What happened to five?

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.

In September, Gwyneth was concerned about the precise use of language. She’s still at it, researching dog breeds on the internet. Hasn’t stopped. Saturday night/Sunday morning, she wanted me to see this:

Remember, her little sister is five.

“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:

A fun conversation, either way.

[Quiz Results] Content knowledge is important at all grade levels

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.

Grade 3/4 Fraction Action

Recently, I was invited into three Grade 3/4 classrooms to introduce fractions.

Cuisenaire rods give children hands-on ways to explore the meaning of fractions. After students built their towers, flowers, and robots, I asked, “If the orange rod is the whole, which rod is one half?” Students explained their thinking: “two yellows make an orange.” I emphasized, or rather, students emphasized that the two parts must be equal.

I asked students to find as many pairs as they could that showed one half. I let ‘em go and they built and recorded the following: Once more, with one third: As children shared their pairs, we discussed the big ideas:

• the denominator tells how many equal parts make the whole (e.g., two purple rods make one brown rod, three light green rods make one blue rod)
• the same fraction can describe different pairs of quantities (e.g., one half can be represented using five different pairs, one third can be represented using three different pairs)
• the same quantity can be used to represent different fractions (e.g., white is one half of red and one third of light green, red is one half of purple and one third of dark green, etc.)

Something interesting and outside the lesson plan happened in each of these three classrooms.

Some students described each pair of rods using equivalent fractions (e.g., 1/2, 2/4, 4/8):I asked the “we’re done” students to represent their own fractions using pairs of rods and determine each other’s mystery fraction. Many students chose fractions like 2/5 or 3/4, not simply unit fractions: After students shared the three pairs of rods for one third, I asked if anyone found any more. “I did,” said one student, unexpectedly. Check this out:I asked her why she chose to combine an orange rod and a red rod to make the whole. She explained that twelve can be divided into three equal parts. Without prompting, the rest of the class starting building these:

adapted from The Super Source

Math Picture Book Post #4: One Is a Snail, Ten Is a Crab

One of my favourite read alouds is One Is a Snail, Ten Is a Crab. In April Pulley Sayre’s “counting by feet book,” one is a snail, two is a person, four is a dog, six is an insect, eight is a spider, and ten is a crab.

The odd numbers to nine and multiples of ten to one-hundred are represented as combinations of animal feet. For example, three is a person and a snail; ninety is nine crabs or ten spiders and a crab.

Last week, Sandra and I visited a Grade 1 classroom in which we asked “How many different ways can you make ten?” Children read a number sentence (e.g., “six and four make ten”) to go with each of their drawings. Some students built the animals using muli-link cubes. Some students wrote addition equations (e.g., 6 + 4 = 10). There were multiple approaches to solving this problem. For example, this student skip counted by twos (I think).

These two students used the ten-fact pair of eight and two to make ten. Ten is a crab and a person (8 + 2) but this can be partitioned further as two snails and two dogs (1 + 1 + 4 + 4) or two dogs and a person (4 + 4 + 2).

Another student (sorry, no photo) broke up ten as five and five and then five as four and one; he drew a dog and a snail twice (4 + 1 + 4 + 1).

These solutions reflect an understanding of “ten-ness.” These students are not (just) counting feet. Gotta be the ten-frames.

It is important to provide opportunities for children to think about numbers as compositions of other numbers. Breaking up numbers, into tens and ones or in other ways, makes computations easier in later grades.

Click here for more math picture book (picture book math?) ideas.

Thanks to Ms. Long and the young mathematicians at Fraser Wood Elementary for inviting us into your classroom. Also, thanks to Pete Nuij and Lesley Tokawa for helping make this happen.

Toblerone Task

“I couldn’t help but admire your large triangular prism,” I wrote. Sadly, this is not the strangest way I have begun an email to a colleague.

“Are you talking about the giant Toblerone-shaped thing? You math guys are weird,” she replied.

Anyway… my three-act math task:

act one

• About how many regular size Toblerone chocolate bars fit inside the giant Toblerone-shaped thing?
• Give an answer that’s too big.
• Give an answer that’s too small.

act two

• What information would be useful to know?

act three

63. Relax. The video is coming soon.

sequel

• If 72 regular size Toblerone chocolate bars fit inside a mega Toblerone-shaped thing, how large would it be?
• If 112 regular size Toblerone chocolate bars fit inside a mega Toblerone-shaped thing, how large would it be?

better still…

• A mega Toblerone-shaped thing is a little bigger than a giant Toblerone-shaped thing. What could its dimensions be?
• How many regular size Toblerone chocolate bars would fit inside?

I like the phrase “a little bigger.” (No TWSS jokes please.) Probably “borrowed” from Marian Small. The ambiguity here allows for multiple solutions. Students could increase the length of the prism or the size of the triangle base. Which has the greater effect?

Also, there’s something interesting happening here with the sum of consecutive odd numbers.

Oh yeah… a shout-out goes to Andrew Stadel for his Couch Coins task.

Plotting Uses of Technology for Learning

How can technology be best used as a tool for learning mathematics? Calculators can assist with computations when learning other mathematics. iPads can help students communicate their learning. I’m asking about something else. I’m asking about the use of technology to develop new mathematical understandings.

Last week, Marc and I explored this question with about twenty math department heads. First, teachers were given time to explore several dynagraphs. In this version, the equation of each function was hidden. This became the problem to solve.

Following this activity, we wanted to discuss the question above. One approach would be to present several different examples and evaluate each, sharing our criteria. Not very effective. It’s our evaluation, our criteria.

Inspired by this

xkcd

we came up with the following:

After generating a list of possible uses, teachers were asked to plot them in the plane. For example:

Quadrant I: The dynagraphs investigation was placed in the first quadrant (active-understanding). The NCTM Illuminations Pan Balance applet can also be placed here. In this quadrant, learners build depth of conceptual understanding, be it of function relationships or algebraic thinking, through problem solving. Learners encounter many, if not all, of the seven mathematical processes identified in the curriculum. They communicate mathematical ideas, make connections among mathematical concepts and to past experiences, reason and justify their mathematical thinking, and use visualization to make sense of mathematics.

Quadrant II: An alternate version of dynagraphs was placed in the second quadrant (passive-understanding). The equation is no longer hidden, thereby replacing problem solving with observation. My GeoGebra material also fits here. You know the type: drag a slider (or, worse, watch the teacher drag the slider); what do you notice?

Quadrant III: Ah, yes, Khan Academy. Enough said? Probably not. Activity is limited to pressing pause and rewind. The “intuition” video comes later, if at all. In the third quadrant (passive-knowledge), learners consume content.

Quadrant IV: In the fourth quadrant (active-knowledge), you will find Math Blaster, an iPad app in which students practice math facts (+, −, ×, ÷) through gameplay.

Many interesting comments were made by the group. Some highlights:

“It’s about teaching, not technology.”

Activities can slide from the first quadrant. Who’s doing the math? If it’s the teacher, then we’ve moved to the left. Is the focus on “how-to’s” or essential understandings? If it’s the former, we’ve shifted down. The same holds true for uses of manipulatives.

“We can’t always be in the first quadrant.”

Fair enough. This activity provides one answer to the opening question. Quadrant I is the ideal. Is there value in quadrants two through four? I think so. A demonstration can be helpful. For example, this applet can help learners make sense of A = πr². So, too, can this low-tech activity. Is there a place for grapefruit? There may be. But this can’t be where we live. FWIW, it’s not just that KA occupies this space. It’s that it goes about it so badly. If you must have a grapefruit…

“You’re axes are wrong.”

At least one teacher suggested that the x-axis be labelled “active learner.” In Math Blaster, children are active in the sense that they are blasting through razor sharp blockades and speeding past the stars on their HyperCycles. A bit of a stretch to call this active learning. Other possibilities for each axis were suggested: student-centred/teacher-centred, conceptual/procedural, process/content, etc. We fully expected this. The intent of this activity was to generate discussion. The imperfection of our labelling of the axes only added to the conversation. The question “How can technology be best used as a tool for learning mathematics?” became “How do students best learn mathematics?”

Mathblogosphere Presentation

Sam Shah is compiling a list of presentations about the mathtwitterblogosphere. Here’s mine:

View this document on Scribd

The workshop was a 4½-hour mixture of problem-solving, show & tell, discussion, and self-directed exploration. This was no ‘sit and git’ workshop (slide 6).

Because teachers at the session came from several different schools, I started with a get to know each other icebreaker. Using the information on their name tags (slide 1), newly introduced teachers created a Venn diagram (slides 8-10) that reflected some aspect of their group.

Presenting on the work of others can be a little odd. Hat tips were given to the mathblogosphere in general (slides 12 & 13) as well as to individual bloggers.

In these groups, teachers solved three problems: stacking cups (slides 14 & 15), LEGO optimization (slides 19 & 20), and visual patterns (slide 23). I connected these problems to related lesson ideas (slides 17 & 21) and teacher-created classroom resource websites (slides 18 & 21).

Next, teachers took part in a couple of activities that could be easily translated to different topics: Pictionary (slide 24) and matching cards (slides 26 & 27). These activities address two of the seven WNCP mathematical processes: visualization (slide 25) and communication (slide 28).

In addition to lesson ideas and teaching strategies, I wanted to draw attention to the mathblogosphere as a place to find conversations (slide 29). Participants chose to read and discuss one of four listed blog posts (slide 30), forming new groups.

Launching off the mathtwitterblogosphere site, teachers were given time to get started using Google Reader, explore on their own, and share their discoveries.

I hope this is helpful to those of you planning presentations on this topic.