Year: 2014 (page 4 of 5)

Words are not ideas, anymore than the picture above is a pipe (it’s a picture of a pipe).

When we communicate about ideas we are forced to use words (or gestures or images, which are also not the ideas themselves), and so consequently we are never communicating ideas directly. We communicate about ideas through the medium of language.

It is possible for someone to learn the words that represent an idea without learning the idea themselves, even so much as to be able to mimic the output expected of someone who understands the idea.

This happened to me. I learned about linear functions in school. I learned about “constant rates of change” and “y = mx + b” and “find the rise over the run” and many other phrases which were connected to finding and subsequently graphing the equation of a line. I even later learned how to think of a specific line as a transformation through a translation and a rotation from any other line.

However, in a fourth year topology class, my professor used a completely different set of words. He said, “Let’s do an easy example. We have line segment AB and line segment CD, and our objective is to find the mapping function from AB to CD. Think about that problem and bring your work to next class.” Well, I spent a week thinking about that problem and could not do it. I spoke to my professor about it, and he gave me a look and told me that it was “not my fault, this is unfortunately due to how you were taught.” Note how I was earlier able to articulate “transform one line into another”, but not able to repurpose this information to create a function. It turns out, the solution is a linear function.

Here I am, many years later, with a much better understanding of the wide range of things in our world which are linear functions. In fact, in only a few minutes, I am now able to “prove” to myself (at least experimentally) a more general version of what my professor asked me to prove all those years ago. I have moved past a “just the words” understanding of linear functions into a more flexible and useful understanding.

This is the objective of good mathematics teaching, and the reason why I think that we need far fewer lists of things to achieve with our students, and far more time spent looking at very similar things but in a very wide range of contexts. We need to move past students learning about the words that represent mathematics and giving them enough time to actually learn about the corresponding ideas.

The house my wife and I live in was recently sold, and so we have started looking for another apartment. Our current lease expires in a year and a half, and so we decided that, given how challenging the rental market is in NYC, that we should start looking right away. We also decided, somewhat arbitrarily, that we would attempt to find an apartment in the next six months, if only because we knew we would get sick of looking pretty quickly.

During the first 5 weeks of our apartment hunt, we found five apartments that we thought were worth looking at. Over six months, I extrapolated that we would get to see about 26 apartments that would satisfy our requirements. What we discovered, with each apartment, is that we basically got to see the apartment and then pretty much decide immediately if we wanted to apply to rent the apartment or not.

It turns out that there is a nifty mathematical algorithm that one can use to optimize one’s chances at picking the best apartment possible. We expect to have 26 apartments to look at, each of which we inspect and then either accept or reject immediately, and we want to optimize our chances of picking the best apartment possible, from the 26 that meet our minimum criteria. This exactly matches a solved problem in mathematics; the Secretary Problem.

In the Secretary Problem, where one has to decide on the best applicant between n randomly ordered applications, an optimal solution is to reject the first n/e applicants, and then accept the next applicant that is better than any applicant you have seen before. The proof of this particular solution is here. I couldn’t reproduce this proof if asked, and there are details in it which are fuzzy for me, but I am pretty sure I understand why it works. Informally, the first n/e applicants act as a sampling space, and this gives you information on how good applicants will be, and that n/e happens to be where you achieve an optimal amount of information on applicant strength, allowing you to make the best determination you can of which next applicant to choose, without raising the probability too high that you’ve already rejected the best applicant.

For our specific apartment hunting problem, with 26 total apartments to view, 26/e (e ≈ 2.71828) ≈ 9.6 ≈ 10 apartments. So, my wife and I looked at 10 different apartments, and while we did this, I informally ranked the apartments based on the criteria my wife and I agreed were important to think about (space, apartment lay-out, cost, location, quality of neighbourhood school, commute time). The 11th apartment either she or I looked was superior in many ways on all of these criteria than any of the others we had looked at, so I whole-heartedly threw in my support for it, knowing that this specific apartment is most likely to be the best apartment we will see.

We’ve submitted an (overly lengthy) application for the apartment. Wish us luck.

When I first started tutoring students, I often noticed that they struggled to add fractions. The addition of fractions just did not make sense to them. Part of this is caused by students having a weak understanding of fractions, and part of this is caused by them not understanding why the typical algorithms used to add fractions make sense.

Here is one model that I developed for myself, so that I could understand why addition algorithms for fractions make sense, and then use this model to help students make sense of adding fractions.

Step 1:

Draw a picture to represent each fraction you want to add. So if I want to add 2/3 and 3/4, I would draw the following, making sure to use rectangles which are the same dimensions, since these two fractions must represent a fraction of the same unit (in this case, the rectangle).

Step 2:

One problem with the above drawing is that it is not the only way to represent fractions. Anticipating that I will want to be able to match units, I realized that it is convenient to draw my fractions with different orientations, as below.

Step 3:

Now, I can’t add the fractions above because the individual pieces of each fraction have different units. I cannot easily see what the total area would be for the two shapes, so I need to divide the two areas again, so that I end up finding the area of each fraction with the same sub-unit of the rectangle (in this case, 1/12).

Step 4:

Now that the units of each fraction are the same, I can simply count the total number of units in the first fraction (9) and add it to the total number of units in the second fraction (8), and I arrive at my answer of 17 units. I have to remember though that the size of this unit, relative to the original unit, is a twelth, hence my answer is 17/12 in terms of the original unit.

What makes this model difficult to understand, and perhaps this lies at the heart of understanding addition of fractions, is that I have switched from counting in one unit (the whole rectangle), to counting in different units (quarters and thirds), to then counting in yet another unit (twelths), and then being able to see this final total in terms of the original unit (the whole rectangle).

This suggests to me that the prerequisites to understanding fraction addition involve an understanding of what a unit is, understanding how to compose and decompose units, understanding different ways of representing fractions, and how different fractions can be equivalent in size, but measured in different units.

The definition of what effective mathematics teaching looks like very much depends on what purpose we assign to teaching mathematics. A classroom where the primary objective is to teach students a specific set of mathematical skills for them to use later will look much different than a classroom where the primary objective is to teach students how to think mathematically, although there is obviously overlap between those two classrooms. For a good description of the type of classroom which achieves the first goal but fails at the second goal, see When Good Teaching Leads to Bad Results by Alan Schoenfeld.

I will describe a classroom where the primary purpose of the classroom is to encourage mathematical reasoning, with a secondary benefit of students practicing mathematics skills they have developed.

What are the students doing?

1. Students are engaged in the standards for math practice.

   The Common Core Standards for Mathematical Practice, which are similar in many ways to the NCTM Process Standards, are a useful tool for understanding the types of activities students should be engaged in within a mathematics classroom.

   In order to really do mathematics, students need opportunities to problem solving, to use mathematics they know to model processes, and to do all of this in the socia-cultural contexts of their classrooms. Both of these sets of process standards do an excellent job of defining what it means to do mathematics, but are flexible enough to allow for a variety of different activities to qualify.

    

2. Spending significant time solving rich mathematics problems.

   Routine problems with limited opportunity for investigation might be acceptable for students to use to practice skills they have learned, but they do not have the breadth necessary to allow students to do the inquiry necessary to learn mathematical reasoning. A significant amount of time in the mathematics classroom investigating, postulating, formulating, deciding, and analyzing mathematical situations is necessary if the habits of mind required for mathematical reasoning are ever going to be adopted by students.

   For excellent examples of rich mathematical tasks (some of which are used for assessment of understanding, and others are used more to prompt student thinking) see some of the web sites linked here.
    

3. Students talk to each other about math.

   While there is definitely value in students spending at least some of their time thinking independently, there is tremendous value in students having opportunities to discuss mathematical ideas and problems with each other. The first is that it is through the repeated access to different linguistic and representative variations on an idea that we come to more than a superficial understanding of that idea. If I say words, you hear the words, and you might even think you can assign some meaning to those words, but it is only when you hear other variations on the formulation of the ideas behind the words, and see other representations (often physical or pictoral) of the ideas represented by those words, that you can come to a full understanding of the concept. For more information on the dangers inherent in a “linguistic-only” understanding of concepts, see Richard Feynman on Education in Brazil.

   There are other benefits of students talking to each about mathematics. One benefit is the person who describes their solution is either likely to see flaws in their reasoning (or at least receive feedback on those flaws) or in the articulation process of their reasoning, come to a better understanding of the concept. Another benefit is that instead of just one person in the room able to give feedback to students, every student in the room becomes a resource for each other. Finally, someone who has just learned a concept, and more importantly recently moved past their own flawed models of that concept, is often more able to explain the concept as compared to someone who learned the concept long ago, and no longer remembers their struggles with it.
    

4. Students have the opportunity to revisit and reflect on mathematics they have learned.

   Human memory is limited. Essentially, our mind trims information from it that is not used frequently (or possibly archives it so that it is difficult to access).

   In a highly effective mathematics classroom, concepts students have learned are revisited, often as embedded practice within the current unit of study. For example, when students create graphs of linear functions by plotting points, they are also practicing plotting points. The skill they practice is used within the context of current problem solving. Note that this practice is only really effective once students have mastered the concept as practice without understanding leads to student confusion.

What is the teacher doing?

1. The teacher uses formative assessment practices on a daily basis within their classroom.

   Aside from engendering the opportunities for “what students do” as described above, an effective teacher gathers evidence of their student’s learning in a systematic way. Formative assessment is a process through which a teacher assesses their students, and then uses this information to inform their teaching. It acts as a feedback loop within the cycle of teaching and learning. If we consider what is to be learned as being like a vast wilderness, then the curriculum the teacher follows is a map through that wilderness, and formative assessment is the process they use when checking their compass so as not to get lost.

   There are three basic indicators teachers can use to collect formative assessment information; what their students write, what their students say, and the body language students use that indicates how they feel. All of these are important markers for teachers.

   Unpacking formative assessment is not the goal of this blog post. For more information, I recommend reading Dylan Wiliam’s Embedded Formative assessment as a good starting place.
    

2. Teachers must build a classroom environment where students want to talk about mathematics and have a growth mindset.

   Developing a positive and productive classroom culture is a critical component of effective teaching. Students must feel that their contributions to the classroom matter, and that they feel safe to make mistakes. Making mistakes, and learning from those mistakes is an important part of learning. The goal of mathematics classrooms should not be to prevent students from making mistakes, but to treat mistakes as opportunities for everyone to learn and to grow.
    

3. An effective teacher uses questioning technique carefully and thoughtfully.

   Teachers ask a lot of questions. For example, in one classroom observation I did this year, a teacher asked 170 questions in a 40 minute period, which averages out to about 1 question every 14 seconds. Given that many teachers ask a great number of questions each class, improvements in questioning technique are therefore likely to improve overall teacher effectiveness, perhaps even dramatically.

   Good questions prompt students to think. Teachers with effective questioning technique do two things well; they have a set of generic questions prepared they can ask students to prompt their thinking which they use frequently enough that students begin to ask these questions of themselves before even talking to the teacher, and they actively listen to, and clarify their understanding of, student reasoning before responding.
    

4. An effective teacher learns about the linguistic and cultural backgrounds of their students and adjusts accordingly¹.

   Mathematics is a cultural activity. Therefore, as mathematics teachers, we are not only teachers of mathematics, but also teachers of the socio-cultural norms of mathematics. In order to do this effectively, we need to understand our students at a more than superficial level. A cautionary note here: This is an area where it is easy to fall prey to cognitive bias and judgemental attitudes. Teachers need to make their best effort to objectively understand their students’ cultures and their linguistic understandings and then make sense of how their students’ backgrounds impact what is effective for their students.
    

5. An effective teacher uses technology to focus students on mathematical reasoning.

   Classroom technology, in an effective mathematics classroom, is used to support student’s mathematical reasoning. Rote practice exercises, even if administered via technology, do little to help students develop their reasoning skills, and because they lack context, have limited ability to help students develop connections between different areas of mathematics.

   Imagine a classroom where students are looking for connections between different forms of a quadratic function. They could plot these functions using pencil and paper, and then look for connections, but during the time students would take to draw the functions, they would lose track of the goal of the graphing. Every time we ask students to do another task in preparation for mathematical study, they lose active cognitive resources to keep track of the overall purpose of the task. Instead, in an effective classroom, the teacher would give students access to a graphing calculator or graphing software, and students would be able to focus on seeing connections between graphs, instead of creating the graphs.

What else would you add to this description of an effective mathematics classroom?

Reference:

1. Suggestion offered by Ilana Horn. See this tweet.

This diagram represents a problem in education which is, by no means, the ONLY problem in education.

How do we change this paradigm?

This is part two of a three part series on formative assessment. This post deals with some things you can do between individual lessons based on formative assessment and during a lesson. You can read part one here.

Introduction

The objective of this post is to describe two possible procedures teachers can use for ongoing, day-to-day formative assessment. The first of these procedures is easier to implement, but gives teachers less information on what students understand. Remember that a primary objective of formative assessment is to create a feedback loop for both teachers and students into the teaching and learning process.

Example 1

At the end of your last class you gave an exit slip. One strategy, which is not too time-consuming, is to take the exit slip and first sort it into No/Yes piles, and then sort these piles into 3-4 solution pathway piles, essentially organizing all of the student work by whether or not it is correct and what strategy students used. It may be useful to have an other group, with students whose strategy which are unable to decode.

These groups of student can be used to decide on student groups (recommendation: group by different strategy) for the following day, decide if you need to try a different strategy for tomorrow, and/or find examples of student work to present to students. It can also be used to decide on re-engagement strategies¹ for the lesson from the previous day, or just decide that you can move onto the next topic in your unit sequence.

Example 2

Part of my current role is to help teachers use formative assessment in their teaching. This has turned out to have some interesting challenges, and has helped me grow tremendously as a teacher.

Dylan Wiliam and Paul Black define formative assessment as “as encompassing all those activities undertaken by teachers, and/or their students, which provide information to be used as feedback to modify [emphasis mine] the teaching and learning activities in which they are engaged.” (Black and Wiliam, 1998a, p7)¹ Another definition I have used is, “A formative assessment or assignment is a tool teachers use to give feedback to students and/or guide their instruction.” Black and Wiliam’s definition is superior because it includes the important “what next” aspect of formative assessment. If the purpose of education is to cause change in a student, formative assessment is the tool that is used to measure and adjust the direction of that change.

My observation is that it depends on how the information collected by teachers is used that ensures if the assessment is formative in nature or not; assessment information not acted on is not formative in nature. The two challenges I have observed in the use of formative assessment are knowing how to act on the information gathered, and being able to find (in an unbiased way) evidence that student thinking has changed based on our instruction.

First, information collected from students can come in a variety of ways. There is the more formal written formative assessment information a teacher can collect, for example: a quiz, an exit slip, a homework assignment, a project, etc… There is also the less formal formative assessment information a teacher can collect (aside: I recommend a clipboard and using a template for quickly recording this kind of information), for example: which students raised their hands to answer a question, how able is a student to explain their reasoning, how does a student respond to another student’s thinking, etc… Each assessment type has its advantages and disadvantages. Formal, written assessment has the advantage that a teacher can look at and think about it when they have time outside of the classroom. The informal assessments have the advantage that a teacher can listen to, make sense of, immediately clarify, and make use of the information.

There are three main ways teachers can effectively respond to assessment information from students.

1. They can use it to guide the next steps of their overall unit planning.
2. They can use it to help plan their next lesson.
3. They can use it immediately to respond to student thinking.

It is most challenging to use formative assessment information immediately, and least challenging to use it to guide an overall unit plan. It is probably worth looking at these opportunities to respond from the most general response, to the most specific.

Formative assessment at the unit level

Imagine you have given a pre-assessment to your students on their knowledge of, and the ability to apply, the Pythagorean theorem to solve for missing lengths in right triangle problems. You discover that the distribution of results for 30 students on the pre-assessment looks something like this, where 4 is given as the cut score².

Remembering that this assessment is at best a proxy for what students have learned³, what do you do? You can see from the distribution of scores that most students did not achieve the cut-score for the task. The situation is more complex however, because a significant percentage of the students did!

You could teach a mini-unit on this topic with the intention of reviewing the material in more depth for students who reached the cut-score for the assessment, and to teach the material as new for students who did not. This is problematic because many students will feel they know the content, and a common reaction to when teachers teach material that students have been told about before (but not necessarily learned) is that students tune out.

If you do choose to teach a mini-unit, you could monitor progress of all students during it. This will help direct your unit toward the most important sub-skills of the unit, and not cover tasks that students are already fairly able to do. For example, if students are all able to consistently apply the Pythagorean theorem to find the hypotenuse of a right triangle, further explicit instruction in this area is not necessary, but it could be a good way to start a task in order to build on strength.

You could also use re-engagement as an alternative to reteaching. This will allow the students who almost meet the proficiency level for this standard a chance to revisit the work that they did and reflect on it and compare their work with other students. All of your students will hopefully increase their understanding of the focus content during re-engagement, but a potential drawback is that not all of them will necessarily improve their mental models to meet our required proficiency level.

You can also use tasks that target the weaknesses of students (or alternatively build upon their strengths) while still allowing all of your students a chance to grow. A task that has a low entry to accessing the task, but has a high ceiling can be an excellent way to differentiate instruction without having to do a huge amount of planning. This also gives students with more background knowledge a chance to deepen their knowledge by thinking about and discussing other people’s misconceptions. During these tasks, it is also possible to confer with individual students and give targeted support to them.

You could return the work to students but give feedback questions instead of scores on the work itself. This way all students, including those that “mastered” the standard have something to work toward. The research suggests that written comments on student work, without numerical grades, are best to produce the desired outcome in students, which is to reflect on their work.

Remember in your unit planning that every time you decide that “enough students get it, and that it is time to move on”, if this concept is critical to understanding future concepts, you’ve left some students with no support. You should plan to find ways to allow students to re-engage with prior concepts and be able to move forward with the rest of the group. You could, for example, spiral back to previous topics through-out the year (which is good practice for all students).

The overall point you should take away from this section is that formative assessment can, and should, be used to modify (or at least justify) your unit plan. If your unit plan is a map through a section of mathematical territory, formative assessment is a bit like your GPS.

In part two of this post, I will outline some examples of day to day formative assessment. 

What other suggestions do you have for teachers who are looking to embed formative assessment in their unit planning process?

Information:

1.  Black, P.J., & Wiliam, D. (1998a). Assessment and classroom learning. Assessment in Education: Principles, Policy & Practice

2.  A cut-score refers to a score which is used in standards-based assessment that indicates what level students need to reach on a performance task to be considered proficient in the associated standard.

3.  All assessment can best measure is a subset of what students are able to do, based on the knowledge and skills they have constructed in their heads. Therefore, all assessments are proxies for what students know. A student who achieves 100% on an assessment does not necessarily understand how to apply a concept 100% of the time, you are best able to say that they were able, on this day, at this time, to write down scribblings we call language which matched your expectations of what those scribblings should look like. This is relevant because you may sometimes find examples of contradictory information, where a student appears on one task to have mastered a standard, and yet appears on another assessment to have not mastered the standard. It also suggests that we can only make the claim that a student has mastered a standard if they have demonstrated proficiency in at least a few contexts.

4.  For other examples of formative assessment, see this presentation that I curated. It has 54 different possible formative assessment strategies in it, some of which are more appropriate for a class focused on literacy skills, and some of which are useful for a mathematics classroom.

Here are seven questions my son asked today.

• Who invented buildings?
• Why don’t we slip on salt?
• When you hold your eyes closed does more water get on the eyeballs than just blinking?
• Why do hummingbirds move so fast?
• Why are butterflies so pretty?
• How did we get the name “people”?
• Why do bees hum?

Kids are scientists. My job with my son is to teach how to answer his own questions.