It is 4:22pm on Wednesday, April 4th. An eccentric billionaire has gathered $5 billion in 500-dollar bills and proceeds to hand them out, one each second, without stopping. What is the exact date and time when the last bill is given away? -A puzzle I got from Mo Page

I’m flabbergasted. I have a number of students—maybe 10? 20?—who determine by division how many bills there are, then figure out by multiplying 60x60x24 how many bills are given away in a day. Fine. But then they start subtracting … after day 1 there are 9,913,600 bills left. After 2 days there are 9,827,200. Almost immediately many students lose interest, but there are a few arithmetic ox that start chugging through it (with calculators, to be sure). 9,740,800. 9,654,400. I watch in disbelief as the markerboards are filled in, line by line. 8,617,600. 8,533,000. After a while I can’t help myself. I casually mention that people sometimes use division to do repeated subtraction, and I countdown from 10 by 2’s and compare to 10/2. They are a little chagrined at not having thought of that, but they try it. Then they face confusion about handling the remainder.

I don’t believe math must be learned in a particular order. My seniors don’t need to model repeated subtraction with division in order to learn the basics of trigonometry. But, really? Should I really keep teaching precalc instead of throwing it out and teaching pre-algebra? I don’t even know *how* to teach pre-algebra! But I’ll learn … if that’s what’s needed.

What should be the primary goal for instruction of students who are placed at one level but who are missing huge chunks of what they were “supposed to have learned” years ago? I feel like I keep asking this same question in different ways on this blog. Instead of elegantly reasoning my way to a solution, I just keep doing what I know how to do, day after day, hopefully making progress, gobsmacked at the glacial slowness.

Brute force. Them and me both.

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>I feel like I keep asking this same question in different ways on this blog.

Keep asking it! It’s the right question to ask. I don’t know a good answer, but I might post about this too some day…

“Brute force. Them and me both”

Love your epilogue!

I have no idea what to do either. I faced this problem last year in Alg2, and this year in Geometry. How am I supposed to get kids up to speed in applied Algebra problems in Geometry when they can’t add signed numbers? When they can’t subtract? When they think that 2x might be 21, 22, 25, 28 or 20?

Another issue is that the kidoos rebel when I spend time working on arithmetic. They tell me it’s baby stuff, and they complain that it’s not a real class. So if I do arithmetic work, it has to be secretly, hidden in Geometry. At least that’s what it feels like.

I have no idea what to do. At all. Here’s what I sort of do now:

1) Hide the arithmetic stuff in good Geometry problems. Like, “what’s the measure of the other four angles in this pentagon.”

2) Build in a little bit of number-sense review into the beginning of class, as a warm up or a game.

3) Spend just a little more time on units that they clearly need complete reviews of (with my group this year, it’s solving basic linear equations and anything that has to do with slope).

There’s not enough time to do both what people want you to do and what kids need. Ultimately, we might need to give up our course-based model of student learning, and move to a more fine-grained competency based model. That has its own challenges, though…

Indeed. One option: Quit and become an elementary school math teacher.

Here’s a slightly more serious and intriguing proposal from Eugene Rutz writing for the American Society for Engineering Education.

Also, if you aren’t already, try reading Brian Frank. He has a remarkable ability to take a process of student reasoning that would make me weep, and use it as a springboard to something remarkable.

“One option: Quit and become an elementary school math teacher.”

I think you would soon find yourself in the same quandary, just with different concepts. I have 4th graders that I’ve been working with all year (and still am) on skip-counting by 2’s and 5’s–a skill they “should” have mastered by the end of 1st grade. If they’re struggling with this prerequisite now, what is the probability they will recognize repeated subtraction as a division situation in high school?

@Sue: I don’t think I’m going to get a final answer (or get tired of asking) anytime soon!

@Thanassis: It’s remarkable how often I feel like what they’re going through is like what I’m going through. I guess I’m always projecting. Or maybe learning is pretty much the same experience for everyone all the time.

@Michael: You and I seem to angst along the same lines. Fortunately we have Mylène to turn to …

@Mylène: I found myself nodding (with agreement, not sleep) as I read Rutz. As a high school teacher I’m most interested in suggestion 2, and it falls along lines we’ve been discussing at school in thinking about the goals of our 4-year curriculum. And that’s a GREAT post from Brian — I’d noticed his writing in comments here and elsewhere but you just got him one more regular reader. Thanks for the steer.

[…] Dan Goldner: I’m flabbergasted. […]

Dan, I feel your pain. It’s the world I teach in as well. I’ll use my 2 cents to reiterate Michael’s option 2 – my daily walk-in procedure consists of a warm-up/bell ringer followed immediately by a timed speed drill which I use to build up procedural fluency on stuff “they were supposed to have learned.” Just the fact that it is timed makes it sufficiently novel that the kids don’t complain about doing “kid stuff.”

Yeah. Brian’s the bomb for sure. I second Mylene’s suggestion that he has something to offer us math teachers.

The situation you have described here is totally maddening, for sure. We’ve all been there. An important role of the math teacher is one we too often forget (or do not really know how to do). That’s to help students step outside their present way of thinking to examine them from a new angle.

In this context, I notice that you had students who divided once in a repeated-subtraction situation: How many groups of $500 in $5 billion? I would want to stop and ask, “What question were you trying to answer there?” and “What question are you trying to answer now?”

At heart, these two questions have the same structure. I want to ask questions that help these students not with their arithmetic, but with seeing the underlying division structure.

Thanks for sharing!

Great post and comments, especially the “brute force” imagery.

Disclaimer: I’ve only taught up to Algebra 1 🙂

Fractions get a lot of attention as the thing that ruins many students on math, but I’ve come to believe that division plants the seed. I’m willing to bet that if you suggested to students that there’s a faster or more efficient way to solve this problem, rather than use division they might start guessing how many days and multiply that by the number of bills they found for one day. They would guess and adjust, even incorporating decimals into their guess until they got sufficiently close. I’ve seen many middle schoolers do this to figure out 7 x ___=30. (of course this might come after convincing them that such a number exists!)

Division is hard. If 6 cookies shared among 3 kids is 2, where did the other 4 cookies go? We’re unfortunately not as explicit as we could be about the rate aspect of this type of problem, and assume that students understand that the result is 2 cookies per kid. No cookies disappear, but it seems like we forget about them. To complicate it, we could pass out the 6 cookies 3 at a time, and the answer is 2 again! This time it’s 2 kids, though. With fractions and decimals and remainders and whatnot, this gets complicated pretty fast. Further, scaling problems aren’t about sharing at all. Does division mean one thing (sharing) or many things?

If students don’t understand division, what could they possibly think about proportions or slope? Or even really believe a fraction and a decimal could be equivalent?

All that is to say that my suggestion is to recognize and mention division every opportunity you can. I did this for a year with the distributive property for 6th-8th grade and was surprised at how many opportunities arose.

I’ll definitely add your blog to my reading list.

As someone who has done math coaching with upper-elementary teachers in Pontiac, middle school math teachers in various parts of NYC, and high school math teachers in Detroit (the past three years), I feel confident in saying that the problem of how to address the myriad Swiss cheese of holes one finds in the mathematical knowledge of many students at every grade band is widespread and pernicious. The ‘traditional’ approach, which I have not found to be effective, amounts to “louder and slower.” Like Americans trying to make themselves understood by non-English speakers, teachers are assigned remedial mathematics classes (or turn their classes into same, at least until the principal or some other authority tells them otherwise) in which they reteach the same arithmetic and other prealgebra topics in the same ways that failed to reach the students before, but with that same phony patience that informs the frustrated American dealing with those damned foreigners who insist upon not understanding or speaking English (the nerve!) – hence, “louder and slower.”

But I don’t really blame the teachers. How can they teach algebra 2 with integrity to kids who can’t do arithmetic?

The “best” solution, to my mind, is not to put the brakes on, taking the first x days, weeks, or months to try to bring everyone up to speed, while letting the official curriculum lie fallow. That’s a sure formula for a lot of grief.

Rather, I’ve counseled finding the many spots in one’s lessons where it should be obvious that students will need a given skill and understanding of some bit of mathematics to succeed with the topic at hand. An obvious example is finding distance on the Cartesian plane. There is a high likelihood that even kids who have some natural understanding of the issue (if asked casually) will struggle to compute with (let alone remember) the distance formula. Part of the problem, of course, is that too often it is simply presented as a formula to be used somewhat mindlessly and in some classrooms remembered regardless of understanding. I don’t think it revolutionary of me to suggest that if students leave high school without memorizing the distance formula, their lives will not be diminished, but if they leave without realizing (or being exposed to) the connections among the distance formula, right triangles, and the Pythagorean Theorem, some several math teachers have been asleep at the switch.

That said, even with all those important conceptual ties made, kids will still fail if the lack the ability to subtract signed numbers. It’s a killer for many students, and it isn’t going to go away in high school math and science and beyond (to say nothing of many ordinary real world tasks). So the opportunity arises here to do a mini-lesson that really tackles their conceptual difficulties with integer arithmetic, to ground their understanding and skills in several models that might help them make better sense of it all than simply repeated a set of rules that they never could keep track of to begin with. And when I do professional development with secondary teachers, I strongly emphasize the sad truth that few if any of them received professional training in elementary mathematics methods when preparing to become math teachers. Hence, they need to admit that they NEED to learn a lot of the models and metaphors from elementary mathematics methods that inform any topic they hope to successfully remediate with middle and high school students. There are quite a few good ones, but I’ve yet to find one that is going to work with all kids. If you are prone to believe, for instance, the “money” is the magic bullet for teaching integer subtraction, you’re likely kidding yourself.

The above example is, I hope, suggestive of how to more organically introduce ‘remediation’ as it arises. And it most assuredly will. But the problem is not well-addressed through your putting off teaching algebra or other subjects and topics until arithmetic is “mastered.” Kids resent being put in strictly remedial classes, so it’s important that they see that what you’re doing is both making essential connections between arithmetic and algebra, and helping them build their capacity in both.

This is how it is explained over in the Victorian education system (in Australia we don’t have the US structure of Algebra 1 & 2 etc). Children move in stages and if one of these stages is missing then trying to build further knowledge on top is like a building without a proper foundation – which we see a lot over here too! The reason students don’t get 2x (why is it a multiply, is it 20 + x or 2 + x ) is because they think additively not multiplicatively. Multiplicative thinking is a level beyond additive thinking (it’s the shift from 5 + 5 + 5 = 15 to 3 x 5 = 15). Students start at make all count all/trusting the count (if I have 15 counters and I add 6 I don’t start at 1,2,3,4,5,6 and then count on another 15 or start at 1 count to 15 and then count on another 6 – I can start at 15 and add on 6). Skip counting implies that they trust the count (they don’t need to count every time), a way to scaffold this is to allow students to make arrays such as these

1 2 3 one three

4 5 6 two threes

7 8 9 three threes

10 11 12 four threes

13 14 15 five threes

16 17 18 six threes

19 20 21 seven threes

22 23 24 eight threes

and so on for all their times tables. This has the implied use of “times” in the vocabulary without the students getting hung up on it and shows count all students that this is always true. Then you can move to 3 times 5 is 15.

Arrays provide a really good structure to show informal multiplication strategies which then make it easier/logical to lead into the formal algorithm and also for algebra.

Eg 15 x 6

10 5

6 Imagine an array of 6 rows by 15 columns here and the multiplications written over the grid

6x 10 = 60 6 x 5 =30

So 15 x 6 = 60 + 90

This moves beautifully on to algebra when the sides don’t have to be 10 it could be x so 6(x + 5) = 6x + 30 and so on for factorising, expanding perfect squares….

Our department of education has created a series of lessons based on the ideas mentioned above which aim to move students on from count all to multiplicative thinkers (called scaffolding numeracy in the middle years, SNMY). This has been the briefest intro so please feel free to email me if you have any further questions 🙂

Thanks, all. I don’t think I’m really considering dropping the curriculum and teaching pre-algebra, though I know I threatened to in my post. I’m feeling quite acutely what “akismet” states: I’ve been prepared to teach secondary math with no knowledge of how primary math can be properly described and taught. It’s great to get the perspectives of folks like Christine and Belinda who have wrestled with teaching these concepts directly.

You ever figure out how to bone up on primary school math? I’d like to start working backwards this summer, and I’m not sure where to start.

@Michael Pershan: Try ANY edition of van de Walle’s Elementary and middle school mathematics: teaching developmentally. There are used copies of the 1994 edition for sale for $0.99 + shipping. Absolutely no need to pay over $100 for the 6th, 7th or forthcoming 8th edition.

@Michael: After my seniors graduated this year I was left with 5 Juniors who will be taking statistics next year. None of them were confident with fractions. So we worked through James Tanton’s Guide to Everything Fractions, and they found it challenging and enjoyable and effective. I suspect Tanton’s book on arithmetic (which Sue just reviewed and from which the fractions guide is excerpted) would be a great source for ways to talk about primary school math that is accessible but challenging (and therefore interesting) to older students.

Another endorsement for Tanton’s books. I have 7 of them.