20. Prototype for lesson construction Touchable Visual Discussion: Makes sense of concept 1 2 Learn to record these ideas V. Faulkner and DPI Task Force adapted from Griffin Symbols Simply record keeping! Mathematical Structure Discussion of the concrete Quantity Concrete display of concept
Using these statistics - How many students are not proficient? Guidelines for discussion: What may contribute to the lower percentage of students who are proficient in mathematics from fourth-grade to eighth-grade? Is this same thing true for the students in your own system? Update 2007--- The average score for fourth-graders increased 27 points over the past 17 years, and the score for eighth-graders increased 19 points. Students at all levels of performance made gains, resulting in higher percentages of students at or above the Basic and Proficient achievement levels. State results show gains since 2005 for both fourth- and eighth-graders in 15 states and jurisdictions. An additional 8 states showed gains for grade 4 only, and 11 states showed gains for grade 8 only.
Have a participate read this slide and then ask for feedback from the group. Some state that this is not a fair comparison because the U.S. educates everyone not just the top students as in Japan and some other countries. The fact is that the TIMSS studies did factor in this difference, to ensure this statement regarding comparison was justified. Why does the U.S. rank near the bottom of this distribution? There have been other studies that have replicated the data found in this study.
Have a participant read this slide and discuss.
Read and discuss. You may want to wait until later on to show the what works site.
Trainer access the web site to show the participants. In 2006 Bush charged panel with determining what students need to be successful in algebra. HANDOUT: Summary of Math Panel report “Foundations for Success” trifold short summary Can be ordered for FREE through http: edpubs.ed.gov
See Trainer Notes Unit 2 Slide 8 Dialogue. An update is provided on page 2 and 3 of the notes.
Calculator vs. No calculator is a red herring---so are real-life vs. naked. These are not the things that created high performing students. High achieving countries also would work with one particular concept for a longitudinal amount of time. Connections between math concepts is a very different thing that connections to the real world. Countries that emphasis real world connections without the math connections perform the worst.
Ask the audience—what is 4 squared? (16) What IS 4 squared? (What’s this geometry term doing in the middle of my prealgebra lesson?) Teachers don’t see this as a connections problem. What we do is to emphasize the procedures. 4 square = 4 x 4 What is the common error pattern? Students tell us 4 x 2. Why does this happen---the kids are concrete---you say something about multiplication and they are connecting to their prior knowledge for multiplication which is to multiply the two numbers. If we give them something else concrete—i.e. the geometry that actually connects the mathematics for them---they will get IT!
How many different forms of the number 16 do you see here? Think about the power for students to understand algebra if they are firmly entrenched in forms of a number. Take your particularity concrete kid---they can now understand this concept using quantity and magnitude We can emphasis for them the equality of the different forms of this number 16 and if the student really needs concrete support to understand the concept—they count the squares to attach to the concept. We are already hooking into our geometric thinking at a very early age! As we talk about this---we are making stronger implementation choices that helps students make stronger connections within the mathematics. In addition, we are differentiating our instruction with various strategies at different places where students can “get it”. Also note--the math panel discusses the need to have a strong foundation in geometry and measurement. This way of teaching exponents supports students’ understanding of linear (1 dimension) and planar (2 dimensional) measurement. Also connects with geometry and Pythagorean Theorem.
Is 1 equal to 1 cubed? What does that mean? Make a sketch—one is a linear measurement and 1 cubed is a volume measurement. They are not the same thing. What does the form of the number stand for? (linear vs.. cubic) Does this mean that they are still equal. We can’t say that they are the “same thing”, because they are not! But they are equal in numeric value! Equal in number dimension but not in the physical dimension.
Implementation choices: Language and accurately and consistently communicating a common language is a huge part of implementation choices. Let’s look at the same situation in a different venue. Are these equal? In what ways are they equal? Are there ways in which they are “not the same”? These two are equal in weight, but they are not the same thing when we consider their physical dimensions. Oftentimes we think we are being more precise for kids by simplifying things when in fact we are communicating misconceptions. In a situation like this, the teacher’s implementation choices should include statements such as they are the same in weight, but they are not the same thing. Classroom example: Second Grade---We are making an exchange of ten with rods and blocks—Are they the same thing? Child already has a sense of sameness and replies “No”. You are right---they are not the same, but they are equal in value—namely they both equal ten units. This is what they found in the TIMSS studies. The countries that are producing stronger math students are the ones where the teachers are making these kinds of habitual implementation choices. They are very aware of how language communicates math accurately for all time. Are we laying down accurate mathematical language that will facilitate future learning and understanding of math as opposed to facilitating getting the right answer in this class at this minute. We are laying down synapses in the student’s brains that will last a lifetime.
Teaching procedures involves a rapid pace, includes modeling, teacher directed product type questions and a smooth transition from demonstration to error free practice. Conceptual instruction connects facts, procedures and ideas and includes having students consciously and purposefully wrestling with the important ideas of the concept. “Students who mastered skills under conceptually supportive conditions acquired different competencies than those trained with a strict focus on developing skills. They were better able to adapt their skills to solve new kinds of tasks.” adapted from James Hiebert and Douglas Grouws in “Effective Teaching fort he Development of Skill and Conceptual Understanding of Number: What is most Effective?”.(NCTM) The key is not the method---expository vs. inquiry, student centered vs. teacher-centered, traditional vs. reform-based, but rather attending to important mathematical relationships.
Give participants time to read the slide on their own. Then review each bullet with comments below: Bullet 1: Problem seems to persist through elementary school mathematics even if computation proficiency is increased. Research not available to say what will help this situation. Bullet 2: Lack of inhibition. If a student were asked what 4 + 8 is, they might say 5 or 9 because those numbers come next in the counting sequence. It is difficult for students to block “irrelevant associations” and focus on the issue at hand. Think alouds and draw out a problem helps these students. Some disagreement on the last three bullets Bullet 3: Difficulty for third graders in visualizing the base ten system (Geary) Bullet 4: Students may learn to manipulate symbols but are not connecting them to meaning. Also, consider the student who is not flexible in interpreting symbols. For instance, does not connect that a division symbol is related to a fraction bar. Also consider the student who does not understand that a number next to a parentheses indicates multiplication yet the parentheses may be just for grouping in another expression. Bullet 5: Working memory problems make multi-step problems difficult. Some researchers have designed interventions that help these students to understand numerical concepts. (Mercer) Next slide
Cecil Mercer’s research supports the idea that students with learning disabilities in math and those who are struggling learners make stronger gains when instructed with a concrete to representational to abstract approach.
Research regarding the struggling learner is consistent in demonstrating that these students need explicit instruction. This includes instruction on how to represent the concrete. We cannot take it for granted that the struggling learner will automatically be able to draw or visualize or even make sense of concrete manipulatives. These students need to be taught how to draw these representations of the concrete. Mercer’s research shows that direct instruction of the mathematical concepts connected with explicit instruction on how to move through these stages is beneficial for these students.
What Sharon Griffin (and others) found in their work with young students was that students must understand quantity and the earliest form of a number line—i.e. counting independently. The first developmental leap in mathematical development is connecting these two concepts by being able to solve problems of quantity through their early understanding of the number line via counting. Next, students are able to connect these understandings with symbolic representations. Griffin found in her research that students who were behind in math in third grade did not have an internalized sense of the number line. This is why in her preventative program (Number Worlds) she emphasizes developing a clear understanding of the mathematical structure of the number line in students learning. This number line can come in the form of a left to right, vertical (thermometer), circular (clock). She also puts them in multiple representations from objects, pictures, lines.
In object land, students count and compare sets of objects or pictures of objects. (Which is bigger or smaller? How many do you have?) In picture land numbers are represented as a set of dot-set patterns such as in a die or tally marks or numerals. )What did you roll/pick? Which has more or less?) In line land, number is represented as a position on a path or a line. (What is your position on the line? Where are you now? How far did you go? Do you go forward or backward?) In Sky land, number is represented as a position on a vertical scale such as a thermometer or a bar graph. (How high or low are you now? What number or amount is higher or lower?) In Circle land number is represented as a point on a dial such as a clock or a sundial. (How many times did you go around the dial? Which number is farther or less far around?)
V. Faulkner developed this model to extend the Griffin model beyond the early years. Dr. Griffin has reviewed and endorsed this model as an extension of her work. The first oval is being able to show a math concept concretely and this includes being able to develop a story problem. The third oval is being able to show how to solve mathematical problems using mathematical symbols i.e. the written language of math. The second oval is perhaps the most difficult and most neglected in instruction in the United States. The second oval demands that the teacher both understands and is able to communicate the most essential mathematical features in the math they are teaching. Again, this connects to other research that indicates that implementation and teacher understanding appear to be the critical piece in improving instruction in the United States for all learners.
Taking Mercer’s research, we can put it into the same framework that Griffin uses to model her research. Let’s look at how these two models fit together?
As you can see, the two models are very similar even though they were developed independently by these two researchers. One of the researchers was a developmental psychologist (Griffin) from Canada while the other researcher was heavily involved with research in learning disabilities. Particularly notice that the first and third ovals are virtually the same. Look at the second ovals. How are these different? (Solicit feedback from the participants.) Make sure to note that Mercer’s second oval emphasizes the explicit instruction of modeling the concrete in the representational stage. Griffin’s second oval emphasizes the mathematical connections that need to be made explicit to the student. Mercer’s approach does not take for granted that the student knows how to draw the concrete, while Griffin’s approach does not take for granted that the student will make the cognitive connections about the mathematics structure. We cannot take either of these things for granted in the instruction of the struggling learner.
Consider what this means for us as instructors. We need to both understand and communicate the mathematical structures that we are teaching and support students in processing these structures. One of the things that helps the student process the mathematical structures is learning to draw representations. Remember though that if you are not communicating the structures to the students with the appropriate mathematical language and appropriate mathematical connections, we cannot assume that the representations themselves are teaching the students an understanding of the math. On the other hand, we need to keep in mind that even if we teach the math structures really well, there will be students who still need explicit instruction and practice in drawing and modeling that mathematics at this representational stage. Ask participants if they can think of any examples where students would need support in transitioning from the concrete to the symbolic. Possible examples: A student can draw a ten rod, but does not really understand base ten or different forms of a number. A student can articulate the idea of equivalent fractions, but struggles when trying to draw models of fractions. A student can articulate that 50 % equals 0.5, but cannot represent the equivalent on a 100s grid. It is only when we have both of these things in place that we can be confident about what is happening in a student’s brain and in their learning. IMPORTANT: Students eventually need to make connection 2 as well. While the first connection has been often overlooked, do not think that we are saying that students only need to work in quantity and structure and representations. Students need to make the connection to how these things are related to mathematical symbols. Fluency will be sacrificed if students do not eventually practice at the abstract level.
Tracy wrote the following subtraction equations. How did she arrive at her solutions? Tracy is very concrete. She sees the subtraction of 5 as simply removing the 5 or the subtraction of 4 removing the 4. This concrete grounding is illustrated very well in the next slide.
I have a red balloon and I subtract a blue balloon---hmmm---I have a red balloon left.
Participants discuss in small group After discussion make sure to enforce these concepts: Remember the findings from the TIMSS studies that strong instruction does make connections for students as a matter of course. Conceptual and mathematical connections are not “extras”. For these students instruction does not have to be as explicit and direct However, we must focus in on the topic to develop a conceptual understanding not just a procedural understanding The way we define our best student may not be a refection of a truly strong math student. Consider the issue of “Academically Gifted” student programming. Sometimes these students are good memorizers. Do they understand the concepts? Oftentimes, they develop the idea that if they can ‘get the answer’ they don’t need to explain their thinking. This is problematic from the perspective of building a strong foundation.
Participants discuss in small group Instruction for these students might include more strategic interventions and collaborative dialogue. There are few research based programs. Assessment guides where these students might need some specific targeted instruction in particular skills or concepts. For skills that are lower, these students may need Tier 2 targeted instruction. This instruction should be more explicit, direct, cumulative, systematic, and multi sensory for these targeted skills. Instruction must also include the clear mathematical connections you will get in this workshop.
Language and reading are often taught very differently than math. Have participants suggest some ways that this statement is true. The next slide will help as a good summary and maybe some more in-depth thought regarding how and why these are taught differently.
See Trainer Note: Unit 2, slide 31 Dialogue
These two statements come from the research in Knowing Mathematics for Teaching out of the University of Michigan. Deborah Ball is the Dean of Education and Liping Ma was one of her doctoral students.
Have the group discuss these terms and come up with conceptual definitions of each in small groups. Groups record definitions on chart paper. One example of instruction that is conceptual. Example: Explicit- Absolute Value number line Possible definitions of the terms: See Slide 33 Trainer Notes for a review of these terms.
Presenter notes: This is a slide to review explicit instruction. Teachers say: How many pieces that represent the total is the larger number? The smaller number is always on top. Teachers communicate this misconception explicitly. There is a transfer to division as well. Smaller number always goes outside the “house”. Unfortunately, many difficulties in mathematics arise because teachers have taught students explicit ‘tricks’ rather than explicitly teaching how the number system is organized and how it works. For instance with 5/4 example. If students were taught simply to ‘color’ in the fraction parts, they will become easily confused by having more than the whole to color. As for last bullet--many of these suggestions are actually not strong mathematical approaches. Key words, “guess and check”, ‘working backwards” are all things that we might do, but they do not explain the number system to students. We will work throughout this training to replace this approach with an approach that emphasizes mathematical structures as your primary approach to attacking a problem, rather than a procedural or more haphazard approach.
Refer back to slide 2 that talks about the large number of students below proficiency. Connect to what works for reading. Direct Instruction programs for reading are not for all students.
We will take each of the concepts above and discuss the concepts in the slides to follow. Bullet one: The sequence of skills and concepts are a critical aspect of teaching math. It is very important that children understand the most basic concepts that will allow for strong connections and understanding later on. Bullet two: We must have a solid understanding of the specific pre-skills to fully understand what a child is ready to learn. Bullet three: It is critical that when we are selecting examples for independent practice that we make sure that we give examples that reflect the true skill being taught – (for example – if we are teaching regrouping and one of our example problems has a 0 in it - this is a much more difficult numeral for students to rename / regroup). It is ONLY after the students have gotten to a point of 95% accuracy in the new skill that we bring in old material for review and discrimination practice. Bullet four: It is critical that children have a chance to practice and review materials in order to keep the skill as well as see how the skill fits into new tasks.
These are some basic guidelines that most of us already know – but do we really use them when we are setting up instruction? Go to board and work 4008 - 9 procedurally. Traditionally we would say “I can’t take nine from 8 and I don’t have any tens and I don’t have any 100s but I have 4 thousands. So I borrow from my thousands, and put my 10 over my first 0 and then I cross it out and make it a nine because I need tens, so now I have 10 tens but need to cross that out and make it a nine and then I put a one next to my 8. Now I can work the problem. What if we actually understand the pre-skills of base-ten, equality and forms of a number. What if we then look at this as Do this problem on the board as you speak so that the participants can see the math. “well we have 400 tens and 8 ones. If we decompose our 400 tens into 399 tens and one ten, we can then take that one ten and decompose into ten ones. Now we have 399 tens and 18 ones. That is still 4008, right?”
Bullet 2: Use the 10% rule. 10% of 80 is 8, so 8 + 8 + 8 + 8 + 4 (½ of 8 since 5 % is ½ of 10%) Component skills are the most generalizable, most explicit, most useful, and are mathematically authentic. If a student really understands powers of ten then they will be able to solve this problem without complicated procedural algorithms.
Carefully scaffolding of new material is critical to the student’s success. Don’t throw in non-examples before student understands the concept. What are you practicing? Communicating the understanding of a concept or drill and kill. We should be practicing the understanding of the concept. Practice is only done with something you have mastery with. If you do not have mastery of a skill-you are not practicing, but learning that skill. How do we achieve that mastery? We achieve that mastery through conceptual understanding.
Homework strips with differentiated problems can be sent home with a student so that each student is receiving practice for a concept that they understand. This will allow for accuracy of practice and will build automaticity. Writing as homework. Have students respond to assignments such as: What is a slope? Bring in a picture of a that illustrates slope, etc. It is all about the language!
Have participants work out both problems and talk about how they are different. The differences are discussed on the next slide.
Discuss the questions above. 3002 – 89 = 2913 364 – 128 = 236 Working memory is a huge part of the differences in the examples. Students with weak working memory actually need to transfer to the symbolic level with meaning, because it is really hard for them to keep track of all of the exchanges if done purely at the concrete level. Regrouping refers to the act of regrouping physically Renaming is regrouping at the symbolic level
Just like reading, automaticity does not develop directly, but through practice in breaking the code of the system. We will see later how the “math fact” 8 + 5 is a composite of several sub-skills that need to be taught and practiced. Review can be evoked through language connections Just because the student can work the 20 problems in 1 minute doesn’t mean that they have any understanding of what they are doing. Understanding what they are doing would slow them down!!!! Does this mean that the fast kids are the smart kids? If a teacher puts stars on the wall for knowing your math facts, Do stars equal understanding? (Refer to charts on the wall with stars to equate knowing the 20 facts within one minute.) Note: fluency is built through practice with mathematical understanding
Have you ever met a student with anxiety like this?
See Trainer Notes: Unit 2 Slide 46
Refer to trainer notes for discussion responses. (Slide 47)
In reading we often look for “maintaining meaning” even if they miss the word. This is not a good idea and not founded in research. These kids often do not learn to decode and their reading falls flat in 3 rd or 4 th grade as decoding demands increase. The same thing happens in math when we teach tricks or mathematically unsound practices to get kids through a grade level. For instance, Guess and Check is not really a mathematical system that helps kids solve problems systematically. If we allow students to learn only the tricks, we are teaching them the wrong thing. We are creating stumbling blocks and even barriers for them in the future. We see children fall behind in reading in the 4 th Grade when texts have no pictures for clues and becomes more difficult. Here text and context clues are more difficult to interpret. The same thing happens with math. If the students do not have a good understanding of the math, they will reach a point when they can no longer do the math.
While we are talking about language---language is not a trick--- You don’t teach a kid to read by grabbing words out of the text willy nilly—you have to keep everything contextualized and draw a picture in your mind of everything that is happening in the paragraph. Research is really clear on this point ---Key words do not hold up to facilitate problem solving.
Key words do not work all of the time, which means that this “strategy” does not work for all time. It is not based in mathematics but in trying to make things “simpler” for the student. But in doing so, we bypass the real mathematics and the real thinking that needs to be done to understand the mathematical constructs. Note: Emphasis on the number line is appropriate with this problem. Draw a vertical number line. Locate Erin at a point 46. This is an assignment sentence. Ask participants: Does Jason have more or less comic books than Erin? (Less) How do you know? (The relationship sentence says that Erin has 18 more than Jason, so Jason has less.) Locate Jason to the left of Erin on the number line. Ask: “What is the difference in the number of comic books that Jason and Erin have?” (18). Show the difference on the number line between Jason and Erin. Ask: “ Why does it make sense to subtract the two quantities rather than add?” (Because we can visually see that in order to find Jason’s number of comic books, we must have a quantity less than Erin’s quantity—a difference of 18.) Jason therefore must have 28 comic books.
We are not engaging sense-making when we ask kids to decontextualize key words. We label the kids “impulsive” when they employ the key word strategy that clearly does not work! We gear kids up for ---”just tell me how to get the answer! Don’t help me to try to understand what is happening mathematically. What is this math problem really asking for?
There are 6 students for every principal. See Trainers Notes in Unit 2 Slide 52 for Dialogue.
In your group list the difficulties struggling Math Students have in Middle school
Counting Principles 1-1 Correspondence: one word tag to each object Stable Order: order of word tags invariant across sets Cardinality: final word tag = quantity of set Abstraction: objects of any kind can be collected and counted Order-Irrelevance: items in set can be tagged in any sequence
Students with math disabilities commonly think that these unessential features of counting are a part of counting. Standard Direction--- “counting proceeds from left to right” Adjacency--- “ must count objects consecutively/contiguously” Pointing--- ”objects are typically pointed at, but only once” Start at an End--- “ counting starts at one end of the array”
First explain that these studies use puppets to see how students understand counting. The puppets would count and make mistakes and the students would correct or praise the puppets. MD = Math Disability RD = Reading Disability A common error with students with math disabilities was thinking that adjacency was required in order to count. One interesting finding was how working memory played into math deficiency for many of these students. When the puppet made a double count near the end of a sequence children generally caught the mistake, however, many of these same students would not recall to correct the error if it happened at the beginning of the sequence.
Again this reinforces the point that if we are only teaching these student with memory based instruction, we may be butting up against the one thing that they struggle with the most. Fluency practice is fine, but it needs to be just a small percentage of what is done. We need to find a different way to access the connections with these students and not continually think that if we can just get them to do things faster than everything will begin to make sense. While that paradigm does seem to be true to a large degree in reading, it does not appear to be true in mathematics in the same way.
These are issues that students who have math difficulties often exhibit.
What does this mean for instruction? How fruitful is it to concentrate on the one thing we know these students may never do well – because of a weak memory system. We may not be able to improve the organic ability to memorize, however, we can move them along with more procedural strategy with a focus in understanding of number. This does not mean we do not do any work with memory but the emphasis should be where we know the student can make meaningful gains. For instance if you work with a student to become more sophisticated with their addition strategies you will likely see more gains than through straight memory work. The Sum, Min, and Max strategies for very young students: The sum strategy is counting all. The min strategy is using the smallest number and counting up. The max strategy is realizing that it is the most efficient to find the biggest quantity and counting up from there. We will also explore other strategies that are even stronger in developing students’ number sense as they get older so that they can operate on number utilizing an understanding of base ten and other critical elements within mathematics.
Summary of research regarding special education. Make note that the meta-cognitive support should not just be thinking out loud about procedures but also thinking out loud and organizing the student around the real mathematics involved. Note also that in units 4-6 we will explore more specifically the Big Ideas of math which develop Number Sense. Also note that Direct Instruction does NOT mean direct instruction of procedures only.
Participants discuss in small group Identify skills that are weak for students. Utilize targeted Tier 2 instructional techniques for students in small groups for specific skill needs. Work on fluency, pre skills, and comprehension. Students who are performing below benchmark in all skills must have direct instruction and time to move from the concrete stage to the representational stage, to the symbolic stage. This may included direct instruction on how to draw a tens bar or a ones cube and discuss it. Cecil Mercer and Susan Miller have a program that will address this “Strategic Math Series” Ideally these student are caught early in pre, k, or first grade. Particularly because perhaps the strongest research tested program available is Number Worlds which is designed for students in the four to seven year old age range. Only those students who do not respond to this intensive targeted instruction for a vast number of skills, would be considered for Tier 3 interventions that continue these intensive interventions. This will be investigated further in Unit 8.
Remember we do not always understand what our students are thinking. As we understand the mathematics and the learner better we equip ourselves to be better instructors. Consider Einstein’s parents.
Jigsaw with Reform by the Book
Go to next two slides to explain the Liping Ma Assignment. Ladder chart for math programs used currently Pass out colored numbers for chapter units
![[object Object],[object Object],[object Object]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)