David Wees's blog
http://davidwees.com/blogs/david-wees
enTeaching Mathematical Language
http://feedproxy.google.com/~r/davidwees/~3/QKniWgovqW4/teaching-mathematical-language
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><h2><strong>The Problem</strong></h2>
<p>Imagine you have a list of possible questions you want students to be able to understand and be able to translate into mathematical symbols, like the following.</p>
<ol><li>The sum of six and a number</li>
<li>Eight more than a number</li>
<li>A number plus five</li>
<li>A number increased by seven</li>
<li>Seven more than a number</li>
<li>The difference of five and a number</li>
<li>Four less than a number</li>
<li>Seven minus a number</li>
<li>A number decreased by nine</li>
<li>The product of nine and a number</li>
</ol><p>One common approach I have seen used is to model a few of the questions on the list and then ask students to attempt the other problems themselves or in small groups. This approach has a serious flaw; <strong>it requires students to know the thing you are trying to teach in order to do the task</strong>.</p>
<p>Let's imagine a very similar task, except now suppose I ask you to translate the list into German. Here are the first two phrases translated into German (<strong><a href="https://docs.google.com/document/d/1WdYGZ_0BopQzcEFCHLWXpX9yvC2SuRmZ5agralpgjBU/edit" target="_blank">thank you anonymous translators</a></strong>).</p>
<ol><li>Die Summe von sechs und eine Zahl.</li>
<li>Acht mehr als eine Zahl.</li>
</ol><p>Now translate the other 8 sentences.</p>
<p>Unless you already know German, you can't do this task. Worse, imagine I gave you the entire list in German and asked you to translate it into Hebrew.</p>
<ol><li>Die Summe von sechs und eine Zahl.</li>
<li>Acht mehr als eine Zahl.</li>
<li>Eine Zahl plus fuenf.</li>
<li>Eine Zahl von sieben vermehrt.</li>
<li>Sieben mehr als eine Zahl.</li>
<li>Der Unterschied von fuenf und einer Zahl.</li>
<li>Vier weniger als eine Zahl.</li>
<li>Sieben minus eine Zahl.</li>
<li>Eine Zahl verringert bei neun.</li>
<li>Das Produkt von neun und eine Zahl.</li>
</ol><p>Here are the first two phrases translated into Hebrew. Translate the rest of the phrases from German.</p>
<ol style="direction: rtl;"><li>החיבור של שש ומספר</li>
<li>שמונה יותר ממספר</li>
</ol><p>Unless you know German well enough to understand the phrases in the first place and Hebrew will enough to translate the German, you cannot do this task. You also cannot do this task if you do not know how these phrases are related to each other in the two different languages.</p>
<p>If your students do not know the vocabulary in the first list I shared and/or they do not know the mathematical symbols, then they cannot do the translation between the two without some intervention.</p>
<p> </p>
<h2><strong>A solution:</strong></h2>
<p>However, students may be able to use their partial knowledge of the symbols or the vocabulary to fill in gaps in either. As an alternative to having them work on the entire list from scratch you could:</p>
<ul><li>Give them a copy of both lists and ask them to individually match as many of the items as possible and then attempt to match the other ones as best as they can. They could then work in groups to refine and improve their lists and then individually try to write out the situations using actual numbers.</li>
<li>Embed the different phrases in a context through which students could make sense of the relationships between the symbols and the text. One issue here with making sense of what these phrases mean is that students do not have sufficient clues as to what they could mean because the phrases are completely contextless. There is not enough information provided as to what these phrases actually mean. <strong><a href="http://schools.nyc.gov/NR/rdonlyres/0D9AA86E-F601-4F26-9598-CF57C4FA7CAB/0/NYCDOEHSAlgebraTheCycleShop_Final.pdf" target="_blank">Here's an example of a task based on equality</a></strong> instead of inequalities but hopefully you can generalize what I mean from it.</li>
</ul><p>If students have insufficient knowledge of either the vocabulary or the mathematical symbols, then they need to build that knowledge <strong>first</strong>. In this case, <strong><a href="http://teacher.scholastic.com/products/readingline/pdfs/ProfessionalPaper.pdf" target="_blank">these ten recommendations on building vocabulary</a></strong> may be useful to consider.</p>
<p> </p>
<h2><strong>Additional point:</strong></h2>
<p>The original goal is probably not a very good goal given that students are rarely, if ever, asked to translate phrases this short into mathematical terminology. Instead of focusing on the small building blocks students might use to translate phrases, it is more useful to start with longer phrases based on meaningful contexts (note: <strong><a href="http://blog.mrmeyer.com/2014/developing-the-question-real-work-v-real-world/" target="_blank">this does not necessarily mean real world</a></strong>) that include more text and to work with students to reduce these phrases to simplest form, and then use these reduced forms to look for mathematical connections between the longer forms of text.</p>
<p> </p>
<p> </p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/QKniWgovqW4" height="1" width="1" alt=""/>Thu, 11 Dec 2014 16:20:11 +0000David Wees990 at http://davidwees.comhttp://davidwees.com/content/teaching-mathematical-language#commentshttp://davidwees.com/content/teaching-mathematical-languageSupporting English language learners with mathematical practices
http://feedproxy.google.com/~r/davidwees/~3/jkt-VMFh2Dg/supporting-english-language-learners-mathematical-practices
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><img alt="SMP relationship" src="/sites/default/files/smp_relationship_1_0.png" style="width: 800px; height: 596px;" /><br />
<small>(Source: <strong><a href="https://docs.google.com/presentation/d/19vuiFQrXIAYNHbdkee46tUJNnO2CRXpIyhqysoMX6DU/pub?start=false&loop=false&delayms=3000" target="_blank">Engaging ALL students in Cognitively Demanding Mathematical Work</a></strong>, November 4th, 2014)</small></p>
<p>On Tuesday, November 4th we had <strong><a href="https://twitter.com/GraceKelemanik" target="_blank">Grace Kelemanik</a></strong> do a presentation and a workshop for the teachers in our project intended to offer ways to use the standards for mathematical practice to support English-language learners (ELL) and students with special needs.</p>
<p> </p>
<h3><strong>Introducing the ideas</strong></h3>
<p>As the picture above suggests, Grace does not consider all the <strong><a href="http://www.corestandards.org/Math/Practice/" target="_blank">mathematical practices from the Common Core</a></strong> as being equal in importance. In particular, she sees MP2 (Reason abstractly and quantitatively), MP7 (Look for and make use of structure), and MP8 (Look for and express regularity in repeated reasoning) as being potential pathways students can use to solve mathematical problems. </p>
<p>If you want to support students in using MP2 to solve problems, consider the following questions:</p>
<ul>
<li>What can I count or measure in this problem situation?</li>
<li>How do the quantities relate to each other?</li>
<li>How can I represent this problem?</li>
<li>What does this (expression, variable, number, shaded region, etc.) represent in the problem context?</li>
</ul>
<p>If you want to support students in using MP7 to solve problems, consider the following questions:</p>
<ul>
<li>What type of problem is this?</li>
<li>How is this (situation, object, process, etc.) connect to another math idea?</li>
<li>Is it behaving like something else I know?</li>
<li>How can I use properties to uncover structure?</li>
<li>How can I change the form of this (number, expression, shape) to surface the underlying structure?</li>
<li>Are there “chunks”?</li>
</ul>
<p>If you want to support students in using MP8 to solve problems, consider the following questions:</p>
<ul>
<li>Am I doing the same thing over and over again?</li>
<li>Am I counting in the same way each time?</li>
<li>Do I keep doing the same set of calculations?</li>
<li>What about the process is repeating?</li>
<li>How can I generalize this repetition?</li>
<li>Have I included every step?</li>
</ul>
<p>These questions come directly from Grace's presentation and are likely to be included (along with more examples) in a book Grace and her co-authors (<strong><a href="https://twitter.com/AmyLucenta" target="_blank">Amy Lucenta</a></strong> and <strong><a href="https://twitter.com/mathgal0921" target="_blank">Susan Creighton</a></strong>) hope to complete soon. The questions remind me of the <strong><a href="http://en.wikipedia.org/wiki/How_to_Solve_It" target="_blank">general problem solving heuristics that George Polya developed</a></strong>, but with a greater level of specificity, perhaps one that students can generalize to any problem context themselves.</p>
<p> </p>
<h3><strong>Modelling the practices</strong></h3>
<p>In the afternoon, Grace modelled some of the practices she talked about in the morning and in particular focued on how a lesson focused on using MP7 could help students make sense of the mathematics being demonstrated. It is worth noting that my description below is based on my memory of what happened (I did not take detailed notes) and so hopefully I have represented Grace's work well.</p>
<p>Grace started by showing something similar to the following expressions:</p>
<ul>
<li><font color="#545454" face="arial, sans-serif" size="2"><span style="line-height: 18.2000007629395px;">3</span></font><span style="color: rgb(84, 84, 84); font-family: arial, sans-serif; font-size: small; line-height: 18.2000007629395px;">×4 + 9×3 + 9×3 + 5×3</span></li>
<li>2<span style="color: rgb(84, 84, 84); font-family: arial, sans-serif; font-size: small; line-height: 18.2000007629395px;">×(9×3) + 9×4</span></li>
<li>5<span style="color: rgb(84, 84, 84); font-family: arial, sans-serif; font-size: small; line-height: 18.2000007629395px;">×10 + 2×(3×4) + 4×4</span></li>
</ul>
<p>She described what the purpose of the activity was, and then uncovered three diagrams (similar to the ones below) which she said corresponded to the calculations of area already given. The fourth diagram she uncovered later.</p>
<p><img alt="Cut 3" src="/sites/default/files/cut_3_-_grace_0.png" style="width: 400px; height: 317px;" /><img alt="Cut 2" src="/sites/default/files/cut_2_-_grace_0.png" style="width: 400px; height: 317px;" /></p>
<p><img alt="Cut 1" src="/sites/default/files/cut_1_-_grace_0.png" /><img alt="No cuts" src="/sites/default/files/grace-drawing_1_0.png" style="width: 400px; height: 317px;" /></p>
<p>First she asked someone in the audience to explain which expression corresponded to the first diagram. While someone explained why the diagram and the expression were related, she carefully underlined the "chunk" of the expression which corresponded to the "chunk" of the diagram and shaded in the related portion of the diagram using the same color.</p>
<p>She then asked students to work with a partner to figure out which expression corresponded to the first diagram and why. Once students (in this case our teachers) had time to work this out, she asked two different pairs of students to come up to explain their reasoning. One partner was allowed to point but not talk and the other partner was allowed to talk but not point. Together they had to explain how they found a relationship between the expression and the diagram and color in the diagram and underline the expression in the same way. Grace repeated this process three times, once for each diagram.</p>
<p>She unveiled the blank diagram and asked participants to think of another way of cutting it up and then constructing the corresponding expression for their version of the diagram. She had someone come up to explain another way of looking at the expression using the blank diagram and construct their own corresponding expression, again asking them to ensure that they used color to relate the diagram and the expression.</p>
<p>Finally she summarized what each group found and asked if anyone saw any generalizations they saw between the different solution methods that were used.</p>
<p> </p>
<h3><strong>Further support for students</strong></h3>
<p>Grace then went on to articulate five instructional strategies she used during the workshop that help support students in understanding the mathematics.</p>
<ul>
<li>Think-Pair-Share</li>
<li>Annotating</li>
<li>Sentence Starters and Frames</li>
<li>Repetition</li>
<li>Meta-reflection</li>
</ul>
<p>She ended her presentation with this terrific comic from <strong><a href="http://www.uvm.edu/~cdci/reports/Final08.pdf" target="_blank">Michael G. Giangreco</a></strong>.</p>
<p><img alt="Inclusion poster" src="/sites/default/files/logs100c_0.jpg" /><br />
<small>(<strong><a href="http://njcie.org/vision-newsletter/" target="_blank">source</a></strong>)</small></p>
<p> </p>
<h3><strong>My reflection</strong></h3>
<p>Grace's workshop took place in a room with about 130 teachers and I observed the diagrams from the opposite corner of the room. What I noticed is that even though I could not hear everything that was said, I was still substantially able to follow the mathematical arguments being made. I attributed my ability to follow the mathematics to the strategic use of color, the repatition in the diagrams, the use of gestures by Grace and the "students" who came up to present, and of course my existing understanding of the principle being focused on.</p>
<p>I also noticed that at no point did anyone actually talk about or calculate the final area. It was not that the final area was not important but that in this context, my suspicion is that it would not have contributed to the conversation, which was focused on the strategies and processes one could use to find the area. Grace was also careful not to introduce "the best way" to cut up the diagram to focus on students making sense of the relationship between the different ways of slicing up the shape given and the expressions given.</p>
<p>Students were not expected to come up with their own way of cutting up the area (which in my experience many students find challenging) or of creating an associated expression until they had listened to three different representations and been required to think about each diagram themselves.</p>
<p> </p>
<h3><strong>A follow-up activity</strong></h3>
<p>As the workshop unfolded during the day, I thought of an activity that one could do on their own to follow up on the first part of the workshop. The purpose of this activity is two-fold; to understand different approaches to solving mathematics problems and to better understand the standards for mathematical practice.</p>
<p>Take a given mathematics problem (<strong><a href="https://drive.google.com/a/newvisions.org/file/d/0BwUlWM5RWHK_bWVSNThDZER5V2c/view" target="_blank">like this one</a></strong>) and construct a solution that emphasizes the use of MP2, another solution that emphasizes the use of MP7, and yet another solution that emphasizes MP8. Look at all three solutions and look for similarities and differences between the solutions. Think about how you would support students in understanding how to construct any one of the solution pathways you constructed (or in supporting students to construct their own solution pathways using MP2, MP7, or MP8).</p>
<p>Update: Here's an example of the same task done in three different ways.</p>
<blockquote class="twitter-tweet" lang="en"><p>I solved the same task three different ways, attempting to use SMP2, 7, and 8 respectively. Thoughts? <a href="https://twitter.com/hashtag/mathchat?src=hash">#mathchat</a> <a href="http://t.co/m8LY3QJQug">pic.twitter.com/m8LY3QJQug</a></p>— David Wees (@davidwees) <a href="https://twitter.com/davidwees/status/531932923088601089">November 10, 2014</a></blockquote>
<script async src="//platform.twitter.com/widgets.js" charset="utf-8"></script>
<p> </p>
<p> </p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/jkt-VMFh2Dg" height="1" width="1" alt=""/>Thu, 06 Nov 2014 20:20:19 +0000David Wees989 at http://davidwees.comhttp://davidwees.com/content/supporting-english-language-learners-mathematical-practices#commentshttp://davidwees.com/content/supporting-english-language-learners-mathematical-practicesSelf-directed Workshops
http://feedproxy.google.com/~r/davidwees/~3/7ksD_cZJp_g/self-directed-workshops
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>I'm facilitating a pair of workshops this weekend in San Francisco, both of which are fairly self-directed workshops. In fact, it occurred to me that a motivated person or small group could probably get a lot of out of what I have constructed without my direct support. So I'm embedding them below. Feel free to use/share these resources (for non-commercial purposes). In each presentation, there is a link to the folder that holds the agenda for the workshop, and that agenda contains a link to the folder of associated resources.</p>
<p><iframe src="https://docs.google.com/presentation/d/1_BgzL6HS9ek-9KQkQ_1yj42cf88hxJgY_qsJDADEF7E/embed?start=false&loop=false&delayms=3000" frameborder="0" width="800" height="479" allowfullscreen="true" mozallowfullscreen="true" webkitallowfullscreen="true"></iframe></p>
<p><iframe src="https://docs.google.com/presentation/d/1IW0LqS36_wrlOPFk37tILUpo2jKnCgNFIXFPULw3sGA/embed?start=false&loop=false&delayms=3000" frameborder="0" width="800" height="479" allowfullscreen="true" mozallowfullscreen="true" webkitallowfullscreen="true"></iframe></p>
<p>If you are reading this blog post and do not see the slides embedded above, <a href="http://davidwees.com/content/self-directed-workshops">try reading it here instead</a>.</p>
<p> </p>
<p> </p></div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/7ksD_cZJp_g" height="1" width="1" alt=""/>Thu, 02 Oct 2014 17:45:53 +0000David Wees988 at http://davidwees.comhttp://davidwees.com/content/self-directed-workshops#commentshttp://davidwees.com/content/self-directed-workshopsWhat do mathematics teachers need to know?
http://feedproxy.google.com/~r/davidwees/~3/xBQ_Gymr1fQ/what-do-mathematics-teachers-need-know
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>I've been thinking a lot recently about what knowledge is needed by mathematics teachers in order to be excellent teachers. It is clear to me that teachers of mathematics must know the mathematics they are to teach, but what else do they need to know?<br /><br /><img alt="Different types of knowledge" src="http://davidwees.com/sites/default/files/screen_shot_2014-09-27_at_8.26.33_pm.png" style="width: 555px; height: 269px;" /><br /><span style="font-size: smaller">(Source: <a href="http://conferences.illinoisstate.edu/nsa/papers/thamesphelps.pdf" target="_blank">Content Knowledge for Teaching: What Makes it Special?)</a></span></p>
<p>I'm not the first person wonder this. In a 2008 paper, Deborah Ball, Mark Thames, and Geoffrey Phelps describe a categorization of the mathematical knowledge needed for teaching into five broad types, described in more detail below.</p>
<p> </p>
<p><u><strong>Knowledge of mathematics</strong></u></p>
<ul><li>
<p><strong>Common mathematical knowledge</strong>:<br /><br />
Deborah Ball et al. <strong><a href="http://conferences.illinoisstate.edu/nsa/papers/thamesphelps.pdf" target="_blank">describes this</a></strong> as "knowledge that we would expect a well-educated adult to know" (Ball, Content Knowledge for Teaching. What makes it special?, 2008). It is closely associated with the curriculum that is being taught and includes knowledge of when students are making mistakes and when there are errors in a textbook.<br /><br />
For example, common mathematical knowledge includes knowing how to subtract a negative number from a positive number. It includes knowing how to multiply multi-digit numbers together. It includes knowing how to solve a quadratic equation by factoring. It includes all of the mathematical knowledge necessary to solve mathematical problems in at least one way.<br /><br />
Note that teachers should at least know all of the common mathematical knowledge they are expected to teach as well as how that knowledge is connected to the mathematics students are likely to learn later in school. If I teach my students that an equals sign (=) means "find the answer" they are likely to find algebra confusing until they learn that the equals sign means "the two expressions on either side of this = sign are equal in size".<br /><br /><img alt="Calculating percents" src="http://davidwees.com/sites/default/files/screen_shot_2014-09-28_at_9.42.36_pm_0.png" style="width: 600px; height: 346px;" /><br /><span style="font-size: smaller">(Source: <a href="https://www.khanacademy.org/math/pre-algebra/decimals-pre-alg/percent-intro-pre-alg/v/finding-percentages-example" target="_blank">Finding a percentage</a>)</span><br /><br />
Notice in the example above, the author of this video is dividing 4 by 16. Instead of making sense of the calculation, he starts by following the algorithm which results in a non-sensical calculation. Someone who understood this division calculation with a stronger conceptional understanding might realize that since 16 is larger than 4, we can skip the first iteration of this division calculation and move straight away into 40 divided by 16. <br /><br />
When teachers lack this mathematical knowledge, they often end up relying on tricks instead of mathematical understandings. In my own practice, I remember when I was first called upon to teach the <strong><a href="http://en.wikipedia.org/wiki/Chi-squared_test" target="_blank">Chi-squared test</a></strong>, which at the time I did not know. I read ahead in my textbook and figured out how to follow the mathematical recipe that produced the Chi-squared calculation well enough to muddle through it in class. However years later I finally connected the expected values to probability and realized that I had for many years missed an opportunity to help students make a connection between this test and other things we were learning that year. <br />
</p>
</li>
<li><strong>Specialized mathematical knowledge</strong>:<br /><br />
This is "knowledge <em>beyond</em> that expected of any well-educated adult but not yet requiring knowledge of students or knowledge of teaching" (Ball, 2008). For example, knowing some of the <strong><a href="http://davidwees.com/content/explaining-subtracting-negative-numbers-my-wife" target="_blank">many different representations of positive and negative numbers</a></strong> is mathematical knowledge that one could not expect every well-educated adult to know, and does not require knowing students or teaching.<br /><br />
An example of specialized mathematical knowledge is knowing how to represent subtracting a negative number from a positive one in <a href="http://davidwees.com/content/explaining-subtracting-negative-numbers-my-wife">a variety of different ways</a>. Since different students are likely to find different models for understanding current topics more or less easy to grasp, it is incredibly useful to know a variety of different ways of representing and approaching each area of mathematics that is taught.<br /><br /><img alt="Models for understanding subtracting negative integers" src="http://davidwees.com/sites/default/files/models_for_understanding_subtracting_negatives_0.png" style="width: 800px; height: 257px;" /><br /><span style="font-size: smaller">Different models for understanding subtracting negative numbers</span><br /><br />
</li>
</ul><p><u><strong>Knowledge of students:</strong></u></p>
<ul><li><strong>Pedagogical content knowledge</strong>
<ul><li><strong>Knowledge of the ways students understand the content</strong>:<br /><br />
This is the knowledge of not just the ways to do mathematics but the ways in which students understand the mathematics. While some mistakes students make are the result of over-taxed memory or carelessness, not all mistakes students make can be explained in these ways.<br /><br /><img alt="Multiplication mistake" src="http://davidwees.com/sites/default/files/img_3364-1024x768_0.jpg" /><br /><span style="font-size: smaller">(Source: <a href="http://mathmistakes.org/?p=1711" target="_blank">Math Mistakes</a>)</span><br /><br />
What mistake did this student make above? What does this mistake mean they were thinking when this did this calculation? How would you help this student? Answering all three of these questions requires an understanding of the ways student understand this mathematical content.<br /><br /><strong><a href="http://www.amazon.com/Teaching-Problems-Magdalene-Lampert/dp/0300099479" target="_blank">Magdalene Lampert</a></strong> once told me the story of an interaction she had with a 5th grader. Near the very end of a class discussion, he said that 0.007 is negative. The class ended and Magdalene was left to try and figure out why the student thought that 0.007 is negative. If teachers have the common mathematical content knowledge they need, they should understand that 0.007 is not negative. If teachers have sufficient knowledge of the many different ways students might understand mathematics, then they will know that there is a fairly logical (although incorrect) train of thought that leads students to the mistaken understanding that 0.007 is negative. Any guesses as to what this student was thinking?<br /><br />
</li>
<li><strong>Knowledge of the ways to teach mathematics</strong>:<br /><br />
This is the knowledge of how to teach mathematics so that it makes sense to someone else. It includes thinking about the sequencing of mathematics concepts as well as knowing rationals for why various pieces of mathematics are correct. Teachers need to make choices about what mathematical representations to use with students and which of those representations are likely to understood and misunderstood by students.<br /><br />
In <strong><a href="http://davidwees.com/content/explaining-subtracting-negative-numbers-my-wife">my previous example</a></strong>, knowing the different models of understanding the mathematics is specialized mathematical knowledge, knowing which models to choose and how to sequence the models is mathematical knowledge for teaching.<br /><br /><img alt="Multiplication models" src="http://davidwees.com/sites/default/files/multiplication_models_0.png" /><br /><span style="font-size: smaller">Different ways of representing 12 times 6</span><br /><br />
These are a few of the <strong><a href="http://davidwees.com/content/different-multiplication-strategies">different models for understanding multiplication</a></strong>. Which of these models do you think work best for initially introducing multiplication? Which of these models is most efficient? How are these models connected to each other?<br /><br /><img alt="Tree x Tree = Line" src="http://davidwees.com/sites/default/files/tree_x_tree_line.jpg" style="width: 599px; height: 457px;" /><br /><span style="font-size: smaller">(Source: Jason Zimba, NCTM 2014)</span><br /><br />
When teachers do not know how to teach a particular idea to students in ways that make sense to students, they may rely on mnemonics or other non-mathematical ways for students to remember mathematical content. If you look at the example above, it is clear that whoever created these diagrams is hoping that these will help students remember the math facts for 3×3 and 3×4. Wouldn't it be better though for students to understand at least these simple multiplication facts based on pictures of what they represent or at least be memorizing the words that represent this relationship?<br /><br />
This area of knowledge also includes how to ask questions in class that get all students thinking, how to give students feedback that helps them move forward, and how to choose <strong><a href="http://instructionalactivities.com/" target="_blank">instructional activities</a></strong> that support students learning mathematics.<br /><br />
</li>
<li><strong>Knowledge of the curriculum</strong>:<br /><br />
Teachers also need to know the scope and range of the mathematics they are teaching. They need to know the standards that are appropriate for their particular course as well as what those standards mean. They need to know what resources they have available as well as how useful those resources are for their teaching. They need access to curricular resources and because of the limited time teachers typically have available to plan, they need to know where to find the resources they need without spending an enormous amount of time searching.<br />
</li>
</ul></li>
</ul><p>There is other knowledge teachers need to know outside of the knowledge related to their content area. </p>
<p> </p>
<p><u><strong>Knowledge of students and schools</strong></u></p>
<ul><li><strong>Knowledge of students' cultures and backgrounds:</strong><br /><br />
While this is not typically included in the lists of knowledge required for teaching, it is pretty clear to me that it is necessary. <a href="http://thejosevilson.com/teacherfolk-kinfolk/" target="_blank">The ignorance that led this group of NYC teachers to not only wear these t-shirts but to also pose for a photo</a> is a clear sign that there is a body of knowledge about students' culture and teachers' impact on it that exists and that not all teachers know.<br />
</li>
<li><strong>Knowledge of the emotional needs of students:</strong><br /><br />
Teachers are also responsible for understanding the emotional needs of the children under their care. These needs change over time and are different for different children. Teachers need to pay attention to the status needs of their students and to understand how status impacts their classroom instruction.<br /><br />
Who can be grouped with whom? Who looks like they need more support today? Which children look like they are being abused or neglected? Who is likely to be excited by today's lesson?<br />
</li>
<li><strong>Knowledge of the rules and procedures related to teaching:</strong><br /><br />
Every school has procedures teachers are expected to follow and every country and state has laws teachers are expected to obey. While knowledge of these rules and procedures is completely insufficient to be able to teach, a lack of knowledge of these rules and procedures has led many educators to disaster.</li>
</ul><p> </p>
<p><u><strong>Conclusion:</strong></u></p>
<p>From my own experience, I learned very little of this before I started teaching, aside from the knowledge of most of the mathematics I was employed to teach. During the course of my career as I attempted to make sense of what students understood and planned lessons to build student mathematical knowledge, I slowly built up my understanding of these areas of knowledge. <br /><br /><img alt="Teacher knowledge" src="http://davidwees.com/sites/default/files/wees-fig_0.png" /><br /><span style="font-size: smaller">(Source: <a href="http://www.amazon.com/Embedded-Formative-Assessment-Dylan-Wiliam/dp/193400930X" target="_blank">Embedded Formative Assessment</a>)</span><br /><br />
The graph above shows a summary of research on the correlation between how many years of service literacy and numeracy teachers have compared to how much their students learn. Early in their careers, both teachers of literacy and numeracy struggle to use the knowledge they have to successfully teach students. One way of interpreting the graph above is that as they progress through their careers, both teachers of literacy and numeracy increase their knowledge of how to teach, and their students benefit. <br /><br />
What of the knowledge above can teachers begin to learn before they start teaching? How can we ensure that every teacher of mathematics learns enough of this content during the course of their career in order to be able to teach? Also, how does this post relate to teachers of other content areas or elementary school teachers?<br />
</p>
<p><strong>References:</strong></p>
<p><span style="color: rgb(34, 34, 34); font-family: Arial, sans-serif; font-size: 13px; line-height: 16.1200008392334px;">Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching what makes it special?. </span><i style="color: rgb(34, 34, 34); font-family: Arial, sans-serif; font-size: 13px; line-height: 16.1200008392334px;">Journal of teacher education</i><span style="color: rgb(34, 34, 34); font-family: Arial, sans-serif; font-size: 13px; line-height: 16.1200008392334px;">, </span><i style="color: rgb(34, 34, 34); font-family: Arial, sans-serif; font-size: 13px; line-height: 16.1200008392334px;">59</i><span style="color: rgb(34, 34, 34); font-family: Arial, sans-serif; font-size: 13px; line-height: 16.1200008392334px;">(5), 389-407.</span></p>
<p><span style="color: rgb(34, 34, 34); font-family: Arial, sans-serif; font-size: 13px; line-height: 16.1200008392334px;">Wiliam, D. (2011). Embedded formative assessment.</span></p>
<p> </p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/xBQ_Gymr1fQ" height="1" width="1" alt=""/>Mon, 29 Sep 2014 01:58:31 +0000David Wees984 at http://davidwees.comhttp://davidwees.com/content/what-do-mathematics-teachers-need-know#commentshttp://davidwees.com/content/what-do-mathematics-teachers-need-know20 things every teacher should do
http://feedproxy.google.com/~r/davidwees/~3/Q3y4iBKQ2po/20-things-every-teacher-should-do
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><img alt="20 things every teacher should do" src="http://davidwees.com/sites/default/files/20_things_every_teacher_should_do.jpg" style="width: 808px; height: 868px;" /></p>
<p>The objective of this list is to identify a core set of teacher practices that every teacher should do. There are some things missing from my list. For example, I have not highlighted the need for every teacher to be aware of issues around the social status of their students, their students emotional needs, or have a certain level of cultural awareness. I think these things are important, but I'm not sure how to describe them in a sentence. If you have any suggestions...</p>
<p>I'm also fully aware that these sentences are probably too brief to be a complete picture of the complexity of teacher practice. There are entire books on giving good feedback and understanding formative assessment. I do not view this list as a simple check-list but rather as a starting place for teachers and teacher-educators to ask questions about effective teaching practice.</p>
<p>I'm sharing this graphic with a Creative Commons Attribution, Non-commercial, Share-alike license. You can <strong><a href="https://docs.google.com/drawings/d/1lRyv2EoTOxEwhX3mpZlIgJZaJmLewda-vbcs3TsJofQ/edit" target="_blank">make a copy</a></strong> to modify it here if you like, provided you make any changes from the original obvious, and provide appropriate attribution.</p>
<p>What else would you include? What do you disagree with? Do you have any clarifying questions about what I mean by anything on this list?</p>
<p> </p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/Q3y4iBKQ2po" height="1" width="1" alt=""/>Mon, 22 Sep 2014 17:03:57 +0000David Wees987 at http://davidwees.comhttp://davidwees.com/content/20-things-every-teacher-should-do#commentshttp://davidwees.com/content/20-things-every-teacher-should-doCould you build a nuclear power plant given the right reward?
http://feedproxy.google.com/~r/davidwees/~3/cCRZ_iRaF34/could-you-build-nuclear-power-plant-given-right-reward
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>From <a href="https://m.facebook.com/WHAS11/photos/a.10150401897545037.633389.198547635036/10153513357435037/?type=1">Facebook:</a></p>
<blockquote>
<p>"my daughter and I were brought in to talk about her " learning disabilities " and how she was not applying herself . They talked about punishments and incentives . After listening to everyone I asked . " If I asked you to build a nuclear power plant could you do it ?" They all answered " no " So I ask well what if i took a way all your free time at work , and did not allow you to go to the ofice party because you could not build it ? Could you build it then ?" Again the answer came back No . So I ask " Okay then how about if I promise you a huge reward , could you build it then ?" Again they all answered no . So I say " what if I read you a manual on how to build a nuclear power plant , gave you people who were educated and worked building nuclear power plants to show you how to build one , gave you every tool , all the materials to build one , could you build one now " one said no still , but the rest said maybe . well I said to them my kid cant do this work if you punish her for not doing it ,or promise her a pizza party. She will only do this work if you put her with people who can help her understand . The world is a power plant to her ." [sic] ~ Adriene Kimiko Pauley</p>
</blockquote>
<p>Could you build a nuclear power plant given the right reward? Or would you need support just to be able to get started?</p>
<p> </p>
<p> </p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/cCRZ_iRaF34" height="1" width="1" alt=""/>Wed, 17 Sep 2014 11:38:57 +0000David Wees986 at http://davidwees.comhttp://davidwees.com/content/could-you-build-nuclear-power-plant-given-right-reward#commentshttp://davidwees.com/content/could-you-build-nuclear-power-plant-given-right-rewardI was not born to teach
http://feedproxy.google.com/~r/davidwees/~3/RTeWpc_y_E8/i-was-not-born-teach
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>I was not born a technology person. </strong></p>
<p class="rteindent1">I am a person who does not accept that he does not know how to do something related to technology and explores how he does it until he figures it out. I am a person who will occasionally search to the fifteenth page of a Google search until I find the right piece of knowledge that helps me figure out my problem. I am a person who believes that I can eventually figure out any technological problem I am given, even if the answer sometimes is to find someone with more expertise. I am a person a who has been exploring the use of technology since I was inspired by the gift of my first personal computer at eight years old.</p>
<p><strong>I was not born a mathematician.</strong></p>
<p class="rteindent1">I am a person who has been exploring numbers since before he started school, supported by a home environment rich in numbers and geometry. I am a person whose parents never told me that math was hard. I am a person who found solace in exploring ideas when he found out that kids can be cruel. I am a person who had a pair of exceptional mathematics teachers from grade four to grade six who taught me that mathematical ideas weren't just meant to be written down and forgotten later but discussed and explored. I am a person who has studied mathematics for nearly 35 years.</p>
<p><strong>I was not born to teach.</strong></p>
<p class="rteindent1">I am a person who often dreaded going to school as a young child either because I was bored or bullied. I am a person who struggled to make friends at school and never really understood other people. I am a person who decided to take musical theatre just to be part of the cool crowd in middle school. I am a person who needed to fill a block in eleventh grade and got pushed into peer tutoring by his guidance counsellor. I am a person who learned that he loved it when he could help other people understand. I am a person who worked hard every moment he was in his teaching program to the point of being called driven by his friends. I am a person who nearly gave up teaching in his first year because of how hard it was. I am a person who only survived his first year of teaching because he found out other people in his school were struggling as much as he was. I am a person who has dedicated a large portion of his life to learning more and more about the relationship between teaching and learning for the last twelve years of my life.</p>
<p>When we see someone who seems to do something well, we have the tendency to assume that they have special talent, rather than they had a set of experiences that prepared them to be successful.</p>
<p> </p>
<p> </p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/RTeWpc_y_E8" height="1" width="1" alt=""/>Sun, 14 Sep 2014 07:44:29 +0000David Wees983 at http://davidwees.comhttp://davidwees.com/content/i-was-not-born-teach#commentshttp://davidwees.com/content/i-was-not-born-teachSharing individualized comments with students with Autocrat
http://feedproxy.google.com/~r/davidwees/~3/ha1uwC5jPek/sharing-individualized-comments-students-autocrat
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong><a href="http://www.ets.org/Media/Research/pdf/RR-08-30.pdf" target="_blank">Much of the research on formative assessment</a></strong> suggests that grades are not effective as feedback because they do not provide students with actionable information they can use to move their learning forward. Comments and questions are much more useful to students when grades are not included.</p>
<p>Unfortunately, if you have 180 students, providing individualized feedback to each student regularly is time-consuming and difficult, maybe even impossible. Grades are much easier to produce. </p>
<p>One partial solution to this is to produce a list of descriptive comments and select the comments you want for each student from this list. Since students often require the same feedback, this provides enough individualization that is still helpful for the student without being over-whelming for teachers to actually do. One way to do this is to create a 1 or 2 page list of the comments and then check-off the comments that actually apply to the individual student, and then give each student their individually selected comments. Unfortunately, while not as time-consuming for teachers, this leaves students having to read a bunch of comments or questions that do not actually apply to them. (Aside: One interesting activity here would be to provide students with the complete list of possible comments and ask them to figure out which comments probably apply to their work.)</p>
<p>It is possible to do both; provide students with an uncluttered list of comments that apply to their work and not have it take an enormous amount of time. The best part of the process I've come up is that you get to keep a copy of the comments you actually gave to students for further reference.</p>
<p>This process uses Google Spreadsheets and the Autocrat add-on from New Visions to turn a template for comments into individual comment documents for each student. If you are using a non-Google Apps for Education account for this process, you should probably not include identifying student information in your personal Google account and just print these documents to share with students.</p>
<p>The first step is to create a Google Spreadsheet and enable the Autocrat add-on. For more information on the Autocrat add-on, <strong><a href="http://cloudlab.newvisions.org/add-ons/autocrat" target="_blank">see this page</a></strong>. I also recommend creating a folder for each assignment to keep all of the comment documents in. You may also find that this process plays well with <strong><a href="http://cloudlab.newvisions.org/add-ons/doctopus" target="_blank">Doctupus</a> </strong>if you are already using it.</p>
<p>The next step is to enter in names and email addresses for each of your students <strong><a href="https://docs.google.com/spreadsheets/d/1vZgmjaAXEw_p_1Lyh6_X7_3vtGU-JsIJshKXzRwHesg/edit?usp=sharing" target="_blank">as in this sample spreadsheet</a></strong>. If you want to avoid storing student information in a Google Spreadsheet, you could use pseudonyms at this stage and enter your own email address, and then print the comments once they are created. Otherwise, you can choose either to print the comments or share them with students electronically.</p>
<p><img alt="Example of initial data entered into the spreadsheet" src="/sites/default/files/screen_shot_2014-09-07_at_1.29.26_pm.png" style="height:388px; width:645px" /></p>
<p> </p>
<p>Now, in this spreadsheet create another tab. Into this new sheet, enter letters from A to Z as necessary into column 1, and enter comments, questions, or other feedback into column 2. These comments should be relatively generic but apply to specific common issues you have noticed in the student work. Remember that every one of your students, even the ones who did very well on the assignment, need constructive feedback to move their learning forward. The comments shown below are associated with <strong><a href="https://docs.google.com/a/newvisions.org/file/d/0ByRs79o6SgLdZFNMTlNweHVCQWc/edit" target="_blank">with this mathematics task</a></strong>.</p>
<p><img alt="Step 2: Adding comments to a separate sheet" src="/sites/default/files/screen_shot_2014-09-07_at_1.37.52_pm_0.png" style="height:458px; width:645px" /></p>
<p> </p>
<p>Now it is time to associate these comments with specific students. Create a number of columns equal to the maximum number of comments you think you will give to any individual student. The title of these columns doesn't matter, but I used Replace 1, Replace 2, Replace 3, etc... to help remind me that these comments are not the ones I intend to share with students. Instead, in these columns place the letters associated with the comments you want to give students instead. You may find it helpful to print the sheet of comments and their associated letters from step 2 to help enter comments when you look at student work. Notice that I have left some of the cells below blank since I only have 2 or 3 comments for those students.</p>
<p><img alt="Step 3" src="/sites/default/files/screen_shot_2014-09-07_at_1.53.38_pm_0.png" style="height:415px; width:645px" /></p>
<p>The next step makes those letters useful for students. Instead of sharing the letters with students, you want to share the comments. Here is where I used a Google formula to take the letters and substitute them for comments in another cell. The actual formula itself will depend on the name you used for the comment sheet you created in step 2 and the number of comments intend to share with each student.</p>
<p><img alt="Step 4" src="/sites/default/files/screen_shot_2014-09-07_at_2.06.37_pm_0.png" style="height:374px; width:645px" /></p>
<p>The formula I used is =if(isblank(E2), "", vlookup(E2,Comments!$A:$B,2)) which you can copy and paste into the top-left most cell of your comment columns as shown above, and then edit to meet your needs. Basically what this formula does is check to see if the associated cell (E2 in this case) is empty, and if it is not empty, it looks up the comment in the sheet you created in step 2 that is next to the letter entered. If you need to edit this formula, <strong>E2</strong> refers to the top-left most cell where you entered the letters corresponding to the comments, and <strong>Comments!</strong> refers to the name of sheet with the comments in it that you entered in step 2, and finally <strong>$A:$B</strong> refers to the two columns from the Comments sheet where you entered the letters and comments. Once you have the formula entered into this cell, you can drag it over and down to fill all of the relevant comments, as shown in the brief video below.</p>
<p><iframe width="640" height="480" src="//www.youtube.com/embed/zXeYGjFJBd4?rel=0" frameborder="0" allowfullscreen></iframe></p>
<p>You will also need <strong><a href="https://docs.google.com/document/d/1qDq3scwA19ngx9ZPIy90KOwSD-jpkWjhbUJQ_c39Hpg/edit?usp=sharing" target="_blank">a template document like this one</a></strong>. Autocrat will take this template document you create, make a copy of it as either a PDF or Google Doc for each student, and replace the variables entered into the template (which look like <<Comment 1>> in this sample) with the comment information you entered into the spreadsheet.</p>
<p>Now you have all of the preparation work done that you need to run Autocrat. Launch Autocrat from the menu above (if you have not added Autocrat to the spreadsheet, you will have to do this now).</p>
<p><img alt="Step 6" src="/sites/default/files/screen_shot_2014-09-07_at_3.03.40_pm_0.png" style="height:448px; width:645px" /></p>
<p> </p>
<p>Click on New Merge Job in the new Autocrat sidebar.</p>
<p><img alt="" src="/sites/default/files/screen_shot_2014-09-07_at_3.07.12_pm.png" style="height:229px; width:386px" /></p>
<p> </p>
<p>Click on Drive in order to select the Google Doc template you created and then give the merge job a name.</p>
<p><img alt="" src="/sites/default/files/screen_shot_2014-09-07_at_3.09.20_pm_0.png" /></p>
<p>Confirm that the <<tag>> and Sheet header information is correctly set. Here Autocrat has attempted to match the headers of this spreadsheet with the variables you entered into the Google Doc template.</p>
<p><img alt="" src="/sites/default/files/screen_shot_2014-09-07_at_3.11.22_pm_0.png" /></p>
<p>Click Save, and then enter the naming format each document to be shared with students should have. You may also want to open up the advanced settings here and select a folder for these documents to be created in.</p>
<p><img alt="" src="/sites/default/files/screen_shot_2014-09-07_at_3.14.06_pm_0.png" /></p>
<p>Now you should be able to click on the Run Merge button on the right. Autocrat will share a copy of the template with each email address entered into the email column with the comments entered instead of the variables. You may want to click on Preview before running the merge, just to see what the documents will actually look like before they are created. Once the merge is done, the documents will exist, be shared with students (assuming you entered their email addresses and not your own), and there will be links to the documents in this spreadsheet.</p>
<p><img alt="Merge done" src="/sites/default/files/screen_shot_2014-09-07_at_3.19.27_pm_0.png" style="height:432px; width:645px" /></p>
<p> </p>
<p>Here's what a folder of these <strong><a href="https://drive.google.com/folderview?id=0BwdARD8jZs6DWEZERjRPcTFVV2M&usp=sharing" target="_blank">comment documents looks like</a></strong>. At this point you can download these documents (you can download all of the files in this folder as a zip archive, and then once you extract them from the zip, print them out for students - you can do batch printing in both Windows and Mac OS) and then print them out for students or students can access the documents themselves online.</p>
<p><img alt="" src="/sites/default/files/screen_shot_2014-09-07_at_3.24.20_pm_0.png" /></p>
<p>The good news is that once you have set up this process once, replicating it again is fairly easy. The part that will always take a while to do is creating the list of feedback comments and/or questions for students. For more examples of good feedback comments and questions, check out the feedback pages of the <strong><a href="http://map.mathshell.org/materials/lessons.php" target="_blank">Classroom Challenge lessons from the Mathematics Assessment Project</a></strong>.</p>
<p>Hopefully this process will make it a bit easier to give students individual feedback in the form of comments. Let me know if you have any questions.</p>
<p> </p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/ha1uwC5jPek" height="1" width="1" alt=""/>Sun, 07 Sep 2014 15:18:03 +0000David Wees985 at http://davidwees.comhttp://davidwees.com/content/sharing-individualized-comments-students-autocrat#commentshttp://davidwees.com/content/sharing-individualized-comments-students-autocratFormative Assessment Responses
http://feedproxy.google.com/~r/davidwees/~3/Ze-Dw1toByM/formative-assessment-responses
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Formative assessment means more than just giving a quiz or an exit ticket. An assessment is only formative if the teacher (or her students) respond to the information gathered.</p>
<p>However coming up with an appropriate response is typically hard to do. After all, the most common finding in formative assessment is that a significant, but perhaps minority, group of students still do not understand a concept, after the teacher gave her best shot at helping students understand. No teachers save their best strategy for teaching a topic until later.</p>
<p>I'm working on <a href="https://docs.google.com/presentation/d/1Rlgc_nTJ47AhCKd5ZlKaEYBtNacuRebZMAujySoKauU/pub?start=false&loop=false&delayms=3000">a menu of possible responses</a> teachers could come up with. Some of these responses depend on the nature of the formative assessment gathered, but most of them can be applied in many different contexts.</p>
<p><iframe src="https://docs.google.com/presentation/d/1Rlgc_nTJ47AhCKd5ZlKaEYBtNacuRebZMAujySoKauU/embed?start=false&loop=false&delayms=3000" frameborder="0" width="800" height="629" allowfullscreen="true" mozallowfullscreen="true" webkitallowfullscreen="true"></iframe></p>
<p>If you have other possible strategies teachers can try, please feel free to <a href="https://docs.google.com/presentation/d/1Rlgc_nTJ47AhCKd5ZlKaEYBtNacuRebZMAujySoKauU/edit?usp=sharing">add them here</a> or comment below.</p>
<p> </p>
<p> </p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/Ze-Dw1toByM" height="1" width="1" alt=""/>Tue, 26 Aug 2014 10:04:03 +0000David Wees982 at http://davidwees.comhttp://davidwees.com/content/formative-assessment-responses#commentshttp://davidwees.com/content/formative-assessment-responsesTwo different approaches to teaching
http://feedproxy.google.com/~r/davidwees/~3/GYFrR5cL4og/two-different-approaches-teaching
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Here's an approach to teaching about the relationships between the different forms of the equation of a line that is based on <a href="http://en.wikipedia.org/wiki/Constructivism_(philosophy_of_education)">constructivism</a>.</p>
<ol><li>Be clear with students what the objective for the lesson is today.</li>
<li><a href="http://davidwees.com/content/using-technology-facilitate-noticing-and-wondering" target="_blank">Give students a tool for looking at the relationships between the different forms of the equation of a line</a>.</li>
<li>Ask them to write down what they notice and what they wonder after they have played around with the tool themselves. Circulate around the room to see what they write down so you can better make use of it in the whole group discussion.</li>
<li>Next, as a group, discuss what <a href="http://mathforum.org/workshops/universal/documents/notice_wonder_intro.pdf" target="_blank">students notice and what students wonder about the graph</a>, or <a href="https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=whole%20group%20discussion%20strategies" target="_blank">use a different talk protocol</a>.</li>
<li>As a group summarize what appear to be the most important observations made, and everyone (including the teacher) writes them down. If important points about these representations of a line are not brought up, either ask questions to prompt students to realize these points themselves or make a note to yourself to structure a task that will help draw out these points next time.</li>
<li>Come back to this concept multiple times in a variety of different ways over the next few weeks, and even find ways to connect to this topic through-out the rest of the year.</li>
</ol><p> </p>
<p>Here's an approach to this same lesson based on <a href="http://en.wikipedia.org/wiki/Cognitive_load" target="_blank">cognitive load theory</a>.</p>
<ol><li>Be clear with students what the objective of the lesson is for today.</li>
<li><a href="http://www.regentsprep.org/Regents/math/geometry/GCG1/EqLines.htm" target="_blank">Give them notes on the different forms of the equation of a line and some worked examples</a>.</li>
<li>Give students twenty or so practice problems to do themselves. Circulate around the room while students do this and give them timely, efficient, and useful feedback.</li>
<li>Quiz students on their understanding of the concept. Use this to guide your planning in subsequent lessons.</li>
<li>Review this concept with students at spaced intervals over the course of the rest of the year.</li>
</ol><p> </p>
<p>My recommendation is that if you are unclear on how either of these approaches is helpful for students, then you should try it at least once. I have in fact tried both of these approaches myself, and I'm quite clear on which approach helps students make better connections to other areas of mathematics, and which approach did not work.</p>
<p>Which approach is more likely to lead to an instrumental understanding of mathematics? Which approach is more likely to lead to a relational understanding of mathematics? For reference, <a href="http://davidwees.com/content/difference-between-instrumental-and-relational-understanding">here's a comparison of instrumental to relational understanding</a>.</p>
<p> </p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Newsletter: </div><div class="field-items"><div class="field-item even"><a href="/category/newsletter/reflective-educator">The Reflective Educator</a></div></div></div><img src="//feeds.feedburner.com/~r/davidwees/~4/GYFrR5cL4og" height="1" width="1" alt=""/>Thu, 14 Aug 2014 09:14:18 +0000David Wees981 at http://davidwees.comhttp://davidwees.com/content/two-different-approaches-teaching#commentshttp://davidwees.com/content/two-different-approaches-teaching