Category Archives: Learning

How do we build on what students can do?

I spent the last week at a lake house with my family.  It was great to be away from the craziness that is New York City.   It was also great for me to spend some time with the younger members of my family.

As I played with them, I was reminded of all the knowledge children bring with them to a classroom.   They have an understanding of who has more or less.  They devise mental math strategies to keep track of the score during a game.  They can estimate how many lily pads are in the lake.   They can figure out how to evenly share 3 brownies with 2 people.   Most of this was not taught to them in school.

When I was teaching, I was constantly thinking about “prior knowledge” when I planned a lesson.  But I think I was somewhat misguided in what I believed prior knowledge entailed.  I thought it meant what the students had learned in prior years in school.  I didn’t think about what intuitive strategies or knowledge students already had from their experiences in the world.

I also didn’t consider how I could build on that in my classroom.  For example, their intuitive ability to share brownies among different numbers of people can be used to begin to teach fractions.

One group of researchers (see the book below) has documented the intuitive knowledge children bring to the classroom as well as ways to use this knowledge in the classroom.   It’s interesting to see how they foster new strategies and understandings by using what children can already do.

Thinking about how to build on what students can do instead of focusing on what they can’t do changed the way I approached teaching.

What about you?  How do you build on what children can do in your classroom?

Want to know more?

Read Children’s Mathematics: Cognitively Guided Instruction.


Do career and technical high schools improve math skills?


Last week I had the opportunity to tag along as the Secretary of Education, Arne Duncan, visited Aviation High School in Long Island City.  He was there to highlight the school as a model for preparing students to become both career and college ready.  You can read about more about the visit in my post on the Department of Education’s blog.

What struck me about the school (besides how neat it was to see the planes up close) was how engaged and motivated the students were as they talked about repairing planes and interning for businesses at nearby airports.

The students talked about how their aviation classes helped them perform better in traditional classes like physics and math, which made perfect sense to me.  In the case of this particular high school, it certainly was the case.

However, I wanted to see what the research had to say, in general, about career and technical education classes and their impact on math achievement.

As is often the case, the research paints a more complicated picture.   Many high school students, including ones who are enrolled in career and technical courses, do not graduate with the math skills necessary for the workplace or for college.

A group of researches (Stone, Alfred & Pearson, 2008) argued that while career and technical courses hold potential for improving math skills, the math that is being used in these courses is not always being made explicit.

They set out to test what would happen if teachers of technical classes (who are not usually trained as math teachers) made the math explicit.  For example, what if the shop teacher talked explicitly about the Pythagorean theorem during a lesson on the T-square in carpentry class?

They found that when technical and math teachers worked together to make the math explicit in these courses, students performed better on traditional and college placement math tests.  In addition, this type of intervention did not negatively affect their content knowledge in their occupational areas.

If I had to guess what was going on in Aviation High School, my guess would be that it is something similar to what the researchers proposed in their intervention.  But I think it’s important to note that while engaging students in hands-on activities and real world situations is part of the story, it’s not the whole story.  How to best connect mathematics and real world learning needs to be thoughtfully considered.

Want to know more?

You can read more about what the students had to say during Secretary Duncan’s visit in my post on Homeroom, the Department of Education’s official blog.

Read the research:  Stone, J. R., Alfeld, C., & Pearson, D. (2008). Rigor and relevance: Enhancing high school students’ math skills through career and technical education. American Educational Research Journal, 45(3), 767-795.

Does everyone understand?

How many times have you asked your students that during a lesson?  I know I asked my students that question too many times to count.  But I never thought about what I meant by the word understand.

Richard Skemp talked about two different types of understanding back in 1976.   The distinction is still important today.  The first type is relational understanding: knowing what to do and why.  The other is instrumental understanding: knowing a rule and how to use it but not understanding why the rule works.

You can probably think of many examples of instrumental understanding in a math class.  Students may know to “flip and multiply” when dividing fractions and may be proficient at computing the answers but they have no idea why the rule works.

Why might it be problematic to only develop instrumental understanding in students?

Skemp explored this by providing a great analogy of visiting a new town.  He talked about learning the route between two points.  He could get from A to B by following step-by-step directions that were given to him.  Or he could explore the town with the goal of developing a mental map of the town.   In the first case, if he makes a mistake, he will be lost.  He needs to rely on outside guidance to help him get back on track.  In the second case, he can find his way around without having to follow step-by-step directions and can choose from multiple paths.  If he turns the wrong way, he can correct his mistake without any outside help and perhaps learn from the mistake.

How does this relate to math class?

If our students only develop instrumental understanding, they need to remember a series of different step-by-step procedures and when to use them.  They are dependent on the teacher for what to do if they make a mistake.  On the other hand, if they develop relational understanding they can use the connections they have developed to begin to correct their own mistakes and use multiple methods to solve a problem.  Skemp further argues that relational understanding is longer lasting and more adaptable to new tasks.

Thinking about what type of understanding about mathematics we want to foster in our students can help us thoughtfully make choices in our classrooms.  Does everyone understand?

Want to know more?

Read Skemp’s article:

Skemp, R. R. (1976). Relational Understanding and Instrumental Understanding. Mathematics teaching, 77, 20-26.

Key words aren’t the key to understanding math

Students in NYC recently finished taking the state math test. As a result, I’ve spent the past couple of weeks watching a lot of test prep going on in classrooms. One of the strategies I see being used over and over is the use of key words.

I admit that I was guilty of using this strategy when I started teaching. I was puzzled when my students could flawlessly perform computations but struggled with word problems. Teaching them to look for key words seemed like an easy fix to this. I had charts in my room listing all the key words and their corresponding operations. Yet, the key words didn’t seem to help them.

So why don’t key words work?

It’s because they don’t allow students to use what they already know to make sense of a situation.

The research backs this up. Drake and Barlow (2007) gave a student the problem below.

There are 3 boxes of chicken nuggets on the table. Each box contains 6 chicken nuggets. How many chicken nuggets are there in all?

Guess what a student who looked for key words answered? 9 chicken nuggets. The student saw the words: “in all” as a signal to add 6 and 3. I would bet that the student could have made sense of this situation and arrived at the correct answer if he drew a picture or reasoned about it. However, using key words led him to an incorrect answer.

Key words encourage students to take a short cut instead of making sense of a situation. If students think about what makes sense, they don’t need shortcuts or key words. They don’t need to worry about what happens when they aren’t any key words or when there are multiple key words in a story.

If we believe that doing mathematics should have meaning for students and make sense to them, then teaching key words doesn’t support those goals. Teaching students to reason about a situation and know why they are performing an operation does.

Have you used key words with your students? What was your experience?

Drake, J. M., & Barlow, A. T. (2007). Assessing Students’ Levels of Understanding Multiplication through Problem Writing. Teaching Children Mathematics, 14(5), 272-277.

My 5 Core Beliefs about Learning

1. All children can learn

They may not learn at the same pace or in the same way.  They may not need to learn the same thing at the same time.   They may not have the same motivation or the same interests.   But they all can learn and they all can learn math.

When I tell people I was a math teacher, they often tell me they just don’t have a brain for math.  I don’t buy it.   We need to rethink the messages we are sending students both in school and at home because I rarely hear anyone say they don’t have a brain for reading.

2.  Things need to make sense to students.

Learning should be a way for students to make sense of the world around them.  In a math classroom, that means taking everyday situations and using math to help make sense of them.

When math is disconnected from the world students live in, it is viewed as a series of random calculations that don’t make sense.  As a result, students resort to tricks and memorization and then struggle when novel situations are presented.    Math needs to make sense to students.   It needs to be a way to organize the world around them.

3.  Build on what students know.      

All students bring experiences about the world to a learning situation.  These intuitive strategies students already have can be built upon to learn math in a classroom.  All too often there is this assumption that what students need to learn about math comes from a textbook or a teacher.  It is as if they have never had any experiences in their lives that could be used to think about mathematics.

We need to think about how we build upon what students know.  For example, I recently designed a lesson on inequalities.  Students have an intuitive sense of whether their brother has more cookies or fewer cookies than them.  They may not have the symbols or tools to represent these situations mathematically but they understand something about that relationship.  This can be built upon so that math becomes a way to record that situation and to make sense of other situations like that.

4.  There is not a single “best way” to foster learning

Some methods may be better than others but there is no one prescription for how to teach.  It would be much easier if there was and I see the temptation for schools or districts to mandate one way of doing things.  However, all students are unique and there can’t possibly be one right way for a variety of individuals to learn.   We need to question policies that demand one way of teaching.

That’s not to say that I don’t think that we need to radically change the way we approach or evaluate good teaching.  We do and we can talk more about what that looks like in later conversations.   But I am nervous when people prescribe one way to teach children.  I always remember someone telling me during my first year of teaching:  “There is more than one way to skin a cat.”    Now I’m not sure why you would want to skin a cat to begin with, but the sentiment stands.

5.  What students understand is not the same as what they are able to do.

Learning is about understanding.   Understanding is sometimes confused with what students can do.  If students can choose the correct answer on a standardized test, we sometimes say they understand a particular mathematical concept.  But all they might know is how to follow a procedure for a particular type of question.

Really thinking about what we want students to understand about a concept is an important step in designing tasks for students that will foster their learning.    It is also an important step in thinking about how we want to assess them.


So there you have it.  Those are the ideas that guide my work as a teacher and as a researcher.  For me, it’s important to be transparent about what my beliefs are.  It allows me to clear about what I value and it allows others to understand my perspective when I talk about teaching and learning.   Being explicit about these ideas also helps me decide who I want to collaborate with, where I want to work and what projects I want to take on.  In future posts, I’ll talk in more detail about some of the ideas and discuss the research that supports them.

What about you?  What are your core beliefs?