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

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MSPnet Academy Discussion
January 11 - January 25, 2012

Ruth Parker, CEO of the Mathematics Education Collaborative (MEC)

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.

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Numerical reasoning

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

Thank you again for attending the webinar. I hope that you'll share your thoughts about the ideas presented. And I'm hoping that together we can think about the questions posed at the end: What would make this an even more compelling argument? Are there ways that we could make this case to a broad national audience? Can we influence the next iteration of the Common Core State Standards? If so, how? How can we make this topic an important part of our national research agenda? Does the practice of 'Number Talks' get us closer to accomplishing the Standards for Mathematical Practice of the Common Core State Standards? Again, I look forward to hearing your thoughts!

post updated by the author 1/11/2012

Broadening the national dialog

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

Ruth --

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

I am a secondary math teacher, and when I taught Algebra 15 years ago I used base ten blocks and Algebra tiles to help my high school students gain better number sense. Even then, I felt that their memorization of algorithms without understanding was impeding their understanding of Algebra, so I was pleased that you brought up the idea of influencing the next iteration of the Common Core State Standards to move away from teaching traditional algorithms. I am now the Director of the Alabama Math, Science, & Technology Initiative (AMSTI) at the University of Montevallo AMSTI site. There are 11 such sites in Alabama and statewide we currently work with approximately 50% of the schools in Alabama.
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

Hy Bass had a paper in the NCTM early grade magazine in which he did a problem like 72 - 26 as one of the ways Ruth Parker did, 70-20=50 and 2-6=-4 so 72-26=50-4=46. I asked him if he thought this should be taught, since nothing was said about that in the article. He said when his students ask him, he says it should not be taught. The reason should be clear, subtraction should be solid before the arithmetic of negative numbers is taught. I said he should have mentioned this in the article. Let me suggest that Ruth Parker add this to her presentation.

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

I'm willing to take a stab at the Russian problem. First, I notice the following two relationships:

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

Another way is to note that each term in the sum can be written as (10 + x)^2 = 100 + 20x + x^2. , where x runs from 0 to 4.

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

The method above is not what I had in mind. Let me give a hint.
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

i believe that the statement "The method above is not what I had in mind" may inadvertantly be a problem [as a response in a typical teaching/learning situation]
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

a geometric representation of the addends seems to reveal a visual means of 'counting'...with relative ease
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

I can do it without paper ( but write it so you can see why)
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

Ben Sayler asked why I put an emphasis on some of the content standards as a first priority over the process standards. I can illustrate with many examples from current books, both school textbooks and college texts for courses for prospective teachers.

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

I am replying to Mr. Askey's statements about something I wrote previously concerning Algebra tiles. Because I was trying to make another point, I did not bother to explain in detail all the ways I used visual representations with Algebra tiles, with drawings, and with many other manipulatives. It is true there are things that can be misleading in using them. But I'm afraid my real point was missed. I was agreeing to Ruth Parker's proposal that teaching traditional algorithms can actually impede students' deep understanding of mathematics and that students may not develop the number sense that they need to be successful in upper level math courses. I am now working with K-12 teachers and administrators and our experiences support that.

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

Anything that is taught poorly will lead to problems. When one claims that teaching traditional algorithms for arithmetic can actually impede
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

Nice question. Here is one stab at a not-messy solution. Recall that in general (a+b)^2 and (a-b)^2 are the same except for opposite sign "middle" term. Thus (a+b)^2 + (a-b)^2 = 2a^2 + 2b^2 since the middle terms cancel.

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

This is the solution I had in mind. There are three very important results about quadratic expressions which students should know backwards and forwards and in disguised forms. One is often used in connection with mental arithmetic: 52*48=(50+2)(50-2)=50^2-2^2
=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

So when I first saw this problem, I saw trying to show that 10^2 +11^2 +12^2 + 13^2 +14^2 is an even number which = 2*365. Great, this seems pretty natural since an even times an even = an even and an even plus an even = an even then 10^2 + 12^2 + 14^2 = 2*something. I also know that and odd times an odd = an odd and an odd plus an odd = an even, so the 11^2 plus the 13^2 = an even = 2*(another thing).
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

Too many interesting threads in one post to reply to...

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

I have a similar approach. 12^2 is the middle term. So the sum of those 5 terms in the denominator must be 5(144) plus something for the 13^2 and 14^2 and minus something for the 11^2 and the 10^2. Now using a^2 - b^2 = (a + b)(a - b) we have
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

I have been working with number talks in a district where for the past 20 years students have struggled with mathematics and the district's test scores have been notoriously low. Helping students to develop number sense K-12 has been a big issue -- identified both within the district and by the state. So, given the opportunity to do some in-depth work with teachers I have come to the realization that sometimes we expect them to change everything at once and in the K-5 arena (maybe K-8) some of the teachers lack the depth of mathematical understanding themselves which leads to rote instruction (show and tell) and lack of success. What I love about number talks is that it doesn't require huge amounts of change on the part of the teacher. They are also learning as they see their students developing mathematical thinking and sense making.

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 want to share some reflections on 'number talks' that have been sent to me. These first two are from two mathematicians at a university here in Washington State.

"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

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