Tag Archives: math methods

Landscape of learning

I wrote quite some time ago about a student of mine in the Math Center (the site of our developmental math courses) and her struggles to learn our textbook’s algorithm for finding least common multiple.

I have brought that post up in personal conversations with students and colleagues several times and I now realize that not everyone sees in the example what I do. So now I will be more explicit.

I see three important themes playing out in my post:

  1. Algorithms (Karen and I developed a new one),
  2. Standard topics in algebra (of which finding LCM of algebraic expressions is one, and which deserve critical reexamination on a regular basis), and
  3. A plea for compassionate practice in mathematics teaching.

What I really had in mind was the third.

Karen was struggling. She was frustrated. I knew she had been working hard and that the textbook explanation simply made no sense to her. Her ideas and the textbook techniques were too far apart.

In their book series Young Mathematicians at Work, Catherine Fosnot and Maarten Dolk write about metaphors for learning mathematics. In particular they contrast a linear metaphor with a richer one.

learning math as a linear process

We have many, many ways of talking about learning mathematics as a linear process. We speak of students being ahead or behind. We talk about accelerating students through material. We speak of filling gaps in student knowledge. We work hard on sequencing material for students.

In this linear metaphor, we presume that one student’s learning process is much like that of each other student. The main difference between students is how quickly they acquire each skill and move on to the next one.

It is as if each student is traveling a highway. The teacher’s job is to keep students moving down this highway at an appropriate speed. When a student, such as Karen in my LCM example, strays from the highway, it is the teacher’s job to get her back on the highway.

Learning math as moving through a landscape

Fosnot and Dolk challenge us to think about learning in a richer way. Their enriched metaphor supposes that there are many ways to know a mathematical topic. They want us to consider the learning process as navigation through a landscape. We want students to head in a particular direction, but we do not assume that there is a single, linear path.

Instead, different students will follow their own paths. It is the teacher’s job to know this landscape very, very well so that when a student is lost, the teacher can help her find a way forward.

A thought experiment

Imagine you are visiting a dear friend in an unfamiliar city. You go out on your own for the day and get lost on your way home. You call your friend. Which of the following two responses will lessen your anxiety and make you trust that you will end up getting where you are going?

  1. Get on I-94…I know you’re lost, that’s why you need to be on I-94; it will take  you right to my house.
  2. Where are you and what do you see? What landmarks do you remember passing recently? Can you see any street signs?

In the linear metaphor, we constantly tell students to get back on the highway (even if they have no idea where the entrance ramp is). In the landscape metaphor, we begin with where they are and help them to get where they need to go.

Karen was way, way off the highway when she was canceling factors. She saw a procedural connection to canceling common factors in rational expressions and she was following that path. I could have tried forcing her back onto the highway (citing the union of the sets of factors). But the more compassionate route was to help her develop an algorithm that was connected to her thinking; one that would always work.


I am pleased to report that “Karen”, after four semesters of beating her head against the Math Center wall, finally passed with a “C”. She then took my College Algebra course where she earned an “A”. I attribute this success to her hard work, and to the College Algebra course being about ideas (the landscape) more than about an arbitrarily chosen set of algorithms (the linear highway).

And she is in someone else’s section of Precalculus where she recently earned an A on her first exam.


Let’s agree not to use these words

Obvious or obviously.


It has to be

It’s the only way



These words and phrases are pervasive in mathematics classrooms, and they are increasingly common the further we go in mathematics.

But each has a toxic effect on student engagement. Consider the following two scenarios.

Scenario 1: I don’t get it

If I don’t understand what others describe as obvious, I feel stupid and I disengage. I feel like I can’t do this.

scenario 2: I do get it

If I am feeling smart because I understand something, and then others describe it as obvious, I don’t get to feel smart anymore. The thing I worked so hard to understand is obvious to everyone else. I am likely to disengage and to feel like I can’t do this.

So what?

As a teacher, I have two roles related to this issue.

The first is to eliminate this language from my own teaching vocabulary. There is no upside to these terms, so I have to stop using them.

The second is to challenge the “obviousness” of what others claim to be obvious. Is it really that obvious? Is that really the only way to do it or to think about it? As the one who sets the tone for discussion in the classroom, I have a responsibility to include everyone. Sometimes that means pretending I don’t understand something I do.

How many ways to roll a 10, continued…

Recall that, in playing and analyzing dice games appropriate for middle school students’ study of probability, I was challenging my secondary methods students (to whom I refer as “483 students” after the course number, not because of how many of them there are) to justify that there are 3 ways, not 2, to roll a 10 when rolling two six-sided dice. The idea is that my future teachers know that there are 3 equally likely possibilities: (1) a 6 and a 4, (2) a 4 and a 6, and (3) two 5’s. But lots of seventh grade students do not know this. Instead, many will view the first two as being the same.

I pushed hard on this point. My students suggested making a 6 by 6 chart, which is useful for some seventh graders. They suggested rolling one die at a time, or rolling two different color dice, or rolling one die twice. Each of these has the same theoretical probability as rolling two identical dice simultaneously. But not all seventh graders know this. I pushed on.

In particular, I was hoping to challenge my 483 students to wrestle with the complicated relationship between theoretical and experimental probability. Most of the time in middle school classrooms we study both of these but we dismiss discrepancies by waving our hands and saying We don’t expect these to be exactly equal; we expect them to be close, and therefore we shouldn’t worry about goofy experimental probabilities.

I was pressing my 483 students to consider whether experimental probabilities can ever provide convincing evidence that our theoretical model is incorrect. A recent article in Mathematics Teaching in the Middle School described a lesson in which seventh graders were asked to decide which dice were loaded and which were fair. I recall a lesson in my educational statistics class in which the professor opened a new deck of cards, shuffled several times and drew cards from the top of the deck. She was curious how many red cards in a row we would have to see before we suspected that something was up.

My challenge to my students was in a similar spirit but I wanted to push them to design statistical tests that would demonstrate that “two ways to roll a 10” is a flawed model. This meant they needed to outline their procedures in full, state the data they would collect and (most importantly) which results would support their theoretical model. I added the additional constraints that the test could not take longer than 10 minutes to run and that they needed to be willing to stake their teaching licenses on the outcome. OK, I was flexible on that last constraint, but it helped lend seriousness to their thinking.

So here are two of the tests they devised:

(1) We will roll two dice 100 times. We will count the number of doubles and the number of non-doubles. If there are only two ways to roll 10, then there are 15 non-doubles and 6 doubles. If there are three ways to roll 10, then there are 30 non-doubles and 6 doubles. In 100 rolls with our theoretical model, we expect 83 non-doubles. With the competing model, we expect 71 non-doubles. We’ll split the difference. If there are 77 or more non-doubles in 100 rolls, then our model is correct.

(2) Keep rolling until you get ten 10’s. If there are only two ways to roll a 10, then we should expect to have to roll 105 times to get ten 10’s. If there are three ways to roll a 10, then we should expect to roll 120 times. Again, we can split the difference; if our test yields more than 112 rolls, this indicates that there are three ways to roll a 10.

BEFORE READING FURTHER, jot down which of these two tests you think is better for demonstrating which model is correct (Hint: one of them is much better than the other).

Notice that test (1) relies on common denominators while test (2) relies on common numerators. That is, test (1) sets the total number of rolls and asks how many 10’s we got, while test (2) sets the number of 10’s and asks how many total rolls we made.

Each of these tests confirmed the correct model in a single trial in class.

But probability isn’t about one-time outcomes. It is about long-term results. So it’s worth asking whether our results in class were typical. In other words, how likely were these tests to work?

I have lately become curious about the potential for the software Fathom to help students to make these connections between experimental and theoretical probability. The software does lots of things well, but what makes it unique is its ability to do probability simulations (see my article in Mathematics Teacher).

We ran each test with dice in class a small number of times. In the time it took to run each test once, I set up a Fathom simulation, which can then be run many, many times. For the record, I think electronic simulations only make sense after collecting real-world data; otherwise they are too abstract for many students to learn from.

In 100 Fathom trials, test (2) only “works” 51 times. That is, the test is no better than a coin flip. Increase the number of 10’s required to 20 and the test still only succeeds 64% of the time.

Test (1) is much better. In 100 Fathom trials, the test “worked” 95 times.

It turns out that devising a good experiment to determine which model is better (order matters vs. order does not matter) is hard. Therefore, we shouldn’t be surprised (1) that middle school students find it challenging to decide which model is correct, (2) that their own models, which are based on their informal observation of experimental probabilities in the world around them, get in the way of analyzing theoretical probabilities, nor (3) that teaching probability is hard.

How many ways are there to roll a 10?

I taught a class at MSU, Mankato titled “Math 483: Advanced Viewpoints on 5-8 Mathematics”. The class had a variety of goals, including pedagogical and mathematical ones. On the pedagogy side of things, we planned lessons, we read The Teaching Gap, we viewed the TIMSS videos and others, etc. On the mathematics side, we worked problems that came directly from middle school curricula, and we investigated questions that go deeper than we would expect middle school students to go, but that form an important foundation for making instructional decisions with middle school students.

In this last category, I tried something new last semester that I wanted to share with a larger audience. I would love critical feedback and questions about the activity and readers’ ideas about the mathematics involved.

My Math 483 students (I will refer to them as my “483 students” from here on out although there are not 483 of them) as a group had quite limited experience with middle school students-this was their first class examining the teaching and learning of mathematics, and most of them have tended to envision becoming high school, not middle school, teachers. One of the roles I played in class was the voice of a middle school student.

We were playing two dice games in class this spring: the Sum Game and the Product Game. For those without experience with these two games, here is how they work. Two players are playing against each other, one is player A, the other is player B. They alternate turns rolling two dice. In the Sum Game, no matter who rolls, if the sum is odd, player A gets 1 point. If the sum is even, player B gets 1 point. The Product Game is the same except we use the product instead of the sum of the dice. In either case, the players roll some set of number of times (say 20) and the person with the most points at the end wins.

In analyzing whether each game is fair (in the sense of each person having the same probability of winning), my students made the claim that the probability of rolling an even sum is 18/36 because there are 36 equally likely outcomes and 18 of them are even.

My inner middle schooler questioned this calculation. My experience with seventh graders and probability is that they commonly consider (4,6) and (6,4) to be the same outcome: a 4 and a 6. The idea that the order of the dice matters is not intuitive to many middle school students. So I posed the question to my 483 students, “How would you convince a seventh grader that (4,6) and (6,4) are different rolls?”

As we worked through a variety of strategies, I came to realize that this wasn’t quite the right question. One of these might be closer to what I intended:

(1) How do you know your model (there are 36 different equally likely rolls of two six-sided dice) is the correct one? How do you really know that?

…or maybe…

(2) What evidence would it take to convince you that your model is incorrect?

…or maybe…

(3) Imagine we were not sure which model was correct, what experiment could we perform that would help us to decide?

In the next post, I’ll share my students’ answers to my original question, and the statistical tests they concocted to answer question number 3.