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High School Students Tell All — Motivation in Math Class

As a strong student with a good deal of intrinsic motivation, I sometimes find it hard to understand my own students’ academic dispositions.  What makes them choose to put forth effort, or not?  What do students find motivating?  Since many studies suggest that time spent practicing is highly correlated with growth, as a teacher I want to know how to structure class to increase time on task.

Which brings me to the panel discussion we had with high school juniors and seniors about their experiences in math class.  None of the students who participated love math or see themselves as strong math students, which is much more typical of the students I teach in 6th grade.  Here are some highlights from our discussion:

What makes you participate during group work?

  • I don’t want to be seen as the weak link / slacker holding the team back.
  • If I don’t know my group members, I’m less likely to participate.
  • If I’m in a group where everyone is low [level], then we’ll all just space out.
  • When the teacher lets us choose our own groups we pick people we can work with.
  • When the teacher won’t answer our questions, we have to rely on each other.
  • Team tests (here’s a great NCTM article) get students to work together.
  • Team roles help give everyone a job that’s unrelated to skill level.

What advice do you have for middle school students / teachers?

  • When I understand something, it makes me want to keep going.  (The seeds of intrinsic motivation?)
  • Parent phone calls would make me act right, because I didn’t want my mom to take away my XBox.
  • Parents should keep their kids busy outside of school so they don’t just play video games or watch TV all day.
  • Make students feel comfortable sharing their ideas.  They need to believe it’s okay to be wrong.
  • Finally, a student said that in non-math classes they participate more because they’re sharing ideas, not answers.  Participating in math class isn’t interesting because either you know the answer (and you’re showing off) or you don’t know the answer (and you don’t want to look dumb).
    • So how can we structure math class so that the discussions are more about ideas, not answers?

 

At one point, a student was talking about logarithms, but couldn’t think of the name.  All he had was a vague sense that they were the “opposite of exponential functions.”  The video below gives students a bit of math history and motivation for learning about logarithms.  Maybe there’s even an interesting discussion in here somewhere 🙂

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Literacy in Math Education – NOT an Oxymoron

In my last post I talked about the use of a pre-assessment in my unit plan teaching percents in context.  The other piece of that unit that was new for me was the infusion of literacy techniques.  After taking a wonderful class by Carol Manocchi at Fordham, I became a literacy convert.

What Is Literacy?

Literacy is much more than the ability to read and write.  When I first started teaching I believed being literate just meant having decoding and comprehension skills.  Students at a low level would be stuck decoding words.  Students at a higher level would be able to read and understand, provided they weren’t hindered with unfamiliar vocabulary.  Thus my view was a blend of emergent and functional literacy, a murky picture at best.

These days I have a much richer understanding.  For one, literacy is impossibly entangled with the specific subject being considered.  One can enjoy Harry Potter or The Hunger Games with a shallow level of functional literacy, but reading the New York Times or The Economist demands a degree of cultural literacy.  Likewise, someone who has access to a range of American cultural allusions may find himself lost while reading a major Spanish newspaper, even if he is fluent in Spanish.  So literacy is not a unitary skill that is acquired and transferred effortlessly across disciplines.

Literacy is the ability to read and understand what others have written, along with the ability to write as a means of recording information and for communicating with others.  Using this definition with regard to a specific discipline yields the idea of content literacy.  As a teacher of mathematics, I am interested in developing students who have content literacy in mathematics.  This ability is predicated on 1) general literacy skills, 2) prior mathematical knowledge, and 3) math-specific literacy (such as familiarity with the notation used in Algebra).

I began to realize more and more the truth that content literacy is not content knowledge during my student teaching this semester.  In eighth grade the most advanced math students are typically those with strong numerical fluency, which allows them to learn procedures and notice patterns while their less savvy classmates are bogged down by the computations.  However, if I were to ask my advanced students to justify their procedures or describe a pattern, they were often unable to do so.  I don’t mean that they lacked the vocabulary, like numerator or quotient or distributive property—use of correct terminology is icing on the cake.  They could not articulate why two different procedures yield the same result.  For example, calculating 7% tax and adding it to the bill versus just multiplying the bill by 107%.  And so it was that I set out to encourage meaning making in math class.

Literacy in Action

My first step was to create a need in students by showing them there were things they could not do and then tell them that I would teach them what they needed to know.  I designed a pre-assessment that included basic mathematical tasks as well as asking them to write how they would explain an idea to a sixth grader.  When we anonymously reviewed the class’s responses, I pointed out that even students who were able to perform the skill were unable to explain it to someone else.  I told them that I would teach them how and help them get better at communicating.

Rather than give the standard lecture (this is how you do X, now let’s all do X, now you do X), I had students read a short bit of text I’d created explaining the idea.  Beforehand, I introduced the concept of marginalia in order to get them engaged in self-cognition as they read.  (Many students read the page without making any marks, which I can only assume meant the words washed over them with little understanding.)  Then I asked them to write how they would do X.  I was surprised by the difficulty students had with what I thought was essentially an exercise in copying.  Just rewrite my explanation with different numbers, right?  Wrong.  A majority of the class was unable to answer the prompt.  So what that told me was that students needed more support in constructing, verifying, and extending meaning as they read.

We did several of this type of lesson throughout the unit, and students gradually improved.  I experimented with prompts; here are a few:

  • Explain how you would do X.
  • Why did both methods end up with the same result?
  • Write what you learned about X from the reading.  Include what makes sense and what you continue to ponder.
  • How would you explain this idea to a classmate?
  • Describe what you noticed from the previous problems.
  • Why do some businesses pay employees with commission?  What are the advantages and disadvantages?

I expected students to balk at this sort of difficult metacognitive, especially when many have conceptualized math as memorizing what to do.  But an anonymous survey a couple weeks in found the majority of students really enjoyed the reading and writing.  I guess it levels the playing field for those who don’t have strong numerical fluency, because we are discussing ideas and not just blindly computing.

Mathematical Discussions & Sentence Stems

Since writing is a time-consuming activity, I also wanted to introduce mathematical discussions for shortest bursts of reasoning.  Much of the time in math classes I’ve observed (and my own as well), the teacher asks a question with one correct answer.  If I were to teach content literacy, students should be able to explain their reasoning, not just produce the correct answer.  So I turned to Zweirs’ Building Academic Language to build a framework for classroom discussions.  It declares that classroom talk is a tool for working with information such that it becomes knowledge and understanding.

The first thing to consider was the balance between closed (or display) questions and open-ended questions.  Display questions, while suitable for activating prior knowledge or displaying what students know, rarely lead to deep discussions.  Zwiers provides four main categories of open-ended question:

  1. Personalizing: thoughts, feelings, opinions, interpretations
  2. Justifying: Why do you think?  What evidence do you have?
  3. Clarifying: What do you mean by…? How do you define…?
  4. Elaborating: ask for more, but may confuse students by sending positive and negative feedback in response to answers

Despite all this, he suggests questions are overused in classroom discussions and often still lead to a teacher-centered pseudo-discussion.

For that reason, I designed a series of four posters with sentence stems for agreeing, disagreeing, observing, and questioning.  Then I created a homework assignment that asked students to solve a problem and write up notes that would help them explain what they did to a classmate the next day.  Before the discussion I had students turn and talk with partners for 2 minutes sharing their answer and how they got it.  Then I said that I noticed many different answers as I walked around checking homework and wanted to have a class discussion about the problem.  I introduced the sentence stems I expected students to use and modeled along with some examples of how not to participate.  And then students discussed, with me merely moderating who was talking.  Occasionally I had to ask students to write down their ideas so that we could come back to them in order to allow each thought to reach its conclusion.  Within 15 minutes we had come to consensus and I sensed there was a high level of understanding from the engagement and what students were saying.  Towards the end I began cold calling students to explain whether they agreed or disagreed with what someone had said, which aided me in checking for understanding and holding everyone accountable for participating.

Even more so than with reading and writing about mathematics, students universally enjoyed our mathematical discussions, and I have been infusing them into class whenever possible.  I have been quite pleased myself with the development of students’ reasoning skills and ability to justify their answers.

Next Steps…

In the future I will focus more on the challenges I faced incorporating literacy into a mathematics curriculum.  At times it felt like I was devoting a lot of the class period to teach through literacy-based instruction what could be direct taught in a fraction of the time.  Asking middle school students to struggle with text and then write their understanding appeared to bear very little fruit at first.  In the future I will slowly introduce the idea of gaining knowledge through text by having students first read about something they are already familiar with.  After they have been successful with that several times, they can begin to use text to study new mathematical concepts.

I also struggle to find a way to consistently include literacy-based components, especially with topics like simplifying algebraic expressions.  There are more mathematical discussions happening, but I find it difficult to find time writing about a skill that they still haven’t mastered.  I think literacy has worked best when the writing was assigned towards the end of the unit.  Discussion may help sort out what to do as students are learning a new skill, but writing in particular forces us to (re-)organize our thoughts in a way that they become crystallized.  I think this may be one of the keys to long-term knowledge retention.  This is perhaps the biggest bane of math teachers, finding students who learned something last year, but have no recollection of what to do now.  Sometimes it doesn’t even take a year—they can’t remember the skills they learned last month.  Taking the time to incorporate literacy strategies to solidify students’ knowledge would be time well spent.

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Using Pre-Assessments in a Unit on Percent

My first year of teaching I was both blessed and cursed by a lack of curriculum.  The previous 8th grade math teacher had left a collection of NY state test questions from the past several years.  But even as an unsophisticated first year teacher, I realized this was a far cry from a well-planned curriculum.  And so I spent about hours and hours every unit scouring the internet for different ways to teach a concept, interesting problems, and standards-driven activities.  This constant quest for good resources has dramatically accelerated my growth as a teacher and is a continual supply of professional development.Now that I have surveyed the landscape, I feel ready to give back to the community.  My initial contribution is a unit on percents that I planned while student teaching in an 8th grade classroom.  There are several features I’d like to highlight that I think were valuable.  In this post I will discuss the use of a pre-assessment.

This was the first time I gave a pre-assessment.  In the past I didn’t gave much thought to prerequisite skills and understanding.  By giving a pre-assessment, I forced myself to consider what students needed to know to learn the new material.  I also sent students the message that I was serious about helping them be successful, and not just blindly following some curriculum.

Following the pre-assessment I collected student work and used the document camera at the front of the room to briefly flip through student responses.  It’s important that this process is anonymous, as the point is to display the range of ideas and not single out students for their mistakes.  The teacher may also want to slip in his or her own work to make a particular point (whether it’s a common misunderstanding, a correct solution, or a controversial answer).  For example, this student’s solution prompted an interesting discussion.

After giving a pre-assessment it’s important to put the data to use.  I found that students had a lot of trouble multiplying decimals and converting between fractions, decimals, and percents.  So I planned a station activity the next day where students could practice two skills.  Each station was led by a teacher or pair of students who had demonstrated mastery of that particular skill.  In this particular instance I allowed students to decide which skill to practice, but you could also assign groups.

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