Author: David Wees (page 15 of 97)

The following are studies which were all featured in the media in 2013. I am posting them here in the hope that they will be read more widely than they are, and that educators will examine the research themselves, and think about how this may affect their current practice.

I’ve included a link to the study as well as either an abstract or the summary of the research as presented by the author of the article linked. These are all studies which either support current hypothesises about the importance of recognizing social and cultural issues in teaching and learning mathematics, relate to the importance of early mathematics education (and the role parents can play), or observe that the style of instruction that is used has an impact on student learning.

Finally, I’ve included a couple of studies I read which were not specifically done in the area of mathematics education itself, but which I think are obviously related.

Socio-emotional and cultural issues
 

Why Students Choose STEM Majors: Motivation, High School Learning, and Postsecondary Context of Support

“This study draws upon social cognitive career theory and higher education literature to test a conceptual framework for understanding the entrance into science, technology, engineering, and mathematics (STEM) majors by recent high school graduates attending 4-year institutions. Results suggest that choosing a STEM major is directly influenced by intent to major in STEM, high school math achievement, and initial postsecondary experiences, such as academic interaction and financial aid receipt. Exerting the largest impact on STEM entrance, intent to major in STEM is directly affected by 12th-grade math achievement, exposure to math and science courses, and math self-efficacy beliefs—all three subject to the influence of early achieve- ment in and attitudes toward math. Multiple-group structural equation modeling analyses indicated heterogeneous effects of math achievement and exposure to math and science across racial groups, with their positive impact on STEM intent accruing most to White students and least to under- represented minority students.”

Women do better on math tests when they fake their names

“Unsurprisingly, and as the title of this post already suggests, women do indeed perform better on math tests when they assume a name other than their own — and this happens regardless of whether they take a male or female name.

As a recent study by Shen Zhang has shown, using another person’s name is a kind of hack to overrule the self-reputational threat — the fear some women have of doing poorly when they’re concerned that it’ll be taken as proof of a stereotype. But removing this pressure seems to alleviate the fear and the distraction.

For the study, Zhang recruited 110 women and 72 men — all of them undergrads — and had them answer 30 multiple-choice math questions. Prior to the test, and in an effort to instill the stereotype threat, all participants were told that men typically outperform women at math. Some of the volunteers were told to write the test under their real name, but some were told to complete the test under one of four different aliases, either Jacob Tyler, Scott Lyons, Jessica Peterson, or Kaitlyn Woods…”

Early Nervousness Over Number Impacts Future Performance

“According to a recent study by Rose Vukovic, NYU Steinhardt professor of teaching and learning, math gives some New York City students stomachaches, headaches, and a quickened heartbeat. In short, math makes these children anxious.

“Math anxiety hasn’t really been looked at in children in early elementary grades,” said Vukovic, a school psychologist and researcher of learning disabilities in mathematics. “The general consensus is that math anxiety doesn’t affect children much before fourth grade. My research indicates that math anxiety does in fact affect children as early as first grade.”

Vukovic’s first study, “Mathematics Anxiety in Young Children,” will be published in the Journal of Experimental Education. It explored mathematics anxiety in a sample of ethnically and linguistically diverse first graders in New York City Title I schools. Vukovic and her colleagues found that many first grade students do experience negative feelings and worry related to math. This math anxiety negatively affects their math performance when it comes to solving math problems in standard arithmetic notation…”

Female teachers’ math anxiety affects girls’ math achievement

“People’s fear and anxiety about doing math—over and above actual math ability—can be an impediment to their math achievement. We show that when the math-anxious individuals are female elementary school teachers, their math anxiety carries negative consequences for the math achievement of their female students. Early elementary school teachers in the United States are almost exclusively female (>90%), and we provide evidence that these female teachers’ anxieties relate to girls’ math achievement via girls’ beliefs about who is good at math. First- and second-grade female teachers completed measures of math anxiety. The math achievement of the students in these teachers’ classrooms was also assessed. There was no relation between a teacher’s math anxiety and her students’ math achievement at the beginning of the school year. By the school year’s end, however, the more anxious teachers were about math, the more likely girls (but not boys) were to endorse the commonly held stereotype that “boys are good at math, and girls are good at reading” and the lower these girls’ math achievement. Indeed, by the end of the school year, girls who endorsed this stereotype had significantly worse math achievement than girls who did not and than boys overall. In early elementary school, where the teachers are almost all female, teachers’ math anxiety carries consequences for girls’ math achievement by influencing girls’ beliefs about who is good at math.”

Early learning and parental involvement
 

Deconstructing Building Blocks: Preschoolers’ Spatial Assembly Performance Relates to Early Mathematical Skills

“This study focuses on three main goals: First, 3-year-olds’ spatial assembly skills are probed using interlocking block constructions (N = 102). A detailed scoring scheme provides insight into early spatial processing and offers information beyond a basic accuracy score. Second, the relation of spatial assembly to early mathemati- cal skills was evaluated. Spatial skill independently predicted a significant amount of the variability in concur- rent mathematical performance. Finally, the relationship between spatial assembly skill and socioeconomic status (SES), gender, and parent-reported spatial language was examined. While children’s performance did not differ by gender, lower-SES children were already lagging behind higher-SES children in block assembly. Furthermore, lower-SES parents reported using significantly fewer spatial words with their children.”

What’s the earliest age that children think abstractly?

“Caren Walker and Alison Gopnik (2013) examined toddlers ability to understand a higher order relation, namely, causality triggered by the concept “same.”

The experimental paradigm worked like this. The toddler was shown a white box and told “some things make my toy play music and some things do not make my toy play music.” The child then observed three pairs of blocks that made the box play music, as shown below. On the fourth trial, the experimenter put one block on the box and asked the child to select another that would make the toy play music. There were three choices: a block that looked the same as the one already on the toy, a block that had previously been part of a pair that made the toy play music, and a completely novel block…”

Quality of early parent input predicts child vocabulary 3 years later

“Children vary greatly in the number of words they know when they enter school, a major factor influencing subsequent school and workplace success. This variability is partially explained by the differential quantity of parental speech to preschoolers. However, the contexts in which young learners hear new words are also likely to vary in referential transparency; that is, in how clearly word meaning can be inferred from the immediate extralinguistic context, an aspect of input quality. To examine this aspect, we asked 218 adult participants to guess 50 parents’ words from (muted) videos of their interactions with their 14- to 18-mo-old children. We found systematic differences in how easily individual parents’ words could be identified purely from this socio-visual context. Differences in this kind of input quality correlated with the size of the children’s vocabulary 3 y later, even after controlling for differences in input quantity. Although input quantity differed as a function of socioeconomic status, input quality (as here mea- sured) did not, suggesting that the quality of nonverbal cues to word meaning that parents offer to their children is an individual matter, widely distributed across the population of parents.”

What counts in the development of young children’s number knowledge?

“Prior studies indicate that children vary widely in their mathematical knowledge by the time they enter preschool and that this variation predicts levels of achievement in elementary school. In a longitudinal study of a diverse sample of 44 preschool children, we examined the extent to which their understanding of the cardinal meanings of the number words (e.g., knowing that the word “four” refers to sets with 4 items) is predicted by the “number talk” they hear from their primary caregiver in the early home environment. Results from 5 visits showed substantial variation in parents’ number talk to children between the ages of 14 and 30 months. Moreover, this variation predicted children’s knowledge of the cardinal meanings of number words at 46 months, even when socioeconomic status and other measures of parent and child talk were controlled. These findings suggest that encouraging parents to talk about number with their toddlers, and providing them with effective ways to do so, may positively impact children’s school achievement…”

Why Mental Arithmetic Counts: Brain Activation during Single Digit Arithmetic Predicts High School Math Scores

“Do individual differences in the brain mechanisms for arithmetic underlie variability in high school mathematical competence? Using functional magnetic resonance imaging, we correlated brain responses to single digit calculation with standard scores on the Preliminary Scholastic Aptitude Test (PSAT) math subtest in high school seniors. PSAT math scores, while controlling for PSAT Critical Reading scores, correlated positively with calculation activation in the left supramarginal gyrus and bilateral anterior cingulate cortex, brain regions known to be engaged during arithmetic fact retrieval. At the same time, greater activation in the right intraparietal sulcus during calculation, a region established to be involved in numerical quantity processing, was related to lower PSAT math scores. These data reveal that the relative engagement of brain mechanisms associated with procedural versus memory-based calculation of single-digit arithmetic problems is related to high school level mathematical competence, highlighting the fundamental role that mental arithmetic fluency plays in the acquisition of higher-level mathematical competence.”

Young Children’s Interpretation of Multi-Digit Number Names: From Emerging Competence to Mastery

“This study assessed whether 207 3- to 7-year-olds could interpret multi-digit numerals using simple identification and comparison tasks. Contrary to the view that young children do not understand place value, even 3-year-olds demonstrated some competence on these tasks. Ceiling was reached by first grade. When training was provided (based on either base-10 blocks or written symbols), there were significant gains, suggesting that children can improve their partial understandings with input. Our findings add to what is known about the processes of symbolic development and the incidental learning that occurs prior to schooling, as well as specifying more precisely what place value misconceptions remain as children enter the educational system.”

Instructional strategies
 

Study shows new teaching method improves math skills, closes gender gap in young students

“When early elementary math teachers ask students to explain their problem-solving strategies and then tailor instruction to address specific gaps in their understanding, students learn significantly more than those taught using a more traditional approach. This was the conclusion of a yearlong study of nearly 5,000 kindergarten and first-grade students conducted by researchers at Florida State University.

The researchers found that “formative assessment,” or the use of ongoing evaluation of student understanding to inform targeted instruction, increased students’ mastery of foundational math concepts that are known to be essential to later achievement in mathematics and science…”

Academic music: music instruction to engage third-grade students in learning basic fraction concepts

“This study examined the effects of an academic music intervention on conceptual understanding of music notation, fraction symbols, fraction size, and equivalency of third graders from a multicultural, mixed socio-economic public school setting. Students (N = 67) were assigned by class to their general education mathematics program or to receive academic music instruction two times/week, 45 min/session, for 6 weeks. Academic music students used their conceptual understanding of music and fraction concepts to inform their solutions to fraction computation problems. Linear regression and t tests revealed statistically significant differences between experimental and comparison students’ music and fraction concepts, and fraction computation at posttest with large effect sizes. Students who came to instruction with less fraction knowledge responded well to instruction and produced posttest scores similar to their higher achieving peers.”

Non-traditional mathematics curriculum results in higher standardized test scores, study finds

“James Tarr, a professor in the MU College of Education, and Doug Grouws, a professor emeritus from MU, studied more than 3,000 high school students around the country to determine whether there is a difference in achievement when students study from an integrated mathematics program or a more traditional curriculum. Integrated mathematics is a curriculum that combines several mathematic topics, such as algebra, geometry and statistics, into single courses. Many countries that currently perform higher than the U.S. in mathematics achievement use a more integrated curriculum. Traditional U.S. mathematics curricula typically organize the content into year-long courses, so that a 9th grade student may take Algebra I, followed by Geometry, followed by Algebra II before a pre-Calculus course.

Tarr and Grouws found that students who studied from an integrated mathematics program scored significantly higher on standardized tests administered to all participating students, after controlling for many teacher and student attributes. Tarr says these findings may challenge some long-standing views on mathematics education in the U.S…”

Duke Study Finds Improving ‘Guesstimating’ Can Sharpen Math Skills

“You may not have heard of it, but it’s a skill you probably use everyday, like when choosing the shortest line at the grocery store or the toll booth with the fewest number of cars. Approximate number math, or ‘guesstimating,’ is the ability to instinctively estimate quantities without counting. Researchers at Duke University set out to discover whether practicing this ability would improve symbolic math skills, like addition and subtraction.

They discovered that study participants who were given approximate number training sessions did dramatically better on symbolic math tests than those who were not. Those who received training also received significantly higher scores on the math tests after the training than before…”

Related research
 

What Science Teachers Need to Know

“The researchers (Sadler et al., 2013) tested 181 7th and 8th grade science teachers for their knowledge of physical science in fall, mid-year, and years end. They also tested their students (about 9,500) with the exact same instrument.

Each was a twenty-item multiple choice test. For 12 of the items, the wrong answers tapped a common misconception that previous research showed middle-schoolers often hold. For example, one common misconception is that burning produces no invisible gases. This question tapped that idea:

But the researchers didn’t just ask the teachers to pick the right answer. They also asked teachers to pick the answer that they thought their students would pick…”

Research: Improving Test Scores Doesn’t Equate to Improving Abstract Reasoning

“A team of neuroscientists at MIT and other institutions has found that even when schools take instructional steps that help raise student scores on high-stakes tests, that influence doesn’t translate to improvements in learners’ abilities to perform abstract reasoning. The research, which took place a couple of years ago, studied 1,367 then-eighth-graders who attended traditional, charter, and exam schools in Boston. (All were public schools.)

The researchers found that while some schools raised their students’ scores on the Massachusetts Comprehensive Assessment System (MCAS) — a sign of “crystallized intelligence” — the same efforts don’t result in comparable gains in “fluid intelligence.” The former refers to the knowledge and skills students acquire in school; the latter describes the ability to analyze abstract problems and think logically…”

Classes should do hands-on exercises before reading and video, Stanford researchers say

“A new study from the Stanford Graduate School of Education flips upside down the notion that students learn best by first independently reading texts or watching online videos before coming to class to engage in hands-on projects. Studying a particular lesson, the Stanford researchers showed that when the order was reversed, students’ performances improved substantially.

While the study has broad implications about how best to employ interactive learning technologies, it also focuses specifically on the teaching of neuroscience and underscores the effectiveness of a new interactive tabletop learning environment, called BrainExplorer, which was developed by Stanford GSE researchers to enhance neuroscience instruction…”

Like many math teachers, I have been following Dan Meyer’s discussion on “fake world” math tasks versus “real world” math tasks with interest, especially since one of my early blog posts was on this subject and one of the presentations I do for teachers is on this very topic.

My observation is that it is not the task that defines whether it is fake or real, it is the person doing the task. Our work then should focus on developing criteria on what makes tasks real for children, and then see which tasks support the criteria we establish. Here are some criteria I think we should consider when developing tasks for students, aside from the obvious; the task should engage students in mathematical thinking.
 

• Relationships:

  Children do mathematics in a socio-emotional context, and virtually always with the support of a mentor (peer, parent, teacher, etc…). A strong relationship with someone who can support their mathematical reasoning is critical. Tasks which the mentor finds interesting or appealing are more likely to be interesting or appealing to the children they support. There are lots of stories of lone mathematicians working in secret for years on developing mathematics, but I do not know of any stories of children doing the same.
   

• Questions:  

  Children are more engaged with tasks that they have questions about. By this, I do not mean the pretend inquiry questions that people sometimes start a unit with (Imagine here a teacher-led discussion that leads to a wall full of questions children made up on the spot to satisfy the “let’s make up our inquiry questions” game…), but actual questions that students have about the world and the objects in it.
   

• Access:  

  Children need to be able to access the task and to do it substantially by themselves. A task where the adult with them has to do most of the work, either physical or otherwise, quickly becomes much less interesting for children. I built (from a kit) a compressed air rocket a couple of months ago with my son. He spent most of the time bored as I fit pieces together and he occasionally got to glue things together. Although he was very interested in the final product (who wouldn’t be interested in something that can shoot paper rockets up 50 metres in the air?), the process of making the rocket was tedious because he only had periphery access to the building process. This is true of mathematical tasks as well. Tasks where children have to rely on a list of “how to” steps provided by someone else are rarely interesting, unless some significant thinking has to occur to make the steps useful.
   

• Challenge:  

  Children often like to do things because they are challenging so we have to be careful not to make things we ask children to do to be too simple when we are ensuring they have access. I once gave the Seven Bridges of Königsberg problem to my 9th grade class. The problem was accessible because every student felt like they have a possible solution path (ie. draw a picture), but it was challenging so the students kept working on it. Working on the problem became infectious, and soon, most of the 9th grade math classes in the school worked on the problem at least a little. Some of my 9th graders spent three weeks trying to solve the problem before finally coming to ask me to prove that it was impossible.
   

• Familiarity:  

  Children do not have questions about things with which they lack familiarity. If the context you are using is completely unfamiliar to children, then they aren’t going to have questions. If one takes the time to develop context around a situation (ie. story-telling), then it is more likely that children will begin to wonder about it. Every good game as a plot that hooks the player into the game, a good math task should do the same thing.

Note that these criteria all lead to an important conclusion; some tasks will be considered fake by some students, and real by others. It is important to note too that because of our shared society and context, there are some tasks which will be real for almost all students, and there are other tasks which will be fake for almost all students.

I just noticed that I can embed recordings of presentations from Big Marker, so I decided to embed my presentation for the Global Math department here. The presentation topic was “Teacher and Student Questions.” You can view the slides here.

I recently facilitated a workshop on social media for a school. I created two short videos to act as discussion starters, and then I created a workshop structure around the video clips that the school could use to facilitate the discussion with their students.

Outline of a workshop for students on social media:

1. Introduce me (Optional):

   David Wees works with New Visions for Public Schools in the a2i project. He spent the last four years working as a educational technology expert for a school in Vancouver, Canada. One interesting fact about David; he has lived and worked in four different countries. He is going to share some information he has learned about social media with us today, with video two short videos he created.

    

2. Show the clip titled: “Social Media Part 1”
   (Optional: Start with students sharing out what they know about the Internet and social media first)

    

3. Discussion (have students work in groups of 3 or 4 students) – 5 minutes or so

   Some possible questions:

   Print the following questions for students to talk about (add more questions as you see fit)

   What kinds of things have I posted online?
   Who can access these things?
   What can I do to limit access to things I post online?
   What could happen to me if I post things online?

    

4. Whole group discussion (5 minutes or so)

   Have different groups share out points from their discussion.

    

5. Show the clip titled: “Social Media Part 2”

    

6. Discussion (have students work in groups of 3 or 4 students) – 5 minutes or so

   What is in my digital footprint?
   What can I do to build a stronger, more positive digital footprint for the future?
   What kinds of things should I avoid doing?
   What kinds of things should I make sure I do?

    

7. Whole group discussion (5 minutes or so)

   Some possible questions:
   (Recommended: Make a chart for the whole room to see of the positive and negative things about sharing stuff online)

   What are some positive things we can share?
   What are the some of the negative things we should avoid sharing?
   What can we do if we find out someone else is sharing negative stuff about us?
   What is the relationship between what we share, and what people think of us?

    

8. Exit slip:

   Have students write down three things they learned from the workshop today. Read what they wrote and use it to inform a future follow-up session on social media.

*Note that I have very much simplified the “how the Internet works” portion of the video as my aim is to get kids talking about it rather than knowing all of the technical details.

As I watched my son over the past few days learn about shape, I am struck by not only how much we need to learn to make sense of the world, but also by how even simple things cannot be taken granted as known by children.

My youngest son, who is about 20 months old now, is learning how to take a small piece of yellow plastic that forms a shape (an oval, a circle, a hexagon, and 4 other shapes) and push it through a hole that is only slightly larger than the shape he is pushing through it. The task requires him to pay close attention to both the shape he has in his hand, the orientation of that shape, and the hole he is trying to force it through.

At first he pretty much chooses shapes at random, and tries to jam into the hole. Eventually he finds a match and gets a shape through the hole, or he gives up. He rarely turns the shape in his hand, or the container the shape goes into.

Eventually he learns that he has to often rotate the shape in his hand, and so he picks up a random shape and tries to shove it in a random hole, and when it doesn’t work the first time, he says, “No…no…no,” and rotates the shape a bit to try again.

Soon he has made a few matches, and is always able to easily find the right hole for the circular shape and the oval shape, and soon after the plus sign shape. For all of the other shapes, he continues to try to randomly match shapes to holes. He looks at the shape to identify it as one of the ones he knows, but he does not seem to connect the general idea that the shape he sees in the piece should match the shape he sees in the hole.

The next day, I am surprised to discover that he can match almost all of the shapes, and frequently looks at the shape to see which one he has, and then which hole he should put the shape in. He often rotates the shape several times when he is sure he has the right hole. When he is not sure, he tends to give up quickly and go back to a shape with which he is very familiar.

In a few days, he goes from basically using a random matching strategy to carefully looking at the shape in order to be able to match it. At this stage the only shapes that stump him regularly are the regular pentagon, and the regular hexagon. 

I notice however that his ability to do other puzzles does not seem noticeably improved. It’s like his learning is restricted to this one very narrow context, and within this narrow context, he has either just learned to match each of the individual patterns or possibly he recognizes a small generalization; for this puzzle the shape should match the hole. As the months progress, I will continue to watch how his understanding of shape grows and develops.

It is fascinating to me to see that shape is a learned concept and that even what seem like simple generalizations are learned. It makes me wonder what concepts my students may not have fully developed, even by the time they arrived to me in high school.

I’m currently working on creating a sample sequence of lessons for teachers to use for a geometry unit. At this stage, students will have been exposed to (but will not necessarily have learned all of) geometric transformations, constructions, and some review on geometric vocabulary.

My objective is to create a sequence of lessons which include:

• embedded formative assessment,
• opportunities for students to discuss student thinking,
• opportunities for students to make sense of the idea of proof,
• opportunities for students to prove geometric proofs,
• opportunities for students to engage in mathematical inquiry,
• engaging tasks that help get kids excited about mathematical arguments.

The last time I taught geometry, I think I maybe managed to to hit 1 or 2 points from this list, so I have given myself a tall challenge. I need help.

One of my university professors once told me, “Something is proven true when everyone stops arguing that it isn’t” so with that in mind, here is a vague idea that I am considering and need help fleshing out:

Basically, students would spend their time during this unit making interesting geometric observations and then attempting to prove to each other that these arguments are true. Part of their time would be exploring geometric objects, possibly through constructions and possibly through looking at (interactive?) diagrams. During this time, when they see something that they think might be true, they create a proof (in everyday language that they understand) that it is true, and then they present their argument, either in small groups or to the whole class.

Here is an example from my imagination. Suppose students have access to this online geometric construction tool (or lots of paper, a compass, and a straight edge) and when playing around, one of them creates this construction.

Student 1: “Oh wow, that’s pretty cool. I made a regular hexagon.”
Student 2: “It sure looks like a hexagon, but remember what Mr. Wees said? It might be just really close to a regular hexagon. How do we know for sure it is one?”
Student 1: “Hrmm. A regular hexagon has all the sides the same length, and this looks like all the sides are the same length, so it must be a hexagon.”
Student 3: “How do you know for sure the sides are all the same length?”
Student 4: “Yah, maybe they are like, one or two pixels off or something.”
Student 2: “How did you draw this? Can you show me how?”
Student 1 shows her group members how she came up with the construction. “See, it’s a regular hexagon.”
Student 3: “I notice that when you made each of the sides you used a circle.”
Student 4: “Yeah, I noticed that too.”
Student 2: “And all of the circles you used were the same size.”
Student 4: “Well, not all of them. There are bigger and smaller circles.”
Student 2: “Okay well all of the smaller circles are the same size.”
Student 1: “How do you know that?”
Student 2: “They all have the same radius, see? These two circles have the same radius, and these two, and these two. They are all connected so they all have the same radius.”
Student 4: “Ooooh, I have an idea…”
Student 1: “Me too! Those radiuses that he pointed out are the same as the lengths of the sides of the hexagon so if they are all the same, then all of the sides of the hexagon are the same size. Done!”
Student 3: “Is it possible to make a hexagon where all the sides are the same length, but the hexagon is still not a regular hexagon?”

 

Obviously this is an idealized situation, and maybe a bit unrealistic but this is where I would like students to end up. What kinds of classroom conditions would lead to students being able to do this? How would the class be structured? What kinds of supports would a teacher have to give to help support students?

Why do we teach math?

(Image source)

It could be because the mathematical procedures that are taught in schools will be useful to students later, but I am pretty sure this is false. Almost everyone forgets those procedures as they get older because most people in our society use virtually none of the procedures they learned in school in their day-to-day life. Obviously there are engineers, mathematicians, and scientists who use the mathematics they have learned, possibly on a daily basis, but I think if you dig deeper into the work they do, many of these people use tools to help to do their work (like Mathematica, for example), look up the finer details of mathematical procedures that they do not use often, or who use only a very specialized portion of their mathematical knowledge regularly.

It could be that we want to expose students to different ways of thinking about the world. In this case we would be less concerned with the exact set of mathematical procedures they have learned, and more concerned with learning mathematics as a way of thinking and knowing. I see little evidence that this is an explicit goal of mathematics instruction given that; the students are assessed only on the procedures, teachers are assessed on their students understandings of those procedures, and that the set of mathematical procedures we want students to know is so prescribed such that it is virtually identical around the world.

It could be that we would like students to learn transferable problem solving skills. In this case, we want to teach mathematics in such a way as to promote the likelihood that students will be able to transfer what they learn to other areas. Cross-disciplinary study would be the norm, rather than the exception. It turns out that “teaching skills that transfer” is not as simple as one thinks. In fact, my understanding is that most of the times when people learn skills in one context, they do not end up transfering those skills to other contexts. Instruction that aims for transferable skills has to provide opportunities for students to make connections between different areas, reflect on what they have learned, and develop metacognitive strategies so that students think about their thinking. What evidence is there that these types of activities are a regular part of math classes?

It could be that we would like students to see the beauty and elegance of mathematics. One way to do this could be through exploring mathematical art. Another might be to look at some famous examples of truly elegant uses of mathematics. We could also ask students to talk about mathematics in the abstract and come to a shared understanding of what elegance and beauty in mathematics mean. As far as I know, none of these activities is a common one in math classes. It is depressing to me that this way of thinking which has so much beauty in it is shared in such a way that almost no one in our society ever gets to experience beautiful mathematics.

If one or more of the reasons I suggested above is something you think is a good reason to teach mathematics, how are you ensuring that you meet this goal with what happens for students in your classroom? 

What other reasons are there to teach mathematics?

I recently read the Common Core standards for Math for grade 6, which is where the concept of variable appears to be introduced. The standard in question reads:

Write, read, and evaluate expressions in which letters stand for numbers.

I tweeted this out, and one person responded with this observation:

@davidwees in d/dx[f(x)]=mx+b …d, f, dx, f, x, m, and b all have completely different meanings, and only m and b “stand for numbers”

— John Chase (@mrchasemath) October 27, 2013

Clearly this definition of variable is limited, and possibly misleading.

How do you define variable with your students? How do you introduce the topic?