Math Test Answers

Last week I posed 12 questions related to teaching elementary school math. The test was specifically designed for those who believe a traditional, "back to basics" approach to math is superior to the type of math we are trying to teach through the Common Core in 2015.

So here are the test questions and answers with the  first answer, the traditional math answer (TMA) and second answer, the 2015 Common Core (CCSSM) answer.


1. What is counting?
TMA - "one, two, three four five etc" with no sense of what the numbers actually mean.
CCSSM -  "one, two, three, four, five etc" understanding that three is one more than two and four is one more than three and that when you count five objects the word "five" refers to all five objects not just the last one counted. This is the idea of cardinality and is key to understanding number,

2. What is addition?
       TMA:  34
                + 48
                   82
       CCSSM The above is only an algorithm, a piece of arithmetic. Knowing just this is useless in
       problem solving. To problem solve you need to know the concepts of joining, separating,
       comparing and part-part-whole if you want to use addition or subtraction in anything useful. .
3. What is multiplication?
       TMA  112
                x    5
       CCSSM Just like addition and subtraction the above is just an algorithm, a piece of procedural   
       knowledge. It doesn't help you decide which operation to use in problem solving. You'll need to
      know the repeated addition, multiplicative comparison concepts as well as a few more to be know
      whether to divide or multiply
4. What is division?
       TMA:  5/115
       CCSSM See above
5. What is Pi?
        TMA: 3.14 etc
        CCSSM Yes, but this doesn't help a whole lot. Pi is a ratio between the diameter and   
        circumference of a circle. The circumference is always just over 3 times the diameter; or, if the
        diameter is 7 the circumference is 22.
6. What does the 0 mean in 308?
       TMA: A place holder
        CSSMIt means there are 0 tens. Place holder is a meaningless term than has no conceptual  
         value. Children need to learn initially that each place in place value has a value.
7. What is 1/2 divided by 1/4?
        TMA:   "Change the sign and invert the second fraction"
        CCSSM  This is virtual nonsense and doesn't help at all. Knowing that the problem is asking   
        how many 1/4s there are in 1/2 is of far more value especially in a world of calculators.
8. Make up a word problem for # 7 above.
        TMA: I can't
        CCSSM: How many 1/4s in the first half of a football game 
9. What do  you get when you reduce 4/8?
        TMA: 1/2  Unfortunately, most children being taught to "reduce" a fraction  think that 1/2
         is  smaller than 4/8 because the word "reduce" means to make smaller and the two numbers.1
         and 2 are  smaller than 4 and 8. I joke about reducing 4/8 to 4/8    
        CCSSM; You don't use the word  "reduce" fractions because children think 1/2 is smaller than
         4/8. You  rename or regroup 4/8.
10. Why do you  put a comma after every three digits in a large number?
         TMA: Because that's what you do
          CCSSM Because it marks the repetition of  100, 10, 1 in each place value referent 
11. What is area?
          TMA: length times width
          CCSSM this only allows you to measure the area of regular shaped objects. Area is the two
         dimensional concept of coverage. You can find the area of your hand by tracing it onto squared
         paper and counting the squares.
12. What does a degree measure in geometry?
         TMA: It measures angles
         CCSSM It measures the rotation of an angle about a point. This is the only way you can explain
         why an angle is 45 degrees 4 inches from its origin and still 45 degrees 4 feet from it;s origin.

Children must understand what they learn at the developmentally appropriate level; this is why the common core is so good.

Maths; The Way It Should Be

Last night marked the last class of probably the best group of graduate students I have ever had the pleasure of working with. The course, Math and Diversity, focused on teaching maths to children with diverse needs including ELL, Special education, poverty, maths disabilities such as dyscalculia, and working with mathematically talented students. The course focused on developing the students' relational understanding of math as well as a growth mindset. Here's a sample journal from Bria, one of the students in the class that is pretty typical of how all the students in the course saw their growth as math teachers this semester.

 "I am so glad that I took this class. My mathematics confidence has skyrocketed as a result of this semester. I used to be told that I was good at math, but I didn’t believe it because I was grouped with people who were notably good at math and did not compare to them. I was one of the weaker links in Mathletes, for example, and I felt that I struggled in my college calculus class. I know that I will never be as strong in math as many others, at least as long as it is not my academic or professional focus. But what I am learning is that I am better at mental calculations and quick consolidation of numerical information than many other people I come across in my everyday life. I was able to comprehend and work with new concepts presented in this class fairly quickly, and enjoyed being able to experiment with my new knowledge. By gaining conceptual understandings of things I had learned procedurally growing up, I am able to approach new math problems that come to me in life more thoughtfully, and I understand those thought processes. My new hobby is doing mental calculations of various operations and then analyzing exactly how I came to my answer. Feeling in control of math is a fairly new feeling, and I love it. It makes me more confident in my mathematics skills, and makes me more comfortable when I find myself out of control of math.

In the past I have not been able to be comfortable with accepting that something was challenging for me; I would admit defeat at the first sign of struggle. I understand that this is a common ailment of the person who has spent their childhood floating through academic requirements and being told that they are smart. But through the education I am receiving in the graduate program at Saint Michael’s, I am finally learning to practice what I preach. I am learning that finding something challenging does not put a blemish on my intelligence, and asking for help does not signify weakness or make the person asked think I am less intelligent than they are. I believe these things wholeheartedly when they come out of my mouth as a teacher, but I continue to struggle with it personally because I always prided myself on being “smart” growing up and worried that people would find me less so if I asked for help. But during the second half of this course, when a concept or problem presented in class was challenging I began to actually feel alright about admitting it and getting help from a neighbor, rather than chastising myself for not understanding something as quickly as my classmates. I think that the confidence I built from taking this class has allowed me to get over any math-related anxiety I used to feel, such that I now know I am “good at math.” I understand that people were not lying when they told me this in the past. With this newfound knowledge, I am comfortable with struggling. Finding math concepts that are difficult for me are not something to avoid, but something to tackle head-on. It’s fun now, and I have this class to thank for it."

Another student in the course felt that, as a child,  she was "a conceptual person trapped in a procedural world". I will miss teaching this course. 

Maths and gender

So finally there is definitive research that shows the human brain of both the male and female of the species is the same. It comprises "a mosaic" of all kinds of characteristics regardless of gender. There is, therefore,  no biological  reason why boys should be better at math than girls.

This means that it is the context in which maths is learned that creates the inequalities, as Jo Boaler  so eloquently points out. It means that we, as teachers, as parents, as a culture need to adjust our expectations of what girls can do mathematically.We need to make sure that girls do not receive subtle or not-so-subtle messages that they cannot do math  and that the types of activities that occur in math class are not gender stereotyped in any way. 

A Maths Test

One of the wonders of the world is why math reform in the US has never seriously happened. It's as if traditional maths is part of the original US Constitution that cannot be changed. Recent commentaries such as this one in the NYTimes  and this one also in the NYTimes  begin to shed light on just what it is that causes this state of affairs. In spite of the overwhelming research evidence (e.g. Jo Boaler) that helping children understand math and not just memorize it is by far the best approach there are constant cries for "back to basics" and the traditional approach of memorizing facts and formulas.
So, for everyone who thinks this return to basics is the way to go here's a test to see how well you know the math that children  are expected to know in 2015.

1. What is counting?
2. What is addition?
3. What is multiplication?
4. What is division?
5. What is Pi?
6. What does the 0 mean in 308?
7. What is 1/2 divided by 1/4?
8. Make up a word problem for # 7 above.
9. What do  you get when you reduce 4/8?
10. Why do you  put a comma after every three digits in a large number?
11. What is area?
12. What is a degree measure in geometry?

Answers next week.     

On-Line Schools Disaster

I've always thought the idea of on-line schools or virtual education was one technological step too far. Now a new study  by the University of Washington, Stanford University and the Mathematica policy research group shows that students enrolled in on-line schools fair poorly when compared with those in traditional schools and classrooms. In the report by the BBC the problems are identified as many and significant and in some cases students have been found to lag as much as a year behind the counterparts in traditional schools. The report also identifies the "digital glitz" associated with the use of the internet and other technological forms of so-called learning.

On another on-line related but completely different topic, I discovered this past week that I have spent a third of the semester so far glued to Canvas. To be more precise, I have spent 160 hours loading my coursework for my three courses onto the Canvas Digital Learning Management System. This is in addition to the hours I spend each week in my classes face to face with my students. I wonder how much better off my students are because of this? I wonder how much better teachers they will become? I wonder what other criteria we should measure the success and value of  an on-line learning management system such as Canvas?

Just because we can, should we?  

Remembering; the Lost R in Education

So, in this world of instant access to virtually anything what should we require our students to remember? What sorts of things do we need to have at our fingertips? What things do we need instant access to that will enrich our lives? What is it that's just plain fun recalling instantly from memory. My son, Andrew, who has Down Syndrome, can recall almost everything that's happened in his life that he has enjoyed in vivid detail. He can remember what he was wearing, who he was with, what the weather was like and what snack he had when he went to the movies to watch a Harry Potter movie ten years ago.

But what about the factual information that makes up our adult lives? Is it important to distinguish a noun from a verb, to know who the Secretary of State is, to know the capital of Holland, or even where Holland is, to know how many ounces there are in a pound and how many pints in a gallon? Or is it important not to clutter our minds with minutia so that we have room for the important stuff like using computers and navigating the endless features of our cell phones?

I know which side I would come down on but I'm from an older generation that simply uses a cell phone to make phone calls and take the occasional picture. But my memory is full of the most incredible things apart from the number  of pints in a gallon, pounds in a ton and meters in a kilometer. I get an incredible sense of joy when I recognize a drumlin, an  erratic or a raised beach when I'm driving around the countryside. It's a genuine thrill when I can recall the name Copland in response to a crossword clue "composer Aaron". The fact that I can remember long-ago-learned facts when I can't remember what I did yesterday is somehow rewarding, reassuring and satisfying.

But what of today's younger generation who do  not seem to be committing things to long-term memory with the same gusto that we did in our youth. What will they have to recall in their old age? 

Making Meaning in Maths with Manipulatives

Ever since I was an undergraduate student learning how to be an elementary school teacher I have always believed passionately in the use of manipulatives to introduce children to new concepts, ideas and skills in mathematics.  It has always seemed the most logical way of starting children on their journey of mathematization, as Bob Wright of Math Recovery would say.

Well for the first time in my professional life someone asked me about the research basis for using manipulatives for teaching math. In the math ed. texts i use in my courses the use of manipulatives is advocated and illustrated with the introduction of every new idea. It's as if it's a no-brainer, something that is as natural as the sun rising each morning or snow falling at some point in a Vermont winter. So I took to Google this morning and tracked down an array of really interesting research-based articles in support of the use of manipulatives for teaching math at the elementary school level. It felt a bit like conducting research to see if more people used raincoats and umbrellas when it was raining than when the sun was shining but the results were startlingly interesting. Here they are;

Here's an NCSM article. NCSM is the national math leadership council and reminds us that we must also include the use of virtual manipulatives. This one is from the Journal of Instructional Pedagogies and gives an overview of the history of using manipulatives as well as the current research.

There are many publishing companies that also produce manipulative materials so it is probably only natural that they should also produce research to support the use of their products. This is a particularly good article from the folks at ETA.

And here's a student research paper on the topic from Marygrove College. Finally, here's a neat article from Sage Publishing that mentions Montessori education which is really where the use of manipulatives in teaching math all began.


 

Thou Shall Not Reduce Fractions

With the increased use of the word "reduce" in the sustainability and recycling movements there is even more reason now not to use the term in reference to fractions.

If we use the word "reduce" to change 4/8 to 1/2 we are giving students two clues to make them think that 1/2 is smaller than 4/8. The word "reduce" clearly means to make smaller while the two numbers in the 1/2 fraction are clearly smaller than the two numbers in the 4/8 fraction. We are, in fact, giving chidlren two clues to make them think 1/2 is smaller than 4/8. To students who have good fraction sense this is probably not a major issue but to the many students who struggle to learn fractions this metaphorical use of the word "reduce" can cause untold misconceptions and misunderstandings.

I have been recommending the use of the term rename or regroup to describe changing a fraction from one form to another. In my math and Diversity grad course last night one of the students suggested using the term "synonym". At first, this seemed quite inappropriate since synonyms do not have exactly the same meaning while 1/2 is exactly the same as 2/4 or 3/6. Then we started to think about about the use of the term and it began to seem more plausible The discussion went something like this.

While two  synonyms may  not exactly be the same it is really the context that requires the use of one over the other. For example, if you walked into a small grocery store and asked for a dozen big eggs you would get a funny look, perhaps, from the store keeper. The usual word to use when referring to eggs is large, a synonym of big. Now apply the same reasoning to fractions. 3/6 is a synonym for 4/8. Even though they are exactly the same size, given that they both refer to the same whole, each one is more appropriate in certain contexts. If you were adding 1/2 + 1/6 it would be more appropriate to rename the 1/2 as 3/6, or use 3/6 as a synonym for 1/2. On the other hand if you  were subtracting 1/8 from 1/2 it would be much more appropriate to use 4/8 as a synonym for the half in just the same way that it is more appropriate to use the word "large" when referring to eggs rather than the synonym "big". 

Math and Poverty

I learn so much from reading my graduate students'
reflections about the readings I assign each week. This is especially true when they relate their own experiences to what we are discussing in class. The thirteen students in my Math and Diversity grad class have such diverse backgrounds that they bring fresh and diverse perspectives to almost everything they read.

One such insight this week really made me stop and think about how we work with children from financially challenged families when it comes to learning math. The student shared an experience where families could sign up their children for a variety of different experiences that were being implemented beyond the normal hours of school. There were series of activities involving art, or science or math, in fact all the disciplines commonly found in the elementary school curriculum.

The observation of particular interest that the graduate student made is that  no children from financially challenged homes were signed up for the math experience. Each different experience had roughly the same number of students signed up but all the students signed up for the math experience came from financially stable or affluent families. The questions arising from this observation are interesting to say the least. Why do financially challenged families not see extra math experiences as beneficial for their children? Why would they rather sign them up for an art or a music experience?

 These are questions I will ask my students in class tonight and ponder for the next few weeks. What do you think?

  

Growth Mindset Maths is a Must

Well, I don"t care what Alfie Kohn says, and I do usually agree with him, but I think  Mindset Theory is the best thing I've added to my teaching repertoire since I discovered John Dewey's ideas of Inspired Vision and Executive Means back in 1969.

I've now introduced it in both my undergraduate and graduate math courses and the results have been great. This is especially true in my undergraduate math class where we've been exploring teaching everyone's seemingly least favorite maths topic, fractions. For some reason, my students nearly always seem to enter this topic with very little relational understanding of fraction concepts or fraction sense. It's as if they've slogged through endless hours of learning nothing but how to add, subtract, multiply, and maybe divide, using archaic, instrumental strategies such as "you can't add apples and oranges", or " invert and multiply"or "cross multiply" to name but a few.

They seem to have one revelation after another when they realize the power of the ONE or referent when when working with fractions. The idea that you can count like fractions the same as you can count anything else and the remarkable patterns fractions make like these two 1/2 2/3 3/4 4/5 5/6 6/7 7/8 8/9 9/10 and 2/1 3/2 4/3 5/4 6/5 7/6 8/7 9/8 10/9. Each forms a pattern approach ONE but never getting there.

Frequently, during class-time, we refer back to the Mindset class we had near the beginning of the course and they all remember Carol Dweck's maxim of "yet".  This idea seems to work well with the Learning Communities in the class where each member of the community bears a responsibility for making sure that every one in their group of 4 or 5 students is developing an understanding of the topic, fraction concepts and skills in this case,

The more I try to develop my Mindset language the more I see the students responding in a positive way. I feel like I am even more "on their side" so to speak than I thought I was before. My job is clearly to help them all succeed in developing the relational understanding of maths  required of being an elementary school teacher. 

How Maths Controls our Lives

Katy, one of the graduate students in my Math and Diversity class has a wealth of experience living, working and studying in a variety of different countries. This gives her a unique perspective on learning maths as she is well aware of how it differs in different countries no only in how it's taught but in the very nature of the maths itself.

Yesterday she shared, with great excitement, some of the remarkable things she was discovering about maths as a result of reading the book Here's Looking at Euclid by Alex Bellos. One story that I only half knew about was how some indigenous folks living in the Amazon rain forest have a very unique outlook on things mathematical.

Their number system for example consists of one and two, which are numerically form followed by threeish, fourish, and fivish, terms that I will need to read the book to fully grasp.

Katy also described how there are no standard units of measurement used to measure anything. For example, time is not divided up into units such as seconds, minutes and hours. There are no length or distance measurements such as feet or meters or miles or kilometers. This made me start wondering about how dependent our lives are on every conceivable unit of standardized measurement. There is almost nothing in our lives that cannot be measured with some standardized unit s is illustrated in this Dictionary of Units of Measurement.

I am sure that the indigenous people of the Amazon have units of measurement for some things that are part of their culture and that they all know but they clearly have no need of standard measures with which to communicate with the rest of the world.

So in my class yesterday I asked my undergraduate students to imagine living without standard units of measurement. I posed the question "Suppose you came to class when you felt like it and left when you felt like it or if there was another way of determining when class started and ended" . We didn't have a whole lot of time to get into a deep discussion and I hadn't really had time to think through where I wanted to go with it but the whole idea does make you start to explore how we live. How mathematically our lives are defined by how we used standardized measures.

Perhaps retirement is life without  the control of standardized measures?

Third Grade Math Problem

Once again the internet is abuzz with hysteria because of a third grade problem that was marked incorrect when, according to many, it was not.

The problem involved using  the repeated addition strategy to solve 5 x 3. When the student wrote 5 + 5 + 5 every one cried foul saying the student's answer was correct. The answer is clearly incorrect because it should be, as shown, 3+3+3+3+3+3.

Understanding the repeated addition concepts is one of the most important ideas in multiplication and division especially in problem solving. There are so many different skills wrapped up in this seemingly simple idea that we, as parents, owe it to our children to take the time to grapple with this issue and understand it so we can help our children and not dismiss  it out of hand becasue it is different fro the way we learned math.

In terms of a simple math procedure 5 x 3 is the same as 3 x 5 since they both = 15. This is called the commutative, or turn around, property of multiplication. But, if you apply this idea to real objects in real situations the two number sentences are not the same. One is three groups of 5 and the other is five groups of three. If you were working say with M&Ms 5 x 3 would be 5 groups of 3 M&Ms which looks quite different from 3 groups of 5 M&Ms. *****  *****  ***** or *** *** *** *** ***

Now, if you apply this to division, the reverse of multiplication, 15 M&Ms divided between 3 children is quite different from 15 M&Ms divided between 5 children. If we teach multiplication as repeated addition we can teach division as the reverse or repeated subtraction.

There are, of course, more concepts related to multiplcaiton and division but that's another story. So my advice to parents is don't knock something you might not understand just because it's different from what you learned 20 - 30 years ago. Take the time to learn it and you'll find that understanding math instead of just memorizing it is so much more rewarding and useful in the long run.

How Much or How Many

One of the wonderful things about teaching is when you find a better way of explaining, presenting or demonstrating something that is difficult for students to grasp.

When teaching about the different types of simple math problems (as defined by Thomnas Carpenter) I use the phrase 'how many" in all of the problems except one in which I use "how much". The students always tell me, when they come to theat problem that I have made a mistake. When I ask why they think that they always say something like "well, it just doesn't make sense to say how much pennies of how much hours, or how much buckets". When I tell them that it  is not I who have made the mistake but them they always look somewhat confused.

To clarify, the problems are set out with no referents next to the numbers ush as "I I  have 6 ___________ and you give me 8 more ________________, how many _____________ do I have now? Their task is to fill in the referents so that the problem makes sense. So the odd problem similar to the one above ends with "how much ____________ do I have now.

The key to success with this particular problem is to change the referent from a discrete one such as pennies or hours or  candies to a continuous variable such as money, time or sand. A discrete variable refers to things that come in single units whereas a continuous variable can be divided in an infinite number of ways.

Yesterday in class I made two vertical  lists on the board ; one of continuous variables and another next to it of related discrete variables. 

Next time you visit your local supermarket check to see if the express lane says "10 items or less" or the more mathematically correct "10 items or fewer.

Taking the Calculating out of Maths

Five years ago Conrad Wolfram made a very compelling argument for taking the calculating out of math education in this TED talk. "Stop teaching calculating and start teaching math" is the slogan that was very attractive to those of us who thought that math was so much more than the drudgery of teaching the four operations of addition, subtraction, multiplication and division. For several years I agreed with him that we really needed to throw out the arithmetic component of elementary school math and focus on problem solving and other more appealing aspects of math, I also realized that this was a radical idea that would probably never fly politically, socially or educationally.

What it did do, however, was to challenge us to think about why we teach these time consuming procedures when children could simply use calculators or computers to complete simple calculations.

I think the answer can be found in Bob Wrights work with the Math Recovery materials in which he advocates for  mathematization as a process for getting children to think mathematically. The focus of Wright's work is clearly on the development of a child's mathematical thinking through the development of numeration and place value understanding. Part of this is process is the application of the algorithmic procedures as a means of applying and practicing this understanding. In other words, the purpose of teaching algorithmic procedures has changed from primarily applying them to problem solving  to applying and practicing conceptual understanding of numeracy, number relations and all the various aspects of what it means to think mathematically.

Wolfram MathWorld is a pretty cool place for everything maths.



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Cell phones and the NGSS Practices

For the past several weeks in my Teaching Elementary Science and Engineering course we've been looking at the three legs that form the three dimensional teaching characteristics of the NGSS program. I recently blogged about the activity we did to illustrate the crosscutting concept of cause and effect. The week before this we explored the disciplinary core ideas by planting 15 Bean Soup beans in hydroponic :gardens" (plastic cups) and the week before that we looked at the science and engineering practices by creating activities with bouncing balls.  The students were able to identify several of the practices that could have been developed through this activity but it was something else they did with the 23 different balls we had  that blew my mind.

As they bounced the balls next to a tape measure taped to the wall they found it difficult to say the exact height of the rebound without any real accuracy. So what did they do? They did what any self-respecting college junior would do in a science class, they pulled out their cell phones and took pictures of it. This would never have occurred to me.

But they didn't stop there. They realized that if they used the slo-mo video feature on their cell phones they could slow the motion down so that they could actually see the elasticity of some of the ball as they hit the ground as well as the exact height of the rebound. Isn't that amazing and wonderful?     

There are 12 Days in a Week!

I have a really wonderful group of students in my Math and Diversity class this semester. All 13 of them seem to revel in the aha moments that happen with great frequency as we explore the wonders of teaching math to students with diverse needs.

There are fiver areas of diversity identified in the course, each with their own particular set of characteristics that need to be considered. The five areas are teaching math to students who are English learners, students who have special needs, students who have a math disability, students in poverty and students who are high fliers. We've just completed the first area where we explored math and the WIDA standards, SIOP and wonderful things like comprehensive input. We even interviewed a group of international students to find out what the math they had learned was like and to explore some of the numerical aspects of their cultures.

Each class comprises an exploration of one of math such as numeracy and one aspect of one of the areas of diversity. Last week we explored place value and completed activities designed to get students to think more deeply about place value than they have ever done so before. We finished up by exploring different bases which is where there were so many aha moments. By the end of the session all the students were able to say why how there could be 12 days in a week and 202 weeks in a year. This is absolutely true if you are using base 5. 12 is "one-two" and not 12 so it is two 1s and one 5 which is 7 in base 10 . 202 is two 1s, no 5s and two  25s which is 52 in base 10. 

I Really Love Maths

 Tomorrow night I'll be giving the key-note presentation at the VMLC (Vermont Math Leadership Council)  annual meeting in Randolph, Vermont. The title of the presentation is I Really Love Maths. I plan to present a sequence of examples of things that have caused me to love the subject I teach to prospective and practicing teachers in three main areas; the theories that have influenced my thinking, the people who have influenced the way I teach, and the life experiences that have caused me to develop a lifelong passion for teaching maths.
  

Perhaps the two most influential theories that I was lucky enough to encounter early in my career were Skemp's idea of instrumental (fragile) versus relational (robust) understanding, and Shulman's conceptual and procedural knowledge. Being able to look at preK-6 maths through these incredible lenses has allowed me to clearly see what matters more and what matters less, as well as the flaws of some of the traditional instructional practices we have had to endure in years past, and sadly still do in some places today.

Two of the people that have most influenced my thinking are smiling at you right now. At least Sir Ken Robinson is. I looked long and hard but could not find a picture of John Dewey smiling. Sir Ken gave me permission to be creative while Mr. Dewey impressed upon me the value of experience in the educational process. To these names I would add Jo Boaler of Standford  whose work in teacher education in math has been completely illuminating in so far as it has shone the light squarely on the need for teachers to help  children  understand the math they are learning. I would also include Vi Hart who's videos on irreverence in math class are so inspiring. And finally I would add Carol Dweck who gave us Mindset theory with the unbelievable idea that everyone can learn math if they have a Growth Mindset.

And finally my experiences working with English Learners has taught me the humility that comes with standing back  and listening to the way people from other countries do math and think mathematically. The diversity  in the ways we count and communicate mathematically are one of the hidden riches of global thinking. I also think of all the students with disabilities I have worked with some of whom think very differently in terms of the maths in their lives. Some function, and very well too, with an almost exclusively  understanding of nominal number as opposed to cardinal number while others can seemingly compute in milliseconds..

I have been lucky indeed to embrace such diversity of thought, experience and practice during the past 50 or so years.   

Cause and Effect in the NGSS

We did a great activity in my Science Ed. class yesterday. In their Learning Communities (LC) (all my classes have learning communities of 4 or 5 students) the students had to find the causes for several effects related to the Earth. They had to explain the cause of day and night, the four seasons, the different time zones and the differences in temperature and hours of sunlight at different places and times on the Earth's surface.

To do this they had a globe, a balloon, markers, a flashlight and access to the internet. Their goal was to make sure that everyone in their LC had a working  understanding of  all these different phenomena by identifying the cause for each effect.  It was a great experience and gave rise to a lot of discussion, mind changing and a host of aha moments.

At the end of class I gave students a picture of my daughter standing astride the Prime Meridian in Greenwich, England. The picture was similar to the one above complete with the all important shadow.  The task for the students was to say which foot was in the eastern hemisphere and which was in the western hemisphere and to say anything else they could about the picture.

Non-Palindromic Numbers are Coolest

OK so this is not a palindromic number but it's way cooler than any palidromic number you can think of. 202 and 1,234,321 are palindromic numbers but they are exceedingly boring. If you reverse a palindromic number and subtract it from itself the difference is always 0.

But....... if you do the same with a non-palindromic number something amazing always happens. The sum of the digits in the difference always equals 9. In this example 3 + 6 + 6 + 3 = 18 and 1 + 8 = 9. This happens for absolutely any non-palindromic number. 7,829,572 and 2759287. The difference is 5070285 which gives a digit total of 27 which
                                                                    together equal 9.  

Professor frustrated by "dads frustrated by the Common Core"

It's bad enough when politicians like Donald Trump say to do away with the Common Core to score political points without probably having read a single word of it we now have dads dissing the Common Core math standards by writing a check using one of the practice activities for helping children develop a sense of number.

Nobody really gets excited about anything with the word 'common' in it. Common  sense, a common noun, the village  common, the Commons in the British parliament. common and garden, and so on. The word just begs to be dismissed. If only they had clalled it something else

But more seriously, the Common Core for State Standards of mathematics are really good and worth sitting down with for a couple of hours. They present a way of looking at mathematics that is based on logic and understanding, a way that can empower children to use and enjoy their math knowledge rather than becoming a slave to it as so many are who have gone before.

Think about the answer to the question; What is pi?  If you answered that it is 3.14 and goes on for ever without recurring then you probably got that question right on a simple math test of recall. But the Common Core requires more than that. We want students now to understand that Pi is a ratio between the diameter and circumference of a circle and that every circle no matter how large or small is just over 3 times around than it is across. Armed with this understanding you can do so much more that simply say it is 3.14.

This is what the Common Core for Math is all about; empowering children with greater understanding.     

Unhappy Children in England

An interesting  article on the BBC website today suggests children in England are among the unhappiest in the group of  15 countries surveyed in the Good Children Report 2015

The primary reason given for the findings is the extent of bullying to be found in English schools but there is another, I think, more compelling explanation for the situation eloquently expressed by  Kevin Courtney, deputy general secretary of the National Union of Teachers in which he "blamed poor mental health on the "narrow curriculum" and "exam factories" culture in schools".

I find this the more compelling and alarming explanation because it's something that is also happening in US schools where the curriculum seems to be focused almost exclusively on those subjects, topics and ideas that can be easily tested, scored and used in the teacher accountability process. The recent growth in "opting out" of testing in New York State is a sign that people have had enough of the endless testing that is afflicting our schools.

If we put the same effort and financial resources into professional development and material resources that we put into the testing process our children would benefit enormously. An SBAC test costs around $35 per student to score depending on the source you use. The Smarter Balance website lists the cost of a test at 33 cents which is somewhat misleading.

I wonder where US children would rate on the unhappiness scale as defined in the Good Children report?

Outrageous Cost of Text Books

For the past several years I have been railing against the outrageous cost of text books. I have used the same text in my math and science education course  for many years as  it is by far the best one available. The problem is that it is reissued every two years with an accompanying dramatic increase in cost.

Subsequent editions contain few changes from the previous one apart from the inclusion in the latest edition of reference to the Common Core. The full price for the text, in paperback form,  is now a remarkable $215 according to Amazon which will sell you a copy for $200.32. Other new copies are available for $180 and a used copy will set you back $130.

To help my students ease the financial burden of college I require them to purchase the 7th edition, available for as little as $15. The fundamental mathematical ideas are exactly the same as in the 8th edition. The only thing missing is the Common Core specifics which are freely available on-line.

It turns out that I am not the only person feeling this way. Today, the Hechinger Report suggests that all educational materials produced with federal funding should be made available at no charge. This would surely spur the publishing companies into some form of price control


How can a paperback book be worth $215? 

Learn Poetry? Then why not Fractals in K-6?

I've said this a million times before and I'll probably say it until I'm laid to rest but we have to include the study of fractals and other kinds of wonderful mathematical patterns in the study of math at the elementary school level. If poetry is part of the CCSS then fractals certainly should be.

Here's a really interesting set of reasons for making poetry part of the study of the English Language Arts from the Atlantic.  One could almost use the same justifications for the inclusion of the study of fractals in the math curriculum.

Cynthia Linus has been promoting the study of fractals for years with a selection of interesting activities on her website; and here's a really cerebral argument for their study by Joe Pagano who even links poems and fractals.

The sad truth about math education is we tend to see it as purely utilitarian. What we learn in math has to be useful, usable, worthwhile or practicable. We never seem to recognize the value of  the aesthetics of mathematical relationships in the same way that linguists recognize the  allure of alliteration, the majesty of metaphor, the perfection of personification. We never stop to marvel at the patterns created by counting by 5s starting at 3, the way equilateral triangles can be divided into four more equilateral triangles, how when 6 circles are placed around one circle a perfect heaxagon can be formed by connecting the centers of the small spaces between the circles, or how magical it is to color in all the even numbers on a Pascall's Triangle and discover you've made a Sierpinski Triangle.

This is what turns children onto math.

Tests Only Measure Other Tests

I have always thought that there is something quite disturbing about testing in the context of education. It's probably because the testing system is used primarily to sort students rather than to develop a sense of whether they understand or know something. In the UK, a test is referred to as an exam which seems to denote better the idea of examining a student's understanding or knowledge of a particular idea. To examine seems a nobler and more useful goal than does to test.


Many years ago when i was a grad student I read somewhere that tests only measure other tests. At first glance this seems like a fairly innocuous statement but the more you think about it the more sense it makes. When I was teaching fourth grade in the UK in the early '70s we changed a reading test one year. The same reading test had been used throughout the school (and all the schools in the entire city of Bristol) for many years so it was decided to use a new test. When the new test results came in it was found that all the students in the city had gained two years in their reading age. Since reading instruction hadn't changed it must have been a much easier reading test but it still made everyone feel really good !!!!!

The same happens eery time you adopt a new test or  testing system such as SBAC. Very seldom re two tests exactly the same and more often than not the new test is more difficult than the old test. This is often done with the somewhat naive belief than making the test more difficult will improve instruction and make the students appear brighter. Usually this results in declining test scores and yet more blame placed upon the education system for falling standards.

But not so in Scotland where a maths test was given  that proved to be too difficult. Instead of blaming teachers and the education system, as we would undoubtedly have done here, the  Scottish Qualifications Authority (SQA) acknowledged the test was too difficult and adjusted the scores accordingly.

We cannot raise standards by making tests, or exams, more difficult. We must improve the way we teach and the way we motivate students to learn. 

Neuroscience and Elementary Education

At first glance the study of neuroscience and elementary education at the undergraduate level seems a little bit like an oxymoron; the pursuit of two extremes; a path to schizophrenia.
But for anyone interested in science and wanting to become an elementary school teacher the combination of these two majors makes incredible sense.

To quote Professor Melissa Vanderkay Tomasulo PhD, Director of the Neuroscience Program and associate professor of psychology; "Neuroscience allows us to explore the world and human existence through biological, psychological, and social lenses. It amazes me and my students to be reminded that a three pound organ the size of a cauliflower enables us to ponder, emote, move, feel, and reason." What is teaching children at the elementary school level if it is not helping them navigate  the biological, psychological and social demands of growing up.

Several elementary education majors students have recently  asked if they can pursue the new  neuroscience major. To make sure this new major falls within the liberal arts second major requirement I contacted the Vermont Agency of Education today.  According to Patrick Halliday, Director of teacher licensure at the VtAoE, the Neuroscience major will fulfill the second major requirement and so we are off and running. 

 

Common Core and Politics

I wonder if the Common Core will survive the 2016 election campaigns? It's been quite astounding how already the Common Core State Standards are mentioned in negative terms almost every time a politician, regardless of her/his party, gets onto the topic of education. The main question to them has to be "HOW MUCH OF THE COMMON CORE HAVE YOU READ?" Then, the next question should be "HOW MUCH OF THE COMMON CORE DO YOU UNDERSTAND?". I wonder how they would respond.

Having become intimately familiar with the Math CCSS I have to say they are pretty good. The authors seem to have included something for everyone. For example they want students to understand what they are learning, a nod to the liberal side of the populace, and the want students to develop rigor in their learning, a nod to the conservative side. Perhaps this is the very reason why both sides are attacking them. The CCSS contain something that everyone can point to as being not what they want in an education system. Instead of seeing them as an entirely complete piece of work they are selecting small pieces to support their argument for dropping the Common Core. It is always much easier, of course, to chop a tree down in 30 seconds that to grow one in 30 years.

Perhaps it is the name "Common Core" that politicians don't like. The word "common" for example has somewhat negative connotations: It's hard to get excited about something called "common". Then there's the word "core". Again it's a word that you tend to find in somewhat negative surrounding such as in the phrase "cut me to the core".

Then, of course, it's all part of "Race To The Top", a phrase clearly not coined by someone involved  intimately in the education of children. I sometimes imagine a kindergarten teacher saying on the first day of the semester "OK kindergartners, now pay attention. I just want to remind you that you are just about to start the "Race To The Top". Since this is a race, some of you will make it to the top first and some of you won't. No pressure, I just want you to be aware of what lies ahead of you".

Here are some interesting links re CCSS and Politics; US News and the Huff Post.
NPR and the Washington Post.


  

 

CFES at SMC in VT in the US

It's always a neat experience doing something you've never done before. This morning I presented an activity on Fractals to a group of students participating in the College For Every Student (CFES) program. CFES is a leading nonprofit designed with the goal of: 
"helping under-served students get to and through college, and ready to enter the 21st century workforce".
CFES currently "supports 20,000 students through partnerships with 200 rural and urban K-12 schools and districts in 27 states and Ireland".

I have worked with many, many different populations of students but this is the first time I have worked with H.S. students considering pursing a college education. We completed a number of activities related to fractals and mathematical patterns using my motto "Math is the science of pattern and the art of making sense". I was so impressed by the quality of their work and their ability to persevere with one particular activity that was not at all easy.

On their website, CEFS describes raising the students' aspirations  since most of them come from families with parents who haven't attended college. Neither of my parents attended college so I have a sense of how they view the whole idea of what it means to attend college and get a degree. I must admit that I think sometimes we are a bit misguided in the way we promote a college education as a way of getting a good job although I do acknowledge this is true.  I'm probably being somewhat idealistic in a material world but I believe a college education for as many people as possible is vital for the promotion and continuation of a civilized world. We need a caring and educated population that values knowledge, understanding and wisdom as they relate to our culture as much as we need skilled workers.

We concluded the activities with my favorite Vi Hart video. 

NGSS - A Worthy Vision

So this week I began my long journey into intimacy with the Next Generation Science Standards. The time has come when I have to know them inside out, be familiar with all the nooks and crannies, find out what resources are available and generally make them part of who I am as a math, science and engineering educator.

My journey began a few months ago when I joined a group of HE faculty working with the science folks at the Vermont Agency of Education to find ways of making sure that the implementation of the NGSS in Vermont schools goes as planned in 2016.

Using John Dewey's pedagogical dualism the NGSS present us with a worthy "inspired vision" of how science education should exist at the elementary school level in the context of the elementary school curriculum which includes ELA, math and social studies. It's interesting that the latter discipline is missing from the Venn diagram above.

The other part of Dewey's dualism, of course, is one's "executive means" or ones ability to put into practice one's "inspired vision" of teaching. According to Dewey's theory there must always be proximity between one's vision and one's executive means but one can never truly achieve one's inspired vision. . The two can never be exactly the same because the practical world of the classroom comprises children who are unique.  The relationships between the students in a class is also unique. Our task as teachers is to continually work to get as close to implementing our "inspired vision" through the use of our "executive means" as we possibly can.

This is a grand task that requires a growth mindset by everyone involved.

  

Common Core - Yes; SBAC testing - No

I've said this before, probably many many times, but it's worth saying again because things are starting to happen.
The Common Core is great. In mathematics, the CCSSM clearly identify the math that students should be expected to learn in each grade level. What is not good are the Smarter Balance Assessment Consortium standardized, computer-based tests that students are expected to take starting in third grade. Having just watched four third grade teachers and their students suffer through these anachronistic evaluation devices (they are clearly not assessments)  I think it's time for parents and teachers and everyone involved in Education to say enough is enough. American students are still the most tested and the least examined.

More and more people are beginning to boycott the tests or opt out such as these students. There is also a growing body of evidence that suggests that the PISA test scores that consistently show the US at a lower ranking than most would like are not a true reflection of the quality of education in American schools. More and more research, and opinion papers by scholars such as this one in the Guardian, are casting doubt upon the authenticity of comparing tests in cultures that are so different. The most odious interpretation of these international results based on false comparisons is to say that higher test scores in many Asian countries are the result of better teaching. There are so many other factors that contribute to differences in test scores.

Perhaps if all the parents of third grade students were to go on-line and take the sample SBAC test items they would see why such testing is so invalid. 

It's a "Testing-Crazy" World.

So this might turn out to be a bit of a "blog-rant" but the constant testing HAS TO STOP!. The BBC reports today that 4-year-olds are to be tested in the UK in reading, writing and maths. It seems that politicians, in particular, see the only solution to the improvement of education is to do more testing; test infants, test children, test students, test pre-service teachers, test teachers. Why do we never test politicians?

As stated in the Beeb article the primary reason for testing 4-year-olds is to provide a baseline by which the education system can be evaluated. This is basically another cry for teacher testing and yet another example of politicians' lack of confidence in teachers and the teaching profession.

And all this happens, of course, at a time of a general election in the UK when each party needs an extra plank in their platform. Education is  an easy target because it is so easy, through the manipulation of numbers, to show how poorly students in one country such as the UK or US are doing compared to students in another country such as Singapore. The same is beginning to happen in the US as we gear up for the 2016 presidential election. International test score comparisons such as TIMMS and PISA are readily available and provide instant numerical comparisons. The unfortunate thing is that politicians, of course,  fail to mention the fine details such as in Singapore there are special schools where all students with special needs are sent so the scores do not reflect the effort that goes into teaching the whole student population.

Tests, such as the SBACS, currently being administered in Vermont schools, test only what is easy to test. They only provide a brief, momentary, narrow snapshot of what children know and understand. The scores are also heavily swayed by a students ability to master the computer skills required to take the test.

Nobody knows a student as well as the teacher who teachers the student. Nobody who lives 300 miles away can possibly be in a position to truly assess what a student knows and understands. 

Why the Bridges 2 Math Program is So Good

This morning I observed one of my student teachers teaching a lesson on remainders from the Bridges 2 Math Program. As the student teacher struggled with her understanding of the mathematics involved it became incredibly clear to me why the program is so good. The activities are so rich with mathematical opportunity  for what Bob Wright called the mathematization of children's minds.

The third grade activity I was observing involved rolling two die and multiplying the subsequent numbers to get a product. Then, using a neat worksheet, the students had to divide the product by 2,3,4,5, and 6 to see what happens. The demonstration roll the student teacher used was 4 x 4 for 16. Dividing by 2 yielded 8, but 8 what? The children were using tiles to make 2 x 8 arrays vertically. Here the 2 referred to the number of columns and the 8 to the number of rows. So the 8 was 8 rows. She could have shown 8 groups of 2 or 2 groups of 8, demonstrating the commutative property of multiplication. This is only true abstractly as the actual arrangement looks quite different. This is known as psychological non-commutativity.

She then went on to divide 16 by 3 getting 5 R1, then divided it by 4 getting 4 (proving 16 is a square number), then by 5 getting 3R1 (showing the commutative property), and finally by 6 getting 2 R4 .  

The interesting thing about the commutative property in division is if you then give the remainder as a fraction. 16 ÷ 3 would give 5 1/3. 16 ÷ 5 would give 3 1/5. So the question is; what do the fractions represent? 1/3 of what and 1/5 of what? A fraction has no value unless you know the size of the one to which it refers. It depends, of course, on what the referents for the 3 and the 5 are. If you had 16 cookies divided between 3 people, each person would get 5 and 1/3 cookies. If you had 16 cookies and divided them  into groups of 5, 3 1/5 people could get a group of cookies; which makes absolutely no sense at all.  

The neat thing about the Bridges Program 2 activities is that they are so rich with mathematical possibilities.

Teachers Need Support with NGSS

We are living in a time of unprecedented change in the teaching profession. While we are still incorporating the changes brought about by the Common Core State Standards we are now getting ready for the implementation of the new Next Generation Science Standards. Current times in Education are also characterized by a significant lack of support for public education brought about primarily by politicians jumping on band wagons and looking for political planks or angles in preparation for the upcoming national election. It is so easy for a politician to compare test score results by waving pieces of paper about which they know nothing, and so difficult for those who are experts in education to correct all the misinformation that is spread in the name of political advantage. 

Amidst all of this, I recently volunteered to teach a new course; a Science Practicum that places students in public schools to learn how to teach science and engineering at the elementary K - 6 level. It will involve introducing students to the new Next Generation Science Standards and helping them learn how to implement inquiry-centered science activities based on these standards.

To do this I am hooking up with the Flynn Elementary School in Burlington, Vermont, which recently became a STEM Academy. It will be the third specialized K - 5 academy in Burlington joining the Integrated Arts and Sustainability Academies. The more I think about this partnership the more excited I become. I want to use this course as a way of supporting teacher and the school as they begin to implement the changes required of becoming a STEM academy. I want my students to be able to gain all sorts of experiences such as creating inquiry activities, science and engineering units of instruction,  bulletin boards and web-sites, as well as working in the after-school STEM program and the anticipated garden project.

I want to be able to support the teachers at the school as they adapt their curriculum materials to incorporate the NGSSs and I can't think of a better way to do this than have my students work to undertake all the time-consuming things that need to be done. The experience of doing this will provide my students with unique opportunities to develop their own skills and understanding of what it means to teach science and engineering.

 

Student Teaching; Spreading One's Wings

At this time of the semester my student teachers are completing their solo classroom experience. This means they are teaching their classes of student by themselves without the assistance of their cooperating teacher. For some it is a time to fly; to stretch the new wings they have been growing in their coursework for the past three and half years. One such student is Miranda who has been working with a class of 24 rambunctious third graders. Interestingly, the solo week always coincides with the onset of Spring, a time of great excitement for young children, who for the first time in months, are able to get outside in the warmer weather.

I want to celebrate Midanda's successful solo week by sharing one of her journal entries:

          "I think this experience also reminded me of why I wanted to be a teacher in the first place. Children, I believe, have a nature and energy that is unmatched by adults. They are curious, energetic, thoughtful, and innocent. They are good-natured, enthusiastic, and empathetic. I think that even on my worst days as a teacher, I need to remember why I wanted to become a teacher in the first place. Children are incredible and they are the key to a better future. I believe that part of my job as a teacher is establishing relationships with my students and helping to mold them into thoughtful, respectful citizens of a community. Tonight I was able to share experiences with my students and their parents outside the classroom that will hopefully remain with them as a positive time with their teacher."  

Isn't that a wonderful way to think about the young students she has been working with this semester?