In the summer, Costco peddles a buttload of educational workbooks. You know the ones: collections of every worksheet necessary for your child to complete <insert grade here> Math. Can’t find them? Look over by the Christmas trees.
I picked up the Grade 3 book. Just browsing. Killing time. I opened to this page:
I’m not a big fan of this approach. Forget about comprehension, just scan for the add or subtract words. See more, think add. But it’s not that easy. More shows up in five of the practice exercises. Try them.
In the picture, how many more 4-legged animals are there than 2-legged ones?
Peter has 39 goats. He wants to have 64 goats. How many more goats should he buy?
Peter has 68 animals on his farm. He buys 23 more. How many animals does he have now?
413 gulls are joined by 311 more. Then 136 more gulls come. How many gulls are there altogether?
There are 576 gulls, but 153 fly away. Then 283 more leave. How many gulls remain?
A mountie (really?!) tells kids (Canadian, no doot) to decide on the operation.
From the answer key:
In the picture, how many more 4-legged animals are there than 2-legged ones? 15 − 12 = 3
Peter has 39 goats. He wants to have 64 goats. How many more goats should he buy? 64 − 39 = 25
Peter has 68 animals on his farm. He buys 23 more. How many animals does he have now? 68 + 23 = 91
413 gulls are joined by 311 more. Then 136 more gulls come. How many gulls are there altogether? 413 + 311 + 136 = 860
There are 576 gulls, but 153 fly away. Then 283 more leave. How many gulls remain?576 − 153 − 283 = 140
Subtraction is used to answer three of five questions with this ‘add’ word. Actually, kids will think addition for the first two questions (12 + 3 = 15 and 39 + 25 = 64) but that’s another post.
My boyfriend came across this while packing for a trip. I thought "What a weird requirement." #anyqs? pic.twitter.com/BgIjec3nSx
— James Cleveland-Tran (@jclevelandtran) July 16, 2013
It is weird. You would think the size limit would be volume, not combined length, right?
The first question that came to my mind was “What are the dimensions of the bag with the greatest volume?”
A “cubey” bag with a length and width of 21 inches and height of 20 inches would have a volume of 8820 cubic inches, or 5.1 cubic feet. The airlines are banking on your bag looking more like the one pictured above. The dimensions are not shown, so let’s assume the golden ratio is at play here:
w + l + h = 62
w + w(1.618) + w(1.618²) = 62
w(1 − 1.618³)/(1 – 1.618) = 62
w = 11.84
w = 12 in, l = 19 in, h = 31 in
A “golden” bag would have a volume of 7068 cubic inches, or 4.1 cubic feet. If passengers were able to check a “cubey” bag, they’d be able to pack about 25% more. Of course, the airlines would still get ’em with the weight limit.
I guess it does make sense to express the size limit in inches rather than inches cubed. After all, a bag with a length and width of 1 inch and height of 7068 inches would also have a volume of 7068 cubic inches.
Math teachers have seen this type of problem before, but never like this. We’ve seen farmers with x feet of fencing faced with the challenge of enclosing the largest possible pig pen. In later grades, we insist that the farmer use the exterior of the barn as one side. Length is given and area is maximized. This can be reversed. That is, given the size of the pen, our farmer must use the least amount of fencing.
The checked baggage problem, on the other hand, jumps a dimension. We’ve never seen problems in which length is given and volume is maximized. I wonder if this opens up some interesting possibilities.
As a student teacher, my mentor teacher gave me this advice. It stuck. For 15 years, it’s been a helpful mantra. A reminder to:
focus on what students are able to do when solving multi-step equations,
recognize some mistakes as being overgeneralizations (e.g., a negative plus a negative is a positive), and
think of contexts in which math mistakes make sense (e.g., 1/3 plus 2/5 does not equal 3/8, except with at-bats in baseball, or powerplays in hockey, or marks in math class, or …)
“You assign grades. Your gradebook offers suggestions.”
Advice given to me as a first year teacher. It stuck. Not helpful day-to-day but invaluable on certain days (i.e., when marks are due). Over the years, remembering this gave me permission to consider other evidence of what a student knew (e.g., classroom observations and conversations with the kid) and assign a higher letter grade when appropriate. Obvious to me now, but as a beginning teacher? Not so much.
Also, it helped me take the top-down policy of “No 46s to 49s” in stride. Bent out of shape, some colleagues took this to mean “45 is the new 50.” Others reacted like a Tim Horton’s franchise owner facing the Canadian government’s phasing out of the penny: rounding down 46s and 47s, bumping up 48s and 49s. Most teachers felt compelled to call students in to finish just enough missing work to reach the magical 49.5. I avoided the silliness. Marks were my decision. Always were. Now I just had fewer options.
#anyqs?
“I ask my students to explain their thinking, and they automatically reach for the eraser.”
Not advice but an observation made by a colleague earlier this year. It stuck. I’ve been working on consistently asking “Why?” both with students in classrooms and with teachers in workshops. It’s easy when students (or teachers) give incorrect answers. Hence the association, built up over time, between “Can you explain?” and the eraser.
It’s also easy when students (or teachers) present an unexpected solution method. But even “I’m curious. Can you explain?” is met with skepticism. “It must be a trick, I must be wrong,” the thinking goes. My reaction to seeing the hand reach for the eraser is often something like “No, it’s right! I just don’t know how you got it. Can you help me make sense of it?” That first part feels cheap. Reassuring the student/teacher may lessen his/her anxiety, but it frees him/her from having to construct a viable mathematical argument. It’s disempowering.
Tougher, for me, is asking “Can you explain?” when I instantly recognize the solution method (e.g., Group A, lowest common multiple, check. Group B, proportion, check. Group C, unit rate, check). But not asking “Why?” here creates the reaction described above.
“How often do you hear a student start a second sentence?” Interesting metric for evaluating how well a math teacher promotes explanation.
“How often do you see a student reach for the eraser?” could be another.
Each sticky quote above (< 140, btw) is probably long forgotten by the speaker. Certainly, they would be surprised to learn that I remember. I’m curious, what are your sticky quotes?
Last week, Nancy Allan, Manitoba’s Education Minister, announced the province’s revised back to basics math curriculum. The move was applauded in a Winnipeg Free Press article.
So, why the changes? From the Winnipeg Free Press: “I heard from parents that their kids were lacking basic arithmetic skills,” Allan said. “It was during the (2011) election, and I picked this up on the doorstep.”
What we know about how children learn mathematics is no match against a politician with a political football. Allan spent “two years of serious work with [the province’s] education partners. We have met with all the math professors, the superintendents have been part of this.” Notably absent from the list: math educators, faculties of education.
The bottom half of the internet gives us a glimpse of Allan’s doorstep. I could have saved her those two serious years. Rather than a revision, her “kids these days” constituents could have been placated with the addition of one prescribed learning outcome, one PLO to rule them all:
Students will be able to make change.
Where an appeal to logic has failed the reform movement, an army of pimply-faced cashiers able to count back change to John Q. Public – even after he’s thrown down an extra nickel after the fact – might just succeed. Of course, this skill involves applying the mental math strategy of thinking addition for subtraction, not the standard subtraction algorithm. But let’s not tell them that.
The Winnipeg Free Press article featured a poll asking readers “Which of these everyday math tasks could you tackle without a calculator?”
Two of my favourites:
Determine how much you should leave for a 20% tip at a restaurant.
14% (6192 of 45357 votes)
Halve a recipe that calls for 2/3 of a cup of an ingredient.
12% (5577 of 45357 votes)
The results are interesting in light of the following: “The minister said the revised curriculum makes Manitoba the first province in Western Canada to go back to placing an emphasis on basic skills previous generations had.” At best, it looks like previous generations have misplaced those basic skills. At worst, they never had ’em. Psst, hey kids… remember this the next time your uncle quizzes you with “What’s 7 times 8?” He may be asking because he doesn’t know.
So, what happened? Previous generations learned to do math, by definition, “the old way.” They were taught the standard pencil and paper algorithms. In the two tasks above, the standard algorithms are far from efficient. In the first task, a more efficient strategy is to divide by ten and double. In the second task, it’s two thirds of a cup… two thirds… twoOOOoooOOOooo thirds.
Buried in the announcement: “Allan said the province will provide parents with a website to help them understand what their kids are learning.” Admirable. But its necessity is an indictment of the parent’s, not the child’s, mathematics education. Not many politicians are going to pick that up on the doorstep.
I wrestled with including the first fifteen seconds of the clip. Will students ask their own questions if they suspect they’re going to answer one of Bourdain’s? Does the remainder of the clip make sense without this? Or, are the first fifteen seconds the first act, the remainder the second? By the way, Bourdain does a pretty good job on his blog of tossing out questions students may have:
Was I doing a good thing? Is it OK to be in the chocolate business? I don’t have any problem with wealthy people who can afford making impulse buys in expensive gourmet shops spending a lot of money on my chocolate. But where does the money go? In fact, where does this chocolate come from anyway? Just about everybody loves the stuff. It’s everywhere. A fundamental element of gastronomy. But I knew so little about it. Where does it come from? How is it made? Most importantly, who does it come from? And are they getting a good piece of the action? Or are the producers, as in so many cases, getting screwed over? I very much hoped to find that whoever was growing our cacao was, at the end of the day, happy about the enterprise — that life after Eric and Tony’s Excellent Chocolate Adventure was, on balance, better than life before.
Act Two
What information would be good to know? I wanted to know, what is a “nosebleed price”? From the man himself:
Thing is, it’s a very boutique-y, very high end, screamingly expensive end of the biz. One of the only 7,000 bars we were able to produce (the whole year’s supply sold off in just a few months) cost the nosebleed price of $18. Even reflecting the remote location, the rarity of the raw ingredient, the long trip from the mountains to the city to Switzerland and then to the States — the whole artisanal process — that’s still a f**k of a lot of money for a chocolate bar.
It looks to me like the producers get 15% of each chocolate ten-frame for the raw cacao, labour another 2.5%. For comparison, the three investors get 5% each.
Act Three
Raw Cacao: $2.70/bar; $18 900 in total
Labour: 45¢/bar; $3150 in total
Doesn’t exactly answer “Are they doing a good thing?” does it? And is it even possible to “show the answer” to this question? Can we adapt this task so that students use proportional reasoning to make a case for our cacao growers rather than just perform a couple of quick calculations? That is, can students use math to answer “How fair?” rather than “How much?” Differences in purchasing power and cost of living between nations now come into play.
Maybe this just doesn’t fit the three-act framework. Too bad. I kinda liked this sequel: How long would a Peruvian cacao grower have to work to purchase a luxury chocolate bar in Manhattan?
123? 12.3? 1.23? One has to ask oneself one question: Which one is one?
Earlier this year, I was invited into a classroom to introduce decimals. We had been representing and describing tenths concretely, pictorially, and symbolically. We finished five minutes short, so I gave the students a blank hundred-frame and asked them to show me one half and express this in as many ways as they could.
As expected, some expressed this as 5/10 and 0.5. They used five of the ten full ten-frames it takes to cover an entire hundred-frame. Others expressed this as 50/100 and 0.50. They covered the blank hundred-frame with fifty dots. I was listening for these answers.
One student expressed this as 2/4. I assumed he just multiplied both the numerator and denominator of 1/2 by 2. And then he showed me this:
One student expressed this as 500/1000 and 0.500. I assumed he was just extending the pattern(s). “Yeahbut where do you see the 500 and 1000?” I asked challenged. “I imagine that inside every one of these *points to a dot* there is one of these *holds up a full ten-frame*,” he explained. As his teacher and I listened to his ideas, our jaws hit the floor.
In my previous post, I discussed fractions, decimals, place value, and language. To come full circle, what if we took a closer look at 0.5, 0.50, and 0.500? These are equivalent decimals. That is, they represent equivalent fractions: “five tenths,” “fifty hundreds,” “five hundred thousandths,” respectively. From a place-value-on-the-left-of-the-decimal-point point of view, 0.5 is five tenths; 0.50 is five tenths and zero hundredths; 0.500 is five tenths, zero hundredths, zero thousandths. Equal, right?
We know about improper fractions. What about improper decimals? #placevaluechat
— Christopher Danielson (@Trianglemancsd) May 7, 2013
If 15/10 is an improper fraction, then shouldn’t 1.5 be an improper decimal? Or is 1.5 a mixed decimal, having more in common with the mixed fraction 1 5/10? Both? Neither?
One definition of decimal:
A fraction whose denominator is a power of ten and whose numerator is expressed by figures placed to the right of a decimal point.
Thus, in 1.5, the implied denominator is 10 and the implied numerator is 5, the figure to the right of the decimal point. We read 1.5 as “one and five tenths,” a mixed decimal. The whole number part is treated separately, making an improper decimal an impossibility.
But what if we didn’t just look at the figures to the right? Nested tenths don’t stop/start at the decimal point. What if we looked at all the figures? We’d read 1.5 as “fifteen tenths,” an improper decimal.
Maybe the improper vs. mixed comparison is throwing me off track. Fractions can be classified as either proper or improper. Why not decimals? Decimals less than one, such as 0.5, would be proper; decimals greater than or equal to one, such as 1.5, would be improper (or, in Britain, top-heavy).
Christopher Danielson wasn’t trying to introduce new vocabulary to the world of math(s). Probably. Rather, he was making a point about place-value.
@mpershan Yeah. That's the conclusion I came to also. I'm thinking we don't spend enough time exploring how many tenths 1.5 is.
— Christopher Danielson (@Trianglemancsd) May 7, 2013
When we teach decimals using ten-frames we do.
If the whole is one full ten-frame, students may build 3.7 like this:
Students will describe 3.7 as “3 ones and 7 tenths,” “37 tenths,” or even “2 wholes and 17 tenths.” This mirrors what students know about place value and whole numbers: 37 can be described as “3 tens and 7 ones,” “37 ones,” or even “2 tens and 17 ones.”
Just like with whole numbers, thinking about place value makes calculations with decimals easier. For example, consider 4.8 + 3.6:
4 and 3 make 7
0.8 (“8 tenths”) and 0.6 (“6 tenths”) make 1.4 (“14 tenths”)
7 and 1.4 (“1 and 4 tenths”) make 8.4 (“8 and 4 tenths”)
Note the shift in thinking, not notation, from 1.4 as “14 tenths” to 1.4 as “1 and 4 tenths.” With fractions, it’s a shift in thinking and notation. Probably why we know about improper fractions but not improper decimals.
At 10:00 pm Saturday I returned home from #NCTMDenver. My daughters Gwyneth (8) and Keira (5) were glued to me for the next two and a half hours. Mostly playing with the Zometool kit I picked up at the exhibit hall, filling me in on the past five days.
“What should I click, Dad?” she asked. I was just about to reply “Doesn’t matter, just pick one” before I stopped myself. Instead, I told her to pick the best wrong answer. I was just curious, not trying to prepare my daughter for future success on bubble tests. “Six to eleven,” she quickly answered. Her confidence surprised me. “Nah, gotta be under 4,” I said.
With some prompting (needling?) she presented three arguments. First, Gwyneth reasoned that since Keira was “five and a bit” her sister was closer to six than four. She argued that it’s less than a year until her sixth birthday and it’s been over a year since her fourth birthday.
Second, she reasoned that “five and a bit” was more than five, the halfway point between four and six.
She gets it. Kids get it. They get that 37 is closer to 40 than 30. They get that 7.3 is less than halfway between 7 and 8. They get it until we ask them to memorize things like “Five and above? Give it a shove.”
Third, Gwyneth argued that since she is eight and her sister is five, the best answer is the one that includes the two of them. A stretch to connect this to measures of central tendency?
I’m not sure if Gwyneth enjoys finding these things for her dad or if she thinks it’s getting her one step closer to this:
Two months ago, I asked, “Which graph best represents the importance of teacher knowledge of mathematical content as a function of grade level taught?”
Twenty-six of thirty-five respondents answered C, matching my answer key. Three out of four math teachers agree: content knowledge is important at all grade levels.
For example:
Sure, you need lots of mathematical knowledge in order to be able to guide students to understanding of the advanced mathematical concepts taught at the upper end of school, but it is also vital that for early years teaching, and throughout elementary school, teachers have a strong knowledge of mathematics. Sure, they might only teach basic number skills, but they need to be able to make connections between ideas, understand the deeper significance of these ideas.
Some picked up on my choice of importance, rather than amount:
You said it’s about the IMPORTANCE of the content knowledge, not the amount they have. For students to develop concepts, they need tasks that help them to engage in and to connect with mathematical big ideas. From the choice or design of tasks, to the good questions that get asked to help students make those connections, the teacher’s content knowledge is critical – in some ways that’s even more important in the early years, but I think an argument could be made that it’s hugely important across the grades.
And again:
I think that teacher knowledge is equally important at every grade level, but a teacher needs to know more mathematics in the higher grades. If the question were about the quantity of knowledge rather than its importance, then I would choose D.
Not all who chose C would buy this amount argument. Not more/less, just different:
But the content is different as grade changes. Calc teachers don’t need to know cognitive structures of place value like K-3 teachers do, for example.
My guess is that those who chose E or its poor cousin D (six in all) would cite complexity. Tom wrote,
The more I learn about high school math (second year teacher, now teaching Alg I, Alg II, Pre-Calc), the more I realize how nuanced upper level topics are. I sat in on a Calculus class and was blown away at the difficulty of it (coming from a math major!) – we’re not just cranking out derivatives here. While TEACHING each grade level requires specific knowledge of HOW students learn each topic, I think the complexity of the math itself increases. Probably not exponentially, but faster than linearly.
Not so fast:
Too frequently it is assumed that elementary teachers don’t need deep knowledge because they’re just teaching kids how to count and add. How hard could it be? But the thing is, elementary teachers are helping very young children build very sophisticated concepts regardless of how easy an algorithm might be to memorize.
Graph A is my take on the complexity question, my response to “Anyone can teach Math 8.” Logarithms in Math 12? Easy peasy lemon squeezy compared to dividing fractions in Math 8. You know the algorithm–just flip it and multiply–but can you answer the 13-year-old who asks why? Then again, maybe I’ve just missed the nuance of logarithms. Thanks for planting that seed, Tom. By the way, nobody chose A.
Only one person chose B. This truly shocked me. I was expecting a much larger number. After all, the role of the teacher has shifted. No longer the primary source of content, no longer…
the sage on the stage.
But here’s the thing: dispensing knowledge requires only a little bit of content knowledge. That and a chisel tip whiteboard marker/Wacom pen. Posing differentiated tasks that will engage students in and help them develop an understanding of the mathematics to be learned? Now that requires content knowledge. It requires that the teacher understands this mathematics deeply. And yes, content is googleable but you need some mad Google-fu skills to get past the procedural.
At the risk of coming across like one of those nutjobs who finds a war on Christmas in “happy holidays,” what importance is placed on content knowledge in “I teach children, not math”? Kids before content. I get that part. Given a choice, I’d pick the pedagogue over the mathematician for my kids. Not even close. But “not mathematics”? To me, it paints a false dichotomy:
Planning and implementing learning tasks, assessing and supporting students’ learning… these must be guided by an understanding of the mathematics at hand (and how this connects to other ideas students see earlier/later).
A better picture:
In fact, some respondents speculated about which graph best matches the importance of PK and PCK across the grades. Most landed on C.
An interesting comment with pro-d implications:
Content knowledge is always important. In the younger grades, teachers need to be able to build and encourage mathematical ability in young students. If they do not have a solid understanding of math, then they themselves can be wary, and students are given Mad Minutes and the like…
Here, the mad minute, a teaching practice, is seen as a symptom of a lack of content, not pedagogical, knowledge. This probably goes against conventional wisdom.
A final comment from David Wees:
What I really wanted to choose was a graph that showed that teachers mathematical content knowledge over time should increase, to demonstrate that they are learning. So while I think C would be ideal, teachers could start anywhere on the scale, provided they are willing to put in the same time exploring mathematics as do their students.
What does this mean? First, “this doesn’t mean elementary teachers need to be versed in differential equations.” Content knowledge can grow with experience… if it’s believed to be important.
Note: I’m wondering if responding to the survey implied anonymity. Please let me know if you wish to have your name attached to your comment.
Chris Hunter 