how to develop arithmetic after negative numbers have been added is a second topic. Hung-Hsi Wu has a nice article on these two topics in American Educator. See http://www.aft.org/newspubs/periodicals/ae/fall2011/index.cfm After material like this is learned, then it is time to see how the Practice Standards are used.

.

### Russian Mental Math

posted by: Ben Sayler on 1/14/2012 3:25 pm

A) 10^2 + 11^2 + 12^2 = 365

B) 10^2 + 11^2 + 12^2 = 13^2 +14^2

I use relationship A to rewrite the denominator of the original expression and relationship B to rewrite the numerator, yielding the following full expression:

2(10^2 + 11^2 +12^2) / (10^2 + 11^2 + 12^2)

From there, I can see the solution is 2.

It took me a couple of pages of fooling around to land on relationships A and B. I don't think I could have gotten there without paper and pencil. Is this sufficiently elegant? Should I have been able to do this in my head? Based on Ruth's presentation, I'm thinking maybe someone could match this with a geometric representation. I think that could be VERY elegant.

I'm left wondering about the underlying point. I liked the process of playing around with this. I drew on some algebra I already knew. Do I understand Richard to be saying that certain content standards must be understood first, before process standards have value?

### Russian Mental Math

posted by: Roy Gould on 1/15/2012 9:36 am

The sum of all five terms will give:

500 for the first term of the binomial.

20*(0+1+2+3+4) = 200 for the second term

0 + 1 + 4 + 9 + 16 = 30 for the third term

So the sum is 730.

This isn't elegant, but it isn't hard to do the sum in your head.

Best,

Roy Gould

### Russian Mental Math

posted by: Richard Askey on 1/15/2012 9:45 am

Symmetry can be useful in other areas of mathematics than just in geometry. There are good mathematical reasons for asking this

question, since the answer uses important results which students should see used in an arithmetic setting.

### a solution...or 'the' solution...or multiple 'solutions'

posted by: Brian Weaver on 1/15/2012 11:31 am

it is a response that possibly promotes convergent thinking (to someone else's preconceived 'solution' or 'rational')

divergent thinking might better be promoted by responses that focus on reinforcing the responder's efforts and 'approximate successes (aspects that are valid to the given task and related/'developmentally-appropriate' facts/concepts/skills/processes)

stated another way, i believe that '20 questions' designed to 'think' of the 'teacher's thinking' is counterproductive to the higher goal of mathematics - to use mathematics as a tool for reasoning, discovering and communicating

...i do enjoy a good riddle though...and look forward to the solution that you are looking for...or that meets the criteria that you have posited

thanks

### geometric...counting

posted by: Brian Weaver on 1/15/2012 11:43 am

i can see students 'seeing' each successive 'square (array)' as 'containing' a 10x10...'some' 10's...and a smaller 'square (array)'

it seems convenient that the 'squares' start with '100' and successively 'develop' by 10's and 'smaller squares'

this 'contiviance' allows counting by 100's, 10's and 'smaller squares'...

(100+100+100+100+100)+(20+40+60+80)+(1+4+9+16)

500 +200 +30

730

730=2(365)

'compatible' numbers

### Russian mental math

posted by: Maria Dworzecka on 1/15/2012 5:09 pm

10^2+(10+1)^2+(10+2)^2 + ( 10+3)^2+(10+4)^2 =

5*10^2+2*10*(1+2+3+4) +1^2+2^2+3^2+4^2=

5*10^2+2*10*10+1+4+9+16=

7*10^2+40=740=2*365

Maria Dworzecka

### Content standards are vital

posted by: Richard Askey on 1/16/2012 12:13 am

How many books show how to solve a linear equation such as 2x+3=7 by subtracting 3 from both sides to get 2x=4 and then dividing by 2 to get x=2, but do not point out that what has been done is to show that if this equation has a solution, then the solution must be 2? That is all that has been shown. To see that x=2 is a solution one goes back to the original equation and replace x by 2 to get 2*2+3=4+3=7, so x=2 is the solution. After a while students have learned more and this is tedious so students should learn that each step in the argument can be reversed so getting x=2 really shows that x=2 is the solution.

Now we skip to trigonometry and the following problem occurs in

quite a few books. Show that (1-cos(x))/sin(x)=sin(x)/(1+cos(x)).

In many instances the authors will cross multiply to get

1-cos^2(x)=sin^2(x) which is the same as 1=sin^2(x) + cos^2(x)

which is true so we are done. The authors then point out that this is not a correct proof since you have assumed what you want to prove to start the proof and that is circular reasoning. This is exactly like that problem of solving 2x+3=7. Every step can be undone. The analogy should be mentioned along with the fact that after one learns enough, it is very tedious to actually undo each step so for a while it would be appropriate to comment that each step can be reversed and that completely the proof, and eventually even that does not need to be said or written just as eventually we do not ask students to show that 5+8=13 by having them count on from 5 or even from 8. The error in saying that one cannot cross multiply to prove an identity is in some books as later as the

seventh edition, and there has not been an uproar from teachers that this is misleading. There was even a relatively recent discussion on this in the AP Calculus discussion on the Math Forum, with a textbook writer claiming this could not be done until someone laid out a careful argument about why this is not true. One sensible teacher had written an earlier comment on this, and I thanked her for what she wrote. She replied that there are teachers who will not accept her argument. You cannot attend to precision if you do not know the mathematics.

Here is another example. When dealing with graphs of linear functions, there are some essential details dealing with slope which are passed over. The first part is that slope is well defined. Somewhat later the question of finding an equation for a line which is parallel to a given line is introduced and the fact that the slopes of these two lines are the same is stated without any argument. One might excuse this since it is ofter hard to prove something which seems so obvious. However, when it comes to perpendicular lines, the criteria for two lines to be perpendicular is not obvious, and there are many books which take the condition of the slopes

being negative reciprocals (or else the lines are parallel to the two different axes) as the definition of perpendicular. That should be a theorem, not a definition. If one does not know the difference between a definition and a theorem, it is impossible to "attend to precision" in circumstances like this.

At a lower grade level, there are books which deal with equivalent fractions by multiplying by "the big one". 1/3 = 1/3 * 1 = 1/3*2/2. The problem here is that equivalent fractions are needed before multiplication of fractions has been done.

### Reply to Mr. Askey

posted by: Deb Ohara on 1/16/2012 1:25 pm

So far, no one has really responded to the question Ruth Parker posed, so I would like to direct attention back to it. I am curious about how much support is out there or how much thought this may have generated.

### Teaching arithmetic

posted by: Richard Askey on 1/16/2012 8:42 pm

students' deep understanding of mathematics, I would object in

two ways. First, the word "deep" is used far too frequently in

current mathematics education. Second, anyone who makes a

claim like this has to explain how understanding is developed.

I gave a reference to a paper by Wu dealing with fractions and negative numbers. Wu has an earlier paper in American Educator which includes information on teaching algorithms. See

http://www.aft.org/pdfs/americaneducator/fall2009/wu.pdf

Here, from "My Pals are Us", the Singapore version from 2002, is

how addition within 1000 is done. First, simple addition with no regrouping. This is followed by simple subtraction within 1000 with no regrouping.Pictures are used to break up the numbers by place value, and the calculations are done from the right, ones, tens, then hundreds. Then addition is done with regrouping in ones, again with pictures and with a vertical alignment of numbers. The warm up example is 347 + 129, and it is done in three steps. In the pictures, the 7 is broken into two parts with one of them combined with the 9 parts and this ten is moved over to the next

group. Numerically below 7 and 9 is placed a 6 and a little 1 is added above and to the left of the 4 in the tens place. Then there is a comment: "Then add the tens". It ends with "Lastly, add the hundreds.I would call this teaching the standard algorithm. After some problems for students to do, there is an example with regrouping in tens, 182 + 93, then some problems, and finally addition with regrouping in tens and ones, 278 + 386. Subtraction is done similarly.

In the older "Primary Mathematics", 5A, first half of grade 5, multiplication and division by two digit numbers is done, and again it is what I would call the standard algorithm.

In TIMSS 2003, the US and Singapore each had one area where the score was significantly better than the scores in the other four areas. For the US it was data, for Singapore it was number. In eighth grade, the US scores in all five areas dropped, in Singapore they went up in all five areas.

I mentioned earlier that Singapore has a short section on mental mathematics in the third grade. I would recommend getting copies of some of these books to see how things are done.

### Messy?

posted by: Eric Hsu on 1/15/2012 12:01 pm

Then you notice the sum is really symmetric around 12 in the sense it's

12^2 +

(12-1)^2 + (12+1)^2 +

(12-2)^2 + (12+2)^2

Combined with the first cancelling trick, these reduce to

12^2 +

12^2 + 1 + 12^2 + 1 +

12^2 + 4 + 12^2 + 4

which is conveniently 5(12^2 + 2). I notice 365 is divisible by 5 and 365/5= 73. Since 12^2+2=146, we see the ratio is 146/73=2.

I consider this a neat solution, possibly not-messy, and pretty straightforward mentally if you've had practice with the canceling trick.

However, given this problem cold, without any practice with the initial simplifying trick and without faith that there is a clean answer, I would probably just sum the squares, as they are all famous values.

### Messy?

posted by: Richard Askey on 1/15/2012 10:56 pm

=2500-4=2496.

The general result as you all know is a^2-b^2=(a-b)(a+b). This is useful in both directions and when it has been partly disguised as

4a^2-9b^2=(2a)^2-(3b)^2=(2a-3b)(2a+3b).

The other two are ones Ruth Parker mentioned (a+b)^2=... and

(a-b)^2=...

One point of this problem is to let students at an appropriate age

see that these can be used. Here it is arithmetic, but later it will

be other topics in mathematics. The related problem with addition replaced by subtraction is (a+b)^2-(a-b)^2=4ab and this is also useful later in mathematics.

In an early posting someone who likes pictures for algebra mentioned having used algebra tiles. Let me suggest that drawings are better than algebra tiles. They are more flexible and do not have

a serious drawback which algebra tiles have of encouraging students to think that multiplication makes things larger. If the small square represents 1, then x^2 is clearly larger than x. Of course one could call the large square 1 and then the small square would be x^2. In the education literature I have never seen a comment on the first concern about algebra tiles strengthening the common student error of thinking that multiplication makes things

larger. My guess is that few if any have tried to use algebra tiles with the large square taken as 1, but if I am wrong I would like to know if my concern about changing what 1 is will further confuse students using algebra tiles is true or not.

If you look at arithmetic books from the late 19th and early 20th century you will see taking square roots motivated by a picture of a square cut into 4 pieces by two lines perpendicular to each other and to the sides of the square with two squares and two congruent rectangles being formed. This was used with numbers to help explain the method, Since it was done before algebra, but with algebra it is just (10a+b)^2=100a^2 + (20a+b)b which some of the older readers will recognize multiplying the initial rough calculation by 20 and then adding and the proposed next term before multiplying by it. The same picture can be used to get the old Babylonian method (now called Newton's method) using

(a+e)^2=a^2 + 2ae + e^2 which is about the same number if e is small since e^2 is much smaller. Then set this to N and solve the linear equation for the new e, which gives a better approximation to square root of N than a is.This was not done in 19th century arithmetic books, but it is easier to explain to prospective primary school teachers. In the old arithmetic books there was also a picture of how to take cube roots by splitting up a cube in a similar way.

In calculus, when one learns how to take the derivative of the product of two functions using the definition of the derivative, it is useful to draw a similar figure with the smaller rectangle having sides f(x) and g(x) and the larger rectangle having sides f(x+h) and g(x+h). We want students to be flexible in their use of mathematics, and pictures are better than algebra tiles for this.

### mental math, a combination of thoughts

posted by: Bob Mastorakis on 1/16/2012 9:04 am

Okay, what's the something. Using some deconstruction mentioned earlier we can see that 10 is (12-2) and 14 is (12+2) so when squaring those amounts the middle terms will cancel which leaves us with two of the (12^2 +2^2) so I now have 2*(12^2+2^2) = 2*(148), so far so good. 12^2 itself is 2*6*12 = 2*(72) so now I have so far 2*148 + 2*72 = 2*(220).

Now I need to deal with finding the "other thing". Again noticing 11 = (12-1) and 13= (12+1) we have (12-1)^2 + (12+1)^2 which again cancels the middle terms leaving us with two of the (12^2 + 1^2).

Now I have 2*(12^2 +1^2) which is 2*(145), adding this to 2(220) gives the result of 2*(365) which is what the question asked.

### Re: messy?

posted by: Eric Hsu on 1/16/2012 11:59 am

Briefly, of course tiles and diagrams can be very powerful as learning tools. There is an inherent issue with tiles, as you say, with representing a range of x values, particularly when x<1, since in most tile sets x is manufactured that way. A computer model where x is an adjustable parameter addresses that (but loses the tactile aspect).

Diagrams are also superior in that one can draw different diagrams for different potential x. However, all area representations have the awkwardness of dealing with negative quantities... It's possible to do, but you add a layer of rules that gets away from the intuitive affordances of area.

I'll have to look into the old arithmetic diagrams you mention... interesting.

I'd like to see every algebra student able to follow and justify the reasoning we've been discussing... but I'm not sure they need fluency and automaticity around it. There are different audiences for algebra -- say, future mathematicians, future scientists, future business workers, and thinking citizens -- and different subgroups will need different fluencies.

### Messy?

posted by: Jim Wilson on 1/16/2012 5:11 am

13^2 - 12^2 = (13 + 12)(13 - 12) = 25

14^2 - 12^2 = (14 + 12)(14 - 12) = (26)(2) = 52

These values need to be added to 5(144) but hold on. Let's see what we need to subtract with the other two terms.

12^2 - 11^2 = (12 + 11)(12 - 11) = 23

12^2 - 10^2 = (12 + 10)(12 - 10) = (22)(2) = 44

If we add 25 and subtract 23 we have 2; if we add 52 and subtract 44 we have 8. The net result is 10.

So our sum in the numerator is 5(144) + 10 = 730, leading to 730/365 = 2. 5(144) is a "messy" calculation but I can do it mentally with 5(100 + 40 + 4) = 500 + 200 + 20 = 720.

Alternatively, consider the sum of the squares of any 5 consecutive counting numbers. Denote them by (x-2)^2 + (x-1)^2 + x^2 + (x+1)^2 + (x+2)^2. Simple algebra leads to the sum = 5x^2 + 10. The mental arithmetic problem could be created with any even number for the value of the middle term. For example:

(2^2 + 3^2 +4^2 + 5^2 + 6^2)/45

(10^2 + 11^2 + 12^2 + 13^2 + 14^2)/365

(18^2 + 19^2 +20^2 + 21^2 + 22^2)/1005

*post updated by the author 1/16/2012*

### Number Talks

posted by: Linda Gojak on 1/16/2012 3:10 pm

After introducing Number Talks to the teachers and spending some time developing some sample NT's, the overall reaction was -=- that's fine, but my kids won't be able to do that. The next time we met, the teachers who had actually tried it were amazed at the strategies students used to solve the examples that were presented. I have had the opportunity to view several number talks in the district and they are amazing.

I could write much more, but if you are unfamiliar with number talks and their use in the classroom, I suggest you examine this strategy for helping students to develop number sense. It is quite powerful and it supports many of the standards for mathematical practice.

LInda Gojak

*post updated by the author 1/16/2012*

### Number Talks

posted by: Ruth Parker on 1/18/2012 1:05 pm

"I teach mathematics at the university level. I want my students to reflect on ideas and relations. I often feel my students pushing back with requests for steps and procedures. I observed 'number talks' in a MEC workshop and was struck by the way number talks create a delightful classroom dynamic. The focus shifts from "right or wrong" to an exchange of ideas about ways to approach the problem, exactly the type of activity I would like to see in my own classroom. It appears to me that each of the following claims is reasonable and worthy of investigation:

"Number talks encourage mathematical discourse: dialog, explanation, and reasoning. The explanations provide informal proofs, creating a habit of mind that foreshadows the more formal proofs students will see in high school and college.

"Number talks provide practice with basic facts and help students develop efficient numerical strategies. They strengthen knowledge of place value and general number sense. Number talks move students from the basic levels of recall and recognition on Bloom's taxonomy toward the higher levels of application and integration of basic number facts.

"Number talks appear to engage ALL the students in the activity. Students seem to feel empowered as the teacher and class accept their contribution as moving the discussion forward. I believe number talks have a positive influence on students' overall perception and attitude toward mathematics."

University Mathematician, Washington State

"It is important to consider that number talks help

1. the teacher to uncover student misconceptions

2. the English Language Learners - narrow the achievement gap."

University Mathematician, Washington State