I hope you have the time to watch.

Thanks!

valerie

THIS JUST IN – Here is a link to an article about the work of NC State’s Patricia Marshall, Jessica DeCuir-Gunby and Allison McCulloch. It addresses the topic of mathematics and multiculturalism in young learners. Take a look if you are interested in learning more about eliminating the opportunity gap!

*Our NC State Mascots are named Mr. and Ms. Wuf and our LIVE wolf mascot is named Tuffy. That’s where the name TUFFTalk came from.

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Let’s start here. This question is from a former student of mine who is now a first year teacher. She is ‘shaking it up’ in her team meetings by asking questions such as this one:

*Hi Dr. Faulkner,*

* A question came up in our meeting today about the “order” of an array. Our workbook that we have for our kiddos explicitly states that the first number in a multiplication equation is the number of rows and the second number is the number of columns. For instance, 3X4 would be 3 rows of 4 columns. Always.*

*I don’t know if there is a specific order to creating a multiplication array or not. I argued on the side of not (but I could be wrong!) The best reason I got from my team to explain why it was rows and then columns was because that’s how it will be when they do area (which made no sense and didn’t satisfy me).*

*Anyway, I was hoping you could shed some light on this problem for me. If I’m right, good for me, if not, I’ll teach it better next year *

*Thanks, TM*

I hear this question a lot and there are several issues involved. My short answer is that I side with TM on this one – it is an artificial idea that an abstract multiplication problem MUST always represent a certain thing. TM’s team does have one decent argument to make, but it is not that this is how we will rigidly address area problems, it instead relates to matrices in algebra. Let’s look at a few things here.

First let’s think about **Practice Standard #2 – Reason Abstractly and Quantitatively ** (* see below for full Practice Standard 2). Creating rigid ‘rules’ ahead of time for students (3×4 always means rows x columns) makes it harder for you and the class to engage around Practice Standard #2. If you have already told me what everything means ahead of time, why would I have to pause and consider what these abstractions mean in my communication to others? In short, I wouldn’t. I just have to memorize that the teacher said this is always rows and this is always columns. The very idea of this practice standard is to understand that numerals by themselves ARE an abstraction. They are abstract. Only a context really brings them to life. As I have heard several people say “numbers are adjectives”. I think that helps. So 3 and 4 are adjectives that describe something. IF you want to convey that 3×4 means, to you, 3 rows x 4 columns, then write that. This encourages you and the class to remember that ‘following your units’ is really important and helpful. You might say something like *“I am in the habit of writing my rows first so I know when I write 3 x 4 that I mean rows x columns – If I know I am communicating to someone else, I would need to have my units in there.” *And something like this as well *“Remember 3 and 4 are just adjectives, descriptors. What are they describing? Well we don’t know that until we contextualize them. That is why units are so important, they help us connect these abstract descriptors to the things they describe. Blue tells you, well, not very much. But a blue cow says a lot! It is the same with numerals…”*

We might also consider whether arrays themselves are merely abstractions and not really very concrete. An array is not exactly the same thing as a real context, right? An array is a mathematical tool to make sense of a multiplication expression, but it is not actually that concrete. What do the rows and columns represent? Rows and Columns for rows and columns’ sake? They are a tool to organize the thinking about what is multiplication. If we want to think of them as more concrete, we need to make them more concrete through a context. A colleague in Asheville, Tima Williams, is currently thinking about the importance of remembering that many of our mathematical tools are not as concrete as they need to be for some students. She sent me this picture to remind us that, for many students the Tool of the Array needs to be made more concrete and that this concrete level needs to be in place for a while for some students. An easy interjection to support the thinking of all students. So this leads us to considering **Practice Standard #5 Use appropriate tools strategically.**** **Can you see that, by determining ahead of time what a given problem represents that we discourage students from considering what it means to use a mathematical tool such as an array (#5)? Rather than thinking* ‘okay so what does my 3 represent in my drawing and how does the array help me make sense of that’ *students will be thinking *‘okay the teacher said this is always the rows – I don’t know why but it just always is’. * Do you see also how this directly affects the student’s sense of purpose and therefore impacts their ability to construct a viable argument for their drawing?

This then impacts **Practice Standard** **#3 Construct Viable Arguments and Critique the Reasoning of Others*** . * As we teach students to construct arguments we must remember that

One of the reasons I love TM’s question is that, to me, she is grappling so clearly with these standards. As a young teacher (or an old one like many of us, for that matter!) it is great to get in the habit of thinking about the math you are teaching in ways that line up with the practice standards. TM is figuring out what it means to use appropriate tools strategically (in this case, arrays) and she is grappling with how to construct a viable argument to bring to her team*.

**Area and Matrices and Practice Standard # 4 Model with Mathematics –** Now, what about the team’s argument? Is there a viable argument for their position? I don’t buy their argument about ‘this is what we will do with area so get used to it now’. Why does area need to be represented in this way always? For some reason we have gotten caught up with some idea that length and width are different things. They really are both lengths. If you want to call the longer length a width, you can certainly do that–but that is a little bit of a non-mathematical artifice. Perhaps we can just say it is a daily usage that doesn’t serve us well in the world of mathematics. In an area representation both 3 and 4 are lengths. We can see just by looking at it which side is longer, we don’t need to name that something different. Where the 3 and 4 go in a certain context determines the orientation of the rectangle. Here we are going to use our mathematical expression to model a patio we want to build (Practice Standard #4). With respect to your front door you could have a 3×4 patio or a 4×3 patio. Maybe one fits better and avoids a drain, maybe one looks better with the flow of the front yard. From there students could be asked to demonstrate their ideas for a patio and then explain which mathematical expression they want to use to describe it and why. Oh, and by the way, here I am talking in meters or yards, not in feet.

Early investigations with PS #4 (Modeling Math) can be fairly straightforward: *In early grades, this might be as simple as writing an addition equation to describe a situation (PS #4).* In our patio example we used a simple multiplication expression to model our patio. If we make a ‘rule’ about these situations ahead of time (‘the ‘length’ always comes first and the ‘width’ second’) we discourage students from connecting meaning to their context and the mathematical expression they want to use to model that context.

For the record, I happened to be at a meeting at NCDPI and was able to consult with both the High School and Elementary Math representatives there on this question. They both agree with the arguments I made above. They also agree that it is artificial to say that this is how we should always address area. They did, however, mention that there is one decent argument for this position. That is that when we get to Algebra there is a formal understanding that the first numeral/adjective/factor in a multiplication expression does represent the row in a matrix and the second numeral/adjective/factor represents the column. But this is still not reason enough to make this a ‘rule’ in an elementary class. Let’s refer to Practice Standard #4 again. In the explanation for this standard it says *“Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.” *The key here is that they are not to be generalized or made formal until later grades. When you formalize ideas into rules too early, you detract from students’ abilities to have discussions and describe their thinking. You also injure their ability to think about the difference between an abstract expression (3 x4), a tool to represent it’s meaning (array – rows and columns), and the actual thing you might want it to represent (a patio).

We want to have these conversations with young students and commanding a rule-from-on-high hurts your efforts to have good math talk. Saying to students *“when you are in high school you will see that mathematicians have an understood habit of having the first product represent the rows, but here we are focusing on whether you can describe your model and connect it to the numerals that represent it” *is very different than saying *“the first factor is always the row and the second factor is always the column”.*

Feel free to add comments and thoughts to what I hope is a viable argument that we should NOT be in the habit of presenting 3 x 4 as an Always Rows Then Columns but instead present it as an opportunity to work on **Practice Standards 2, 3, 4 & 5**.

*Postscript – Just like TM, I am grappling with the standards as I write this as well. I sent this blog post to some colleagues and they suggested that I make a clearer distinction between practice standards 4 (models) and 5 (tools).

This is what my colleague Temple Walkowiak suggested I consider regarding my interpretation of Standard 4 (models) versus 5 (tools) (which I did and made changes to my blog above : ).

Temple argues that TMs question is more about Standard 5 than about 4:

*Here’s why….as defined by the CCSS-M, modeling WITH mathematics is when we use mathematics to model real-world situations. As written in your blog post, it sounds like they were grappling with just arrays absent of any real-world context. So, I might say, “TM is figuring out what it means to use appropriate tools strategically (in this case, arrays) in her classroom and she is grappling with how to construct a viable argument to bring to her team.” *

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to *decontextualize*—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to *contextualize*, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

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This week Tacey Miller ’15 and Maryam Khan ’13 both made the news. Pretty cool stuff.

**Here Tacey does what so many of us have been trying to do **– articulate how bizarre it is that the Teaching Fellows program was cut in NC while the TFA program was maintained. I could not be more proud of her. Kudos to the author of the story as well.

Maryam is also amazing. She is one of the happiest, most generous people I have ever met. She is humble and dynamic both at the same time.** Check out the clap mob she received.**

And thanks to every one of the hard-working pre-service and in-service teachers I have been fortunate enough to teach. Keep up the great work!

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So, it is sometimes really hard for me to talk frankly about the unfortunate working conditions that have developed for teachers in the United States. I don’t want it to be true. I want to know that I am sending these amazing young practitioners out in to a world that embraces their passion and supports them in becoming great. That inducts them into the workplace by providing them leadership and time. That includes the space for honing their craft. At many schools, principals are working hard to create that space for teachers, but again and again those efforts are limited by the budgets and the expectations that teachers are to be with students throughout their entire working day, as if planning and preparing for lessons was not an actual part of their work. Again and again teachers are identified as the problem and not the solution. And somehow, most of them stay. Most of them continue to do their great work.

Countries that want to make a difference in Educational outcomes make two basic changes: They address economic inequities and they entrust teachers to do the work of teaching. In the United States we have ignored the underlying issue of economic inequity and we have created a culture that distrusts the work of the classroom teacher.

This distrust has led to the belief that we need to ‘find the great teachers’ and ‘remove the deadwood’. This approach is , I daresay, ignorant. The teachers we have in the United States are certainly good enough to create school systems that are great. But getting rid of bad teachers can’t be bad, right? Wrong. Putting increasing energy and resources into hyper-evaluating teacher performance creates a dark and depressing work environment filled with layers of policy that work against creating better classrooms for students. The work environment becomes increasingly about evaluating and less about supporting teachers. So our teachers, as a whole, have less and less opportunity to develop their craft. Trust me when I tell you that teachers, more than any group of people I have ever met, WANT to improve. They want to engage in staff development, they want to be the best they can be for their students. But the system is not built for it. There is no time in the day for it. The somewhat inelegant way I discuss this with my graduate students is this: It doesn’t matter what meat you put into a sausage maker, in the end it will come out as sausage. We are worrying about the meat (teachers) when we should be looking at the sausage maker (the systems and assumptions we have in place about schooling).

The system teachers enter is so rife with economic inequity and misguided policy that student outcomes are weakened yet the teachers are blamed for the bad taste. Teachers have no control over child poverty in the United States, and yet they take the blame for the ravages of poverty. Teachers have no control over the measurement-happy policies that destroy the potential for rich academic exploration, yet they take the blame when their intended student-outcomes are disrupted by these top-down multiple choice tests.

Do you want to make schools in the US the best in the world? Do two things: Reduce economic inequity and give teachers more time to plan. That’s it. Okay, so you don’t want to reduce economic inequity? That’s too big of a political leap? Let’s say we take a temporary pass on that. Then just give teachers more time to plan. More time to plan will create better working conditions. Better working conditions will increase teacher retention. Increased teacher retention will keep us from losing talent to other occupations. Maintaining our talent level will increase the impact of the lesson study teachers engage in with the time they are given. This collegial interaction will boost the effectiveness of teachers and the academic outcomes of students.

Delicious.

**5 Myths about our schools that fall apart when you look closer, The American Federation of Teachers.**

This video from the AFT is the best I have seen on this topic. In only 5 minutes it covers an amazing amount of territory.

Enjoy!

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*The Digit Song aka The Numberbet Song *

*(sung to the tune of Twinkle, Twinkle Little Star; * *Lyrics Valerie Faulkner © 2014)* * *

*C F C*

0, 1, and 2 and 3

*F C G C*

These are digits don’t you see

*C F C G*

4, 5, 6, 7, 8 and 9,

*C F C G*

These are digits oh so fine

*C F C*

Digits, Digits, numbers not

*F C G C*

‘til you tell them where’s their spot.

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Link to Article on Math Language: Why the common core changes math instruction.

vnf

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Remember when you were hating algebra and you had to deal with things like 3x – 36 = 0 ? You immediately dug your nose into your book and said to yourself, ‘okay, I’ll add 36 to both sides because Mr. Brook told me to (and wow, his polyester shirt is really sticking to his back today, I wonder if he can feel that?)’. But you could have just said to yourself, ‘well I know that 3x must be 36 because that’s the only way that side of the equation could equal 0 (and that must itch)’ (Note: remember that you get 0 by taking away all of whatever you have – I know you know this because you have played poker before and lost your entire pile of chips). Again, I know you maybe don’t care now, but if you know a kid who is suffering through this stuff get him or her to think about the fact that the equals sign matters and that all it means is that two things have the same value, not that they are ‘the same thing’. Start young. The example I use in work shops is two trucks being equal in weight to an elephant. I have no idea, actually, what trucks and elephants weigh, but let’s just pretend. If I know that those two things are equal in weight, does that mean they are the same thing? Really? Do you see what I mean? Won’t that play with a third grader’s mind?

So, instead of saying yes! You are so wonderful, 3 + 4 is the same as 7. Say instead, well no, 3+4 is not exactly the same thing as 7, 3 + 4 is decomposed into two parts (trust me they can handle the word decomposed and anything else accurate that you throw at them) and the 7 is in simplified form. But they do have the same *value*, and we call that equal. Quick – would you rather have 3 quarters and 4 dimes for a total of 7 coins or 4 quarters and 3 dimes for a total of 7 coins? Oh, yeah, that’s right, it doesn’t matter, they’re just the same thing…

Also, kids in the United States think an equals sign means ‘put your answer here’. So if they are given the problem 9 + 4 = ____ + 7, they confidently put in the blank spot, ’13’ (if you don’t believe me, try it on three elementary school kids and see what happens – if they get it right, write a thank you note to their teacher, or go to a beer festival). This happens because we only ever show kids equations like this 3 + 4 = 7 and never like this 7 = 3 + 4. Whoops. Turns out they were paying attention to us and we blew it by just doing with them what adults did with us. But, we can’t do that anymore. We have to think more about how we talk about math with kids. And we have to get them to see that when two things are equal it is an important piece of information. But we can write numbers and think about the same values in many different forms. And that’s when math gets interesting–deciding what form of a value you want. Is it the same thing to say that Susie O made 365/489 foul shots as to say that she shot them at 75%. They are considered equal, but the two different forms tell you different things because, well dammit, they are not the same. The first one tells you she got fouled a lot and is strong from the free throw line, the second form just helps you to see better that she makes 75 out of every 100 shots she takes (or 3 out of 4 if you want to think of another form altogether). It doesn’t actually mean she took that many shots, and in fact, someone could shoot 75% and literally only have taken 4 shots and made 3. oh. that’s weird. I thought they were the same thing because they were equal? math is stupid.

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To me, performance and sense of aptitude is driven by the things we know from expert performance research,(Ericsson, 2000) which is more or less: Effective Time in = Increased Ability out.

What is odd, is that, at the same time, students in the United States actually like math better than their peers in higher performing countries. In fact, there is a clear inverse relationship: Countries where students like math more function below average, Countries where students like math less excel at math.

Maybe math is about persistence, critique and justification, and hard work with effort. I believe that we can perhaps make math less obviously ‘fun’ and yet more effective in the long run. If we believe that it is a certain type of fun to learn, maybe that is okay.

U.S. academic performance compared to other countries. The U.S. scores comparatively lower on PISA measures than on TIMSS measures. PISA video PISA 2009 Scores TIMSS 2011 4th Grade TIMSS 2011 8th Grade

Note: Some argue that the scores underestimate U.S. scores. When I read these arguments they seem to bury the real questions in number. The real question is ‘are we performing to our potential as a country and are we using our human resources well?’ The answer to that seems to be ‘no’. We are not leveraging our human resources well. Here is what the argument tends to look like when arguing ‘against’ paying attention to these scores.

For instance from Carnoy and Rothstein, 2013 (Economic Policy Institute, online)

These facts are laid out in separate spots:

**Because in every country, students at the bottom of the social class distribution perform worse than students higher in that distribution, U.S. average performance appears to be relatively low partly because we have so many more test takers from the bottom of the social class distribution. (underline added)*

**In math, disadvantaged and lower-middle-class U.S. students perform about the same as comparable students in similar post-industrial countries. (Carnoy and Rothstein, 2013)*

If we disentangle economics from education then the conclusion is – We have more poor kids so we should not be held accountable for these lower scores. If we look at the data and just compare poor kids to poor kids and wealthier kids to wealthier kids, we are doing better than reported. – This appears to be, or at least is my interpretation of, what Carnoy and Rothstein argue.

If we DON’T disentangle economics from education and we consider that making sure kids are not in poverty is a PART of our responsibility as a society when we consider the education of our students, then those two facts are re-interpreted in a different way. Now they say – Wow, we have more poor kids and it is bringing our overall scores down. It indicates that we are not utilizing our human resources well. We need to reduce the amount of poor kids and this includes ensuring that more students have access to better education and thus perform better. Maybe social issues and education can’t be disconnected if we want to improve our human capital.- This is how I read these facts.

For more on the second way to interpret these facts, see Linda Darling-Hammond, The Flat World and Education. She seems to interpret these facts in this way as well.

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As part of this site, I will be blogging about once a month and my topics will likely be of two sorts. Type one will be making completely geeky math stuff accessible to everyone. Here I hope to leave the reader with more answers than questions. Type two will be pondering issues of policy and culture in relation to mathematics. With these blogs I hope to leave the reader with more questions than answers.

I am extremely excited to have this site up and running. Please take a look around. I have already posted a lot of information, but am working with many friends and colleagues to get more videos, links, and information available to you all!

For now, take a look around the site and start giving me suggestions on what you want to see. Ask questions, provide feedback and let me know what you do and don’t understand and what you need.

Thank you!

valerie

*For those of you who did not grow up in the 60’s-70’s there is a very powerful Brady Bunch episode where Cindy Brady is totally prepared for a game show she is on and then, when the red light goes on to get started, she goes blank. I am literally thinking about that as I get started with this talk. I knew that cute little Cindy was bad news…

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